Electromagnetic Theory and Applications for Photonic Crystals

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Electromagnetic Theory and Applications for Photonic Crystals

edited by

Kiyotoshi Yasumoto Kyushu University Fukuoka, Japan

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

© 2006 by Taylor & Francis Group, LLC

Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-3677-5 (Hardcover) International Standard Book Number-13: 978-0-8493-3677-5 (Hardcover) Library of Congress Card Number 2005041895 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data Electromagnetic theory and applications for photonic crystals / edited by Kiyotoshi Yasumoto. p. cm. -- (Optical engineering) Includes bibliographical references and index. ISBN 0-8493-3677-5 (alk. paper) 1. Photonics--Materials. 2. Crystal optics--Materials. 3. Electrooptics. I. Yasumoto, Kiyotoshi. II. Optical engineering (Marcel Dekker, Inc.) TA1522.E24 2005 621.36--dc22

2005041895

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Preface Photonic crystals are periodic dielectric or metallic structures that are artificially designed to control and manipulate the propagation of light. A photonic crystal can be made either by arranging a lattice of air holes on a transparent background dielectric or by forming a lattice of high refractive index material embedded in a transparent medium with a lower refractive index. The lattice size may be roughly estimated to be the wavelength of light in the background medium. The behavior of light propagating in a photonic crystal can be intuitively understood by comparing it to that of electrons in solid-state materials. The electrons passing through a lattice of the atoms interact with a periodic potential. This results in the formation of allowed and forbidden energy states of electrons. The light propagating in a photonic crystal interacts with the periodic modulation of refractive index. This results in the formation of allowed bands and forbidden bands in optical wavelengths. The photonic crystal prohibits any propagation of light with wavelengths in the forbidden bands, i.e., the photonic bandgaps, while allowing other wavelengths to propagate freely. The band structures depend on the specific geometry and composition of the photonic crystal such as the lattice size, the diameter of the lattice elements, and the contrast in refractive index. It is possible to create allowed bands within the photonic bandgaps by introducing point defects or line defects in the lattice of photonic crystals. Light will be strongly confined within the defects for wavelengths in the bandgap of the surrounding photonic crystals. The point defects and line defects can be used to make optical resonators and photonic crystal waveguides, respectively. The photonic crystals with bandgaps are expected to be new materials for future optical circuits and devices, which can control the behavior of light in a micronsized scale. Although there is continuing interest in new findings of photonic bandgap structures associated with particular lattice configurations, recent attention has been focused on the engineering applications of the photonic crystals. To be able to create photonic-crystal-based optical circuits and devices, their electromagnetic modeling has become a much more important area of research. From the viewpoint of electromagnetic field theory, the photonic crystals are optical materials with periodic perturbation of macroscopic material constants. Fortunately we have a great deal of knowledge about the electromagnetic theory for periodic structures. During the past few decades, various analytical or computational techniques have been developed to formulate the electromagnetic scattering, guiding, and coupling problems in periodic structures. The aim of this book is to provide the electromagnetic theoretical methods that can be effectively applied to the modeling of photonic crystals and related optical devices. This book consists of eight chapters that are ordered in a reasonably logical manner from analytical methods to computational methods. Each chapter starts with a brief introduction and a description of the method, followed by detailed formulations for practical applications. Chapter 1 describes the scattering matrix method based on multipole expansions, its extension combined with the method

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of fictitious sources, and the Bloch modes approach. The applications of the methods and numerical examples are presented with a particular emphasis on phenomena of anomalous refraction and control of light emission in photonic crystals. In Chapter 2, the multipole theory of scattering by a finite cluster of cylinders is discussed to investigate the propagation of light in photonic crystal fibers and the radiation dynamics of photonic crystals. The multipole method combined with the notions of lattice sums and Bloch modes is presented to model the scattering and guidance in various photonic crystal devices. The scattering and guidance by photonic crystals are formulated in Chapter 3, using a model of multilayered periodic parallel or crossed arrays of circular cylinders standing in free-space or embedded in a dielectric slab. The method uses the aggregate transition matrix for a cluster of cylinders within a unit cell, the lattice sums, and the generalized reflection and transmission matrices in a layered system. Chapter 4 is devoted to the method of multiple multipole program applied to the simulation of photonic crystal devices. The method comprises a modeling of periodic structures using the concepts of fictitious boundaries and periodic boundary conditions, novel eigenvalue solvers, a so-called connections scheme that is a unique macro feature of the method, and eigenvalue and parameter estimation techniques. In Chapter 5, the mode-matching method for periodic metallic structures is reexamined. A novel technique for mode matching combined with the generalized scattering matrix method is presented to deal with the scattering and guidance by metallic photonic crystals with lattice elements of arbitrary cross sections. Chapter 6 describes the method of lines, which is one of the efficient numerical algorithms for solving electromagnetic guiding problems. The mathematical formulation and analysis procedure based on the generalized transmission line equations are discussed. The results of applications are demonstrated for photonic crystal devices consisting of various bends, junctions, and their concatenations. In Chapter 7, the full-vectorial finite-difference frequency-domain method is treated. The absorbing boundary conditions, the periodic boundary conditions, and an interface condition for dielectric interfaces with curvature are implemented in the finite-difference scheme. The method is applied to the analysis of photonic crystal fibers, photonic crystal planar waveguides, and bandgap structures. Chapter 8 describes the finite-difference time-domain method based on the principles of multidimensional wave digital filters. The method employs the finite difference schemes using the trapezoidal rule for discretizing Maxwell’s equations that has advantages with regard to numerical stability and robustness. Numerical examples are presented for various photonic crystal waveguide devices. It is hoped that the material is sufficiently detailed both for readers involved with the physics of photonic bandgap structures and for those working on the applications of photonic crystals to optical circuits and devices. Finally, I would like to thank the authors for their excellent contributions. It is also a pleasure for me to acknowledge Jill J. Jurgensen and Taisuke Soda of CRC Press, Taylor & Francis Group, for their help throughout the preparation of this book. Kiyotoshi Yasumoto © 2006 by Taylor & Francis Group, LLC

The Editor Kiyotoshi Yasumoto earned the B.E., M.E., and D.E. degrees in communication engineering from Kyushu University, Fukuoka, Japan, in 1967, 1969, and 1977, respectively. In 1969, he joined the faculty of engineering of Kyushu University, where since 1988 he has been a professor of the Department of Computer Science and Communication Engineering. He was a visiting professor at the Department of Electrical and Computer Engineering, University of Wisconsin in Madison in 1989 and a visiting fellow at the Institute of Solid State Physics, Bulgarian Academy of Science and Institute of Radiophysics and Electronics, Czechoslovakian Academy of Science, in 1990. He is a fellow of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, a fellow of the Optical Society of America (OSA), a fellow of the Chinese Institute of Electronics (CIE), a senior member of IEEE AP, MTT, and LEOS Societies, and a member of IEE Japan and the Electromagnetic Academy. His research interests are in electromagnetic wave theory, analytical and numerical techniques in microwave and photonics, and radiation and scattering in electron beam–plasma systems. He has served as a member of organizing, steering, technical program, and international advisory committees and as a session organizer for various international conferences. He has published more than 250 papers in various international journals and conference proceedings.

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Contributors Ara A. Asatryan Centre for Ultrahigh-Bandwidth Devices for Optical Systems and Department of Mathematical Sciences University of Technology Sydney, Australia

Christian Hafner Computational Optics Group Laboratory for Electromagnetic Fields and Microwave Electronics ETH Zentrum Zurich, Switzerland

Lindsay C. Botten Centre for Ultrahigh-Bandwidth Devices for Optical Systems and Department of Mathematical Sciences University of Technology Sydney, Australia

Stefan F. Helfert Allgemeine und Theoretische Elektrotechnik University of Hagen Hagen, Germany

Hung-Chun Chang Department of Electrical Engineering Graduate Institute of Communication Engineering National Taiwan University Taipei, Taiwan, Republic of China Stefan Enoch Faculté des Sciences de Saint Jérôme Institut Fresnel Université Paul Cézanne-AixMarseille III Marseille, France Daniel Erni Communication Photonics Group Laboratory for Electromagnetic Fields and Microwave Electronics ETH Zentrum Zurich, Switzerland David P. Fussell Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics University of Sydney Sydney, Australia

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Hiroyoshi Ikuno Department of Electrical and Computer Engineering Kumamoto University Kurokami, Japan Hongting Jia Department of Computer Science and Communication Engineering Kyushu University Fukuoka, Japan Boris T. Kuhlmey Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics University of Sydney Sydney, Australia Timothy N. Langtry Centre for Ultrahigh-Bandwidth Devices for Optical Systems and Department of Mathematical Sciences University of Technology Sydney, Australia

Daniel Maystre Faculté des Sciences de Saint Jérôme Institut Fresnel Université Paul Cézanne-AixMarseille III Marseille, France

Geoffrey H. Smith Centre for Ultrahigh-Bandwidth Devices for Optical Systems and Department of Mathematical Sciences University of Technology Sydney, Australia

Ross C. McPhedran Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics University of Sydney Sydney, Australia

C. Martijn de Sterke Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics University of Sydney Sydney, Australia

Yoshihiro Naka Department of Electrical and Computer Engineering Kumamoto University Kurokami, Japan

Gérard Tayeb Faculté des Sciences de Saint Jérôme Institut Fresnel Université Paul Cézanne-AixMarseille III Marseille, France

Nicolae A. Nicorovici Centre for Ultrahigh-Bandwidth Devices for Optical Systems and Department of Mathematical Sciences University of Technology Sydney, Australia

Thomas P. White Centre for Ultrahigh-Bandwidth Devices for Optical Systems and School of Physics University of Sydney Sydney, Australia

Reinhold Pregla Allgemeine und Theoretische Elektrotechnik University of Hagen Hagen, Germany Jasmin Smajic ABB Switzerland, Ltd Corporate Research Baden-Dättwil, Switzerland

© 2006 by Taylor & Francis Group, LLC

Kiyotoshi Yasumoto Department of Computer Science and Communication Engineering Kyushu University Fukuoka, Japan Chin-Ping Yu Graduate Institute of Electro-Optical Engineering National Taiwan University Taipei, Taiwan, Republic of China

Table of Contents Chapter 1

Scattering Matrix Method Applied to Photonic Crystals ...........................................................................1 Daniel Maystre, Stefan Enoch, and Gérard Tayeb

Chapter 2

From Multipole Methods to Photonic Crystal Device Modeling ..........................................................................47 Lindsay C. Botten, Ross C. McPhedran, C. Martijn de Sterke, Nicolae A. Nicorovici, Ara A. Asatryan, Geoffrey H. Smith, Timothy N. Langtry, Thomas P. White, David P. Fussell, and Boris T. Kuhlmey

Chapter 3

Modeling of Photonic Crystals by Multilayered Periodic Arrays of Circular Cylinders .......................................123 Kiyotoshi Yasumoto and Hongting Jia

Chapter 4

Simulation and Optimization of Photonic Crystals Using the Multiple Multipole Program........................191 Christian Hafner, Jasmin Smajic, and Daniel Erni

Chapter 5

Mode-Matching Technique Applied to Metallic Photonic Crystals .........................................................225 Hongting Jia and Kiyotoshi Yasumoto

Chapter 6

The Method of Lines for the Analysis of Photonic Bandgap Structures .....................................................295 Reinhold Pregla and Stefan F. Helfert

Chapter 7

Applications of the Finite-Difference Frequency-Domain Mode Solution Method to Photonic Crystal Structures........................................................351 Chin-Ping Yu and Hung-Chun Chang

Chapter 8

Finite-Difference Time-Domain Method Applied to Photonic Crystals ...................................................................401 Hiroyoshi Ikuno and Yoshihiro Naka

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Matrix Method 1 Scattering Applied to Photonic Crystals Daniel Maystre, Stefan Enoch, and Gérard Tayeb

CONTENTS 1.1 Introduction ..................................................................................................2 1.2 Scattering Matrix Method ............................................................................3 1.2.1 Presentation of the Problem and Notation ......................................3 1.2.2 Fourier–Bessel Expansions of the Field inside the Cylinders ........5 1.2.3 Fourier–Bessel Expansions of the Field outside the Cylinders........7 1.2.4 First Set of Equations: Causality Property for Each Cylinder......11 1.2.5 Second Set of Equations: Introducing the Coupling between Cylinders ........................................................................12 1.2.6 Final Equation ..............................................................................15 1.3 Combination of Scattering Matrix and Fictitious Sources Methods..........16 1.3.1 Introduction....................................................................................16 1.3.2 Setting of the Problem ..................................................................17 1.3.3 The Method of Fictitious Sources (MFS) ....................................18 1.3.4 Implementation of the Scattering Matrix Method (SMM)............22 1.3.5 Hybrid Method Using MFS and SMM ........................................23 1.3.6 Numerical Example ......................................................................24 1.4 Dispersion Relations of Bloch Modes ......................................................25 1.4.1 Infinite Structure............................................................................26 1.4.2 Finite-Size Photonic Crystals ........................................................30 1.5 Theoretical and Numerical Studies of Photonic Crystal Properties ..........35 1.5.1 Ultrarefraction with Dielectric Photonic Crystals ........................35 1.5.2 Ultrarefraction with Metallic Photonic Crystals ..........................36 1.5.3 Negative Refraction by a Dielectric Slab Riddled with Galleries ................................................................................39 1.6 Conclusion..................................................................................................42 References ..........................................................................................................43 1

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Electromagnetic Theory and Applications for Photonic Crystals

1.1 INTRODUCTION The scattering matrix method (SMM) is one of the most efficient methods for solving a problem of scattering by a large but finite number of objects. It basically takes into account separately the specific scattering properties of each object and then evaluates the coupling phenomena between them. Even though it can be presented in a quite rigorous form, it is based on a physically intuitive approach to the problem of scattering from a set of objects. One of its advantages is to be accessible to postgraduate students. Moreover, the numerical implementation does not present major difficulties. In contrast to other classical methods like Finite Difference Time Domain method (FDTD) or the finite element method, it becomes much simpler in the case of two-dimensional (2D) photonic crystals with circular cross sections or 3D photonic crystals formed by spherical inclusions. It deals with crystals of finite size regardless of whether or not they have defects of periodicity. The first achievement relating to that method should be attributed to Lord Rayleigh, who dealt with the electrostatic case [1]. The electromagnetic version of this method has been developed since the 1980s in various forms by different groups working independently [2–9]. Their studies deal with 2D or 3D, dielectric, metallic, or perfectly conducting objects placed in space in a periodic or random way, but the essence of each of the approaches remains the same. However, the SMM is not able to deal with some interesting configurations, especially when the set of scatterers is surrounded by a jacket. To extend the method to more complicated structures, it is possible to combine the SMM with the method of fictitious sources (MFS). The MFS is another rigorous method that is able to solve the problem of scattering from arbitrary scatterers. In MFS, the field in each medium is represented as the field radiated by a set of fictitious sources with initially unknown intensities. These intensities are obtained by imposing the boundary conditions for the fields on the surfaces of the scatterers. By combining these two methods, we concurrently procure their advantages. The method is described in Section 1.3 in a 2D case and can, for instance, address structures such as a finite dielectric body pierced to form galleries, such as a photonic crystal made with macroporous silicon. More generally, the method could be useful for the study of problems dealing with small clusters of buried objects. These two methods enable one to deal with a large range of two-dimensional photonic crystals. However, the phenomena generated by photonic crystals are so surprising and so complex that a theoretician needs a preliminary phenomenological approach. For this purpose, the notion of Bloch modes provides a valuable tool. We will describe such modes, and we will show that their dispersion curves enable one to predict most of the properties of photonic crystals. Indeed, these dispersion curves enlighten us on quantities such as the average energy velocity and provide an intuitive way of grasping the vital notion of effective optical index in the case of a heterogeneous material. A special emphasis will be put on anomalous refraction phenomena and the control of light emission. Two kinds of anomalous refraction phenomena can be distinguished. In the phenomenon of ultrarefraction, a photonic crystal simulates

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Scattering Matrix Method Applied to Photonic Crystals

3

a material having an optical index between zero and unity, while in negative refraction, it acts like a material having a negative index. These phenomena will be predicted from three-dimensional dispersion diagrams and illustrated using the two methods. It will be shown that these phenomena can lead to new optical components: directive sources, microlenses, and so on.

1.2 SCATTERING MATRIX METHOD The method discussed here can deal with collections of objects having different electromagnetic parameters or different shapes. However, for simplicity, we will limit ourselves to a set of identical objects, which is in general the case for photonic crystals. Furthermore, we will confine the theory to the particular case of 2D photonic crystals with identical circular cylinders illuminated with s-polarized light (electric field parallel to the cylinder axes). The generalization to p-polarized light (magnetic field parallel to the cylinder axes) or to the case in which the cylinders are different is straightforward, while the extension to 3D photonic crystals leads to a considerable increase in the complexity of the algebraic developments.

1.2.1 PRESENTATION

OF THE

PROBLEM AND NOTATION

The scattering problem is represented in Figure 1.1. An incident monochromatic plane wave with wavelength 0  2/k0 in vacuum propagates in a homogeneous y
inc

Incident field

Z O

x

FIGURE 1.1 Presentation of the problem of scattering: an s-polarized incident plane wave illuminates N cylinders (here N  6).

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4

Electromagnetic Theory and Applications for Photonic Crystals

material of relative permittivity r,ext. The generalization to an incident beam or to a field generated by a 2D antenna is not problematic. It illuminates under s-polarization and incidence angle inc a set of N identical dielectric cylinders of radius R and relative permittivity r,int. We work under the assumption that all the materials are nonmagnetic, and the optical indices of the materials outside and inside the cylinders are denoted respectively by next   r,ext and nint   r,int . Using a time dependence in exp(it) and denoting by xˆ, yˆ and zˆ the unit vectors of the three axes in the Cartesian coordinate system xyz, the incident electric field is given by: Ei  Ezi zˆ (1.1) with: Ezi  exp(ik0 next(x sin inc  y cos inc))

(1.2)

From Maxwell’s equations, the total electric field satisfies a Helmholtz equation in the sense of distributions: 2 Ez  k02er ( x , y) Ez  0

(1.3)

er , ext outside the rods er (x , y)    e  r , int inside the rods

(1.4)

with:

The reader who is not well acquainted with the theory of distributions may consider that Equation (1.3) is valid in the sense of functions but includes in addition the boundary conditions at the interfaces between different materials (continuity of Ez and of its normal derivative). In outline, the method can be divided into three steps. The first step consists of showing that the total field can be expressed in the form of Fourier–Bessel series. Outside a cylinder, the series can be separated into two parts: the first part describes the total incident field on the cylinder. The total incident field includes not only the incident plane wave given by Equation (1.2), but also the field scattered by the other cylinders in the direction of the cylinder that is considered. The second part represents the field scattered by the cylinder. The second step is achieved by requiring that a relation of causality exists between the field scattered by a cylinder and the total incident field that illuminates the same cylinder. This relationship can be expressed in terms of the Fourier–Bessel coefficients through the notion of the scattering matrix. The third step addresses the fact that the total incident field on a cylinder is the sum of a known component (the incident plane wave) and of the fields scattered by the other cylinders. A second relation between these two parts of the field surrounding each cylinder can thereby be obtained. In contrast to the second step, the last step expresses the coupling phenomena between all the cylinders.

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Scattering Matrix Method Applied to Photonic Crystals

1.2.2 FOURIER–BESSEL EXPANSIONS

OF THE

5

FIELD

INSIDE THE

CYLINDERS

The expansion of the field in Fourier–Bessel series inside and outside the cylinders constitutes the basis of the formalism, allowing the initial problem, namely the determination of the field at any point of space, to be reduced to the evaluation of a set of complex coefficients. First, let us establish that the field inside each cylinder is described by a Fourier–Bessel series. This can be demonstrated using a local system of polar coordinates (rj, j) with an origin located at the center Oj of each cylinder (Figure 1.2). It can be shown that this property extends to noncircular cylinders, but in that case the domain of validity of the Fourier–Bessel series does not include the entire cross section of the cylinder. Obviously, the electric field is, for a given value of rj, a periodic function of j with period 2, and thus it can be represented by a Fourier series: Ez (rj , u j ) 

∑

mZ

(1.5)

Ez , m (rj ) exp(imu j )

Inside the cylinder, the total field obeys the equation: 2 Ez  k02 er ,int Ez  0


inc

(1.6)

y

Incident field P)

r j(

l

j

j

l

Oj

P

j

Ol rj

j th rod j

O

x

FIGURE 1.2 Domains of validity of the Fourier–Bessel series inside and outside the jth cylinder. The deep gray region represents the domains of validity of the Fourier–Bessel series inside the cylinder (with Bessel functions of the first kind). The light gray domain represents the domain of validity of the Fourier–Bessel series around the cylinder (with Hankel functions and Bessel functions of the first kind). The arrows l → j and j → l show the fields scattered by the lth and jth cylinders, respectively, which propagate toward the interior (and exterior) of the light gray ring.

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6

Electromagnetic Theory and Applications for Photonic Crystals

Using the expression of the scalar Laplacian in polar coordinates: 2V 

2V 1 V 1 2V   2 2 rj rj rj  2j  rj

(1.7)

Then substituting the expression of the field given by Equation (1.5) into Equation (1.6), a straightforward calculation shows that the Fourier coefficients satisfy the following equation: d 2 Ez , m

∀m,

drj2



m 2  1 Ez , m  2   k0 er ,int  2  Ez , m  0  rj  rj rj 

(1.8)

Defining rj  k0nintrj, this equation becomes: ∀m,

d 2 Ez , m drj′2



  1 Ez ,m  m2   1  2  Ez , m  0  rj′ rj′ rj′  

(1.9)

Thus, Ez,m satisfies the Bessel equation, whose general solution [10] is given by: Ez , m  c ′j ,m Jm (rj′)  d ′j , mYm (rj′)  c ′j , m Jm (k0 nint rj )  d ′j , mYm (k0 nint rj )

(1.10)

with Jm and Ym being Bessel functions of the mth order, respectively of the first and second kinds. It is interesting to express the right-hand member of Equation (1.10) in a different form: Ez ,m  (c ′j , m  id ′j , m )Jm (k0 nint rj )  id ′j , m (Jm (k0 nint rj )  iYm (k0 nint rj ))

(1.11)

and bearing in mind the definition of the Hankel function of the mth order and of the first kind: H m(1) (u)  Jm (u)  iYm (u) (1.12) Equation 1.10 can be rewritten: Ez , m  c j ,m Jm (k0 nint rj )  d j ,m H m(1) (k0 nint rj )

(1.13)

Introducing the expression of Ez,m in Equation (1.5), we are led to an expression of the total field in the cylinder in the form of a Fourier–Bessel expansion: Ez (rj , u j ) 

∑ [c j ,m Jm (k0 nint rj )  d j ,m Hm(1) (k0 nint rj )] exp(im j )

(1.14)

mZ

The expression of the field given by Equation (1.14) is best adapted to an intuitive physical interpretation of the two terms on the right-hand side. Indeed, inside

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Scattering Matrix Method Applied to Photonic Crystals

7

the cylinder, all the coefficients dj,m vanish. This is a straightforward consequence of the fact that the Hankel function Hm(1)(s) has a singularity for s  0 and that the field should not be singular inside the cylinder. Thus, if rj R,

Ez (rj , u j ) 

∑ c j ,m Jm (k0 nint rj ) exp(imu j )

(1.15)

mZ

Finally, the determination of the field inside the cylinders reduces to the calculation of the coefficients cj,m of the Fourier–Bessel expansion on the right-hand side of Equation (1.15).

1.2.3 FOURIER–BESSEL EXPANSIONS OF THE FIELD OUTSIDE THE CYLINDERS The calculations we have done for the expression of the field inside a cylinder can be reproduced almost identically for the regions surrounding the cylinders. The field outside the cylinders satisfies the following Helmholtz equation: 2 Ez  k02 er , ext Ez  0

(1.16)

We consider the ring surrounding the jth cylinder and extending to the nearest point of the closest cylinder (the light gray region in Figure 1.2). Using the same coordinate system as in the preceding section, following the same mathematical lines, and remarking that for a given value of rj, the value of the optical index remains equal to next, it can be shown that inside this region the field takes the form of another Fourier–Bessel series having the same form as that given in Equation (1.14): Ez (rj , u j ) 

∑ a j , m Jm (k0 next rj ) exp(imu j )  ∑ b j , m H m(1) (k0 next rj ) exp(imu j ) mZ

mZ

(1.17)

By contrast with the expansions inside the cylinders, it is not possible to show that the coefficients bj,m vanish, because no physical argument prevents the expression of the field in the light gray ring from being singular at the center of the cylinder, due to the fact that this center does not belong to the ring. Now, it turns out that the properties of Bessel functions allow us to give a physical meaning to each of the terms on the right-hand side of Equation (1.17). Indeed, from an intuitive physical viewpoint, the field in the ring can be separated into three different parts: • The incident plane wave (source) given by Equation (1.2). • The fields scattered by all the other cylinders toward the jth cylinder, which behave for this cylinder as incident fields. These fields add to the incident plane wave (source) to constitute the total incident field on the jth cylinder. • The field scattered by the jth cylinder, which according to the radiation condition must propagate away from the cylinder.

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8

Electromagnetic Theory and Applications for Photonic Crystals

In order to identify in Equation (1.17) each physical component of the field, let us note first that, obviously, the total incident field on the jth cylinder has been generated by sources located outside this cylinder (at infinity for the incident plane wave and inside the other cylinders for the complementary part). Thus, it must satisfy inside this cylinder a Helmholtz equation and cannot have any singularity. This property allows us to deduce that the total incident field is contained inside the first sum of Equation (1.17). Consequently the second sum represents a scattered field of the jth cylinder. The question that now arises is whether this second sum represents the totality or only a part of the field scattered by the jth cylinder. In the first case, the first sum represents the total incident field on this cylinder (and only the total incident field on this cylinder). In the second case, the first sum represents not only the total incident field on the cylinder but also a part of the scattered field. Causality properties provide the answer to this question. The field scattered by the jth cylinder must propagate away from it. It emerges that the only Fourier–Bessel functions that satisfy this condition are those containing Hankel functions of the first kind. Indeed, the asymptotic expression at infinity of Hankel functions of the first kind is given by [10]: H q(1) (k0 next rj ) 

  2 p p  exp  i  k0 next rj  q     pk0 next rj 2 4 

(1.18)

and they satisfy, at any order q, the radiation condition at infinity. So the second sum of Equation (1.17) does indeed represent a field scattered by the jth cylinder. Conversely, an arbitrary term of the first sum cannot represent a scattered field because, from Equation (1.12), Jm(s) is given by: Jm (s)  H m(1) (s)  H m(1) (s)

(1.19)

– – with H m(1) complex conjugate of H m(1)(s). Obviously, Equation (1.18) shows that –(1) H m (s) does not satisfy the radiation condition at infinity. As a consequence, the first sum in Equation (1.17) actually represents the total incident field illuminating the jth cylinder, and the second sum represents the field scattered by the same cylinder. Now let us calculate the contribution in the total incident field of the incident plane wave (source), as given by Equation (1.2). With this aim, let us express the amplitude of this field at an arbitrary point P of space (Figure 1.2) in the form: Ezi (P)  exp(i k.OP )

(1.20)

where the incident wavevector k has components (k0next sin inc, k0next cos inc). Expressing OP as the sum of OOj et OjP yields: Ezi (P)  exp(ik(OO j  O j P ))  exp(ik0next r j sin(inc  u j )) exp(ik0next r j(P) sin(inc  u j (P))) (1.21)

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Scattering Matrix Method Applied to Photonic Crystals

9

r j,  j denoting the polar coordinates of the origin Oj of the local coordinate system in the general system xy (Figure 1.2). Using Equation (1.21) and the following expansion [10]: exp(ij cos(u)) 

∑ (i)m Jm (j) exp(imu)

(1.22)

mZ

it emerges that: E zi(P)  exp(ik0next r j sin(inc  u j ))

∑ exp(iminc)Jm (k0 nextrj (P)) exp(imj(P))

(1.23)

m∈Z

Bearing in mind that Jm (s)  (1)mJm(s), m can be replaced by m in Equation (1.23), enabling us to obtain an expression that can be compared directly with the first sum of Equation (1.17): E zi(P)  exp(ik0next r j sin(inc  u j ))

∑ ()m exp(iminc)Jm (k0 nextrj (P)) exp(imj(P))

(1.24)

m∈Z

Equations (1.23) and (1.24) provide the development of the source incident field in Fourier–Bessel series in the coordinate system linked to the jth cylinder. Coming back to Equation (1.17), it becomes possible to distinguish the coefficients of the Fourier–Bessel series associated with the source field from those associated with the field scattered by the other cylinders. They will be denoted by source rods aj,m and aj,m , respectively, their sum being equal to aj,m. Equation (1.24) allows source us to identify aj,m : asource  (1)m exp(ik0 next r j sin(inc  u j )iminc) j ,m

(1.25)

Finally, the three components of the field in the ring surrounding the jth cylinder can be expressed respectively by the following Fourier–Bessel series: • The source term (incident plane wave): Ezsource (rj , u j )  ,j

∑ asource j , m J m (k0 next rj ) exp(imu j )

(1.26)

mZ

• The field scattered by the other cylinders toward the jth cylinder: Ezrods , j (rj , u j ) 

© 2006 by Taylor & Francis Group, LLC

∑ a rods j , m J m (k0 next r j ) exp(imu j )

mZ

(1.27)

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Electromagnetic Theory and Applications for Photonic Crystals

• The field scattered by the jth cylinder: Ezd, j (rj , u j ) 

∑ b j , m Hm(1) (k0 next rj ) exp(imu j )

(1.28)

mZ

It has been established that the total field in the ring can be expressed from source rods three series of coefficients: aj,m (known), aj,m (unknown) and bj,m (unknown). Now let us demonstrate a vital property of the field scattered by a cylinder. The expression of the field scattered by the jth cylinder in the ring around this cylinder given by Equation (1.28) extends in fact to the entire space surrounding this cylinder. This fundamental property is very intuitive: the field scattered by the jth cylinder is produced by sources located inside this cylinder and thus can be defined in the entire space around the cylinder. In order to give a mathematical demonstration of this property, let us rewrite Equation (1.3) in the form: 2 Ez  k02 er , ext Ez  (k02 er , ext  k02 er ( x , y)) Ez

(1.29)

Bearing in mind that the incident field satisfies the Helmholtz equation 2Eiz  k20r,ext, Ezi  0 and subtracting this equation from Equation (1.29), it can be derived that the field Edz scattered by the entire set of cylinders, defined at any point of space as the difference between the total field and the source incident field (the incident plane wave), satisfies the equation: 2 Ezd  k02 er , ext Ezd  (k02 er , ext  k02 er ( x , y)) Ez

(1.30)

Hence, the scattered field at any point P of space outside the cylinders can be expressed using Green’s theorem: Ezd ( P) 

i 2 k H (1) (k n PM )(er , ext  er ( M )) Ez ( M ) dx dy 4 0 ∫∫ 0 0 ext

(1.31)

with x and y coordinates of M in the general coordinate system. Notice that the integral on the right-hand side of Equation (1.31) can be restricted to the set of cylinders since r,ext  r(M) vanishes outside these cylinders. Consequently, the scattered field can be represented as a sum of integrals on the cylinders: Ezd ( P) 

Ezd, j ( P) 

ik02 (er , int  er , ext ) 4

∑

j1, 2 ,… , N

∫∫

Ezd, j ( P)

(1.32)

H 0(1) (k0 next PM ) Ez ( M ) dx dy (1.33)

j th cylinder

The field Edz,j can be identified as the field scattered by the jth cylinder. Equation (1.33) clearly shows that it can be defined in the entire space and satisfies outside

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Scattering Matrix Method Applied to Photonic Crystals

11

the jth cylinder the Helmholtz equation: 2 Ezd, j  k02 er , ext Ezd, j  0

(1.34)

As a consequence, the expression of the field scattered by the jth cylinder given by Equation (1.28) extends to the entire space surrounding this cylinder, and from Equation (1.32), the total scattered field is obtained by summing the fields scattered by the entire set of cylinders: N

Ezd ( P)  ∑

∑ b j , m Hm(1) (k0 next rj ) exp(imu j )

(1.35)

j1 mZ

1.2.4 FIRST SET OF EQUATIONS: CAUSALITY PROPERTY CYLINDER

FOR

EACH

We have given an expression for the field surrounding a given cylinder (Equation (1.17)) that separates the total incident field on this cylinder and the field scattered by the same cylinder. These two fields are linked by causality, and this link can be expressed mathematically through the use of a scattering matrix: b j  S ja j

(1.36)

In this equation, we have introduced the infinite column matrices aj and bj with components aj,m and bj,m, and Sj is a square matrix of infinite dimension. When the cylinders have circular cross section s, the scattering matrix can be calculated in closed form by writing the boundary conditions on the periphery of a cylinder. Equation (1.3) being valid in the sense of distributions, the electric field and its normal derivative are continuous across this boundary (this property can also be proved from the continuity of the tangential components of the electric and magnetic fields). Using the expansions of the electric field inside and outside the cylinder given by Equations (1.15) and (1.17), then identifying the Fourier coefficients on both sides for rj  R, it can be deduced that: a j , m Jm (k0 next R)  b j , m H m(1) (k0 next R)  c j , m J m (k0 nint R)

a j,m

d ( Jm (k0 next rj ))

 c j,m

drj

 b j,m r j R

d ( Jm (k0 nint rj ))

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drj

(1.37)

d ( H m(1) (k0 next rj )) drj

r j R

(1.38) r j R

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Electromagnetic Theory and Applications for Photonic Crystals

Eliminating coefficients cj,m between Equations (1.37) and (1.38), it turns out that: Jm (k0 next R) b j , m  H m(1) (k0 next R)

d ( Jm (k0 nint rj )) drj

 Jm (k0 nint R)

drj

r j R

d ( Jm (k0 nint rj )) drj

d ( Jm (k0 next rj ))

 Jm (k0 nint R)

r j R

a j,m

d ( H m(1) (k0 next rj ))

r j R

drj

r j R

(1.39) Note that the variable rj in this equation can be replaced by a variable r independent of the cylinder. Indeed, since the cylinders are identical, this relation of causality is independent of the particular cylinder considered. Thus, all the matrices Sj defined by Equation (1.36) are diagonal and equal to the matrix S defined by: Jm (k0 next R) Sm , m  H m(1) (k0 next R)

d ( Jm (k0 nint rj )) drj

 Jm (k0 nint R) rj R

d ( Jm (k0 nint rj )) drj

 Jm (k0 nint R) rj  R

d ( Jm (k0 next rj )) drj

rj R

d ( H m(1) (k0 next rj )) drj

rj R

(1.40) and Sn , m  0

if n ≠ m

(1.41)

Obtaining the scattering matrix for noncircular cylinders is much more difficult because this calculation is no longer analytic. In this case, numerical techniques can be employed instead, for example the FDTD method, a finite element method, or the method of fictitious sources. The primary advantage of the scattering matrix method is that the use of a classical way of considering scattering is restricted to a single cylinder rather than to the entire set of cylinders. When the cylinders do not have the same radii, or the same permittivities, the scattering matrices cease to be identical, but they nevertheless remain diagonal if the cylinders are circular. The extension of the method to the p-polarization case is straightforward: the only change is in the calculation of the Sj matrices.

1.2.5 SECOND SET OF EQUATIONS: INTRODUCING THE COUPLING BETWEEN CYLINDERS First let us give the intuitive physical background of the introduction of the coupling phenomena between cylinders. The total incident field contains two components: the source term (known) and the field scattered by the other cylinders toward the jth rods cylinder (unknown). The latter, represented by coefficients aj,m , is nothing but the field scattered by all the other cylinders toward the jth cylinder. Now, the field

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Scattering Matrix Method Applied to Photonic Crystals

13

scattered by an arbitrary cylinder is given by the second series of Equation (1.17); rods therefore, we can deduce easily that it is possible to express the coefficients aj,m in terms of the set of coefficients bl,m with l  j. However, we must overcome a mathematical problem: the incident field on the jth cylinder is expressed in a local coordinate system linked to this cylinder, while the fields scattered by the other cylinders are expressed in the local coordinate system associated with those other cylinders. To solve this problem, the fields will be expressed in a unique coordinate system, that associated with the jth cylinder. With this aim, an adequate mathematical tool is provided by the Graf formula [10]. This formula gives in a mathematical form a very intuitive result: the field scattered by the l th cylinder toward the j th cylinder satisfies around this last cylinder a Helmholtz equation and does not possess singularities. As a consequence, it can be expressed in the form of a Fourier–Bessel series involving exclusively Bessel functions of the first kind. This formula can be written using the notation of Figure 1.3 as: if rj(P) rjl  OjOl, H m(1) (k0 next rl ( P)) exp(imul (P))  ∑ exp(i(m  q)ulj )H q(1) m (k0 next rjl ) Jq (k0 next rj ( P)) exp(iqu j (P)) (1.42) q∈Z

y rj (P)

P

j (P )

Oj
inc

j l

l

j rj rl =

rj r

Ol j

O

FIGURE 1.3 The coordinate systems.

© 2006 by Taylor & Francis Group, LLC



x

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Electromagnetic Theory and Applications for Photonic Crystals

Equations (1.17) and (1.42) enable us to express the field scattered by the lth cylinder toward the jth cylinder in the coordinate system of origin Oj. Thus, if rj(P) rjl  R: Ezd,l ( P) =

∑ bl , m ∑ exp(i(m − q) lj )Hq(1−) m (k0 next rjl )Jq

m ∈Z

q∈ Z

(k0 next rj ( p)) exp(iq j ( P ))

(1.43)

or, by exchanging indices m and q: Ezd,l ( P)  ∑ bl , q ∑ exp(i(q  m)ulj )H m(1) q (k0 next rjl ) Jm (k0 next rj ( P)) q∈ Z

m ∈Z

exp(imu j ( P))

(1.44)

By summing the contributions of the scattered fields Edz,l, with l j, we must rods reconstitute the component Ez,j on the jth cylinder, given by Equation (1.27). The identification leads to the following relation: a rods j,m  ∑

∑ bl , q exp(i(q  m)ulj )Hm(1) q (k0 next rjl )

(1.45)

l ≠ j q∈ Z

source of the source field (Equation (1.25)) to the coeffiBy adding the coefficient aj,m rods cients aj,m , it is possible to express aj,m in terms of the coefficients bl,q of the fields scattered by the other cylinders:

a j , m  (i)m exp(ik0 next r j sin(inc  u j )iminc) ∑

∑ bl , q exp(i(q  m)ulj )Hm(1) q (k0 next rjl )

(1.46)

l ≠ j q∈ Z

In matrix form, Equation (1.46) can be written a j  Q j  ∑ Tj ,l bl

(1.47)

l≠ j

where Qj is a column matrix with mth component Qj,m given by: Q j , m  (1)m exp(ik0 next r j sin(inc  u j )iminc)

(1.48)

and Tj,l is a square matrix with (m, q)th component Tj,l,m,q given by: T j ,l , m , q  exp(i(q  m)ulj ) H m(1) q (k0 next rjl )

(1.49)

Equation (1.47) is a set of N matrix equations relating the 2N column matrices aj and bj.

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Scattering Matrix Method Applied to Photonic Crystals

15

1.2.6 FINAL EQUATION Equations (1.36) and (1.47) constitute a set of 2N matrix equations with 2N unknown column matrices aj and bj. A single matrix equation can be obtained by multiplying both sides of Equation (1.47) by Sj  S, Equation (1.36) being used for simplifying the left-hand side. We thus obtain b j  ∑ STj ,l bl  SQ j

(1.50)

l, j

or, in a more explicit form:  I ST1,2 ST1,3   I ST2,3 ST2,1  ST I 3,1 ST3,2    … … …   … … … 

… …  b1   SQ1      … …  b2   SQ2      … …  b3    SQ3      … …  …   …      … …  …   … 

(1.51)

In this way, we get an infinite linear system of equations, I being the infinite unit matrix. In order to limit the size of the system, matrices S, Tj,l, Qj and bl are truncated by restricting the values of m and q between M and M, the final size of the system being equal to N(2M + 1). The field at infinity can be deduced easily from the column matrices bl. With this aim, the expression of the field scattered outside each cylinder (right-hand side of Equation (1.28)) is transformed to the unique coordinate system xy. The change of coordinate system from the local system linked with a cylinder to xy can be done by applying Graf’s formula [10]: if r  rj, H m(1) (k0 next rj (P)) exp(imu j (P ))

 ∑ exp(i(m  q)u j )Jqm (k0 next r j )H q(1) (k0 next r ) exp(iqu)

(1.52)

qZ

with r and  being the polar coordinates of a point P in space in the xy coordinate system. Thus it turns out that at a large distance from the cylinders: Ezd (P)  ∑ bq H q(1) (k0 next r ) exp(iqu)

(1.53)

qZ

where the bq coefficients (not to be confused with the bj column matrix defined before) are: bq 

∑ ∑ b j , m exp(i(m  q)u j )Jqm (k0 next r j )

j1,N mZ

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(1.54)

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Electromagnetic Theory and Applications for Photonic Crystals

The field at infinity is obtained by using the asymptotic form of the Hankel function at infinity, given by Equation (1.18). Equation (1.53) then becomes: Ezd ( P)  g( )

exp(ikr )

(1.55)

r

with: g( ) 

 p  2 p exp i  ∑ bq exp iq  exp(iqu) pk0 next 2   4  q , 

(1.56)

The bistatic differential cross section, which represents the intensity of the field at infinity, is given by: D(u)  2p|g(u)|2

(1.57)

When all the materials (cylinders and exterior) are lossless, the energy balance criterion (also called the optical theorem) can be expressed in the form: 2p

∫0

|g(u)|2 du  2



  p 2p Re  exp  i  g(inc  p)  0 k0 next  4  

(1.58)

This theorem can be valuable for testing the validity of a numerical code, although like casting out nines, it does not provide a rigorous verification of validity.

1.3 COMBINATION OF SCATTERING MATRIX AND FICTITIOUS SOURCES METHODS 1.3.1 INTRODUCTION In this section, we present a method that combines the scattering matrix method (SMM) described in Section 1.2 with the method of fictitious sources (MFS) in order to deal with more complicated structures. Although the basic ideas can be generalized to 3D cases, we will consider only 2D cases for simplicity. Note that the SMM presented in Section 1.2 cannot deal with structures in which the set of cylinders is embedded inside a jacket consisting of a dielectric medium that is itself surrounded by another external medium (Figure 1.4). This limitation can be made to disappear by using the results of the present section. The MFS method [11] can solve the problem of scattering from arbitrarily shaped scatterers. The space is divided into different regions in which the field is represented as the field radiated by a set of fictitious sources with unknown intensities. These intensities are obtained by matching the fields at the boundaries of the regions using a least squares technique. Returning to Figure 1.4, the basic idea of the method proposed in this section is to use the SMM in order to build a set of functions that correctly represent the

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Scattering Matrix Method Applied to Photonic Crystals

17

ui Ωe Ωi

0 ne

2 1 ni

3

y

x

FIGURE 1.4 Description of the problem: a set of cylinders inside a jacket (shaded region).

field inside the jacket. These functions are then used to solve a fictitious sources problem on the boundary 0 of the jacket. Combining the SMM and MFS methods offers the advantages of both rigorous methods and enables us efficiently to address new classes of problems, such as a finite dielectric body riddled by galleries.

1.3.2 SETTING

OF THE

PROBLEM

The cylindrical scatterer is delimited by its external boundary 0. The exterior of 0 is the domain e, filled with a medium of optical index ne (ne may be complex, and we define ke  k0ne). The interior of 0 is the domain i. It is filled with a medium of optical index ni (shaded region in Figure 1.4; ni may be complex, and we put ki  k0ni ). The domain i also contains cylinders with boundaries j ( j  1, 2, 3, . . .), filled with arbitrary media. The structure is illuminated by an incident field coming from the exterior (this assumption is made for clarity, but the method also can deal with an excitation from inside 0 with very little change). This incident field is also assumed to be z-independent. For instance, it can be a plane wave, or the field emitted by one (or several) line source(s) placed outside 0. It is well known that in this case the problem can be reduced to two independent problems: the s-polarization case where the electric field is parallel to the z axis, and the p-polarization case where the magnetic field is parallel to the z axis. Each of these cases leads to a scalar problem in which the unknown u is the z component of either E or H: u  Ez (for s-polarization) or u  Hz (for p-polarization). We denote by ui the incident field, and by ud the scattered field, in such a way that the total field is: u  ui  u d  u  u int 

© 2006 by Taylor & Francis Group, LLC

in e (outside  0 ) in i (inside  0 )

(1.59)

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Electromagnetic Theory and Applications for Photonic Crystals

1.3.3 THE METHOD

OF

FICTITIOUS SOURCES (MFS)

The MFS is a versatile and reliable method to deal with many scattering problems. It relies upon a simple idea: the electromagnetic field in the various domains of the diffracting structure can be expressed as a combination of fields radiated by suitable electromagnetic sources. These sources have no physical existence, and this is why they are denoted as fictitious sources. They are located in homogeneous regions and not on the interfaces. In other words, one can consider that they generate electromagnetic fields that taken together faithfully reproduce the actual field and as such form a convenient basis for this field. From a numerical point of view, proper bases are those capable of representing the solution with the fewest possible functions. Obviously, the quality of the bases is closely linked with the nature of the sources and their location. The freedom in the choice of the sources provides a great adaptability to various complex problems. The MFS has been developed in our laboratory in the last decade from both theoretical and numerical points of view [11–15]. Almost at the same time and independently, other groups have worked on the same basic ideas [16–21], but their approaches are slightly different from ours. In fact, one of the first attempts at using this method is probably due to Kupradze [22]. The method has been developed and applied to a large collection of problems, and a good review can be found in [23]. It is not our intention to depict here all the details of the MFS, and it will be sufficient for our present purposes to give an outline of the general principles. Let us consider the same situation as in Figure 1.4 but without the inclusions: the interior region i is thus filled with a homogeneous material of optical index ni (Figure 1.5). Let us introduce a set of Ns fictitious sources Se,n (n  1, 2, …, Ns) Ωe ud ui

S i,n

S e,n

0 Ωi u int

FIGURE 1.5 The sources Se,n (represented by dots) radiate the fields Fe,n(r) used to represent the scattered field ud in e, whereas the sources Si,n (represented by stars) radiate the fields Fi,n(r) used to represent the total field uint in i.

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Scattering Matrix Method Applied to Photonic Crystals

19

located at Ns points re,n in i, which are supposed to radiate in a homogeneous space filled with a medium of index ne. Let us denote by Fe,n(r) the field radiated by the source Se,n. By construction, the fields Fe,n(r) satisfy Maxwell’s equations in e, and a radiation condition at infinity. If the coefficients ce,n are conveniently chosen, a linear combination ∑nce,n Fe,n(r) can thus be regarded as an approximation d u (r) of the diffracted field ud(r) in e, where ce,n can be understood as the complex amplitude of the source Se,n. So we obtain an approximation for the field in e: def N s

u d (r )  ∑ ce , n Fe , n (r ), n1

in e

(1.60)

and thus: Ns

u(r )  ui (r )  ∑ ce , n Fe , n (r),

in e

n1

(1.61)

In the same way, we imagine another set of fictitious sources Si,n (n  1, 2, …, Ns) located at Ns points ri,n in e and supposed to radiate in a space filled with a medium of index ni. Let us denote by Fi,n(r) the field radiated by the source Si,n. The functions Fi,n(r) satisfy Maxwell’s equations in i and can be used to get an approximate expansion uint(r) of the total field uint(r) in i: def N s

u int (r )  ∑ ci , n Fi , n (r ), n1

in i

(1.62)

where ci,n can be understood as the complex amplitude of the source Si,n. Note that the nature of the sources can be chosen arbitrarily. In our case, we choose infinitely thin line sources parallel to the z axis: Se,n(r)  4i (r  re,n) and Si,n(r)  4i (r  ri,n). In that case, they radiate the fields: Fe, n (r )  H 0(1) (ke |r  re , n |)

(1.63)

Fi , n (r )  H 0(1) (ki |r  ri , n |)

(1.64)

The determination of the coefficients ce,n and ci,n is accomplished by matching the boundary conditions on the cylinder surface 0. For a given function (r), let us denote by D (r) the value of its normal derivative on 0. The exact solution satisfies ui (r )  u d (r )  u int (r )  0   Dui (r )  Du d (r )  pDu int (r )  0 

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on  0 on  0

(1.65)

20

Electromagnetic Theory and Applications for Photonic Crystals

where p is a polarization-dependent constant equal to  p  1   p  (n / n )2 e i 

in s-polarization case in p-polarization case

(1.66)

The coefficients ce,n and ci,n that give the best approximation for the fields given by Equations (1.60) and (1.62) are those that match the boundary conditions most satisfactorily. They are obtained by minimizing the two expressions V1 and V2 derived from Equation (1.65) and defined on 0: Ns Ns   i (r )  V  u c F ( r )  ∑ e, n e, n ∑ ci , n Fi , n (r)  1 n1 n1    Ns Ns V  Dui (r )  c DF (r )  p c DF (r ) ∑ ∑ e, n e, n i,n i,n  2 n1 n1 

(1.67)

This can be done in several ways. The simplest is to use a point-matching method that enforces the vanishing of these two expressions on sample points on 0. In this way, we can get a system of 2Ns equations for the 2Ns unknowns ce,n and ci,n. From our numerical experiments, it has emerged that it is preferable to use an overdetermined system and to solve it in the least-squares sense [24]. Thus, for a given computation time, we obtain a better approximation for ud and uint. Of course, the efficiency of the method depends on the locations and the number Ns of the fictitious sources. It can be shown that the precision of the method is related to the least-squares remainder obtained in the last step. This remainder can thus be used to quantify the quality of the solution, and this quantity is quite helpful in the numerical implementation. The interested reader will find more details in [11]. We have also developed some tricks in order to place the sources automatically. The general idea used in these tricks is to increase the density of sources in regions where the radius of curvature of 0 is lower. An example is given in Figure 1.6. The cross section of the cylinder 0 mimics a rounded F letter (first letter of Fresnel Institute), and it is given by the parametric equation z (t )  x (t )  iy(t ) 

∑

n5, 5

cn exp(in 2pt )

(1.68)

The values of the coefficients cn are given in the caption to Figure 1.6. We use Ns  200 sources in each region e and i, and 2Ns sample points on 0. This cylinder of index ni  1.5 lies in a vacuum (ne  1) and is illuminated with an incidence angle inc  45° by a plane wave of wavelength 0  2 and unit amplitude. Note that, contrary to Section 1.2, the angles are measured with respect to the x axis. Figure 1.7 gives the intensity D() scattered at infinity in the direction . Let us recall the definition of D() already given in Section 1.2. Due to the asymptotic

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Scattering Matrix Method Applied to Photonic Crystals

21

5

y axis

 0

 inc

−5

−5

0 x axis

5

FIGURE 1.6 Cross section of the cylinder and the two sets of sources. The profile is given by Equation (1.68) and the values c5  0.1134  i0.1310, c4  0.0297  i0.3238, c3  0.4117  i0.0973, c2  0.1260  i1.4149, c1  2.3936  i2.4031, c0  0.5714  i0.5000. c1  1.5568  i0.1876, c2  0.1212  i0.0197, c3  0.8158  i0.2155, c4  0.2772  i0.1039, c5  0.1532  i0.0102.

1000

Intensity at infinity D ()

s -polarization p -polarization 100

10

1

0.1 0

30

60

90 120 150 180 210 240 270 300 330 360 Diffraction angle 

FIGURE 1.7 Scattered intensity at infinity for both polarizations.

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22

Electromagnetic Theory and Applications for Photonic Crystals

8 6 4

y axis

2



0

 inc

−2 −4 −6 −8

−8

−6

−4

−2

0 x axis

2

4

6

8

FIGURE 1.8 Modulus of the total field in p-polarization.

behavior of the Hankel function, the scattered field at infinity can be written from Equations (1.60 and 1.63) as: u d (r )  g(u)

exp(ike r )

(1.69)

r

and the intensity scattered at infinity is D(u)  2p|g(u)|2

(1.70)

Finally, Figure 1.8 shows the resultant field map in the vicinity of the scatterer.

1.3.4 IMPLEMENTATION

OF THE

SCATTERING MATRIX METHOD (SMM)

We apply the SMM to the N cylinders j, as shown in Figure 1.9. In order to lay the groundwork for section 1.3.5, we assume that the medium outside the cylinders has index ni defined in section 1.3.2. Contrary to our assumption in Section 1.2, we do not suppose here that the cylinders have a circular cross section. All the developments done in Section 1.2 are still valid, and the only difference is as follows. For any cylinder j, we consider a circle j with center Oj, in such a way that the cylinder is completely inside j (Figure 1.10). Via Equation (1.35), the SMM

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Scattering Matrix Method Applied to Photonic Crystals

23

Line source

ni

2 1

n2

3

n1 n3

FIGURE 1.9 A set of N  3 parallel cylinders in a medium with index ni, in which case the incident field is created by a line source. yj

j xj

Oj

y

j

x

O

FIGURE 1.10 Circle j surrounding the cylinder j and local coordinate system.

provides the set of coefficients bj,m that enables us to write the total field u everywhere outside the circles j: N

u( P )  u i ( P )  ∑

∑ b j , m H m(1) (ki rj (P)) exp(imu j (P))

(1.71)

j1 mZ

The consequence is that everywhere outside the circles j, the total field u and also its derivatives are known closed form from Equation (1.71).

1.3.5 HYBRID METHOD USING MFS AND SMM Let us come back to the original problem described in section 1.3.2. This problem can be solved by a slight modification of the method of fictitious sources described in section 1.3.3. Indeed, the scattered field can still be expressed as in Equation (1.60) using the same fictitious sources Se,n (line sources that radiate Fe,n(r) fields expressed as Hankel functions exactly as in Equation (1.63)). However, in that case, the Fi,n(r)

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24

Electromagnetic Theory and Applications for Photonic Crystals Si,n

ni

0 2 1 ni

3

FIGURE 1.11 Posing of the problem in order to get the functions Fi,n(r).

functions used to expand the field in i (inside 0) must be changed. Let us consider the problem depicted in Figure 1.11. The inclusions j are immersed in a medium with index ni (the boundary 0 of the external scatterer is suppressed). We keep the same line sources Si,n as in Section 1.3.3. The new function Fi,n(r) is the total field when the structure of Figure 1.11 is excited by the source Si,n. By solving this problem using the SMM as proposed in Section 1.3.4, Fi,n(r) can be expressed by the series provided by Equation (1.71). In other words, Fi,n(r) (n  1,2, … ,Ns) is a set of solutions for the total field inside 0 that are available in closed form and that can be used to expand the field in i following Equation (1.62). Using Equation (1.71), we can compute the value of Fi,n(r) and its normal derivative on 0 and get the righthand sides of Equation (1.67) to be minimized. This minimization gives the coefficients ce,n and ci,n, and we finally get the expressions of the total field in closed form everywhere using Equation (1.61) in e and Equation (1.62) in i.

1.3.6 NUMERICAL EXAMPLE Let us now illustrate the capabilities of the method by applying it to a concrete example. As in section 1.3.3, the cross section of the cylinder 0 is a letter F, but now with sharp edges. The reason for this choice is only to prove that the method also works pretty well in that case, which is more difficult to solve than a cylinder with rounded boundaries. All the coordinates of 0 corners in the (x, y) plane have integer values that can be deduced from Figure 1.12. The profile 0 is described by a series identical to Equation (1.68), but in this case we use a large number of cn coefficients (n  100, 100) in order to get a quasi-polygonal shape. This scatterer of index ni  1.5 lies in vacuum (ne  1). There are also four inclusions inside 0. The elliptical inclusion has principal axes with half dimensions equal to 1 and 0.5; its center is placed at coordinates (1, 4); its principal axes are rotated 45° away from the (x, y) axes; and it is made of infinitely conducting material. The rectangular inclusion is

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Scattering Matrix Method Applied to Photonic Crystals

25

10 8 6 4

y axis

2 0 −2 −4  inc

−6 −8 −10

−10

−8

−6

−4

−2

0 2 x axis

4

6

8

10

FIGURE 1.12 Modulus of the total field for a scatterer with four inclusions.

placed in the region 1.5 x 0.5 and 2 y 0 and is filled with vacuum. One of the circular inclusions has its center in (2,1) and a radius of 0.5, and it is filled with vacuum. The second circular inclusion has its center in (3,4), a radius of 0.5, and it is filled with a lossy material of optical index 0.5  2i (typical value for a metal in the optical range). This structure is illuminated with an incidence angle inc  45° by a plane wave of wavelength 0  2 and p-polarization. In that case, the number of sources in each region e and i is chosen to be Ns  500. The total Hz field map is shown in Figure 1.12. Note that the present version of our numerical code does not allow the computation of the field inside a circle that includes elliptical or rectangular bodies. This is why dark areas appear around these two inclusions.

1.4 DISPERSION RELATIONS OF BLOCH MODES In this section, we develop useful theoretical tools to understand and design the properties of devices based on photonic crystals. We summarize the properties of the propagating Bloch modes in infinite photonic crystals and pay special attention to the propagation of energy. Our aim is to predict the behavior of the finite structure from the knowledge of the properties of the infinite crystal. We then apply these tools to understand how a photonic crystal can behave as a homogeneous material with an optical index smaller than one (to illustrate ultrarefraction) or even with a negative optical index (to illustrate negative refraction). We consider

© 2006 by Taylor & Francis Group, LLC

26

Electromagnetic Theory and Applications for Photonic Crystals y

x

FIGURE 1.13 Finite-size two-dimensional photonic crystals made of a finite number of cylinders. The structures could be illuminated by an external beam. The coordinate system used is also represented.

two-dimensional photonic crystals made of a finite number of dielectric cylinders placed in a vacuum. The cylinders are assumed to be infinite along the z-axis (the coordinate system is defined in Figure 1.13).

1.4.1 INFINITE STRUCTURE We first review the properties of the allowed electromagnetic propagating modes in the infinite photonic crystal. Of course, if there are no such propagative modes for a frequency range we have detected a bandgap. We assume that the crystal fills the whole space and that there is no incident field. To be consistent with the previous parts, we consider two-dimensional problems. Consequently, two fundamental cases of polarization exist and the examples are given for the s-polarization case; that is, when the electric field is parallel to the cylinder axes. If the periodicity of the structure is defined by the vectors d1 and d2, then for any integer values of l and m: er (r  ld1  md 2 )  er (r )

(1.72)

where r(r) is the relative permittivity. It is now well known that the allowed propagative modes in photonic crystals are Bloch modes (as for electrons in crystals). As in the previous parts we assume an exp(it) time dependence and use the usual complex amplitudes for harmonic fields. Thus, for any Bloch mode, the z-component of the electric field can be written in the form: uk (r )  exp(ik  r )v(r )

© 2006 by Taylor & Francis Group, LLC

(1.73)

Scattering Matrix Method Applied to Photonic Crystals

27

0.6

0.5

d /

0.4

0.3 ky

M (/a,/a)

0.2 kx

0.1

Γ

X

0.0 Γ

M

X

M

FIGURE 1.14 Dispersion relation of the Bloch modes in a photonic crystal made of dielectric cylinders with permittivity r  9 lying in a vacuum. The circular cross section of the cylinders has a radius equal to 0.475. The cylinders are arranged on a square lattice with periods d  d1  d2  1.27. The field is s-polarized. The small insert represents the reduced first Brillouin zone.

where v(r) is a periodic function: v(r  ld1  md 2 )  v(r )

(1.74)

and k is the Bloch wavevector, which is usually assumed to be real. Several methods are now available for calculating the dispersion relation (k) of the Bloch modes [7, 25–27]. Irrespective of the method used, one can obtain a dispersion diagram of the sort found in the literature, such as that shown in Figure 1.14 (the parameters of the structure are given in the figure caption). On this diagram, the normalized frequency d/(2c)  d/0(d  ||d1||  ||d2||) represented as a function of the Bloch wavevector k, whose termination point makes a path along the edges of the irreducible portion (see insert) of the first Brillouin zone of the crystal considered. For our purposes it is convenient to give a more complete representation of the dispersion relation as shown on Figure 1.15. Again the normalized frequency is plotted as a function of the Bloch wavevector, but now the full area of the first Brillouin zone is taken into account explicitly. Note that the diagram in Figure 1.14 is nothing but the intersection between the surfaces of Figure 1.15 and vertical planes passing through the edges of the irreducible portion of the first Brillouin zone (represented by the lines in the (kx, ky) plane).

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Electromagnetic Theory and Applications for Photonic Crystals 0.60 0.55 0.50 0.45 0.40

d/()

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.0

2.0 1.5 0.5

1.0

1.0

0.5

1.5

Kx

2.0

Ky

0.0

FIGURE 1.15 Three-dimensional dispersion diagram. The parameters are identical to those of Figure 1.14. The lines in the (kx, ky) plane represent the reduced first Brillouin zone.

We consider now the frequency region just above the second bandgap. Figure 1.16 is an enlarged view of the dispersion relation in this region. Two bands represented by two surfaces on this figure exist: one that ascends and another that descends when the Bloch wavevector components increase. Let us now consider a harmonic problem, that is to say at a given frequency. The Bloch modes that can exist are given by the lines of intersection between the surfaces and the horizontal plane corresponding to this frequency. In our example we consider the plane corresponding to the value 0  2.545 (passing through the boundary between the two differently shaded regions on the surfaces of Figure 1.16). If we represent these intersections in the (kx, ky) plane and complete the diagram using the symmetries of the reciprocal lattice we obtain the constant frequency dispersion diagram of Figure 1.17. In order to study the transport of energy inside the crystal, let us recall a fundamental result linking the dispersion relation and the energy propagation [28,29]. For a given Bloch wave, we denote by Ve the averaged velocity of the energy flow (the average is taken over a lattice cell):

Ve 

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∫∫cell (Poynting vector) dr ∫∫cell (energy density) dr

(1.75)

Scattering Matrix Method Applied to Photonic Crystals

29

0.53 0.52

d/()

0.51 0.50 0.49 0.48 0.47 0.46 0.0

2.0 0.5

1.5 1.0 K

1.0

1.5

x

0.5

2.0

Ky

0.0

FIGURE 1.16 Enlarged view of the three-dimensional dispersion diagram. ky

y

2.0

1.0

x −2.0

−1.0

Dispersion curves

1.0

2.0

kx

−1.0

−2.0

FIGURE 1.17 Constant-frequency dispersion diagram for 0  2.545. The arrows indicate the direction of the energy velocity for two specific Bloch waves.

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30

Electromagnetic Theory and Applications for Photonic Crystals

and by Vg the group velocity of the same Bloch wave, deduced from the dispersion relation (k) by: Vg  grad k (v) 

v v xˆ  yˆ kx k y

(1.76)

The result is that Ve  Vg

(1.77)

Thus, in Figure 1.17 the direction of Vg is perpendicular to the curves and points toward the ascending side of the sheets. Note that, of course, the direction of average phase velocity is given by k and its modulus is /||k||. Thus it can be completely different from the group velocity.

1.4.2 FINITE-SIZE PHOTONIC CRYSTALS We now show how we can use the dispersion relations of Bloch modes to predict or to design the properties of photonic crystal based devices. The key point is that the tangential component (in this case, the x-component) of the Bloch wavevector is conserved at the boundaries of a slice of photonic crystal. This result is rigorously satisfied for an infinite slice of photonic crystal along the x-axis [29,30]. Consider a crystal with finite thickness modeled as a stack of N grids. The structure is still infinite along the x-direction. Thus, each grid is equivalent to a grating, and we denote by d the period of this grating along the x axis. We assume that this crystal is illuminated by an incident plane wave: ui ( x , y)  exp(iax  iby)

(1.78)

As is well known, the total field u(x, y) is a pseudoperiodic function of x [31] with pseudoperiodicity coefficient ; that is: ua ( x  d , y)  exp(iad ) ua ( x , y)

(1.79)

In this case, it is possible to find a relationship between the allowed Bloch wavevector in the infinite photonic crystal and the pseudoperiodicity coefficient . A detailed explanation of this link is given in Reference [30]. Here, we only give the conclusion and a more intuitive interpretation. From Equation (1.73), a Bloch mode (in an infinite crystal) is characterized by the fact that the elementary translations ld1  md2 only give rise to phase shifts on the fields. As regards the grating case (crystal with finite thickness), the phase shift due to the first translation along the x-axis is automatically taken into account by the pseudoperiodicity. Equations (1.73) and (1.79) match when (1.80) k a x

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Scattering Matrix Method Applied to Photonic Crystals

31

Now, we assume that this crystal is illuminated by an arbitrary incident electromagnetic field. For instance, it can be excited by a localized source. Then, the relevant field component can be written as a Fourier integral: u ( x , y)  ∫





uˆ (a, y) exp(iax )da

(1.81)

By splitting the integration interval ], [ into subintervals [(n  (1/2))(2/d), (n  1/2))(2/d)], a simple change of variable leads to the alternative expression: u ( x , y)  ∫

p/d

p/d

ua ( x , y)da

(1.82)

where the integrand ua ( x , y) 



∑ uˆ ( a  m 2dp , y ) exp ( i ( a  m 2dp ) x )

(1.83)

m

is a pseudoperiodic function of x with pseudoperiodicity coefficient . Consequently, the study of the general field u(x, y) reduces to the study of its pseudoperiodic components u(x, y) for all values of  in the first Brillouin zone [/d, /d] of the x-periodic problem. Then we are led to the geometric construction shown in Figure 1.18. The large circle is the constant frequency dispersion diagram of the surrounding medium (with radius /c, for example, for the vacuum) while the other curves represent the constant frequency dispersion diagram for the photonic crystal considered. Let us suppose that an incident plane wave illuminates a slice of this photonic crystal. We assume that the interfaces of the slice are perpendicular to the y-axis. If the angle of incidence of the plane wave is such that its wavevector ki is in the direction given by the solid line arrow (see Figure 1.18), then the tangential component of this wavevector is given by the vertical dashed line. Thus the two Bloch modes that can be excited in the crystal are given by the intersection of this line with the constant frequency dispersion diagram. Given the ascending nature of the surface corresponding to the central feature in Figure 1.18 (see Figure 1.15) the energy flows in the directions shown by the solid-line arrows (starting from the dispersion curves) in this case. Notice that to predict the field behavior in the illuminated crystal, we consider only the propagating Bloch modes. One should be aware of the empirical aspect of this hypothesis that neglects the evanescent solutions [32]. One knows that similar assumptions in grating theory can lead to erroneous results. Nevertheless, we will see that this method gives an interesting intuitive description of the energy flow in the crystal that proves to be in good agreement with our rigorous numerical checks.

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32

Electromagnetic Theory and Applications for Photonic Crystals

ky i

K

Circle: vacuum dispersion curve

kx

Crystal dispersion curves

FIGURE 1.18 Geometrical construction based on the conservation of the tangential components of the wave vector. The large circle with radius k0  /c represents the constant frequency dispersion diagram of the vacuum. The angle of incidence of the plane wave is 6.4° (represented by the arrow coming from the top of the figure). The two Bloch modes that can be excited are deduced from the conservation of the tangential component of the wave vector (vertical dashed line). The arrows represent the direction of the associated energy flows.

Figure 1.19 shows a map of the electric field modulus resulting from a rigorous numerical simulation using the scattering matrix method. A finite size photonic crystal made of 483 cylinders is illuminated by a limited Gaussian beam coming from the top of the figure. The exact definition of this incident field is: u i ( x , y)  ∫





A(a) exp(iax  ib(a)y)d a

(1.84)

with ()2  k02  2, and with a Gaussian amplitude: A(a) 

 (a  a )2 W 2  0  exp    4 2 p   W

(1.85)

The mean incidence angle 0 of the beam is such that 0  k0 sin 0. The parameter W appearing in Equation (1.85) is directly linked to the incident beam width.

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Scattering Matrix Method Applied to Photonic Crystals

33

50 40 30 20 10 0 −10 −20 −30 −40 −50 −50

−40

−30

−20

−10

0

10

20

30

40

50

FIGURE 1.19 Map of the electric field modulus for a photonic crystal made of 483 cylinders with the parameters of Figure 1.14. The crystal is illuminated by a Gaussian beam with mean angle of incidence 0  6.4° and W  10. The lines show the locus of the maximum of the beams.

The fringes above the crystal are due to interferences between the incident beam and the reflected beam on the first interface. The beam inside the photonic crystal propagates in the exact direction given by the previous geometrical construction. If we interpret this diagram in terms of Snell-Descartes laws, it corresponds to an effective optical index smaller than one. Let us now increase the angle of incidence such that the tangential component of the wavevector is given by the vertical dashed line in Figure 1.20. Then the two Bloch modes that can be excited are given by the intersections with the constant frequency dispersion diagram, but now the surfaces accessed ascend as one approaches the center of the diagram (see Figures 1.16 and 1.17), and the energy propagates as represented by the arrows. Figure 1.21 shows the electric field modulus map when a finite size photonic crystal is illuminated by a finite size beam and again inside the photonic crystal the energy propagates in the direction given by the geometrical construction. The refracted beam in the crystal behaves as in a homogeneous medium with negative optical index, which leads us to call this phenomenon negative refraction. Notice that for these Bloch modes the average energy velocity is almost the opposite of the average phase velocity and with a slight change of angle of incidence

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34

Electromagnetic Theory and Applications for Photonic Crystals

ky

Ki

kx Crystal dispersion curves

FIGURE 1.20 Same as Figure 1.18 but with 0  40°. 50 40 30 20 10 0 −10 −20 −30 −40 −50 −50

−40

−30

−20

−10

0

10

FIGURE 1.21 Same as Figure 1.19 but with 0  40°.

© 2006 by Taylor & Francis Group, LLC

20

30

40

50

Scattering Matrix Method Applied to Photonic Crystals

35

these two velocities can be exactly opposite. This and the fact that negative refraction can be observed make the link with the so-called left-handed material, or double negative material, and also material with negative refractive index [33–37]. These are artificial materials that consist of periodic metallic structures (and can also be considered as metallic photonic crystals). These structures are actually the subject of intensive research and controversy. Several properties, such as their ability to focus even the evanescent waves to make a perfect lens, have been studied [35,37,38]. Their realization in the optical domain will be difficult because of the losses of the metals for optical wavelengths. Here, dielectric photonic crystals could be a valuable alternative. Note that negative refraction has been observed experimentally in a very nice experiment by Kosaka et al. [39].

1.5 THEORETICAL AND NUMERICAL STUDIES OF PHOTONIC CRYSTAL PROPERTIES 1.5.1 ULTRAREFRACTION WITH DIELECTRIC PHOTONIC CRYSTALS Ultrarefractive properties of photonic crystals have been suggested in previous papers [40]. We show here that the formalism described in the previous sections is well suited to determine the parameters that enable one to engineer devices based on this phenomenon (for instance, an ultrarefractive microlens). The aim is to design an artificial material whose effective index is less than 1 and if possible close to zero. This means that the constant-frequency dispersion diagram should follow a circle 2 k x2  k y2  k02 neff

(1.86)

That is, a small circle centered on the origin in the (kx, ky ) plane. In order to obtain this property, let us decrease the frequency associated with the horizontal intersection plane of Figure 1.16. The parameters of the photonic crystal are still those defined in the caption for Figure 1.14. Figure 1.22 shows the constant-frequency dispersion diagram for two wavelengths close to the kx  ky  0 limit of the sheet of Figure 1.16. The curve for 0  2.56 is very close to a circle, and in this case the effective index given by Equation (1.86) is neff  0.086. Notice that even if the crystal behaves as a homogeneous material, the physical situation is very different from that generally studied in homogenization works where one considers quasi-static limits [41,42]. Since the index contrast with other materials is very large, new optical elements can be proposed. For instance, let us design a microlens composed of 295 rods of the same photonic crystal. The width of this lens is equal to 64 (i.e., 250), and the radius of curvature of the concave face is R  50. When illuminated by a Gaussian beam of width W  40 at normal incidence, the transmitted field focuses at a point situated at f  53 from the concave face (Figure 1.23). Notice that the same lens built from classical material (optical index greater than 1) is

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36

Electromagnetic Theory and Applications for Photonic Crystals

0.4

0.3 ky

0 = 2.545

0.2

0 = 2.56

0.1

0.0 0.0

0.1

0.2

0.3

0.4

kx

FIGURE 1.22 Same as Figure 1.17 for two different wavelengths. Only the central region is presented.

divergent. On the sides of Figure 1.23, the field is principally due to the diffraction by the edges of the lens. A direct application of geometrical optics gives the focal length: f

1 R  54.7 1  neff

(1.87)

which agrees well with the actual value f  53 and demonstrates the very good similarity between the crystal and an isotropic homogeneous material. If the constant-frequency dispersion diagram does not look like a circle, then there is no clear focusing of the light.

1.5.2 ULTRAREFRACTION WITH METALLIC PHOTONIC CRYSTALS In this section we describe a theoretical and numerical work concerning a two-dimensional approach to the design of directive antennas. One of the aims is to obtain antennas that are much more compact than classical solutions. Another interesting feature is that these antennas can be excited by a single feeding device (patch, monopole). Such antennas could be useful for microwave telecommunications. The antenna presented here uses a metamaterial that is a composite stack of metallic grids and foam layers. This metamaterial is analyzed using the techniques

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Scattering Matrix Method Applied to Photonic Crystals

37

10 0 −10 −20 −30 −40 −50 −60 −70 −80 −90 −50

−40

−30

−20

−10

0

10

20

30

40

50

FIGURE 1.23 The microlens is illuminated in normal incidence configuration from the top by a Gaussian beam in s-polarization.

that we have developed in Section 1.4. Near the band edge, this material behaves as a homogeneous medium with low optical index (much less than unity). Under proper conditions, the energy radiated by a source embedded inside the structure is concentrated in a narrow emission lobe. Based on this analysis, we have built a device working in the microwave domain: a ground plane is covered by the metamaterial, with a monopole source embedded inside. The experimental measurements are in good agreement with the theoretical predictions [43]. How can we apply the results of Section 1.4 to the design of directive sources? The aim is to realize a device that radiates energy in a very narrow angular range. Our objective is to embed a localized source inside a metamaterial (that is in fact a photonic crystal) and to restrict the radiated field inside a small angular range centered around the yˆ direction. Equation (1.81) implies that in the homogeneous medium outside the crystal any radiated field can be expressed as a sum of plane waves. In this sum, directive radiation requires that the plane waves with significant amplitude should correspond to small values of  (let us assume [max, max]). Then, from Equation (1.80), the allowed Bloch modes of the photonic crystal should lay inside the region kx [max, max]. In Figure 1.24, we represent the dispersion curve of the homogeneous medium outside the crystal. We assume that this medium is vacuum and this curve is the circle defined by k2x  ky2  k02. The angular range around the vertical direction provides the value of max. This means that the

© 2006 by Taylor & Francis Group, LLC

38

Electromagnetic Theory and Applications for Photonic Crystals Allowed propagation directions outside the metamaterial ky Vacuum dispersion curve circle with radius kx2 + ky2 = k02

Dispersion curve of the meta material radius k0 neff −max

max

kx

FIGURE 1.24 Schematic construction linking the propagation direction outside a photonic crystal with the Bloch wave vector.

curves corresponding to the Bloch modes in the constant-frequency dispersion diagram should lay inside the region between the two dashed lines in Figure 1.24. We are now concerned with the problem of finding a photonic crystal whose constant-frequency dispersion diagram is entirely included in this region. This property can be obtained by using a metallic photonic crystal. It is well known that a periodic set of parallel metallic wires has filtering properties for low frequencies and for s-polarization. This means that there exists a full bandgap for low frequencies. For frequencies slightly higher than the cutoff, the constant-frequency dispersion diagram is then a small curve in the (kx, ky) plane. The structure is modeled as a stack of metallic grids made of infinitely conducting wires. The infinitely conducting metal is indeed a reliable approximation in the range of wavelengths of interest. Let us now turn to a model directly linked with the experimental study that we have done. In the experimental structure, we found it convenient to make the grids by etching a thin copper plate. Consequently, the cross section of the wires is not a circle but a rectangle. After optimization of the parameters in order to match fabrication constraints and to get interesting properties around 14 GHz, it appears that convenient parameters are the following: the cross section of the metallic wires is 0.014 cm wide along the y axis and 0.071 cm wide along the x axis, dx  0.58 cm, dy  0.63 cm. This study is achieved using the SMM. Figure 1.25 shows the modulus and the lines of equal phase of the total field radiated by the structure made with 40 6 of these wires above a ground plane located at y  0. The source is a wire antenna parallel to the z axis and placed in the middle of the metamaterial with a working vacuum wavelength 0  2.07 cm. The most striking fact is the very slow variation of the phase inside the metamaterial, which is a proof that the effective index in this material is quite small. Notice that the phase of the emitted field is nearly constant on planes parallel to the emitting surface. The radiation pattern exhibits a narrow lobe (Figure 1.26) with a half-power beamwidth of 2 3.8°.

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Scattering Matrix Method Applied to Photonic Crystals

39

80 60 40 20 −100

−50

0

50

100

150

−100

−50

0

50

100

150

80 60 40 20

FIGURE 1.25 Field modulus (top) and lines of equal phase (bottom) of the field radiated by the 2D antenna.

90 120

60

30

20

150

30

10

0

180

0

FIGURE 1.26 Radiation pattern for the device of Figure 1.25 (dB scale). The half-power beamwidth is 2 3.8°.

1.5.3 NEGATIVE REFRACTION GALLERIES

BY A

DIELECTRIC SLAB RIDDLED WITH

Metamaterials and negative refraction are the subjects of considerable attention due to their attractive potential applications, such as the controversial perfect lens proposed by Pendry [35–38]. In the microwave range, some solutions are proposed that use metallo-dielectric arrays of split-ring resonators and wires [34]. For

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40

Electromagnetic Theory and Applications for Photonic Crystals 0.8

d/2c

0.6

0.4

0.2

0.0

X

Γ

J

X

FIGURE 1.27 Dispersion diagram of a hexagonal photonic crystal for both s-polarization (heavy lines) and p-polarization (lighter lines).

applications in optics, the losses inherent to metals are unavoidable. As shown in Section 1.4, dielectric photonic crystals can also produce negative refraction. In the same way as before, we can use the dispersion diagrams of Bloch modes in order to find suitable parameters. Let us consider the case of a dielectric slab periodically drilled with air holes. According to the notation of Section 1.3, the permittivity of the slab is n2i  12; the radius of the holes is 0.294; and they are placed with hexagonal symmetry. The distance between the centers of two neighboring holes is equal to d  0.68. The dispersion diagram of this photonic crystal is plotted in Figure 1.27. The oblique straight line is the light line   ck0. This line intersects the second s-polarized band for d/(2c)  0.337, that is, 0  2.02. Figure 1.28 shows the constant frequency dispersion diagram at this wavelength. The curve is almost a circle and simulates a homogeneous material with optical index neff  1, at least for propagative waves. Let us now consider a finite structure: a finite slab whose faces are defined by x  13.944 and y  2.6996 (see Figure 1.29) is drilled with 364 circular air holes. The structure lies in vacuum and is illuminated by a Gaussian beam coming from the top with an incidence angle 0  30° (see Equation (1.84)) and W  5. Figure 1.29 shows the resulting field map and the negative refraction inside the crystal. In order to show the focusing capabilities of such a device, let us assume that the structure is excited by a point source located at (x  0, y  5.4). The resulting field map is shown in Figure 1.30. It is clear that a focus spot appears below the crystal. Obviously, this lens is not perfect: the transmitted energy is weak, and the width of the focus spot is exactly 0 /2. The improvement of this aspect would require a control of the evanescent waves in order to restore their amplitude in the image.

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Scattering Matrix Method Applied to Photonic Crystals

41

ky

3

2

1

0 0

1

2

3

4

kx

FIGURE 1.28 Constant frequency dispersion diagram at 0  2.02 for the photonic crystal in s-polarization (solid line), and for the vacuum (dashed line). The arrows give the direction of Vg in both cases. 15

10

5

0

−5

−10

−15 −15

−10

−5

0

5

10

15

FIGURE 1.29 Modulus of the total field. Above the slab, the black line shows the locus of the maximum of the Gaussian incident beam. Below the slab, it shows the locus of the maximum of the transmitted field. Above the slab, the structure of the field is due to the interference between the incident and the reflected fields.

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Electromagnetic Theory and Applications for Photonic Crystals 15

10

5

0

−5

−10

−15 −15

−10

−5

0

5

10

15

FIGURE 1.30 Modulus of the total field when the slab of Figure 1.29 is illuminated by a point source.

1.6 CONCLUSION It has been shown that the scattering matrix method is a valuable tool for the numerical study of finite-size photonic crystals. It is especially efficient when the shape of the objects is circular (2D crystals) or spherical (3D crystals) because in those cases half of the calculation (determination of the scattering matrices) can be made in closed form, contrary to the classical general methods. Another feature of the method is that it is based on a detailed physical analysis of the multiscattering phenomena occurring between the objects, the Graf formula being the mathematical tool that conciliates the rigor and the simplicity. In this chapter, the solution is obtained through the inversion of a matrix, the size of which increases with the number of objects. Thus, there is a limitation on the number of objects that can be handled. For 2D photonic crystals, a number of cylinders of the order of 1,000 can typically be treated using a simple personal computer. Nevertheless, it is possible to avoid a direct inversion by using iterative methods (see for example Reference 3). The combination of the scattering matrix method with the method of fictitious sources permits a significant extension of the domain of application of the first method. For example, it permits dealing with the case in which the set of cylinders is surrounded by a jacket. Thanks to this extension, it is possible to deal with some important applications of photonic crystals: in particular, photonic

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crystal fibers and microstructured optical fibers surrounded by a jacket and photonic crystal slabs. Finally, it has been shown that the notions of Bloch waves and their corresponding dispersion curves allows one to understand and predict the surprising phenomena generated by photonic crystals. Nowadays great attention is paid to phenomena arising near the edge of the photonic bandgaps, such as ultrarefraction and negative refraction. In these phenomena, it is necessary to distinguish the notions of phase and group velocities. It is very important to know whether or not the dielectric photonic crystals that generate negative refraction phenomena are left-handed materials and if they can amplify the evanescent waves. Indeed, a positive answer to this question could lead to a breakthrough: the realization of lenses more powerful than classical lenses in the visible region.

REFERENCES [1] Lord Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Philos. Mag., 34, 481–502, 1892. [2] W.C. Chew, L. Gürel, Y.M. Wang, G. Otto, R.L. Wagner, and Q.H. Liu, A generalized recursive algorithm for wave-scattering solutions in two dimensions, IEEE Trans. on Microwave Theory and Techniques, 40, 716–723, 1992. [3] M. Defos du Rau, Electromagnetic diffusion in heterogeneous dense media: presentation of a hybrid model for the study of heterogeneous materials, PhD dissertation, Bordeaux University, Bordeaux, France, 1997. [4] A.Z. Elsherbeni and A. Kishk, Modeling of cylindrical objects by circular dielectric and conducting cylinders, IEEE Trans. on Antennas and Propagation, 40, 96–99, 1992. [5] D. Felbacq, G. Tayeb, and D. Maystre, Scattering by a random set of parallel cylinders, J. Opt. Soc. Am. A, 11, 2526–2538, 1994. [6] D. Felbacq, Theoretical and numerical study of light scattering by a set of parallel rods, PhD dissertation, University of Aix-Marseille III, Marseille, France, 1994. [7] N.A. Nicorovici, R.C. McPhedran, and L.C. Botten, Photonic band gaps for arrays of perfectly conducting cylinders, Phys. Rev. E, 52, 1135–1145, 1995. [8] M. Vlassis, R.C. McPhedran, and N.A. Nicorovici, Radiation modes of multiple core fibres, Opt. Comm., 129, 256–272, 1996. [9] H.A. Youssif and S. Köhler, Scattering by two penetrable cylinders at oblique incidence: Part 1. The analytical solution, J. Opt. Soc. Am. A, 5, 1085–1096, 1988. [10] M. Abramovitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1970. [11] D. Maystre, M. Saillard, and G. Tayeb. Special methods of wave diffraction, in P. Sabatier and E.R. Pike, eds., Scattering, Academic Press, London, 2001. [12] G. Tayeb, R. Petit, and M. Cadilhac, Synthesis method applied to the problem of diffraction by gratings: the method of fictitious sources, Proceedings of the International Conference on the Application and Theory of Periodic Structures, J.M. Lerner and W.R. McKinney eds., Proc. SPIE 1545, 95–105, 1991. [13] G. Tayeb, The method of fictitious sources applied to diffraction gratings, Special Issue on Generalized Multipole Techniques (GMT) of App. Computational Electromagnetics Soc. J., 9, 90–100, 1994.

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[14] F. Zolla., R. Petit, and M. Cadilhac, Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources, J. Opt. Soc. Am. A, 11, 1087–1096, 1994. [15] F. Zolla and R. Petit, Method of fictitious sources as applied to the electromagnetic diffraction of a plane wave by a grating in conical diffraction mounts, J. Opt. Soc. Am. A, 13, 796–802, 1996. [16] Y. Leviatan and A. Boag, Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model, IEEE Trans. Ant. Prop., AP-35, 1119–1127, 1987. [17] A. Boag, Y. Leviatan, and A. Boag, Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model, Radio Science, 23, 612–624, 1988. [18] A. Boag, Y. Leviatan, and A. Boag, Analysis of diffraction from echelette gratings, using a strip-current model, J. Opt. Soc. Am. A, 6, 543–549, 1989. [19] A. Boag, Y. Leviatan, and A. Boag, Analysis of electromagnetic scattering from doubly-periodic nonplanar surfaces using a patch-current model, IEEE Trans. Ant. Prop., AP-41, 732–738, 1993. [20] C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics, Artech House Books, Boston, 1990. [21] C. Hafner, Multiple multipole program computation of periodic structures, J. Opt. Soc. Am. A, 12, 1057–1067, 1995. [22] V.D. Kupradze, On the approximate solution of problems in mathematical physics, original (Russian): Uspekhi Mat. Nauk, 22, 59–107, 1967. English translation: Russian Mathematical Surveys 22, 58–108, 1967. [23] D. Kaklamani and H. Anastassiu, Aspects of the method of auxiliary sources (MAS) in computational electromagnetics, IEEE Ant. and Prop. Magazine 44, no. 3, 48–64, June 2002. [24] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in Fortran: The Art of Scientific Computing, (2nd ed.), Cambridge University Press, Cambridge, 1992. [25] R.D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, Accurate theoretical analysis of photonic band-gap material, Phys. Rev. B, 48, 8434–8437, 1993. [26] A. Moroz, Density-of-states calculations and multiple-scattering theory for photons, Phys. Rev. B, 51, 2068–2087, 1995. [27] B. Gralak, S. Enoch, and G. Tayeb, From scattering or impedance matrices to Bloch modes of photonic crystals, J. Opt. Soc. Am. A, 19, 1547–1554, 2002. [28] P. Yeh, Electromagnetic propagation in birefringent layered media, J. Opt. Soc. Am., 69, 742–756, 1979. [29] B. Gralak, Etude théorique et numérique des propriétés des structures à bandes interdites photoniques, PhD dissertation, Université d’Aix-Marseille III, Marseille, France, 2001. [30] B. Gralak, S. Enoch, and G. Tayeb, Anomalous refractive properties of photonic crystals, J. Opt. Soc. Am. A, 17, 1012–1020, 2000. [31] R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980. [32] D. Felbacq and R. Smaâli, Bloch modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals, Phys. Rev. Lett., 92, 193902, 2004. [33] V.G. Veselago, The electrodynamics of substance with simultaneously negative values of  and , Sov. Phys. Uspekhi, 10, 509–514, 1968.

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[34] R.A. Shelby, D.R. Smith, and S. Schultz, Experimental verification of negative index refraction, Science, 292, 77–79, 2001. [35] J.B. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85, 3966–3969, 2000. [36] P.M. Valanju, R.M. Walser, and A.P. Valanju, Wave refraction in negative-index media: always positive and very inhomogeneous, Phys. Rev. Lett., 88, 187401, 2002. [37] N. Garcia and M. Nieto-Vesperinas, Left-handed materials do not make a perfect lens, Phys. Rev. Lett., 88, 207403, 2002. [38] D. Maystre and S. Enoch, Perfect lenses made with left-handed materials: Alice’s mirror? J. Opt. Soc. Am. A, 21, 122–131, 2004. [39] H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Superprism phenomena in photonic crystals, Phys. Rev. B, 58, 10096–10099, 1998. [40] J.P Dowling and C.M. Bowden, Anomalous index of refraction in photonic bandgap materials, J. Mod. Opt., 41, 345–351, 1994. [41] R.C. McPhedran, N.A. Nicorovici, and L.C. Botten, The TEM mode and homogenization of doubly periodic structures, J. Electrom. Waves and Appl., 11, 981–1012, 1997. [42] D. Felbacq and G. Bouchitté, Homogenization of a set of parallel fibers, Waves in Random Media, 7, 245–256, 1997. [43] S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, A metamaterial for directive emission, Phys. Rev. Lett., 89, 213902, 2002.

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Multipole Methods 2 From to Photonic Crystal Device Modeling Lindsay C. Botten, Ross C. McPhedran, C. Martijn de Sterke, Nicolae A. Nicorovici, Ara A. Asatryan, Geoffrey H. Smith, Timothy N. Langtry, Thomas P. White, David P. Fussell, and Boris T. Kuhlmey

CONTENTS 2.1 Introduction ................................................................................................48 2.2 Multipole Theory for Finite and Infinite Structures ..................................51 2.2.1 General Framework ......................................................................51 2.2.2 Field Representation ......................................................................52 2.2.3 Rayleigh Field Identity ..................................................................55 2.2.4 Field Coupling and Continuity Conditions....................................58 2.2.5 Field Problem of Microstructured Optical Fibers ........................60 2.2.6 Infinite Periodic Structures ............................................................62 2.2.7 Gratings..........................................................................................68 2.3 Multipole Modeling of Photonic Crystal Fibers ........................................73 2.3.1 Background ....................................................................................73 2.3.2 Guiding Mechanisms and the “Finger Diagram” ..........................73 2.3.3 Implementation of the Multipole Method ....................................75 2.3.4 Effective Index-Guided Modes......................................................76 2.3.5 Air-Guided Modes ........................................................................80 2.4 Radiation Dynamics and the Local Density of States................................82 2.4.1 Background ....................................................................................82 2.4.2 2D Green Tensor and LDOS ........................................................83 2.4.3 2.5D Green Tensor and LDOS ......................................................86 2.5 Bloch Mode Analysis of Composite PC Devices ......................................92 2.5.1 Background and Nomenclature ....................................................92 47

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2.5.2 2.5.3 2.5.4 2.5.5

Formulation of the Eigenvalue Problem........................................95 Calculation of Band Diagrams and Band Surfaces ......................97 Classification of the Eigenvalues ................................................100 Defect Mode Modeling of Extended Photonic Crystal Devices ........................................................................................104 2.5.6 Interfacing Photonic Crystal Media and Fresnel Coefficients ..................................................................................105 2.5.7 Recursively Combining PC Stack Segments ..............................107 2.6 Modeling of Photonic Crystal Devices ....................................................108 2.6.1 Background ..................................................................................108 2.6.2 Straight PC Waveguides ..............................................................109 2.6.3 Resonant Filters and Junctions ....................................................110 2.6.4 Mach–Zehnder Interferometer ....................................................114 2.7 Discussion and Conclusions ....................................................................116 Acknowledgment ..............................................................................................119 References ........................................................................................................119

2.1 INTRODUCTION The scattering of waves of all types (electromagnetic, acoustic, matter, etc.) is one of the fundamental problems of physics. The field has a long and rich literature with contributions from many prominent physicists. Lord Rayleigh [1] was one of the early initiators of a systematic mathematical analysis of wave scattering and was the first to derive a rigorous solution to the problem of wave scattering from cylindrical structures — a theory that spawned a class of methods that has been particularly successful. These methods are variously referred to as Rayleigh or multipole methods, with the latter term reflecting the structure and basis of the field expansions that are strongly tailored to the geometry and the form of the scatterers. Consequently, such methods have received wide attention because of their computational efficiency and accuracy as well as their analytic tractability. This class of methods dates back to the classic 1892 paper by Rayleigh [2], in which he developed a method for solving electrostatic problems for lattices of spheres or arrays of cylinders. At the time, Rayleigh was aiming to demonstrate the limits of the validity of the Lorentz–Lorenz equation — a fundamental equation of optics that provides a bridge between the microscopic/atomic model of materials, and the continuous model of a homogeneous isotropic dielectric through which an electromagnetic wave propagates. Through the multipole model, Rayleigh showed that the Lorentz–Lorenz equation was a dipolar approximation and was able to exhibit higher order corrections. The essential feature of any multipole method is the application of an ingenious field identity that relates the regular field in the vicinity of any scatterer to fields radiated by other scatterers and external sources. While its origins lie in

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electrostatics and periodic systems, the technique has evolved to become an important tool for dynamic problems in a range of fields involving both finite and periodic systems — in particular, in electromagnetism, solid state physics, and solid mechanics. Largely, however, its widespread computational application has occurred only since the mid-1970s when computers of sufficient power allowed its full potential to be realized. Among the milestones of the method is the work of Ignatowsky (1914) [3] who considered the scattering of plane waves from two-dimensional gratings, and Wait [4] who first generalized the treatment to three dimensions in 1955. The multipole method has been used extensively in solid-state physics through the work of Korringa [5], and Kohn and Rostoker [6], who separately formulated what is now known as the KKR method and used it to calculate the band structure of solids. Applications of this approach to electromagnetism also have been developed [7]. A key feature of the application of multipole methods to periodic systems is the appearance of lattice sums, namely sums of multipole terms, evaluated at each lattice point of the structure, characterizing the scattering contribution of the entire lattice to each multipole term. Their evaluation is an important and highly subtle aspect of the method, with difficulties in their evaluation arising through the occurrence of conditionally convergent series. Thus, there has been much research devoted to the development of efficient and accurate methods for their calculation. Among the pioneers of this field is Twersky [8], who derived the first closed form expressions for grating lattice sums. Unlike the earlier Ewald summation methods [9], the accuracy of which can be questionable, Twersky-type methods may be accelerated by a range of techniques [10] to yield well-conditioned and accurate sums. Our group has also demonstrated relationships [11] between the lattice sums of a two-dimensional (2D) array and a one-dimensional (1D) grating that are computationally advantageous, and that can be extended to higher dimensions and different lattice geometries. Our particular interest in multipole methods derives from recent research in photonic crystals [12–14], including microstructured optical fibers [15–17], the study of the radiation properties of sources embedded in such structures [18], and the design of complex/composite devices (e.g., splitters, couplers, interferometers, etc.) constructed in photonic crystals [19]. Such structures form a strongly scattering environment due to the high material contrast and the scale and geometry of the structure. Accordingly, approximate methods, such as the Born approximation, and the coherent potential approximation, are not applicable, while the Feynman diagram method is computationally too cumbersome. Because of this, the electromagnetic research community has developed a wide range of modeling tools including the plane wave method [20], the finite-difference time domain (FDTD) method [21], differential and boundary integral methods [22], the finite element method [23], and multipole methods [24–27]. The bulk of these are omnibus, computational methods that handle the full generality of problems. For particular geometrical configurations and symmetries, however, a technique such as the multipole method has much to offer, in that it can analytically preserve symmetry, and exploit the natural basis of functions for the scatterers (e.g., cylindrical

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harmonic functions for cylindrical scatterers). Accordingly, this can yield important physical insight into the scattering processes not otherwise possible with purely numerical methods. The implementation of the method is both efficient and accurate, and can avoid numerical artifacts by explicitly embedding the symmetry of the structure within the formulation. The sole proviso is that one is able to construct a suitable basis of multipole functions for the particular scatterer geometry. This chapter is built around the role of the multipole method in three paradigm applications: • Microstructured optical fibers, in which we show that the multipole theory answers fundamental questions about the symmetry of modes and facilitates the most accurate calculations of losses in such fibers (Section 2.3) • Studies of the radiation dynamics of 2D and 3D sources embedded in finite 2D photonic crystals, for which the multipole method analytically avoids problems in the calculation of the local density of states (LDOS) associated with the divergence of Green’s function at source points, and leads to highly accurate and efficient calculations of the LDOS and the Lamb shift (Section 2.4) • Scattering matrix methods for solving propagation problems in extended photonic crystal devices that exploit the natural Bloch mode basis, yielding design tools that are accurate and efficient, and that simultaneously provide substantial insight into the propagation mechanisms by mapping the analysis onto familiar concepts of thin film optics (Sections 2.5 and 2.6). The last of these applications is based on transfer matrix methods that model the structure as a stack of gratings, the scattering matrices for which rely on efficient and accurate computation using multipole techniques. Throughout this chapter we work with cylindrical structures, and in Section 2.2 we provide an integrated tutorial on the multipole method, commencing with the analysis of a single scatterer and its extension to a finite cluster. We treat the problem in both 2D and 3D, and go on to develop the theory for calculating the modes of microstructured optical fibers (MOFs). We extend the theory of finite clusters to allow for a 2D (line) source and further extend this in Section 2.4 to consider a point dipole source, thus enabling the study of the radiation dynamics of PC clusters. Section 2.2 also develops the multipole theory of periodic structures considering 1D cylinder gratings and the 2D crossed layers that constitute a woodpile PC, and focuses on the calculation of the plane wave scattering matrices that underpin the later Bloch mode analysis of PC devices. Section 2.3 builds on the theoretical introduction and goes on to survey some of our recent work in the area of photonic crystal fibers including the implementation of the method, guiding mechanisms in both air-cored and conventional dense-cored MOFs, the calculation of losses, and the development of the phase diagram that characterizes modal cutoff.

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Section 2.4 is concerned with the calculation of the local density of states in 2D and 2.5D photonic crystals. Here, the term 2.5D refers to a 3D (point) source located in a 2D photonic crystal (with infinitely long scatterers). We show the strong suppression of the local density of states (LDOS) for both polarizations in 2D photonic crystals for gap wavelengths, while for the 2.5D modeling the achievable suppression of the LDOS, relative to its free space value, is more modest. Section 2.5 builds on our experience with diffraction grating formulations to develop transfer matrix methods for computing Bloch modes in PC devices. The nature of the resulting eigensystem is closely analyzed, and important modal reciprocity and orthogonality properties are derived. Fresnel reflection and transmission scattering matrices, the analogues of the Fresnel coefficient of thin films, are introduced to characterize propagation across an interface and are used to build the general theory of propagation in extended PC devices that closely mirrors the corresponding theory in thin film optics. Finally, in Section 2.6 we make use of the tools of Section 2.5 to design a range of compact and efficient PC devices including the “folded” directional coupler, an efficient Y-splitter, and its application to the design of a Mach–Zehnder interferometer. A feature of this section is its focus on the key role of the photonic crystal in their design.

2.2 MULTIPOLE THEORY FOR FINITE AND INFINITE STRUCTURES 2.2.1 GENERAL FRAMEWORK The purpose of this section is to develop a general framework that enables us to handle the gamut of field problems for cylinder photonic crystals and finite photonic clusters, in both two and three dimensions, using the multipole method. In particular, we outline, in a systematic and integrated manner, the key steps that allow us to formulate the solution of a diverse variety of scattering problems in a unified framework. Before commencing the detailed derivation of the theory, we begin by setting in place some of the key nomenclature used in this chapter. In the 2D problems we consider, the cylinder axes are aligned with the z axis; however, we also allow alignment parallel to the x axis to form a woodpile structure comprising crossed 1D grating layers. We must also allow for general conical incidence characterized by an incident wave vector k0  (k0x, k0y, k0z), with k0z  0. In z-invariant environments in which both the structure and the field sources are two dimensional (i.e., infinitely extended in the z-direction) we may prescribe the z-dependence of various physical quantities to be exp(ik0zz). The complete geometry is defined by the dielectric function (r). In the context of a plane wave of wavelength l incident from free space with polar angle w and azimuthal angle c, we may set k0  kf (sin w cos c, sin w sin c, cos w) where kf  2p/l  v/c is the free space wavenumber, assuming a harmonic time dependence of exp(ivt). In source-free regions, the Maxwell curl equations can be used to express the transverse components of the electromagnetic field (i.e., those perpendicular to

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the cylinder axes) in terms of the longitudinal field components Ez and Kz [28]. Thus, Et 

i [k E  k f zˆ  K z ], k⊥2 0 z z

Kt 

i [k e(r )zˆ  Ez  k0 z K z ] k⊥2 f

(2.1)

2 with k2  k2f (r)  k2  k0z . For convenience, we introduce a rescaled magnetic —— field K  Z0H, where Z0  m0/0  377  is the free space impedance. With this transformation, Kz has the dimension of an electric field. Then, by writing each of the longitudinal components in the form

Ez  V E exp(ik0 z z ),

K z  V K exp(ik0z z )

(2.2)

it can be shown using Maxwell’s equations that the scalar “potentials” VE and VK each must satisfy a Helmholtz equation [2  k⊥2 (r )]V  0

(2.3)

where k2  k2f (r)  k20z . Here, (r) denotes the relative permittivity, which is related to the refractive index n(r) by (r)  n 2(r). Equations (2.2) give the general form of the longitudinal fields in an environment in which the structure is z-invariant and the field sources (and resulting fields) have a harmonic longitudinal dependence exp(ik0z). In Section 2.4, which looks at the effect of a photonic crystal environment on the emission from line or point sources, it is necessary to generalize the field representation to take into account both structural invariance in the z-direction and the z-variation of the source. In this case, we must express Ez and Kz in a Fourier integral form

Ez (r , z )  ∫ V E (r , k0 z )eik0 z z dk0 z , 



K z (r ,z )  ∫ V K (r , k0 z )eik0 z z dk0 z 

(2.4)

thus allowing us to treat scattering problems involving localized sources.

2.2.2 FIELD REPRESENTATION Here we develop the representation of the electromagnetic fields in the scattering problem involving a finite cluster of nontouching cylinders (Figure 2.1[a]), with the physical structure, sources, and fields being invariant in the z-direction. We begin in a simple manner, deriving the fields in the vicinity of a single cylinder Cl in order to understand the physical origin of the various terms. In the region exterior to Cl (Figure 2.1[b]), the field quantity V (i.e., either—— V E or—— V K) satisfies 2 e 2 e 2 2 the Helmholtz equation [  (k) ]V  0, where k  k f e  k20z denotes the transverse wavenumber applicable to the (exterior) background region of

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A Cl r

rj

rl

s rs

Cext

o (a)

(b)

FIGURE 2.1 (a) The geometry of a cluster of cylinders, (b) the integration contour for Green’s theorem in Equation (2.6).

refractive index ne. This leads us to introduce the Green function satisfying the inhomogeneous equation [r2  (k⊥e )2 ] G0 (r; r ′)  d(r  r ′)

(2.5)

the solution of which is G0(r; r )  iH0(ke |r  r |)/4. Here, and throughout this chapter, we will denote the Hankel functions of the first kind Hn(1) by Hn, to simplify the notation. Applying Green’s theorem to the domain A of Figure 2.1(b) bounded by the perimeter Cl of the cylinder and the exterior boundary Cext

∫ A [V (r ′)r2′G0 (r; r ′)  G0 (r; r ′)r2′V (r ′)] dAr′

  ∂ ∂  V (r )  ∫  V (r ′) G0 (r; r ′)  G0 (r; r ′) V (r ′)  dsr′  ∂A  ∂r ′ ∂r ′  

(2.6)

we see that V(r) can be expressed as the sum of two line integrals, respectively, around Cl and Cext. With the field point r located in A between the two boundaries, we note that when the source point r  Cl then |r  rl| |r  rl|, while for r  Cext then |r  rl|  |r  rl|. This has a profound effect on the representation of the Green function when expanded using Graf’s addition theorem [29] to change the origin of coordinates to rl, the center of Cl. That is, G0 (r; r ′) 



1 4i

n

1 4i

∑

∑

H n (k⊥e |r  rl |) Jn (k⊥e |r ′  rl |)ein[arg(rrl )arg(r′rl )]

(2.7)

Jn (k⊥e |r  rl |) H n (k⊥e |r ′  rl |)ein[arg(rrl )arg(r′rl )]

(2.8)

for r  Cl, and G0 (r; r ′) 



n

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for r  Cext. Accordingly, the contributions to V(r) due to the integrals around the inner (Cl) and outer (Cext) contours are expanded respectively, in multipole e series comprising outgoing wave (Hn(k |r  rl|) exp(in arg(r  rl))) and standing e wave (Jn(k|r  rl|) exp(in arg(r  rl))) terms. Thus, Vl E (r )  Vl K (r ) 



∑

[ AnE ,l Jn (k⊥e |r  rl |)  BnE ,l H n (k⊥e |r  rl |)] ein arg(rrl )

(2.9)

[ AnK ,l Jn (k⊥e |r  rl |)  BnK ,l H n (k⊥e |r  rl |)] ein arg(rrl )

(2.10)

n



∑

n

While at first sight the results of Equations (2.9) and (2.10) may appear obvious and little more than the general solution of the Helmholtz equation, the above analysis is significant in that it points to the local and global representations of the field. Firstly, the representations (2.9) and (2.10) are valid provided that Cext contains only a single cylinder (or source). From this we may deduce that a sufficient condition for the validity and convergence of what we call the local expansions about Cl is that r lies in an annulus bounded by Cl and the largest contour Cext that does not intersect with any other source or cylinder. Secondly, the derivation identifies the respective sources of the field — with outgoing (Hankel H) waves emanating from Cl and with standing (Bessel J) waves being sourced on Cext. As we now show, this generalizes immediately to give rise to what is known as the Wijngaard expansion [30] — the global representation of the field that converges everywhere in the domain A. We thus consider the field problem of Figure 2.2 associated with a set of Nc cylinders located at positions rl and an explicit z-invariant line source at rs. Again using Green’s theorem, this time with the area A defined by the c interior of the region bounded by A  C0 ∪ Cs ∪lN1 Cl (Figure 2.2), we deduce Nc

V E (r )  ∑



∑ BnE ,l Hn (k⊥e |r  rl |)ein arg(rr )  W E (r)  S E (r) l

(2.11)

l1 n

together with a corresponding expression for VK. Here, W E (r ) 



∑

n

AnE , 0 Jn (k⊥e |r|)ein arg(r )

(2.12)

is the field due to sources exterior to C0, and SE(r) is the contribution to the field due to the line source, where S E (r ) 



Cs

   G0 (r; r ′) ∂ V E (r ′)  V E (r ′) ∂ G0 (r; r ′)  dsr′   ∂r ′ ∂r ′

(2.13)

and Cs is a circle that encloses the actual source. Note that the contour Cs need not be concentric with the source point. Evaluating the line integral in Equation

© 2006 by Taylor & Francis Group, LLC

From Multipole Methods to Photonic Crystal Device Modeling

A

CI

55

C0

Cs

FIGURE 2.2 The integration path for Green’s theorem for a cluster of cylinders and a source. Also shown is the annular region about Cl in which the local field expansion of the form (2.9) converges.

(2.13) gives, in general, a multipole representation of the form S E (r ) 



∑

n

BnE , Se H n (k⊥e |r  rs |)ein arg(rrs )

(2.14)

where rs is the center of the circle that contains the source. Thus, the contribution to the field due to an explicit source is indistinguishable from the contribution due to the scattered radiation from any cylinder.

2.2.3 RAYLEIGH FIELD IDENTITY The derivation of the field identity requires the matching of the local field expansion in the vicinity of cylinder l with the field due to the outgoing sources on all other cylinders j and the explicit sources s. That is, we set

∑

n

[ AnE ,l Jn (k⊥e |r  rl |)  BnE ,l H n (k⊥e |r  rl |)]ein arg(rrl ) Nc

∑



∑

j1 n

r j )

BnE , j H n (k⊥e |r  r j |)]ein arg(r

(2.15)

 W E (r )  S E (r ) (2.16)

an expression that is valid inside an annular region extending from the perimeter of cylinder l to the nearest source or cylinder. It is thus necessary to express all

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Electromagnetic Theory and Applications for Photonic Crystals

field and source expansions (2.11) in a basis of functions associated with a coordinate origin at the center of cylinder l. We express r  rj  (r  rl)  (rj  rl) and implicitly take the field point r to be in the vicinity of the surface of cylinder l, thereby assuming that |r  rl|  |rj  rl|. This change of coordinate origin necessitates the use of Graf’s addition theorem in the form [29] H m (k⊥e |r  r j |)eim arg(rrj ) 



∑

p

H pm (k⊥e |r j  rl |)ei ( pm ) arg(rj rl )

 J p (k⊥e |r  rl |)eip arg(rrl )

(2.17)

Thus, in the vicinity of cylinder l the regular part of the field due to all the other cylinders is

∑

n

Nc

AnE ,l Jn (k⊥e |r  rl |)ein arg(rrl )  ∑



∑

j1 n

j ≠l

BnE , j H n (k⊥e |r  r j |)e

in arg ( rr j )

(2.18)

Using Graf’s addition theorem, we deduce AnE ,lj 



∑

m

lj B E , j , H nm m

i ( nm ) arg ( r j rl ) lj  H e where H nm (2.19) nm (k⊥ |r j  rl |)e

An identical form to (2.19) also applies to explicit multipole source(s) which, in such formulations, are equivalent to additional cylinder(s). Accordingly, the contribution to the regular field in the vicinity of cylinder l due to the presence of either a cylinder j, or an explicit source j, may be expressed in matrix form as A E ,lj  Hlj B E , j

lj ] where Hlj  [ H nm

(2.20)

Here the notation X  [Xn] defines the vector X with elements Xn. Similarly, variables with two subscripts, as in (2.20), define a matrix. Similarly, the explicit external source contribution BE,Se to the regular field, in the vicinity of cylinder l, is obtained by applying Graf’s addition theorem to (2.14) for r  rs  (r  rl)  (rs  rl). That is,

∑

m

AmE ,lSe Jm (k⊥e |r  rl |) eim arg(rrl )

where AmE ,lSe 



∑

n

lSe H nm BnE , Se

(2.21)

The exterior field WE(r) (2.11) that originates at the outer boundary, and is defined with respect to the global coordinate system, needs to be recast in the natural basis of cylinder l. Again, we apply Graf’s theorem to change the origin of the field expansion, this time using Jm (k⊥e |r|)eim arg(r ) 



∑ Jmn (k⊥e |rl |)ei (mn) arg(r ) l

n

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Jn (k⊥e |r  rl |)ein arg(rrl )

(2.22)

From Multipole Methods to Photonic Crystal Device Modeling

57

and proceed to equate the field representations in the global and local bases



m

n

∑ AmE ,0 Jm (k⊥e |r|)eim arg(r )  ∑ AnE ,l 0 Jn (k⊥e |r  rl |)ein arg(rr )

(2.23)

l

to derive the new coordinates arising from the change of basis. That is, i ( mn ) arg( rl ) l 0 ], J l 0  J e where Jl 0  [ Jnm nm mn (k⊥ |rl |)e

A E, l 0  J l 0 A E ,0

(2.24)

The field identity, which is commonly referred to as the Rayleigh identity, is obtained by matching the local field expansion to the global Wijngaard expansion (2.16); i.e., A E ,l  ∑ A E ,lj  A E ,lSe  A E ,l 0

(2.25)

j ≠l

 ∑ Hlj B E , j  HlSe B E ,Se  Jl 0 A E ,0

(2.26)

j ≠l

A similar expansion also applies to the magnetic field component. Combining these, we obtain the vector form of the Rayleigh identity A l  ∑ Hlj B j  HlSe B Se  Jl 0 A 0

(2.27)

j ≠l

where A0



 E ,0  A       K ,0   A 

B0

,



 E ,0  B       K ,0   B 

,

Al



 E ,l  A       K ,l   A 

and

 E ,l  B       K ,l   B 

Bl





 l0 J     0

and Hlj



 lj H     0

    lj H 

0

,

H

lSe



       

HlSe 0

     lSe  

0 H

,

and

J

l0

    Jl 0 

0

Note that in (2.27) the source terms have been labeled as BSe rather than as BS to emphasize that the source is exterior to the cylinders. Finally, combining the equations (2.27) for all l, we derive the full vector form in which (2.27) becomes A  HB  HSe B Se  JB 0 A 0

(2.28)

where A  [A l ],

B  [Bl ],

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H  [Hlj ],

HSe  [HlSe ]

and

JB 0  [J l 0 ]

58

Electromagnetic Theory and Applications for Photonic Crystals

and where [HlSe] and [J l0] denote partitioned column vectors with components HlSe and J l0, respectively.

2.2.4 FIELD COUPLING AND CONTINUITY CONDITIONS Previously, we emphasized that a circular boundary can be regarded as a source of outgoing waves (Hankel H functions) for points outside it, and of regular standing waves (Bessel J functions) for points inside. Accordingly, as shown in Figure 2.3 the “outgoing” waves that are sourced at the boundary are caused by either of two fields that impinge on the boundary: (a) standing waves (i.e., Bessel J functions) from outside or (b) outgoing waves (i.e., Hankel H functions) due to possible sources inside the boundary. With this formulation in mind, we may characterize the action of the fields at this boundary in terms of cylindrical harmonic reflection and transmission matriˆ l and Tˆ l . The labels  and  classify the matrices as either ces denoted by R   exterior or interior forms respectively. Accordingly, on the exterior to the boundary, we express outgoing field coefficients (Bel ) as the sum of a reflection ˆ l A l ) of the exterior standing wave impinging on the interface and a trans(R  e l mission (Tˆ  Bil) of an outgoing field sourced from within the boundary. Similarly, the field inside the cylinder may be written as the sum of the transmission of the exterior standing wave component transmitted through the boundary and a reflection from the boundary of the interior source. Thus, Bel  Rˆl Ael  Tˆl Bil

(2.29)

Ail  Tˆl Ael  Rˆl Bil

(2.30)

Here the subscripts e and i indicate that the respective field quantities are associated with regions exterior, or interior, to the boundary. Note that the Bel and Ale are

FIGURE 2.3 The fields at the surface of a cylinder Cl.

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From Multipole Methods to Photonic Crystal Device Modeling

59

precisely the Bl and Al of (2.27). See the treatment in [31–33] for a complete derivation of these reflection and transmission matrices. ˆ l is a partitioned Now, suppressing the  subscripts, we observe that R matrix  ˆ EE ,l ˆ EK ,l  l R R   Rˆ   KE ,l KK ,l  R  ˆ ˆ R

(2.31)

This implies that, in general, the electric and magnetic fields are coupled rather than independent. However, note that this coupling occurs via the application of the boundary conditions (2.31), rather than through the Rayleigh identity (Section ˆ ab,l, Tˆ ab,l are diagonal, with the 2.2.3). For circular inclusions, all matrices R elements derived by solving the field continuity equations for the four tangential field components Ez, Kz, Eu, and Ku. In general, the field components Ez and Kz are ˆ EK and R ˆ KE do not vanish. However, in the case coupled because the matrices R of in-plane incidence, when k0z  0, for all types of cylinders (i.e., perfectly conducting, dielectric, finitely conducting), or for arbitrary k0z in the special case of perfectly conducting metal cylinders, for which the tangential components of the ˆ EK  R ˆ KE  0, and thus the general conical problem electric field vanish, R decouples [34] into the two disjoint E|| and H|| polarizations (also called TM and TE polarizations, respectively). For noncircular inclusions, Equations (2.29) and (2.30) still hold, this time respectively on the inscribing and circumscribing circles of the lth inclusion. To ˆ ab and T ˆ ab in this case, a full integration of the wave equaderive the elements of R tion is required — typically numerically [35]. The resulting forms are no longer diagonal, but instead are dense. In forming the field identity, only (2.29) is of immediate interest. In vector form, combining the results for all cylinders, it becomes ˆ  TB ˆ Si B  RA

(2.32)

where BSi refers to the sources inside the cylinders. For a single source contained within a particular cylinder p, BSi  [BlSidlp], all the source multipole coefficients are zero except for those corresponding to the source inside the cylinder p. Also, ˆ l) and Tˆ  diag(Tˆ l) are block diagonal matrices of the matrices R ˆl Rˆ  diag(R l ˆ and T , respectively. For the field problem associated with the solution of problems in radiation dynamics (see Section 2.4) which involves explicit sources exterior and interior to the scatterers, the solution of (2.28) and (2.32) yields the relevant identity ˆ )1 (RH ˆ SeB Se  TB ˆ Si ) B  (I  RH

(2.33)

which is exploited in Section 2.4. Note that in this equation there is no explicit exterior driving field, i.e., A0  0. Also, in the case of the radiation dynamics

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Electromagnetic Theory and Applications for Photonic Crystals

modeling, the single source is either an interior source or an exterior source, and hence only one of the two terms in the right-hand side of (2.33) can be nonzero for any given problem.

2.2.5 FIELD PROBLEM

OF

MICROSTRUCTURED OPTICAL FIBERS

Here we discuss how the theory developed in Sections 2.2.2–2.2.4 applies to microstructured optical fibers (MOFs) — also known as photonic crystal fibers or holey fibers. The results of this analysis are discussed in Section 2.3. There are two main types of MOFs: one type consists of pure silica, with a solid core and a cladding that has a regular pattern of air holes that run the length of the fiber. Typically, the holes are hexagonally packed so that they form a finite photonic crystal having the cross section shown in Figure 2.4(a). For this type of MOF, the light is guided through the core by total internal reflections, somewhat similar to the guiding mechanism in conventional fibers. The second class of MOFs has an air core (Figure 2.4[b]), through which the light is guided only by Bragg reflections due to the photonic crystal cladding. From a theoretical point of view, a MOF can be regarded as a finite set of infinite parallel cylinders, for which we set up a field problem in order to determine the modes of the structure. More precisely, we seek the values of k0z that yield a nontrivial solution of the homogeneous wave equation (2.3), i.e., in the absence of either explicit sources or exterior driving fields. We begin with a MOF surrounded by a jacket and note that this cladding boundary acts as a source of standing (Bessel J) waves that arise due to reflections of outgoing fields originating on the cylinders. The modeling of a MOF using the multipole method requires two types of systems of coordinates: a global system having the origin at the center of the fiber, and local systems attached to the centre of each cylinder. In the case of a solid core MOF the key equation of our formulation is (2.28), but with BSe  0

Λ d

(a)

(b)

FIGURE 2.4 The cross section of a MOF with a hexagonal arrangement of holes. (a) Solid core MOF, (b) air core MOF.

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From Multipole Methods to Photonic Crystal Device Modeling

61

(i.e., without any exterior source). Thus, each cylinder l gives rise to outgoing waves, which are respectively represented in the global and local systems of coordinates for the electric field component Ez as

∑ BnE 0l Hn (k⊥e |r|)einu  ∑ BmEl Hm (k⊥e |r  rl |)eim arg(rr ) l

(2.34)

m

n

Applying Graf’s addition theorem to the Hankel functions on the right-hand side of Equation (2.34) in the vicinity of the jacket boundary where r rl, we may write [29] H m (k⊥e |r  rl |)eim arg(rrl ) 



∑

n

H n (k⊥e |r|)ein arg r Jnm (k⊥e |rl |)ei ( nm ) arg rl

(2.35)

and, in turn, express (2.34) in the matrix representation B E 0 l  J 0 l B El

i ( nm ) arg rl 0 l ], J 0 l  J e where J 0 l  [ Jnm nm nm (k⊥ |rl |)e

(2.36)

Here, J0l can be regarded as the matrix of a transformation that changes the representation in the local basis associated with cylinder l into the global representation. Note also that in this nomenclature the addition theorem (2.35) can be expressed as H m (k⊥e |r  rl |)eim arg(rrl ) 



∑

n

0 l H (k e |r|)e in arg( r ) Jnm n ⊥

(2.37)

The sum of all outgoing electric fields sourced on the cylinders represents the total electric field incident on the jacket. In global coordinates, this has the functional form VJE 0 (r ) 



∑

n

BnE 0 H n (k⊥e |r|)ein arg(r )

(2.38)

where, in vector form, Nc

Nc

l1

l1

B E 0  ∑ B E 0 l  ∑ J 0l B El

(2.39)

Combining this form (2.39) and the analogous expression for the magnetic field, leads to the vector representation of the outgoing multipole field that is incident on the jacket B0

N c  J 0 l    l1  0

∑

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    0 l J 

0

 El B    Kl  B

     

Nc

 ∑ J 0 l B l  J0 BB l =1

(2.40)

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Electromagnetic Theory and Applications for Photonic Crystals

where J0B represents a 1  Nc partitioned vector [J 0l], the components of which are 2  2 block diagonal matrices J 0l  diag(J0l, J0l). We complete the procedure by observing that the regular field sourced by the jacket arises because of the reflection of the outgoing radiation (B0) from the cylinders (2.40). Accordingly, from Equation (2.30), we deduce that A 0  Rˆ 0 B 0

(2.41)

ˆ 0 is the interior (cylindrical harmonic) reflection matrix for the jacket. where R The MOF field identity then follows by combining Equations (2.28), (2.32) with no interior source, (2.40), and (2.41) to derive the homogeneous equation [I  Rˆ (H  JB 0 Rˆ 0 J0 B )]B  0

(2.42)

For an infinite cladding, corresponding to the absence of a jacket, we simply set ˆ 0  0 and derive R ˆ )B  0 MB  (I  RH

(2.43)

The nontrivial solutions of (2.42), or the derived form (2.43), correspond to the modes of the MOF. To locate the modes, we seek those values of k0z for which the determinant of (2.42) or (2.43) vanishes. Then, for each root k0z the multipole coefficients of the corresponding mode are determined from the singular value decomposition of the matrix in (2.42). A detailed discussion of the implementation of the multipole method is contained in Section 2.3.3.

2.2.6 INFINITE PERIODIC STRUCTURES We now detail the application of the multipole method to periodic structures consisting of regular two-dimensional lattices (arrays) of parallel cylinders. With the lattice specified by basis vectors a1 and a2, the location of the center of unit cell l  (l1, l2) is written as rl  l1a1  l2a2. In the case of the simplest such structure, that is with a single cylinder centered in each unit cell, the regular field incident on each cylinder must be equal to the sum of the outgoing fields from all the other cylinders in the lattice. Thus for a particular cylinder l in the lattice, it follows that in the local coordinate system of cylinder l, the electric field multipole coefficients satisfy AnEl  ∑ ∑ Hnljm BmEj j ≠l

(2.44)

m

Combining (2.44) with its magnetic field counterpart then yields A l  ∑ Hlj B j j ≠l

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(2.45)

From Multipole Methods to Photonic Crystal Device Modeling

63

O′ rl ′


l ′′

O rl
l ′′

FIGURE 2.5 Two supercells showing the position of a cylinder l  (l , l).

This form is identical to that of (2.27), with the exception of the interior or exterior source terms. Before dealing with this problem further and, in particular, the characterization of the periodicity, we generalize the treatment to consider a lattice of identical unit cells, each of which comprises a common, finite set of identically placed cylinders (Figure 2.5). For such composite structures, it is common to refer to the unit cell as a supercell. The characterization of the geometry of this structure requires two vector bases: 1. A basis of lattice vectors {a1, a2} that generates the supercells with the vector rl defining the position of the center of supercell l  (l 1, l 2) given by rl  l 1a1  l 2a2. 2. A local basis within each supercell, where the vector Rl determines the position of cylinder l relative to a reference point in each supercell (typically the center of the supercell). Thus, each cylinder l is characterized by the pair l  (l , l), which denotes the cylinder belonging to supercell l (associated with rl ) and having a local position Rl within the l supercell. The position of cylinder l relative to global coordinate origin is thus rl  rl  Rl. One of the supercells is designated as the primary supercell and is labeled with a superscripted 0, corresponding to the choice l  0 with the global coordinate origin set at the center of this supercell. Each cylinder l within the primary supercell (l  0) is matched by a corresponding cylinder l  (l , l) in each of the supercells l  0. Because the structure is infinite and periodic, the field, and thus the multipole coefficients, must satisfy the constraints A El  A El ′′ eik0  rl ′ ,

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B El  B El ′′ eik0  rl ′

(2.46)

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Electromagnetic Theory and Applications for Photonic Crystals

imposed by the Bloch condition, V(r  rl)  V(r) exp(ik0  rl). Note that in (2.46) we have simplified the notation for the primary supercell quantities by referring to AE(0,l) and BE(0,l), respectively, by AEl and BEl. To obtain the field identity equivalent to (2.44) for a unit cell with multiple cylinders, we consider a fixed cylinder l  (l , l) and partition the sum on the right-hand side of (2.44) into two parts: 1. Over the periodic replicates j  (j , l) of the local cylinder l  (l , l) in supercells j  l , 2. Over cylinders j  ( j , j) for all j  l and in all supercells j (including j  l ). Using the Bloch condition (2.46), the multipole source coefficients arising in these two sums are BmEj  BmE ( j′,l ′′)  BmEl ′′ eik0 rj′

for j ′ ≠ l ′

BmEj  BmE ( j′, j′′)  BmEj′′ eik0 rj′

for j ′′ ≠ l ′′

and

Thus, the field identity (2.44) reduces to AnEl ′′ 



∑

m

Snl ′′l ′′m BmEl ′′ 



∑ ∑

m j′′ ≠ l ′′

Snl ′′j′′m BmEj′′

(2.47)

where Snl′′l ′′  ∑ H n (k⊥e |r j′ |)e−in arg(rj′ ) eik0  rj′

(2.48)

j′ ≠ 0

Snl′′j′′  ∑ H n (k⊥e |r j′  R j′′  Rl ′′ |)ein arg(rj′

+ R j ′′ −Rl ′′ ) ik0  r j ′

e

(2.49)

j′

are the lattice sums discussed below. Note that in deriving (2.48) and (2.49) we have used the fact that the difference of two lattice vectors is itself a lattice vector, hence replacing (rj  rl ) by rj and making the appropriate adjustments to the summation indices, i.e., changing them from j  l to j  0. More generally, this shows the invariance of the lattice sums with respect to the transformations of the subgroup of discrete translations from the lattice symmetry group. Also, the derivation of the lattice sums in Equations (2.48) and (2.49) has relied solely on Bloch’s theorem with no presumption of the dimensionality of the lattice. In the case of 2D arrays, the summation over j is two dimensional, resulting in what we will refer to as array sums.

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From Multipole Methods to Photonic Crystal Device Modeling

65

For diffraction gratings that are considered in the following section, the summation runs over a one-dimensional lattice and leads to grating sums. The quantities Sll n (2.48) are referred to as the global lattice sums and are an essential feature of the application of multipole methods to periodic systems. For lattices with complex unit cells with more than one cylinder, the global lattice sums need to be augmented by the relative or shifted lattice sums (2.49). Note that for j  0 we have |Rj  Rl|  |rj |, and thus by applying the addition theorem for Hankel functions (2.17), we can express the relative lattice sums (2.49) as a series in terms of the global lattice sums (2.48) (e.g., [36]): Snl′′j′′ 



∑

m

Sml′′l ′′ Jn−m (k⊥e |R j′′  Rl ′′ |) ei ( nm ) arg( R j′′Rl ′′ )

(2.50)

The construction of both global and relative lattice sums relies on the existence of a suitable addition theorem (e.g., Graf’s theorem) for the basis of functions (e.g., cylindrical harmonics in this case) in which fields are expanded. In general, the global sums are sums of multipole source terms over the entire lattice (with the exception of the primary supercell) and characterize the field of a phased array of multipole sources at the origin of the primary cell. In passing, we mention an alternative and widely used definition of the global sums that appears in the literature in arg ( r j ′ ) ik0  r j ′ Snl′′l ′′  ∑ H n (k⊥e rj′ )e e

(2.51)

j′ ≠ 0

and which is related to the form (2.48) used throughout this chapter by ~ ll  (1)nSnll. Sn While it is beyond the scope of this chapter to give a comprehensive description of the computation of the lattice sums, we nevertheless outline the techniques involved and refer interested readers to the references [10] and [36–38]. In what follows, we consider global array (2D) sums (2.48) (which for simplicity we denoted by SnA  Snll) and discuss the difficulties in their evaluation caused by convergence problems in the defining series, the terms of which are summed over the direct lattice (2.48). For this series, Cauchy’s integral test shows that the SnA are conditionally convergent for all orders n. To overcome these difficulties, we have devised two summation methods. The first converts the sum over the direct lattice to an absolutely convergent sum over the reciprocal lattice, the convergence of which may be accelerated by successive integrations [10,37]. The second technique is based on the observation that a 2D array may be regarded as an infinite stack of 1D gratings, leading to the representation of the array sum in terms of a grating sum [8] and exponentially convergent correction terms [11,36]. The derivation of the former method commences with the comparison of the spatial and spectral domain forms of the quasiperiodic Green’s function [10,37], i.e., G (J) 

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i 4

∑ H0 (k⊥e |J  rj |)eik  r 0

j

j

(2.52)

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Electromagnetic Theory and Applications for Photonic Crystals

and G (J) 

1 A WS

∑ l

eiQl  J

(2.53)

Ql2  k⊥e 2

respectively, where J is an arbitrary vector inside the circle inscribed within the primary Wigner–Seitz cell, AWS is the area of the unit cell, Ql  Kl  k0, and {Kl} are reciprocal lattice vectors corresponding to the direct lattice vectors {rj} [10,37]. We apply Graf’s theorem to (2.52) and identify the array sums (2.48) in the result G (J) 

i i H (k e j)  4 0 ⊥ 4



∑

n

SnA Jn (k⊥e j)ein arg(J )

(2.54)

Here, the first term represents the source due to the central cylinder, while the terms in the series describe the multipole sources of order n, with the array sums giving the contribution of all the other cylinders of the array. Next, we use the Bessel series expansion for the exponential in (2.53): eiQl  J 



∑

m

i m Jm (Ql J)eim arg(Ql ) eim arg(J )

(2.55)

and by means of Poisson’s summation formula we may show that (2.54) and (2.53) are equal [10,37]. We then identify the coefficients of equal powers of exp[i arg(J)] to obtain the expressions for the array sums SnA Jn (k⊥e J) H 0 (k⊥e J)dn 0 

4i n1 A WS

∑ l

Jn (Ql j) Ql2

 k⊥e 2

ein arg(Ql )

(2.56)

Note that in (2.56) J is an arbitrary vector inside the primary Wigner–Seitz cell. While the series in (2.56) converges absolutely as O(Ql2.5), its convergence may be accelerated by integrating over j, exploiting the identity [znJn(z)]  znJn1(z), to increase the order of the Bessel functions and thereby increasing the rate of convergence to O(Qlm2.5) after m such integrations. Hence, our summation method [10,37] is far superior to Ewald’s method [9] in both accuracy and speed of convergence. The derivation of the second summation method is based on relationships between the global array and grating sums [11]. The index j in the definition of global array sums (2.48) represents a pair of integers so that j  (p, q), and j  0 means (p, q)  (0, 0). The terms corresponding to q  0 are associated with cylinders located on a particular line of the array that can be regarded as a grating. Consequently, for an array defined by the vectors rpq  pa1  qa2, with a1  (D, 0), we may rewrite (2.48) in the form [11] SnA 

∑

( p, q ) ≠ ( 0 , 0 )

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H n (k⊥e |rpq |)ein arg rpq eik0  rpq  SnG  Sn (

)

(2.57)

From Multipole Methods to Photonic Crystal Device Modeling

67

where SnG 

∑ Hn (k⊥e |rp0 |) ein arg(r

p0 )

eipa0 D

(2.58)

p≠0

 Sn  ∑ m q q≠0



∑

p

in arg ( rpq )

H n (k⊥e |rpq |) e

eipa0 D

(2.59)

with m  exp(ik0  a2), a0  k0x. Here, D is the array constant, in the direction of a1, of an array of supercells. The grating sums SnG of Equation (2.58), which are conditionally convergent for all orders n, have been studied by Twersky [8], who using Sommerfeld’s integral representation of Hankel functions and Euler’s summation formula obtained an absolutely convergent sum over the reciprocal lattice, in terms of elementary functions. The correction factor Sn in (2.59) converges exponentially [11] so that the summation method (2.57) is computationally advantageous. It is also much faster than the previous method (2.56), which relied on accelerated convergence techniques, and can be extended to higher dimensions and different lattice geometries. Returning now to the field problem, we express (2.47) in vector form A El  ∑ Slj B Ej ,

lj ], where Slj  [ Snm

and

j

lj  S lj Snm nm

(2.60)

Note that an identical expression holds for the multipole coefficients of magnetic field. Then, combining the corresponding equations for the electric and magnetic fields in a matrix form we derive A l  ∑ S lj B j

or

A  SB

(2.61)

j

the Rayleigh identity for the periodic system. In (2.61), Slj  diag(Slj, Slj) is a 2  2 block diagonal matrix, and S  [Slj] is a rectangular matrix of partitions Slj. The band diagram of a lattice of parallel cylinders may then be generated from the solution of a propagation problem that derives from (2.61) and the boundary conditions B  RA derived from (2.32) in the absence of sources. That is, ˆ )B  0 MB  (I  RS

(2.62)

e and the resolute of the Bloch The matrix M is a function of both wavenumber k vector in the plane perpendicular to the axis of the cylinders. In the case of a strictly 2D problem, this vector is k0, while for a 2.5D problem with specified z-dependence of exp(ik0z z), the Bloch vector resolute is k0  k0  k0zzˆ. Accordingly, the propagation problem is solved by prescribing the Bloch vector k0 or k0, as appropriate, and determining the eigenvalue ke that render M in (2.62) singular. Once a zero of the determinant is found, we may use a singular value decomposition to determine the null vectors B which characterize the modes.

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Electromagnetic Theory and Applications for Photonic Crystals

2.2.7 GRATINGS With the Rayleigh identity and lattice sums now established for infinite periodic media, we turn our attention to the theory of gratings (1D and 2D), the scattering action of which we characterize by plane wave scattering matrices. Here, we consider a layer of supercells in the xz plane, which in the case of a 1D cylinder grating (with cylinder axis parallel to the z-axis) is generated by the lattice vector a1  Dxˆ. Throughout this section we take the cylinders defining the reference origin in each supercell located with centers at rj  j Dxˆ . In the calculation of the lattice sums, the summations over j in Equations (2.48) and (2.49) run over the 1D array of supercells. We will also use two representations for the reciprocal lattice vectors Qs (s  Z ) namely, Qs  Qts  Qys yˆ  Qs  Qzs zˆ

(2.63)

The first form is necessary in plane wave expansions and, in particular, in the solution of the diffraction problem in which transverse field components are matched at the interface between free space and the grating region. The second form is necessary in the prescription of the multipole representations of the longitudinal field components Ez and Kz in the grating region, for which the z-dependence is exp(iQzsz). The focus of our calculations is the solution of an inhomogeneous scattering problem, driven by incident plane wave fields respectively from above and below:   E    Einc   E   z     Ds  eiQys y   Ds  eiQys y  ei (Qxs xQzs z )  ∑  inc    K    D K    K z   Ds  s   s 

(2.64)

where Qs  Qts  Qys yˆ ,

Qts  (Qxs , 0, Qzs )  (Qxs , 0, Qz )  k0 t 

2ps xˆ , s ∈  D (2.65)

—— —— Here, k0t  (k0x, 0, k0z), Qys  k2  Q2ts, and k  kfve denotes the background wavenumber. In the case of a single grating layer driven by a plane wave of wavelength l incident on the grating with polar and azimuthal angles w and c, respectively, the vector Q0 takes the form k0  k(sin w cos c, cos w, sin w cos c). Here, however, we are concerned with not just one plane wave but a general incident field that may comprise an arbitrary linear combination of all plane waves with direction sines (Qxs /k and Qzs /k) that belong to the family defined by k0. Later, in Section 2.5 dealing with photonic crystals, we will adopt a different interpretation regarding the transverse resolute k0t as the transverse resolute of the Bloch vector.

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69

The derivation of the field identity in terms of multipole coefficients follows along similar lines to that for Equation (2.25) but without the explicit source term and this time in the context of a periodic array (2.61). We thus write A l  ∑ S lj B j  J l D  J l D

(2.66)

j

in which the final two terms are multipole representations of the plane wave driving fields (from above and below) and are analogous to the term J l0A0 in Equation (2.27). To see this, let us consider a grating with a unit cell comprising Nc collinear cylinders centered at Rl  rlxˆ and write (for cylinder l) ei (Qxs xQys yQz z )  eiQxsrl eiQz z ei[Qxs ( x l ) r

Qys y ]

(2.67)

Then, expanding the last exponential in terms of Bessel functions of the first kind we form ei[Qxs ( xrl )Qys y ] 



∑

n

(1)n Jn (Qs sl ) ein (us wl )

(2.68)

where Qs  (Qxs, Qys, 0), us  arg(Qys  iQxs), Sl  r  Rl, and wl  arg(Sl), from which we obtain the cylindrical harmonic representation of the backward and forward going plane waves ei (Qxs xQys yQz z )  eiQz z



∑

n

l Jns Jn (Qs sl )einwl

(2.69)

l l where Jl  [Jns ], Jns  (1)neinuseiQxsrl. The matrices Jl are then used to change the basis of the incoming plane waves D from above and below to their cylindrical harmonic representations J l D  J l D where J l  diag(Jl, Jl) and D  [(DE)T (dK)T]T. Finally, combining Equations (2.66) for all cylinders into a single matrix equation in the same manner as on previous occasions, we arrive at

A  SB  JD  JD

(2.70)

in which J is a column vector formed from the J l. Using Equation (2.70), we derive the field identity ˆ )1Rˆ (JD  JD ) B  (I  RS after the boundary conditions have been applied.

© 2006 by Taylor & Francis Group, LLC

(2.71)

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Electromagnetic Theory and Applications for Photonic Crystals

Equation (2.71) expresses the outgoing multipole field B generated by the incident plane wave field D. From this, we may reconstruct the plane wave fields radiated by the grating and, in turn, form the plane wave scattering matrices. The sketch of the derivation given here is necessarily brief and we refer the interested reader to [36] and [39] for a comprehensive treatment. This derivation of plane wave quantities begins with the Cartesian form of the free space Green’s function G0 (r , r ′) 

1 2D



∑

s

1 i[Qts .(r−r′)+Qys | y− y′|] e Qys

(2.72)

Following an application of Green’s theorem [36,39] we may compute the outgoing fields above and below the grating in terms of the multipole source coefficients B. Thus, we may show that the outgoing fields are  E  z   Kz 

      

  f E    f E    ∑   s  eiQys y   s  eiQys y  ei (Qsx xQz z )  K  fsK  s    fs   

(2.73)

where   f        f   



  D         D 





2 1  K  Q  B D y  K  

(2.74)



 f E∓  , K  [K,l], a row vector of matrices K,l  In Equation (2.74) f    K∓  f  iQxsrl ,l diag(K,l, K,l), K,l  [K,l (1)neius. Also, Qy  sn ], and Ksn  e diag(Qys), Qy  diag(Qy, Qy), and Qy  diag(Qy, Qy). In Equation (2.74), K performs a change from the basis of cylindrical harmonics to the basis of plane waves, analogous to the change from the plane wave to the cylindrical harmonic basis performed by the J. We also observe that Equation (2.74) comprises two terms — an undeviated incident beam D and a diffracted/scattered field caused by the presence of the grating. Combining Equations (2.71) and (2.74) yields   f        f   

   I   



      K    D  2    ˆ ˆ  1   Q y   (I  RS) R[J J ]       D  K   D    

(2.75)



The final step is to convert to the more usual TE/TM system of plane wave modes where the term “transverse” refers to the fields being transverse to the plane of

© 2006 by Taylor & Francis Group, LLC

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71

grating. Here, we follow the nomenclature of Smith, et al. [40], in which the TE and TM plane wave modes are, respectively, R sE (r ) 

yˆ  Qt ,s Qt , s

eiQt ,s ⋅r ,

R sK (r ) 

Qt , s Qt ,s

eiQt ,s ⋅r

(2.76)

On this basis, the standard representations of the electric and magnetic fields are 1/2 {F E eiQy ,s y  F E e iQy ,s y }R E Et  ∑ [j s s s s s

j1s/2 {FsK eiQy ,s y  FsK eiQys y }R sK ]

(2.77)

zˆ  K t  ∑ [j1s/2 {FsE eiQy ,s y  FsE eiQy ,s y }R sE s

1/2 {F K eiQy ,s y  F K e iQys y }R K ]   s s s s

(2.78)

with js  Qys /k. Equations (2.77) and (2.78) apply both above and below the grating. Following the derivation in [40] and using the subscripts 1 to denote fields above the grating and 2 to denote fields below the grating, we convert the Cartesian form (2.75) into a TE/TM form    F  2,       F1,   

S

  F   1,       F2,   

where Fj, 

 E  F       K±   F 

(2.79)

and S has the form  s s    K LJ K sLJa    k    S  T1 I  2 X   X T  k D  K aLJs K aLJa       

(2.80)

ˆ S)1R ˆ , k2  k2  Q2  Q2  Q2 , and T is a In Equation (2.80), L  (I  R  z xs ys transformation that reflects the symmetry relationships between electric and magnetic quantities that are imposed by Maxwell’s equations. The matrices Ja and Js (respectively Ka and Ks) are linear combinations of the matrices J (respectively K). Here Ja and Js represent a change of basis from plane waves into cylindrical harmonics, while Ka and Ks represent the inverse transformation. The indices s and a refer to a generalization of the symmetric and antisymmetric problems discussed in [36]. If the grating is up-down symmetric, then the terms KsLJa and KaLJs vanish and the system decouples completely. That is, the solution of the diffraction problem is reduced to two independent problems (whose matrices are of half the dimension of the original) — the symmetrized and antisymmetrized solutions that correspond to a grating, embedded into which (through the center line) is a

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Electromagnetic Theory and Applications for Photonic Crystals

magnetic or electric mirror. While the plane wave diffraction problem is best formulated in terms of TE and TM modes, the multipole scattering problem as developed in [39] is best handled in terms of principal Cartesian field components parallel to the cylinder axes. These two representations are related by the matrix X. We next turn to 3D photonic crystals. One such device is the woodpile photonic crystal, which is a structure consisting of layers of cylindrical rods with a stacking sequence that repeats itself every four layers. Within each layer, the rods are parallel and separated by a constant distance D. The distance between successive layer centers is also constant, and the rod axes in adjacent layers are orthogonal. To obtain a periodicity of four layers in the stacking direction, rods separated by one intermediate layer are offset by a distance of D/2 in the direction perpendicular to the rod axes (see Figure 2.17[b]). In order to obtain reflection and transmission scattering matrices for the woodpile structure, we first derive the scattering matrices for the basic unit, namely a pair of crossed gratings, and then use a well-established algorithm [36] to form a stack of such crossed pairs. The plane wave diffracted orders in a pair of crossed gratings are a doubly infinite set indexed by pairs ( p, q) corresponding to diffraction in the plane of each grating. Here, q enumerates the diffracted orders of the grating with generators parallel to the z axis, while p enumerates the orders of the orthogonal grating. For either grating, there is dispersion in only one direction. Thus the 2D scattering matrix for a single grating is essentially a block diagonal matrix with each block being the scattering matrix for a 1D problem indexed over, say, channels q and driven with incidence parameters corresponding to channel p of the orthogonal grating. The scattering matrix for the bottom grating is computed by a change in coordinates x  z, y  y, and z  x, in which the rod axes of the bottom layer are parallel to the new z axis. This is a subtle and intricate process, and the reader is referred to [40] for details. We can also derive terms for the reflection and transmission scattering matrices for both a single layer of parallel rods and a structure consisting of a pair of crossed gratings. Let R and R , respectively, denote reflection scattering matrices for incidence above and below the grating, and let T and T denote the corresponding transmission scattering matrices. Clearly the matrix S of Equation (2.79) can be expressed in the form  T R′   S    ′  R T 

so that

  F  2      F 1  

 T R ′   F     1      ′  R T   F2 

(2.81)

In the 1D case, a comparison between Equations (2.79), (2.80), and (2.81) can thus be used to yield explicit expressions for the transmission and reflection matrices [40]. There is an analogous method for the 2D case of a pair of crossed gratings. We have to replace the scattering matrix S with the corresponding scattering matrix for the crossed pair. Equation (2.81) provides the basis for the Bloch mode transfer matrix method (Section 2.5), which provides an efficient method for calculating the band structure of 1D and 2D PCs.

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73

This concludes our development of the underlying multipole theory for photonic crystal structures, providing the essential and integrated theoretical foundation for the study of applications in three paradigm areas — propagation in microstructured optical fibers (Section 2.3), radiation dynamics of photonic clusters (Section 2.4), and a Bloch mode analysis of composite PC devices (Section 2.5). In the subsequent sections various extensions to the theory will be made. In all cases, however, this will be undertaken in the context of the particular application.

2.3 MULTIPOLE MODELING OF PHOTONIC CRYSTAL FIBERS 2.3.1 BACKGROUND Photonic crystal fibers (PCFs), or microstructured optical fibers (MOFs), consisting of an arrangement of dielectric cylinders running down the length of an optical fiber have generated much interest in recent years. These fibers in their various forms and geometries have a number of potential applications in nonlinear optics, high-power laser delivery, dispersion compensation, and tunable optical devices. Since in many cases the ideal geometry of a MOF involves circular dielectric or air cylinders in a homogeneous background, the multipole method provides an efficient and accurate technique for modeling such fibers. In addition, it handles finite collections of scatterers exactly. It does not discretize the problem using a grid and so is not affected by numerical convergence problems or the use of periodic boundary conditions that afflict plane wave and other methods that can introduce errors such as the numerical birefringence of modes. By virtue of solving for the propagation constant b  k0z as a function of kf , the method’s handling of material dispersion has advantages over techniques (such as plane wave eigenvalue methods) that solve for kf as a function of b. Furthermore, the method is capable of yielding the geometric loss of MOFs accurately (see Section 2.3.3). The most common MOFs have circular air holes in a silica background, arranged in a number of rings about a core region that can be formed by either a missing hole or a larger hole, as shown in Figure 2.4. Although guiding is still possible if the holes are arranged irregularly around a solid core region [41], we consider here the case where the holes form a regular hexagonal pattern characterized by a hole separation of  and diameter of d. The dimensionless ratios d/ and l/ are used widely to describe MOF geometries.

2.3.2 GUIDING MECHANISMS AND THE “FINGER DIAGRAM” The multipole method and associated Bloch techniques provide several useful tools for a thorough study of MOFs. We begin by considering the properties of the holey cladding region surrounding the core. The cladding essentially provides a mirror for reflecting light back into the core, one of two requirements for the existence of an optical fiber mode. The second condition for a mode is that a phase condition is satisfied. In conventional fibers the mirror is a result of total internal reflection at the core–cladding interface, and thus the calculation of modes essentially involves only

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Electromagnetic Theory and Applications for Photonic Crystals

the phase condition being satisfied. The cladding of a MOF can behave as a mirror in a number of different ways. If the longitudinal propagation constant b is such that total internal reflection occurs between the core region and the cladding, in which the average refractive index is between that of air and glass, guidance is similar to that in conventional fibers, and light can be guided in a solid core (“effective index guiding”). However, for a MOF with an air-core, this condition cannot be satisfied, and reflection is provided by Bragg reflection from the regular arrangement of cylinders in the transverse direction. Since this case is much more sensitive to propagation conditions, one must first identify the parameters for which the cladding behaves as a Bragg reflector before searching for modes that satisfy the required phase relation. A third confinement mechanism, similar to that in antiresonant reflecting optical waveguide (ARROW) structures is not discussed here [42,43]. In Section 2.5, we describe the application of the Bloch method for computing band diagrams for conical incidence on an array of cylinders. Although the refractive index difference between the air holes (nc  1) and the (typically) silica background (ne  1.45) is insufficient to achieve a full bandgap for transverse propagation, bandgaps open up for conical incidence. Locating the bandgaps of the holey MOF cladding requires traversing kf  b parameter space and assessing whether propagating states exist. For each b, the perimeter of the irreducible part of the Brillouin zone is traversed for all frequencies, to produce, in effect, a transverse band diagram for every b, which we refer to as a finger diagram. Strictly speaking, one would have to include the entire irreducible part of the Brillouin zone and not just its perimeter. However, in practice band minima and maxima occur only on high symmetry points or lines. Accordingly, it is unlikely for the interior of the Brillouin zone to limit the extent of the bandgap, and thus it is sufficient to compute the finger diagram using only the perimeter of the Brillouin zone. See Section 2.5.3 for a discussion of the computation of band and finger diagrams using the Bloch transfer matrix method. The results of such a scan are shown in Figure 2.6 for a hexagonal lattice of air holes with d/  0.696 in a background of index ne  1.39. The plot color density is proportional to the number of propagating states, and thus complete bangaps are represented by the white fingers in which the mirror condition is satisfied. The region of interest for air-guided MOF modes lies near the intersection of the light line for air kf  b and the bandgap fingers. If the same cladding were used to surround a solid-core, effective index guiding MOF, one would expect to find guided modes in the region kf vc  b  kf ve bounded by the air and silica light lines marked on Figure 2.6. Thus the finger diagram is of most benefit when studying bandgap guided modes. At this point we discuss one other feature of the finger diagram. Note that at each frequency there is a maximum value for b. This b is associated with the fundamental space filling mode (FSM), the mode with the largest propagation constant that can propagate in the cladding for a given frequency [44]. Therefore, we know that any MOF modes that we find with larger b must be confined, since the field cannot propagate through the cladding. These modes, which can be found between the FSM and the light line for the background medium, rely for their confinement on

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From Multipole Methods to Photonic Crystal Device Modeling 18

75

11.5

16 11

12

k0Λ

k0Λ

14

10 8 6

FSM

e

lin

ht

10

ig rl

ai

Effective index ine mode region htl lig a ic l i Air core mode region s

8

10.5

10

12 14
Λ

16

18

9.5 9.5

10

10.5

11

11.5


Λ

(a)

(b)

FIGURE 2.6 (a) The “finger” diagram for a hexagonal air hole lattice with dD/  0.696 and background refractive index n  1.39. (b) The air core mode region with the light line (dashed) and the modal dispersion curve (solid) displayed.

total internal reflection. Modes that are located in the finger diagram in one of the white fingers, are confined by Bragg reflection. In the following sections we describe some typical applications of the multipole method to studying both index guiding and photonic bandgap (PBG) guiding microstructured optical fibers.

2.3.3 IMPLEMENTATION

OF THE

MULTIPOLE METHOD

In Section 2.2.5, the Rayleigh identity was derived for a finite array of dielectric cylinders under conical incidence. The application of this to MOFs is relatively straightforward. Since there are no external sources, we must simply search for solutions to the equation det(I  RH)  0 (2.43) by fixing the frequency of the light and scanning over the complex propagation constant b. As discussed in more detail in Section 2.3.4, the propagation constant is in general complex, with the imaginary part indicating the modal loss. This contrasts with conventional optical fibers where the propagation constants of the bound modes are real. The difference is that in conventional fibers the cladding can be taken to be infinite to a good approximation, so that the field strength in the outer cladding is essentially zero. In contrast, in MOFs the size of the cladding is determined by the position of the holes and is finite. Therefore there is always a finite probability for the light to traverse the cladding region leading to modal losses. This argument is sufficiently general not to rely on the specific guiding mechanism and is thus applied to the study of both index guidance and photonic bandgap guidance. Because the propagation constant is a complex quantity, the search for the modes can be time consuming and requires an algorithm capable of finding all the zeros of the determinant of (2.43) in a region of the complex neff plane. The algorithm should be economical in function calls, as each evaluation of the determinant is

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Electromagnetic Theory and Applications for Photonic Crystals

computationally expensive for large structures. Since the zeros are usually very sharp, a highly accurate first estimate of the zero is necessary for most simple root-finding routines. More specific algorithms for finding zeros of analytic or meromorphic functions have good convergence for simple structures (with six cylinders, for example) but fail for more complex structures, even with good initial guesses. Our present approach to root finding seems computationally efficient. We first compute a map of the modulus of the determinant over the region of interest and then use the local minima of this map as initial points for a mixed zooming and modified Broyden algorithm (an iterative minimization algorithm, guaranteed to converge for parabolic minima) [45]. The initial scanning region has to be chosen in accordance with the physical problem studied: if the fiber is air-cored and air-guided modes are sought, we choose Re(neff)  1, whereas if the fiber has a solid core we usually choose to search for modes in a region where Re(neff) lies between the optical indices of the background and the fundamental space filling mode (FSM) (Section 2.3.2). Note that for a given propagation constant, the FSM is the mode of lowest frequency that can propagate in the bulk photonic crystal (that comprises the cladding) and thus nfsm defines the lower bound for the search. In the latter case, for small Re(neff), there may exist hundreds of modes that are of little interest because of their high losses. We therefore often concentrate on a smaller neff scanning region near the highest index of the structure. A scanning region for Im(neff) that gives good results in almost any case is 1012  Im(neff)  103. For large MOFs with reflection and/or rotational symmetry, the mode calculation time can be reduced significantly by applying a knowledge of the mode symmetries determined by the symmetry properties of the MOF structure [46,47]. This can be achieved because only multipole coefficients for inclusions lying in the minimum sector [46] need be specified; those for holes outside the minimum sector can be obtained by multiplying by the appropriate geometric phase factor. The resultant reduction in the order of matrix I  RH in (2.43) depends on the type of mode that is addressed. For the most common six-fold symmetric MOFs, the matrix size is reduced by a factor of 6 for nondegenerate modes and by a factor of 3.5 for degenerate modes, leading to considerable reductions in processing time.

2.3.4 EFFECTIVE INDEX-GUIDED MODES The guiding mechanism for effective index-guided modes is somewhat similar to that of conventional fibers. As discussed, in the finger diagram they are located between the light line for silica and the FSM. Since the existence of these modes does not rely on the periodicity of the inclusions in the cladding, they are quite robust. Here we discuss some of the key properties of these modes. 2.3.4.1 Number of Modes Consider the structure of Figure 2.4(a). As already mentioned, because the cladding formed by the successive layers of holes around the solid core is of finite

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width, tunneling losses through the cladding are unavoidable. All modes carried by the MOF are hence leaky. In fact, it can be shown that there is a discrete infinity of such leaky modes, with Equation (2.43) being satisfied for each of these. The fraction of power lost through these effects per unit length, the confinement loss, is given by the imaginary part of the effective index through:

L

20 2p Im(neff )  106 ln(10) l

(2.82)

with the wavelength l in m. However, most of these modes are not localized in the core but have field distributions extending over the entire cladding region. This is not dissimilar to the situation of conventional step index fibers with a finite cladding, for which there are a limited number of modes that are guided in the core but a large number of cladding modes. In the case of MOFs, because the core and cladding regions are not separated by a well-defined boundary, it is a much more delicate problem to distinguish the “core modes” from the “cladding modes”. The distinction between core modes and cladding modes is a fundamental problem and is linked to many others. Indeed, generally only those modes that are guided in the core are technologically relevant. It is well known that in step index fibers, truly guided core modes become leaky modes of the core — which become confined by the cladding if the latter is of finite diameter — after they have been cutoff. This cutoff of a mode can be seen as a transition between two states of the one mode, one being confined in the core and the other in the cladding. Hence, the precise definitions of core modes and cladding modes can be seen as being equivalent to the definition of the cutoff of modes. Several approaches to such a definition have been suggested, relying on the fraction of power being carried in the core [48], the behavior of losses versus the width of the cladding [49], or the value of a mode’s effective index neff compared to the index associated with the cladding’s fundamental space filling mode (bFSM/k0)[44]. The latter argument relies on an analogy between MOFs and step index fibers but assumes the cladding to be infinite: if the propagation constant of a mode is smaller than the largest propagation constant allowed in the cladding, the mode can couple to propagating modes of the cladding and is no longer confined in the core. The first two arguments are closely related and are more specific to claddings of finite size. If a mode is confined in the core, leakage through the cladding occurs through tunneling and its losses should decay exponentially with increasing cladding width, while its modal diameter should be dictated by the physical size of the core. Once it is no longer confined in the core, the mode becomes confined by the cladding/jacket boundary. The modal diameter is then dictated by the size of the cladding (and will be largely independent of the size of the core), and one can show that losses decay following an approximate power law with increasing cladding width (the leakage being no longer due to tunneling but to imperfect reflection at the cladding/jacket boundary).

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Electromagnetic Theory and Applications for Photonic Crystals

It appears that all of the above definitions are largely equivalent and give close agreement for the cutoff wavelengths. Studying the number of core modes using any of these definitions shows that at any wavelength and regardless of the diameter on pitch ratio d/, only a finite number of modes are guided in the core. More surprisingly, it also has been shown that below a critical value of d/, only one mode, the fundamental mode, is confined in the core, with all other modes of the MOF being cladding modes regardless of the wavelength. Figure 2.7 shows the “phase-diagram” for the second mode of solid core MOFs obtained using the multipole method and the definition of cutoff relying on loss properties above and below cutoff [49]. The solid curve in Figure 2.7 represents the locus of the cutoff of the second mode in the MOF parameter space of normalized wavelength and normalized diameter of holes. For normalized wavelengths below the solid curve, the second mode is confined and MOFs are hence multimode. For wavelengths above that curve, the second mode becomes a cladding mode, and the fundamental mode is the only mode confined in the core. The cutoff curve of the second mode intersects the l/  0 axis at d/  0.406. For normalized diameters smaller than this, the second mode remains a cladding mode regardless of the wavelength; the fiber is thus said to be endlessly single-mode [44]. 2.3.4.2 Properties of the Fundamental Mode In most applications, single-mode behavior is sought, so that the only relevant mode is the fundamental mode. The fundamental mode of solid core MOFs, such as that depicted in Figure 2.4, is degenerate and has field distributions resembling 0.406

10

Single mode fiber

/Λ

1

0.1 Multimode fiber 0.01

0.001

0

0.2

0.4

0.6

0.8

d /Λ

FIGURE 2.7 “Phase-diagram” for the second mode of solid-core silica/air MOFs with a hexagonal array of circular cylinders. Insets: Power density distribution of the second mode in the confined (right) and unconfined (left) states.

© 2006 by Taylor & Francis Group, LLC

From Multipole Methods to Photonic Crystal Device Modeling

79

that of the HE11 mode of step index fibers. Note that if the symmetry of the MOF structure is broken by design or because of the fabrication process, this degeneracy is lifted and the fiber becomes birefringent. At short wavelengths and for large holes it is very well confined within the core, and losses tend to be negligible even with few rings of air holes. At longer wavelengths or with smaller holes, the field distribution of the fundamental mode starts to extend into the cladding and losses can increase very rapidly. Figure 2.8, resulting from multipole simulations, shows the limit above which losses of the fundamental mode become larger than 1 dB/m for MOFs with different numbers of rings of holes. Above each curve, losses are larger than 1 dB/m for the given MOF. In the vicinity of that curve, the losses tend to be a very steep function of wavelength, and MOFs basically become unusable for wavelengths above the curve. The equivalent curves for the second mode are shown as well; they closely follow the second mode cutoff. The similarity between the abrupt change of losses of the second mode near its cutoff and the steep change of losses for the fundamental mode led to a suggestion that the fundamental mode has a cutoff as well [49]. However, recent work using the newly developed fictitious source superposition (FSS) method [50] for computing defect modes in structures with infinite cladding has shown that the fundamental mode has no cutoff. Using this technique, we have been able to demonstrate that the effective index of the mode satisfies neff nfsm, with neff approaching nfsm asymptotically with increasing wavelength. Because neff nfsm, the field in the cladding remains evanescent and the mode is guided with no cutoff. Even though the fundamental mode has no cutoff, the existence of the transition region, found from the properties of MOFs with finite

10

/Λ

1

0.1

(8) (8)

(6) 0.01 (4) 0.001

0

0.1

(4)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/Λ

FIGURE 2.8 1 dB/m loss curves for the fundamental (solid curves) and second (dashed curves) mode for MOFs with different numbers of rings (in brackets) around the solid core for l  1.55 m. The dotted curve represents the phase diagram from Figure 2.7.

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Electromagnetic Theory and Applications for Photonic Crystals

cladding [51], is genuine and may be interpreted as the wavelength interval over which the spatial extent of the fundamental mode increases significantly. Because of the relatively large index contrast between air and silica, the chromatic dispersion of the fundamental mode of MOFs is dominated by waveguide dispersion, which can be controlled largely through the proper design of the fiber. Indeed, spectacular dispersion properties that are not feasible with conventional fibers have been demonstrated in [52–55]. However, since most of these dispersion properties appear in regions of parameter space where the fundamental mode starts leaking into the cladding, it is often crucial to take into account confinement loss when designing MOF dispersion properties [55,56].

2.3.5 AIR-GUIDED MODES In Section 2.3.2 it was shown that application of the Bloch method to the cladding structure of an air-core MOF can be a very useful tool for identifying the regions where air-guiding is likely to occur or specifically where the cladding behaves as a strong reflector. The presence or absence of a fiber mode in this region of kf  b space is determined by a number of other factors, such as the size and shape of the core, both of which determine the phase condition that must be satisfied by the mode. In addition, since the finger diagram in Figure 2.6(a) is calculated for an infinite hexagonal lattice of cylinders, whereas the fiber cross section is finite, there is still considerable loss due to leakage through a finite number of rings of holes even if the parameters lie in a white band of Figure 2.6(a). The marked curve in Figure 2.6(b) gives the dispersion relation for the fundamental mode of a MOF with four rings of holes in the arrangement described above. It is seen to extend into the region where cladding modes do exist; this is because the mode calculation was performed for a structure with a finite number of holes, for which the cladding reflectivity is a smooth function of wavelength. This should be contrasted to the band diagram calculations, in which the use of suitable lattice sums implies an infinite system. The dispersion curve is located on the high frequency side of the light line, so that neff  b/kf  1. This means that the guiding mechanism cannot possibly be total internal reflection. Of course, an effective index less than unity is not unphysical since this relates to a phase index rather than a group index. As mentioned in Section 2.2.5, the multipole method is well suited to estimating the loss in MOFs with a solid core. The calculation can, of course, also be applied to the air-core fibers that we are considering here, with Im (neff) again indicating the confinement loss through (2.82). However, it has been shown by West et al. that this is not the dominant loss mechanism in these fibers [57]. Rather, the losses are mostly associated with the coupling between the core mode and modes that reside on the interface between the core and the cladding. The importance of Re (neff) is not affected by this observation. However, a discussion of all these issues is outside the scope of this chapter. The spatial variations of the magnitudes of the longitudinal components of the electric and magnetic fields are shown in Figure 2.9, together with the

© 2006 by Taylor & Francis Group, LLC

From Multipole Methods to Photonic Crystal Device Modeling |Ez l

81

|Kz l 0.2

40 0.15

20 0

0.1

–20

40

0.25

20

0.2

0

0.15 0.1

–20 0.05

–40

0.05

–40 –40

–20

0

20

40

–40

–20

0

Sz

20

40

Sz 0.6

40

0.5 20

1 0.5

0.4 0

0

0.3 0.2

–40

0.1

y

–20

40 20 0 –20 –40

–40

–20

0

20

40

–40

–20

0

20

40

x

FIGURE 2.9 Longitudinal field components of fundamental air-guided mode and longitudinal Poynting vector Sz for six-ring MOF. The fiber has a central core diameter d /  2.27, while the diameter of the holes in the photonic crystal is d/  0.696. The wavelength of the incident radiation is l/  0.593, and the background has a refractive index 1.39.

z-component of the Poynting vector. Since the latter of these is quadratically dependent on the fields, it is better confined than either Ez or Hz. Nevertheless, there is a significant flux leakage into the regions between the central hole and the first ring of smaller holes and the region between the first and second ring of holes. The results reported here require the solution of the multipole coefficients for each cylinder, with the accurate characterization of the field around the large central hole requiring additional terms in the multipole series used in the implementation (characterized by M0, the symmetric truncation order) [33]. Since we have independent coefficients for the electric and magnetic fields, the resulting matrix M, defined in (2.43), has dimension 2[Nc(2M  1)  (2M0  1)] for an Nc  1 hole system. Here M0 is the truncation order of the central hole, and M is that for all others. For the six-ring structure considered here, with M  5 and M0  19, the matrix M has dimension 3642 [33]. The dimension of the matrix can be reduced, however, by using symmetry considerations, as discussed in Section 2.3.3. Nevertheless, even with these reductions in matrix size, the computational demands are near the limits of standard workstations, and the quality of the bandgap confinement that can be modeled on such systems is still somewhat less than desirable.

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Electromagnetic Theory and Applications for Photonic Crystals

2.4 RADIATION DYNAMICS AND THE LOCAL DENSITY OF STATES 2.4.1 BACKGROUND The notion of spontaneous emission was introduced in 1917 by Einstein [58] in his derivation of Plank’s black body radiation formula. Subsequently, it was realized that the coefficient of spontaneous emission rate A is not a constant but depends on the environment in which the radiation sources or atoms are embedded [59,60]. Indeed, the variation of A is particularly strong in photonic crystals [61]. For example, for wavelengths in a bandgap, the density of states is strongly diminished, leading to a reduced rate of spontaneous emission. Some new quantum phenomena have also been predicted: the localization of super-radiance near a photonic band edge [62], the control of the spontaneous emission rate [63] and the nonMarkovian character of radiative decay [64] have been reported. The key parameter that determines the radiative dynamics of sources is the spatially resolved or local density of states (LDOS) for light, r(r, v) [18]. According to Fermi’s golden rule, the spontaneous emission coefficient A is proportional to the LDOS r(r, v) and, in general, is a function of not only frequency but also position. This is a generalization of the usual density of states that depends on frequency only. The spatial dependence at a frequency v is associated with the spatial dependence of the eigenfunctions at this frequency and needs to be included because the size of the individual emitters (atoms) is much smaller than the wavelength of light. Consequently, if this eigenfunction has a node at a particular position, the electric field for an emitter at this position vanishes as well, and r(r, v)  0. Therefore, at a given frequency in a band, the spontaneous emission rate may be increased close to antinodes of the field and decreased near the nodes. Although the calculation of the LDOS is entirely classical in that it derives from the solution of a classical electromagnetic problem, its main applications are associated with quantum problems. In general, the LDOS can be calculated from the imaginary part of Green’s tensor [65] r(r ; v) 

2v Im Tr[G E (r , r ; v)] pc 2

(2.83)

where c is the speed of the light, and GE(r, rs; v) is the electric field Green’s tensor for a source at rs and observation point at r. The Green tensor G  xx  G E   Gyx    Gzx

Gxy Gxz    Gyy Gyz    Gzy Gzz 

(2.84)

is a 3  3 second rank tensor [66]. The elements in column u represent the comE E E ponents of the electric field vector (Gxu , Gyu , Gzu ) generated by a dipole source radiating parallel to the u  x, y, or z axes, respectively. © 2006 by Taylor & Francis Group, LLC

From Multipole Methods to Photonic Crystal Device Modeling

83

In what follows, we consider the calculation of the LDOS in 2D corresponding to a z-invariant structure and a line source, and in 2.5D, defined as a 2D z-invariant structure with a 3D (point) source.

2.4.2 2D GREEN TENSOR AND LDOS For the strictly 2D problem, our source is an infinite line antenna located at rs and we take k0z  0. We consider cylinders of refractive index nl located in a background of refractive index ne. For in-plane incidence, the solution of the problem decouples into the two principal E|| and H|| polarizations, which enables the form of the Green tensor to be simplified as follows [67]. For E|| polarization, it reduces to a form involving only a single nontrivial scalar GE  Gzzduz, where u  x, y, z, while for H|| polarization it reduces to a 2  2 tensor G  xx  G E  G yx    0

0    0    0 

Gxy Gyy 0

(2.85)

In Section 2.2.3, we have already given the field identity (2.33) for the cases of exterior and interior sources. In this we note differences in the way in which the source terms enter the identity: in the case of an exterior source, the source term enters directly, while for an interior source it enters indirectly through the transmission of the source field across the boundary of the cylinder. In turn, this affects the representation of the source coefficients that we present immediately below. We begin by noting that source term SE in (2.14) for an E|| polarized line source is given by S E (r ) 

1 H (k |r  rs |) 4i 0 f

(2.86)

where rs is the position of the source, n  ne for an exterior source, and n  nl for a source inside the cylinder l. Then, after applying the procedure described in Sections 2.2.2 and 2.2.3, we derive an identity of the form (2.33) for the electric field only. From Graf’s addition theorem applied to (2.86), we can find the explicit expression of the source multipole coefficients for the cylinder l. In the case of an exterior source, we obtain an expression of the form (2.14), while for a source inside cylinder l, we apply Graf’s addition theorem in the form 1 1 H (k |r  rs |)  H 0 (k f l |(r  rl )  (rs  rl )|) 4i 0 f l 4i 1

 ∑ H (k |r  rl |)Jm (k f l |rs  rl |)eim arg(rrl )eim arg(rsrl ) 4i m m f l 



∑ BmE ,lS Hm (k f l |r  rl |)eim arg(rr ) i

m

© 2006 by Taylor & Francis Group, LLC

l

(2.87)

84

Electromagnetic Theory and Applications for Photonic Crystals

Accordingly, the source multipole coefficients for the cylinder l are 1 H (k |r  rl |)eim arg(rs rl ) , for sources outside the cylinders 4i m f e s (2.88) 1  Jm (k f l |rs  rl |)eim arg(rs rl ) , for a source inside the cylinder l 4i

BmE ,lSe  BmE ,lSi

Finally, by calculating the multipole coefficients BE using an analogue of Equation (2.33) for this polarization, the field may be reconstructed using Equation (2.11) and the LDOS from Equation (2.83). The real part of Green’s tensor diverges when the source and field points coincide (according to the definition of the LDOS). Accordingly, care is required in its numerical evaluation and, as we see, the multipole method is ideally suited for this type of calculation since the divergent behavior can be avoided analytically. To exemplify the method, we consider a cluster of hexagonally packed circular air voids with common radius al /D  0.48 in a background of dielectric constant e  n2e  13. Note that throughout this section the symbol D denotes the lattice constant of the hexagonally packed clusters. The corresponding infinite structure has bandgaps for 1.94  l/D  2.31 for E|| polarization and for 1.94  l/D  2.83 for H|| polarization and thus exhibits a full bandgap in the wavelength range 1.94  l/D  2.31. The cluster is constructed as a sequence of concentric shells of air voids with a central cylinder placed at the origin. Note that this differs from the MOF geometries from Section 2.3.5, which have a defect in the center. For display purposes, we plot a normalized form of the LDOS, namely r (r , v) 

pc 2 r(r , v) 2v e2

(2.89)

where the normalizations have been chosen such that in free space ~ v)  0.25. In Figure 2.10 we show a density plot of r~ as a function of r(r, position for E|| polarization and for two wavelengths — a gap wavelength l/D  2.25 in Figure 2.10(a), and a propagating band wavelength l/D  3.0 in Figure 2.10(b). Note that in the bandgap, the LDOS is small everywhere in the interior of the structure. In the middle of the central unit cell, r~ is approximately 107 times its value r~  0.25 in the background medium. Thus the emission by a line antenna located in this region is drastically reduced. Outside the cluster, the LDOS rapidly approaches its free space value. In a pass band, as in Figure 2.10(b), the LDOS does not decrease inside the cluster but fluctuates around the value of 0.25 in the uniform background. In Figure 2.11(a) we plot the LDOS along the y-axis for clusters of Nc  19, 61, and 127 cylinders for a wavelength of l/D  2.25. Note that the LDOS decreases, on average, exponentially inside the cluster, though oscillations are clearly visible. As we increase the size of the crystal, the positions of the maxima and minima of the LDOS remain approximately the same because they are

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From Multipole Methods to Photonic Crystal Device Modeling 8

8

0 −2

4

85

0 −2

4

−4 0

−6 −8

−4

−10 −8

−8

−4

0

4

−4 0

−8

−4 −8

8

−6

−10 −8

−4

(a)

0

4

8

(b)

FIGURE 2.10 (a) Density plot of log10 r~(r, v) for E|| polarization versus position at l/D  2.25 (stop band) for E|| polarization. (b) As for part (a) but for l/D  3.0, corresponding to the pass band.

0 log10 [ (y,)]

log10 [ (y,)]

0 −2 −4 −6 −8 −10

−6

−2

2

6

10

−1

−2 −10

−6

−2

2

y/d

y/d

(a)

(b)

6

10

FIGURE 2.11 (a) Normalized LDOS r~(r, v) along the y axis at l/D  2.25 in the stop band for E|| polarization. The dotted curve corresponds to a cluster with Nc  19 cylinders, dashed for Nc  61, and solid for Nc  127. The right panel (b) is the same as in (a) but for l/D  3.0, Nc  19 (dotted line), and Nc  127 (solid line).

determined by the cylinder positions. The horizontal straight line indicates the free space LDOS of r~  0.25. The LDOS rapidly approaches this free space value as we leave the cluster. Figure 2.11(b) is the same as Figure 2.11(a), but for the pass band wavelength l/D  3.0 and for a cluster of Nc  19, 127 cylinders. The LDOS now oscillates inside the cluster and approaches the free space value for points exterior to the cluster. For H|| polarization, the source term SK(r) in (2.14) is S xK (r )  S yK (r ) 

© 2006 by Taylor & Francis Group, LLC

e H (k |r  rs |) sin arg(r  rs ) 4 1 f e

e H (k |r  rs |) cos arg(r  rs ) 4 1 f e

(2.90)

86

Electromagnetic Theory and Applications for Photonic Crystals

for orientation of the source along the x- and y-directions respectively. After applying the procedure described in Section 2.2.2 we derive the field identity in the form (2.25), where the explicit expressions for the source multipole coefficients BK,Se and BK,Si for the orientation of the source along the x-axis are BmK ,lSe 

e [ H (k r )ei ( m1)uls  H m1 (k f e rls )ei ( m1) uls ] (2.91) 8i m1 f e ls

BmK ,lSi 

l [ J (k r )ei ( m1)us  Jm1 (k f l rs )ei ( m1) us ] 8i m1 f l s

(2.92)

In Equations (2.91) and (2.92), (rls, uls)  rs  rl. For the source dipole oriented parallel to the y axis, the dipole source multipole coefficients for BK,Se and BK,Si have the form BmK ,lSe  BmK ,lSi 

e [ H m1 (k f e rls )ei ( m1)uls  H m1 (k f e rls )ei ( m1) uls ] 8

l [ J (k r )ei ( m1)us  Jm1 (k f l rs )ei ( m1) us ] 8 m1 f l s

(2.93) (2.94)

Figure 2.12 displays the H|| polarization equivalents of the E|| polarization results shown in Figure 2.10. The general features of the two cases are similar: a reduction of the LDOS for the gap wavelength in Figure 2.10(a), while for a wavelength in the pass band in Figure 2.10(b), the LDOS oscillates around the free space value. Note, in addition, that for H|| polarization, the LDOS is discontinuous across the cylinder boundaries, as illustrated in Figure 2.13, which shows a cross section as in Figure 2.11.

2.4.3 2.5D GREEN TENSOR AND LDOS The 2.5D Green tensor applies to a dipole point source that radiates in 3D, embedded in the 2D scattering geometry of a 2D photonic crystal [68]. The whole scattering problem can again be formulated in terms of Ez and Kz components of the field in a similar way to that described in Section 2.2.2. The free space electric and magnetic fields [69] for a dipole oriented in the direction u (where u is a unit vector) are ik R  (u ⋅ )  e f G E ( R)   u  2 2  , k f  4 pR  

1 e f (u  ) ik f 4 pR ik

G K ( R) 

R

(2.95)

where n is a piecewise constant function n(r) — the refractive index of the medium in which the source is located. Also, R  |R| and R  r  zzˆ , with

© 2006 by Taylor & Francis Group, LLC

From Multipole Methods to Photonic Crystal Device Modeling 8

87

8

0

0 −2

4

−2

4

−4

−4 0

0

−6 −8

−4

−6 −8

−4

−10

−10 −8

−8

−4

0 (a)

4

−8

8

−8

−4

0 (b)

4

8

FIGURE 2.12 The same as Figure 2.10 but for H|| polarization.

1 log10 [ (y,)]

log10 [ (y,)]

0 −2 −4 −6

−6

−2

2

0 −1 −2

6

−6

−2

2

y/d

y/d

(a)

(b)

6

FIGURE 2.13 The same as Figure 2.11 but for H|| polarization. Note that the LDOS is discontinuous across the surface of the cylinders.

r  (x, y, 0). The source term in (2.95) is then written as a Fourier integral (i.e., a spectral decomposition in terms of cylindrical waves [69]), eik f e R i  4 pR 8p



∫ H0 (ke |r  rs |)e

ik0 z z

dk0 z

(2.96)

—–— ——– where ke  k2f ve2  k20z and, without loss of generality, we may assume that the source is located on the plane zs  0. Then, we substitute (2.96) into (2.95) to form GE/K(R), the z-component of which allows us to infer the source terms SE/K (2.14) from GzE/K ( R)  ∫





© 2006 by Taylor & Francis Group, LLC

SeE/K (k0 z )eik0 z z dk0 z

(2.97)

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Electromagnetic Theory and Applications for Photonic Crystals

We thus derive [68] the z-components of the source terms (2.14) as follows S xE (r )  S xK (r )  S yE (r )  S yK (r ) 

k0 z ke 8pk 2f ne2

H1 (ke |r  rs |) cos arg(r  rs )

ke H (k e |r  rs |) sin arg(r  rs ) 8pk f 1  k0 z ke 8pk 2f ne2

H1 (ke |r  rs |) sin arg(r  rs )

(2.98)

ke H (k e |r  rs |) cos arg(r  rs ) 8pk f 1 

SzE (r ) 

ike 2 H (k e |r  rs |) 8pk 2f ne2 0 

SzK (r )  0 for the source orientation u along the xˆ, yˆ, zˆ directions, respectively. The transverse components of the source field can be found using (2.1). After applying the procedure [68] described in Section 2.2.2, we form the field identity (2.27), where the multipole source field coefficients in the field identity have the form  k02z  BzE,m,lSe  1  H (k e r )eimuls (k f ne )2  m  ls   BxE,,mlSe  ByE,,mlSe 

ik0 z ke  H (k e r )ei ( m1) uls  H m1 (ke rls )ei ( m1) uls   2(k f e )2  m1  ls k0 z ke

 H (k e r )ei ( m1) uls  H (k e r )ei ( m1)ls  m1  ls  2(k f ne )2  m1  ls

(2.99)

BzK,m,lSe  0 BxK,m,lSe 

ke  H (k e r )ei ( m1)uls  H m1 (ke rls )ei ( m1) uls   2i  m1  ls

ByK,m,lSe  

ke 2

 H (k e r )ei ( m1)uls  H (k e r )ei ( m1) uls  m1  ls  m1  ls 

for sources outside the cylinders. In Equations (2.99), (rls, uls)  rs  rl denotes the position of the source relative to the center of cylinder l. For interior sources, these terms take the same form with the replacement of ne by nl (for cylinder l) and Hm by Jm. The unknown B coefficients are again found by solving the field identity (2.33), enabling the field reconstruction (2.11) for the spectral components k0z.

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From Multipole Methods to Photonic Crystal Device Modeling

89

The full fields are finally obtained by the inverse Fourier transforms (2.4). These integrals need to be evaluated in the complex k0z-plane using a suitable contour [68] to avoid numerical convergence problems. This is because the integrands have poles on or near the Re(k0z)-axis due to bound modes in the range kf min (nl, ne)  Re(k0z)  kf max(nl, ne). Causality is enforced by deforming the contour above these poles for Re(k0z)  0 and below for Re(k0z) 0. The contour can then be returned to the Re(k0z)-axis when | Re(k0z) | kf max(nl, ne). The process of computing the LDOS may then be summarized as follows. For each of the three source orientations, u  xˆ, yˆ, zˆ , we must solve the problem of generating each of the three columns u of the Green tensor. For a given u and k0z we generate the source coefficients according to Equations (2.99) and solve the field problem for B (2.27) from which we reconstruct the total field. We then integrate this over k0z to generate the relevant column of the Green tensor. Finally, with three field problems solved, we may compute the LDOS from (2.83). The 2.5D Green tensor has been applied to the calculation of the 2.5D LDOS (2.83) in the 2D macroporous silicon photonic crystal (2D MSPC) [70]. Fabricated by electrochemically etching hexagonal arrays of cylindrical air-pores with an aspect ratio (pore height to pore diameter) of 100–500 in silicon (ne  3.4), the 2D MSPC is an essentially ideal 2D PC system. The specific sample modeled is composed of four concentric hexagonal rings of air-voids (Nc  61) of radius al /D  0.45 enclosed by an air-jacket of radius a0 /D  5.0 in a fiber-like arrangement. Although a complete understanding of the LDOS results requires modal field patterns, a qualitative understanding is provided by the band structure. The outof-plane band structure (k0z 0) for the underlying lattice of the 2D MSPC is characterized in Figure 2.14. We note that this figure is similar to the finger diagram (Figure 2.6). However, the parameters of the structure are quite different; in particular, the high refractive index contrast between silicon and air considered here leads to an in-plane gap, whereas the low contrast in Section 2.3 only leads to narrow out-of-plane gaps (the “fingers”). The in-plane band structure (k0z  0) has a narrow E|| bandgap in the frequency (normalized) range vD/2pc  [0.405, 0.445] within a wide H|| bandgap in the frequency range vD/2pc  [0.303, 0.495] to form a complete in-plane bandgap. The out-of-plane band structure is determined by a projection effect because the in-plane component of the wave vector ———— — 2 k  k2f n e2  k0z needs to satisfy the Bragg condition. As k0z increases out-ofplane, the frequency of modes increases to compensate, resulting in a backward sloping band structure. To the left of the light line (k0z  kf) lie the radiation modes that are oscillatory in air and resemble in-plane modes at the projected wave vector. To the right of the light line (k0z kf) lie bound modes that are evanescent in air. These modes are largely confined to the silicon background by total internal reflection. For a given frequency, the integral over k0z is represented by a horizontal line across the diagram. This amounts to a rough summation over in-plane modes at and below the frequency. While there is no complete gap, a pseudogap covers the widest range of k0z at the top of the complete in-plane bandgap; as we see below, the structure affects the LDOS most strongly at this frequency.

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Evanescent-modes

Oscillatory-modes

0.5 pseudo-gap d / 2c

0.4 0.3 0.2

Light-line

0.1 0 0

0.2

0.4

0.6

0.8

1


d / 2c

FIGURE 2.14 Out-of-plane band structure for the 2D MSPC. The pseudogap is formed by the out-of-plane trajectories of the edges of the E|| bandgap. The dashed lines indicate the |out-of-plane trajectories of the edges of the H|| bandgap. Radiation modes lie to the left of the light line, while bound modes lie between the light line and the fundamental spacefilling modes. The horizontal line indicates where the pseudogap covers the widest k0z range.

The spatial dependence of the 2.5D LDOS in the 2D MSPC at the top of the complete in-plane bandgap is plotted in Figure 2.15(a). The integral over k0z washes out the more dramatic positional variations seen in the 2D LDOS, and the 2.5D LDOS is basically the same in each unit cell inside the two outermost rings. At the center of the central air-pores, the 2.5D LDOS is suppressed by one order of magnitude and increases smoothly toward the pore wall. The dependence on cluster size is shown in Figure 2.15(b). Limited by a pseudogap, the suppression of the 2.5D LDOS inside the air-pores saturates rapidly with cluster size. This is again in contrast to the behavior of the 2D LDOS, for which the suppression grows exponentially with cluster size (i.e., the LDOS decreases exponentially with cluster size), in view of the existence of a complete bandgap. The frequency dependence of the 2.5D LDOS at the cluster center is plotted in Figure 2.16(a). Suppression grows with frequency inside the in-plane bandgaps as the corresponding out-of-plane pseudogaps cover a widening range of k0z values. Maximum suppression of one order of magnitude is reached at the top of the complete in-plane gap. At frequencies immediately above the gap, the 2.5D LDOS is a factor 20 larger as in-plane modes now exist (see Figure 2.14). This asymmetric frequency dependence about the upper edge of the in-plane bandgap is the signature response of point source spontaneous emission in a 2D PC. Wubs and Lagendijk [71] noted that the LDOS for a point source in a finite structure that is periodic in 1D or 2D can be separated into a radiation component, which is detected outside the structure in the plane of periodicity, and a bound component, which is detected at the ends of the structure. The radiation component is

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91

0.5 4 0.25 log10[ / 0 ]

y/d

2 0 −2

0 −0.25 −0.5 −0.75

−4

−1 −4

−2

0

2

−4

4

−2

x/d

0

2

4

x /d

log10[/0 ] (a)

(b)

0

1

−1

0.5

log10[/0 ]

log10[/0]

FIGURE 2.15 (a) 2.5D LDOS across the plane of the 2D MSPC at the top of the complete in-plane bandgap, normalized to its free-space value, r0  v2/p2c3. (b) Cluster size dependence: 2.5D LDOS along the x-axis with Nc  1 (dash), 7 (dot), 19 (dash-dot), and 61 (solid).

−2 −3

0 −0.5 −1 −1.5

0.2

0.3

0.4 d/2c (a)

0.5

0.6

0.2

0.3

0.4

0.5

0.6

d/2c (b)

FIGURE 2.16 (a) 2.5D LDOS vs. frequency in 2D MSPC at cluster center (x, y)/D  (0.0, 0.0). Total 2.5D LDOS (solid), radiation component (dash). (b) 2.5D PLDOS for u  zˆ at the center of the cluster (solid) and in silicon background (x, y)/D  (0.5, 0.0) (dash). Solid vertical lines indicate E|| bandgap, and dashed vertical lines indicate H|| bandgap.

also included in Figure 2.16(a). Determined by small k0z values, the radiation component is naturally more affected by the periodicity of the structure and undergoes more dramatic variations. Fortuitously, the pseudogap of the 2D-MSPC almost completely covers all radiation modes at the top of the complete in-plane gap. As a result, the radiation component of the 2.5D LDOS is heavily suppressed by a factor of 5000. In-plane modes are encountered above the top of the in-plane gap, and the radiation component rises rapidly. The sudden burst into the radiation component about the top of the in-plane gap is perhaps the most striking effect of a 2D-PC on spontaneous emission.

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While the LDOS gives the spontaneous emission rate averaged over dipole orientation, the projected LDOS (PLDOS) is for a particular dipole orientation and can be obtained by contracting the Green tensor with the dipole unit vector u (i.e., an inner product with u). The 2.5D PLDOS for u  zˆ is plotted in Figure 2.16(b). No longer averaging over orientation, variations in the 2.5D PLDOS are stronger than in the 2.5D LDOS. As the 2.5D PLDOS for u  zˆ is determined by the Ez component of the fields, the result is closely tied to the in-plane E|| band structure and the associated modal field patterns. Suppression begins to grow inside the H|| gap because the modes of the second E|| band, which lies between the bottom of the H|| bandgap and the bottom of the E|| bandgap, have nodes at the center of the air-pores and are not contributing to the 2.5D PLDOS. The 2.5D PLDOS then jumps sharply higher at the top of the E|| bandgap as the first modes encountered have antinodes at the air-pore center.

2.5 BLOCH MODE ANALYSIS OF COMPOSITE PC DEVICES 2.5.1 BACKGROUND AND NOMENCLATURE The modeling of photonic crystals and composite devices fabricated from photonic crystals is challenging, not only because of the strongly scattering environment that is a consequence of the geometry and scale of the structure, but also because of the high contrast optical materials that are used. Over the years, a range of computational methods have evolved to model PC structures including finite difference time domain (FDTD) methods [21], plane wave methods [20], beam propagation methods [72], transfer matrix methods [73], layer KorringaKohn-Rostoker (KKR) methods [7,36], Wannier function methods [74], and finite element methods [23]. Although these methods lead to accurate results, the bulk of them are entirely numerical in their approach and do not readily provide insight into the essential underlying physics of propagation. Despite the importance of this, it is only recently that real efforts to develop tools that can answer such questions have begun to emerge. Of particular interest are the methods based on Bloch modes, transfer matrices, and related techniques, that are now generating significant interest (see [40,74–81]). It is in this area that our group has been particularly active, developing the Bloch mode scattering matrix formalism and applying this semianalytic approach to the modeling of photonic crystals. In such methods, the photonic crystal lattice is regarded as a periodically stratified medium (in the vertical direction, see Figure 2.17) in which each 1D or 2D layer is a grating that is periodic in the transverse plane. In this, the unit cell of the transverse period(s) may range from a simple cell that includes only a single scatterer (i.e., cylinder) to a complex supercell that has a number of cylinders suitably arranged to define a photonic crystal device. The fields at the upper and lower interfaces of each grating layer are represented in plane wave expansions characterized by the transverse component of the Bloch vector. For a 1D grating,

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From Multipole Methods to Photonic Crystal Device Modeling y

93 y

f −a

Pa

a2

f +a x

a1 f −b

Pb

(a)

a2 a1

x

f +b

(b)

FIGURE 2.17 (a) Geometry of the unit cell defined by the fundamental translation vectors a1  (D, 0) and a2 for the Bloch method calculations. The phase origins Pa and Pb of  the fields f a and f b above and below the grating, respectively, are shown. (b) The fundamental translation vectors for a woodpile structure.

the plane wave expansion at the interface y  yj takes the form V f( j ) (r ) 



xs 1/2  fs( j )eixs ( yy j )  fs( j )eixs ( yy j )  eias x   s

∑

(2.100)

———– where as  2ps/D  a0, xs  k2  a2s, with D denoting the lateral or transverse period of the structure (which can be a superlattice) and with a0  k0x denoting the transverse component of the Bloch vector, and where k denotes the wavenumber in the background (i.e., k  kfne). In (2.100), the expansion has been normalized with the inclusion of the factor xs1/2 to enable energy fluxes (for propagating orders) to be computed from the square magnitude of the associated amplitude. The action of each grating layer, in turn, is characterized by reflection (R) and transmission (T) scattering matrices with, for example, Rpq denoting the amplitude of the field reflected into the pth channel (or diffraction order) corresponding to unit amplitude incidence in channel q. The reflection and transmission scattering matrices may be computed with a variety of techniques including differential and integral methods, finite element techniques, and the multipole method, which has been an important feature of our work. As is elaborated in the following section, the reflection and transmission scattering matrices that characterize the layer then allow us to define a transfer matrix (T) that relates the fields on either side of the layer. Finally, the vertical periodicity of structure and the associated Bloch condition lead to an eigenvalue problem for the transfer matrix T, the solution of which yields the modes (both propagating and evanescent) of the structure. The strength and utility of the method lie in its use of the natural Bloch mode basis of functions to describe the properties of photonic crystals. If we consider a plane wave incident upon a photonic crystal, the incident field excites a field within the PC that can be expanded as a superposition of Bloch modes. In most

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practical situations, however, only a single, or at most a few, propagating Bloch states actually contribute to the far field of the device. This, in turn, can reduce the complexity of the treatment of the multiple scattering problem, in many cases transforming the analysis of the PC structure to that of well-known devices that arise in thin film optics, e.g., generalizations of the Fabry–Perot interferometer. In this section, we outline the rigorous theoretical framework of the method. We go on to study various applications of the method in Section 2.6. The use of plane wave scattering matrices and transfer matrix methods in the determination of band structures dates back to the work of McRae [82] in low energy electron diffraction. In recent years, the method has been extended [40,75,76] to determine band structures and to enable the use of Bloch mode methods in structures that exploit defect modes. Our analysis begins with the reflection and transmission scattering matrices, computed by the multipole method, for a constituent grating layer (Figure 2.17), relative to some phase origin denoted by the superscript 0. This layer may be a one-dimensional structure, such as a single grating, or a composite multilayer grating, or a two-dimensional structure, such as a woodpile, which comprises crossed gratings. The grating layers are taken parallel to the xz plane and are characterized either by the direct lattice basis vector a1 for a 1D structure or by a pair of basis vectors {a1, a3} for a 2D structure. Accordingly, the summation indices in the general plane wave expansions (2.77) and (2.78) are either one- or two-dimensional, summing over the set of all vectors in the reciprocal lattice, viz. {s1b1} and {s1b1  s3b3}, where {bj} denotes the basis of the reciprocal lattice vectors corresponding to the direct lattice vectors {aj}. This, in turn, is reflected in the structure and dimension of the scattering matrices introduced in (2.79). For incidence from above and below, let these be denoted by (R0a, T 0a ) and (R0b, T 0b ), respectively, with the superscript 0 referring to a particular phase origin. A lattice is then formed by stacking individual layers, offsetting each from its predecessor by the basis vector a2. Following the treatment in [40,75], the reflection and transmission matrices relative to the phase origins shown in Figure 2.17(a) are  0 0 T R   a b   QP  T a Rb  PQ  0 0    Ra T b   Ra T b  1

Q  diag(Q

2

1

1

1

, Q 2 ), P  diag(P 2 , P 2 )

(2.101)

for a two-dimensional grating. In (2.101), the matrices Q and P respectively translate the phase origin in the transverse and vertical directions in accordance with the out-of-plane lattice vector a2. Here, both Q and P are written as block diagonal matrices to handle the E|| and H|| field components. Naturally, each of the diagonal blocks contains identical elements, i.e., {(eiQt,sa2)} and {(eiQ||,sa2)}, respectively, for the matrices Q and P, which are used to shift the phase origins for the scattering matrices respectively for incidence from above and below to the

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points Pa and Pb of Figure 2.17. For a one-dimensional grating, the corresponding expressions are  T  a    Ra 

Rb   Tb 

 0 T a

 QP 

 R0  a



Rb0 

1

Q  diag(Q

 PQ, Tb0 

2

1

1

1

,Q 2 ), P  diag(P 2 ,P 2 )

(2.102)

where Q  diag(eiQ t, sa2), P  diag(eiQ||,sa2), Qt,s  (Qxs, 0, Qzs), and Q||,s  (0, Qys, 0).

2.5.2 FORMULATION

EIGENVALUE PROBLEM

OF THE

In the notation of Section 2.2.7 and Figure 2.17(a), we may express the outgoing fields in terms of the incoming fields fa  Ra fa  T b fb ,

fb  T a fa  Rb fb

(2.103)

The Bloch condition requires that fields in an infinite lattice satisfy V (r  a j )  exp(ik0 ⋅ a j )V (r )  for a lattice vector aj. This, in turn, requires that f b  mf a where m  exp (ik0  a2). For a general lattice in which there is no explicit symmetry to exploit, we solve (2.103) to generate the eigenvalue equation in terms of the interlayer transfer matrix T

Tfa  fb  mfa ,

  f   j     f  j  

where f j  

and

 T  R T 1R a b b a T    1  T b Ra

Rb T b1   T b1  (2.104)

In (2.104), the subscripts j  a, b respectively denote the interfaces above and below the layer and also designate the reflection and transmission scattering matrices for incidence from above and below. Since the scattering matrices from which the transfer matrix T is calculated explicitly embed both the wavenumber k and the transverse component of the Bloch vector k0t in their calculation through the solution of a diffraction problem for a single layer, it follows that T  T(k, k0t). This therefore leaves the component of the Bloch vector in the direction of translation, i.e., k0  a2, to be determined from (2.104). While Equation (2.104) formally defines the modes, this particular formulation of the problem is not well conditioned, particularly for matrices of large dimension that arise when working with 2D crossed structures such as woodpiles. The essential difficulty is associated with the inversion of the

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matrices Tj, the elements of which decay with increasing plane wave order toward the edge of the matrix [39,75,76]. One technique for addressing this ill-conditioning [75], developed for 2D lattices of high symmetry (e.g., square, rectangular and hexagonal lattices), involves symmetrizing and antisymmetrizing the original eigenvalue equations. In the case of the square symmetric lattice, this leads to a modified eigenvalue equation 1 g, S  I  (Ra  T a )(Ra  T a ) 2c m  m1 1  (f a  f b ), c  g 2 2

S1Tg 

(2.105)

While (2.105) overcomes the conditioning problems of the original T-matrix method, its applicability is restricted to lattices of high symmetry where symmetrization of the eigenvalue equations is possible. Accordingly, we outline a general technique developed independently in [39,76]. Following the treatment in [39], we introduce the R-matrix that relates the electric and magnetic fields on both sides of the layer by  e  a   eb 

      

  R k  R12  11 1/2  a   X1/2  X    kb   R21 R22   

(2.106)

Here, X1/2  diag(X 1/2, X 1/2), and the vectors of plane wave coefficients for the total electric and magnetic fields are given by ej  X 1/2 (fj  fj ) and kj  X1/2  1 (f j  f j ), with X  j or X  diag(j, j ), respectively, for the cases of 1D and 2D gratings. Explicit forms for the partition elements of the X and R matrices are given in [39] and may be derived by substituting the representations for ej and kj in terms of the fj into (2.106). With these, the cross-layer transfer equation (2.104) may be recast as g b  Mg a

(2.107)

where I R  11 I  R11 g j    fj , I   I

and

R  R   R1 11 I   12 M   22   I 0   0 

    R21 

0

(2.108)  The quasiperiodicity f b  mf a imposed by the Bloch condition in turn requires   that g b  mg a , resulting in the eigenvalue problem Mga  mga. This form, however, is little more than a reformulation of the original method (2.104) and is thus no better conditioned, specifically because the problems afflicting the inversion

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97

of the matrices Tj also arise in the inversion of R12. However, the use of the R-matrix nomenclature has simplified the structure of the resulting eigenvalue equation and allows us to convert easily a problem involving an ill-conditioned matrix into one that is well conditioned. While this can be done in two ways [39,76], we deal here with only one, defining a derived matrix  R D 12

M′  (I  zM)1    

D

    D[ R12  ( R22  R11 )] 

R12 DR21

(2.109)

where D  {R12  z2R21  z(R22  R11)}1. The matrix M has eigenvalues (1  zm)1, where m is an eigenvalue of M, and hence the eigenvalues of M can be inferred from those of M . While z is chosen to avoid singular behavior, the values z  1 generally suffice. An alternative treatment, analogous to the symmetrization process that led to (2.105), leads to the introduction of an alternative derived matrix M  (zM  z1M1)1, whose form can be calculated algebraically and which is free of conditioning problems. Setting z  1 leads to a matrix whose eigenvalues (m  1/m)  1/(2c) occur in precisely the same form as those of [75].

2.5.3 CALCULATION

OF

BAND DIAGRAMS AND BAND SURFACES

Crucial to the study of photonic crystal devices is the capacity to compute the band diagram of the underlying lattice and dispersion diagrams for defect states. The most widely used tool for such calculations is the plane wave method for solving the operator eigenvalue equation [14]  1  v2 H(r )       H(r )   2 H(r )  c  e(r )

(2.110)

For a given choice of Bloch vector k0, the eigenvalue equation (2.110) is discretized into a plane wave basis to yield an algebraic eigenvalue problem that is solved for the permissible frequencies v of the modes, which, in turn, are characterized by the eigenvectors. By scanning k0 over the Brillouin zone, we can generate a band surface diagram. In a similar manner, by scanning the Bloch vector k0 around the exterior of the irreducible part of the Brillouin zone, we may generate photonic band diagrams in their usual form. While the capacity to select explicitly the value of the Bloch vector serves to facilitate the computation of band diagrams, it simultaneously introduces a serious drawback for working with dispersive media having material constants that depend on frequency — the eigenvalue output of the numerical problem. It is in dealing with this problem that transfer matrix methods show their strengths, building into the problems from the outset the actual material constants for the chosen frequency. The scan of the Brillouin zone thus commences with the choice of frequency and transverse component of the Bloch vector and produces, from the solution of an algebraic eigenvalue problem (2.104), the out-of-plane component of the Bloch vector.

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Electromagnetic Theory and Applications for Photonic Crystals 0.8

0.6

0.6

0.4

kd /2

kd/2

98

0.4

0.2

0.2 Γ

M a2

y

Y b2 Y

a1

x

Γ (a)

M

Γ

M

K

y

ky M

a2

b1 kx

a1 x

M

b2 k y M K kx

Γ (b)

b1

FIGURE 2.18 (a) Band diagram for the square lattice (cylindrical rods of radii a/D  0.3 and refractive index n  3.0, E|| polarization); (b) band diagram for the hexagonal lattice (cylindrical rods of radii a/D  0.3 and refractive index n  3.6, E|| polarization). In both cases, the Wigner–Seitz cell, the first Brillouin zone, and irreducible part of the first Brillouin zone are shown.

The downside of this approach is some complication in the drawing of the band surfaces and band diagrams. In what follows, we discuss the use of transfer matrix methods in band diagram calculations that, for brevity, we restrict to 2D lattices characterized by lattice basis vectors a1 and a2 and reciprocal lattice vectors b1 and b2, normalized such that bi  aj  2pdij (Figure 2.18). Any point in the directed lattice may thus be written r  h1a1  h2a2, and correspondingly any point in the first Brillouin zone of the reciprocal lattice can be specified by k0  j1b1  j2b2 for j1, j2  [1/2, 1/2]. To build up the band surface, we must consider a sequence of diffraction problems associated with the lattice composed of layers of gratings generated by a1 and translated by a2. For a given frequency v  kc and Bloch vector component parallel to the plane of the grating k0g  k0  a1  2pj1, we must run our grating code to produce scattering matrices that lead to the transfer matrix, the eigenvalues m of which yield j2  arg(m)/(2p). Iterating in both frequency (or wavenumber k) and j1 then leads to the construction of the band surface diagram. Of course, the basic grating could have been alternatively chosen as that generated by a2 or, for that matter, any grating generated by an arbitrary choice of Miller indices. The only requirement is that we can devise an efficient procedure by which we can scan the first Brillouin zone to search for propagating eigenstates. In the case of high symmetry lattices, it is usual to plot band diagrams that are generated by traversing the perimeter of the irreducible part of first Brillouin zone. In the case of a square symmetric lattice (Figure 2.18[a]) with direct lattice vectors a1  (d, 0) and a2  (0, d), the corresponding reciprocal lattice vectors are b1  (2p/d, 0), b2  (0, 2p/d), and the perimeter of the irreducible part of the first Brillouin zone is designated as YM with three segments respectively parameter– ized as Y: k0  0b1  j2b2, YM: k0  j1b1  b2/2, M: k0  j(b1  b2)/2.

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To expedite the calculations, we need to choose one or more grating configurations that exploit the lattice symmetry and generate the band diagram using only a sequence of frequency scans. Two preferred configurations are the normal incidence and Littrow mounts which, in the language of lattices and photonic crystals, correspond to periodic (Brillouin zone center) and antiperiodic (Brillouin zone edge) boundary conditions. In the case of the Y segment, we see that the grating generated by a1 (i.e., the grating whose plane is parallel to a1), sets k0g  k0  a1  0, thus enabling this segment to be generated with a frequency scan corresponding to periodic boundary conditions. For the YM segment, we ask what grating, characterized by some generating vector a, is ideal in the sense of reducing k0g  k0  a  j1b1  a  -12 b2  a to a suitably convenient value. Indeed, it is clear that the choice of a  a2 will yield k0g  p, and so the YM segment can be generated with a frequency scan with a constituent grating (a2) operated in an antiperiodic configuration. Finally, for the M segment, we can set k0g  0 by choosing a grating plane a  a1  a2 because this is perpendicular to the vector b1  b2 that generates the M segment of the band diagram. The constructions needed for the band diagram of a hexagonal lattice may be derived in a similar manner. From Figure 2.18(b), we see that the M segment is parallel to b2, which is orthogonal to the generator a1 of the primary grating. Thus, the M segment of the band diagram can be generated by a frequency scan based on the primary grating operated at normal incidence, i.e., with a periodic boundary condition. In turn, the MK segment is parallel to a1 and thus is orthogonal to b2. Hence, a grating generated by a lattice vector parallel to b2 operated with a periodic boundary condition can generate the MK segment of the band diagram. Finally, we observe that the K segment is parallel to a2 and orthogonal to b1. However, in a hexagonal lattice the two directions b1 and b2 are equivalent, and thus the entire MK segment can be generated by a single grating operated in normal incidence. Figure 2.18(b) illustrates the band structure of this lattice by plotting the frequency of propagating states for each point on the perimeter MYM of the irreducible Brillouin zone. Two bandgaps are visible as shaded regions in the figure. The occurrence of bandgaps and propagating states may be visualized in various ways appropriate to a range of different applications. In the case of air-cored microstructured optical fibers (MOFs), in which the light is guided axially through the air hole by photonic bandgap effects, we are interested primarily in locating a complete bandgap of the bulk crystal that simultaneously corresponds to a guided mode. Here, it is convenient to characterize the bulk crystal by a projected form of the band diagram, i.e., the finger diagram (Figure 2.6). The finger diagram derives from a three-dimensional dispersion diagram with axes corresponding to frequency v, axial propagation constant b, and a traversal of the in-plane Brillouin zone. The actual 2D finger diagram involves a projection onto the vb plane to form a density plot in which the intensity of the coloration is proportional to the number of propagating states. Figure 2.19 shows a projected band diagram for an infinite woodpile [40,83]. The vertical axis is proportional to the wavenumber, while the horizontal axis

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Electromagnetic Theory and Applications for Photonic Crystals 0.8

y

x

z

kd /2

0.6 0.4 0.2 0

M X

Γ Γ

X

M

Γ

k0 (a)

(b)

FIGURE 2.19 (a) Two unit cells of a woodpile structure. (b) The two-dimensional projection of the band structure for a woodpile: cylindrical rods have radii of 0.1 m and index 3.6. The pitch is 0.711 m, and the separation between layers is 0.21 m. (Reprinted with permission from G.H. Smith, et al., Phys. Rev. E, 67, 056620, 2003.)

traverses the irreducible part of the projected Brillouin zone, as shown in the inset. For the woodpile, the projected Brillouin zone is a square and, from symmetry arguments, one need consider only the irreducible octant labeled XM in Figure 2.19. The projection is from the k0z direction, which derives from the solution of the eigenvalue equation, again with the coloration set proportional to the number of propagating states. The completely white regions correspond to an absence of propagating states.

2.5.4 CLASSIFICATION

OF THE

EIGENVALUES

A necessary precursor to the modeling of photonic crystal devices using Bloch mode expansions is a thorough understanding of the modes and their properties. Accordingly, in this section, we focus on the classification of the modes and the derivation of key properties such as orthogonality relations. The outline here is necessarily brief, and we refer readers to [81] for a detailed treatment. The eigenvalues m of T (2.104) may be partitioned according to the direction of mode propagation. The evanescent or nonpropagating states (i.e., those eigenvalues associated with bandgap states) have |m|  1 and are readily classified according to their magnitudes. Those with |m|  1 are regarded as forward propagating, while those with |m| 1 are backward propagating. The classification of the propagating states for which |m|  1 is more subtle and must be ascertained from the direction of energy propagation, i.e., the direction of the y-component of the group velocity vg,y  dk/dk0y  F, y/(kD), where e F ,y 

© 2006 by Taylor & Francis Group, LLC

 H H f f 

  

 I iI   f  p   p      iI p I p   f 

(2.111)

From Multipole Methods to Photonic Crystal Device Modeling 

r

c−2

101

M1 +

M2

c2

M3

t D

FIGURE 2.20 A typical three-segment photonic crystal device showing the component regions M1, M2, and M3, three lateral supercells of the model, a constituent grating bounded by dashed lines, and the field quantities used in Equations (2.135) and (2.136).

denotes the net vertical (y) energy flux through the unit cell, and D denotes the (positive) energy density per unit cell. The sign of F,y thus determines the propagation direction, and we interpret F,y 0 as a forward propagating state and correspondingly F,y  0 as a backward propagating state. In (2.111), the derivation of which [38,40] involves a Poynting vector integral, the flux comprises terms that arise from both the backward and forward plane waves {f} that comprise the Bloch mode (2.104). The components of these vectors are associated with the diffraction orders of the E|| /H|| fields, the terms of which may be either propagating or evanescent. In (2.111), for the woodpile and other 2D layers, Ip  diag(Ip, Ip,) and Ip–  diag(Ip–, Ip–,) with Ip  [drsr], I p–  [drs(1  r)], and r taking the values 1 and 0 respectively for propagating (i.e., js real) or evanescent (i.e., js imaginary) plane wave orders s. For 1D layers operated in a principal polarization, the form of (2.111) is unchanged, although the dimensions of the vectors differ since they comprise coefficients of only the electric or only the magnetic field. In this case we have [75] e F ,y 

  H  f

H f  

 I  p    iI  p 

iI p   f     

 

I p   f  

(2.112)



Lattice symmetry further constrains the form of the forward and backward modes. In the case of up-down symmetric structures such as 2D arrays formed from a rectangular lattice of 1D cylinder gratings for which Ra  Rb and Ta  Tb, it is easily shown [75] that the eigenvalues m and m1 are paired, respectively corresponding to eigenvectors [fT fT]T and [fT fT]T. For simplicity in what follows, we restrict the discussion to a study of twodimensional photonic crystal devices [79,81,84,85,86] operated in either of their principal polarizations. In preparation for the Bloch mode modeling of extended photonic crystal devices of the type displayed in Figure 2.20, we consider a segment of a 2D photonic crystal comprising a number of cylinders per unit cell. The structures that we model, i.e., waveguides, splitters, couplers, interferometers, etc., are devices that rely on defect modes of the bulk crystal. Accordingly, we must choose the period of our supercell to be sufficiently large to ensure effective

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isolation between adjacent periods to avoid undesirable cross-talk between adjacent supercells. In such circumstances, the calculations become independent of k0x, the lateral component of the Bloch vector, and it is convenient to choose k0x  0. With the modes partitioned into forward/backward sets, the transfer matrix can be diagonalized T  FLF1 ,

 F F′  , F     F F′ 

 0   L     0  ′ 

(2.113)

a result that encapsulates the entire set of eigenvalue equations TF  FL. The columns of the matrix F are the eigenvectors of T, and L is a diagonal matrix, the entries of which are the corresponding eigenvalues m. In light of the mode partitioning, the columns of the matrices F, F  are formed from the f components of the eigenvectors, respectively, for the forward and backward modes, with  and  being diagonal matrices that contain the corresponding mj. 2.5.4.1 Formal Properties of Bloch Modes For structures of the kind that we are considering here, namely those that can be constructed from general lattices of multicylinder gratings, the physical concepts of reciprocity and energy conservation introduce other important constraints on the modes. The origin of such properties requires a careful analysis, which we now summarize [81]. Firstly we consider reciprocity, a geometrical constraint that is independent of material properties and constrains the transfer matrix to be symplectic [87], thereby pairing the forward and backward modes. Its derivation in 2D follows from an application of Green’s theorem 

V f



n

∫ (Vg 2V f  V f 2Vg )dA  ∫ Vg

 Vf

Vg   ds n 

(2.114)

around a supercell layer for fields Vf and Vg, each satisfying the same Helmholtz equation (for a given k), and which on the upper ( j  a) and lower ( j  b) boundaries of the grating layer are expressed in plane wave expansions V f( j ) (r ) 



∑ xs 1/2 [ fs( j )eix ( yy )  fs( j )eix ( yy ) ] eia x s

j

s

j

s

(2.115)

s

From this it follows that the bilinear product gTj Qpwfj is invariant under translation by a grating layer, i.e., gTa Qpw fa  gTb Qpw fb

© 2006 by Taylor & Francis Group, LLC

where

 0 Q  Qpw    Q 0 

(2.116)

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103

the skew-symmetry of which, i.e., QTpw  Qpw, follows from the corresponding antisymmetry of the cross-product inherited from Maxwell’s equations. In (2.116), Q denotes the reversing permutation derived from the identity matrix I by reversing the order of the rows. The layer independence of the bilinear product, together with the transfer matrix relations fb  Tfa and gb  Tga, allows us to show that T is symplectic [87], i.e., TT Qpw T  TQpw TT  Qpw

(2.117)

Because TT and T1 are related by a similarity transformation TT  QpwT1Q1 pw (from (2.117)), the eigenvalues of TT, i.e., m, and T1, i.e., m1, must be paired. The eigenstates are thus arranged into forward and backward propagating pairs, respectively associated with eigenvalues m and 1/m, and in turn this relates the eigenvalue partitions of L (2.113) according to   1. The second of the constraints follows from energy conservation and hence is appropriate only for lossless systems, in contrast to reciprocity, which is universally applicable. Its derivation follows from the application of Green’s theorem  V V f  g 0  ∫ (V f 2Vg  Vg 2V f )dA  ∫ V f  Vg  n n

  ds  

(2.118)

to a grating layer within the photonic crystal, the area integral of which vanishes only for lossless systems. This, together with the quasiperiodicity of the field, means that the bilinear product gHIpwf is conserved. That is,

g aHIpw fa



g bHIpw fb

where

  I p Ipw     iI p

iI p    

I p 

(2.119)

Note that with g  f, the bilinear form gHIpwf is simply a generalization of the forward flux F,y  fHIpwf (2.111) of the plane wave field. The conservation of the bilinear form (2.119) together with the transfer relations fb  Tfa and gb  Tfg leads to a new conservation relation satisfied by the transfer matrix, namely T HIpw T  Ipw

or

T H  Ipw T1Ipw1

(2.120)

– and m With TH and T1 related by a similarity transformation, the eigenvalues m must also be paired. Combining this with the pairing relations implied by the symplectic nature of T, we see that the eigenvalues must always occur in a –, m1, m –1}. quadruple {m, m In our subsequent treatment of propagation of light in extended photonic crystal devices, it is convenient to work with modes that are normalized in such a way that they enable physical quantities to be expressed in their simplest and

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most amenable forms. This requires that the modes represented by the columns of F be scaled appropriately. Following the treatment in [81], we may derive the respective modal reciprocity and modal orthogonality relations FT QpwF  Q bm ,

 0 I    Q bm   I 0    I m

FHIpwF  Ibm ,

(2.121)

iIm 

Ibm  

 

(2.122)

Im 

  iI m

from the diagonalized form of T (2.113) and the conservations relations (2.117) and (2.120). In (2.122), the matrices Im and Im– are diagonal with unit entries on the diagonal respectively only for propagating and nonpropagating Bloch modes. It thus follows that the plane wave and modal “flux” matrices, respectively Ipw and Ibm, are isomorphic. The second of the results in (2.122) expresses the orthogonality of the modes — a result that we exploit in Section 2.5.6.

2.5.5 DEFECT MODE MODELING CRYSTAL DEVICES

OF

EXTENDED PHOTONIC

The extended devices that we wish to model comprise an arbitrary number of PC segments — a semi-infinite input and output segment at each end and a finite number of intermediate segments, each having a finite number of layers. For simplicity, the analysis that we present here is for a three-segment device — see Figure 2.20, which is structurally similar to Fabry–Perot interferometer — which can be easily generalized to an arbitrary number of layers by recursion [81]. In each of the regions Mn, n  1, 2, 3 (labeled from top to bottom), the field can be expanded in terms of forward and backward Bloch modes, the amplitudes of which are given by vectors c n. Relative to the respective phase origins at the top and bottom of the segment (comprising Ln layers) for each of the forward (c n ) and backward (cn ) propagating modal fields, the field at some interface l  [0, Ln] can be expressed as a plane wave expansion, the forward and backward components of which are represented by vectors of plane wave coefficients f  n (l) where    f  (l )   F   F′  n n  ( L l )   n  l     n fn (l )   c n  Fc n (l )     n c n    n      F′   fn (l )   Fn   n  def

(2.123)

with  def  c n (l )    L (l )c c n (l )    n n   c n (l ) 

where

 l  n

Ln (l )    

0

    ( Ln l )   n 

0

  c  n     c   n 

, c n  

(2.124)

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105

The importance of working with appropriately normalized modes becomes apparent when we wish to transform the forward flux expressed in the plane wave basis to a more useful form in the modal basis. Exploiting the normalized form (2.122), the forward flux at any level l in medium Mn can be converted from the plane wave representation to the equivalent modal form by en (l )  fnH (l )Ipw fn (l )  c nH (l )FnHIpwFn c n (l )  c nH (l )Ibm,n c n (l )

(2.125)

where Ibm,n denotes the flux matrix with respect to the Bloch mode basis of medium Mn. In turn, en (l )  c nHLHn (l )Ibm,nLHn (l )c n



 H H c cn  n

  

  I m    L  iI  n  m





iImLn   cn  Im

   

    c   n 

(2.126)

Naturally, the resulting expression for the flux (2.126) is independent of layer l, the proof of which relies on the eigenvalue pairings discussed in the previous section. Note also that as the length of the segment Ln → , the flux approaches 2  2 mm |c m |  |c m | , the difference between the forward and backward fluxes computed over only the propagating states m. For small separations, however, evanescent mode terms can play a significant role and typically must be taken into account for Ln   5.

2.5.6 INTERFACING PHOTONIC CRYSTAL MEDIA AND FRESNEL COEFFICIENTS Having established the formal properties of the modes and their representation in matrix form, we can turn now to the study of propagation of Bloch modes through photonic crystal structures formed from stacks of identical layers (see Figure 2.20). The interfacing of two semi-infinite PC media requires the application of field continuity conditions at the matching boundary and is best represented by interface reflection and transmission matrices, which are the matrix analogues of the familiar scalar Fresnel coefficients in optics. For the sake of both brevity and simplicity of nomenclature, we demonstrate this with a simplified geometry comprising up-down symmetric layers arranged in a square or rectangular lattice. In this case, the forward and backward modes are related by F n  F n and F n  Fn . The generalized Fresnel reflection and transmission coefficients are derived by matching the fields along the interface between the two PC media, Mn and Mm. Here, we consider an incident field of modal amplitudes c n incident from above within a semi-infinite region Mn giving rise to a reflected field with modal amplitudes cn in Mn and a transmitted field c m in Mm. Assuming a common background medium in which each of Mn and Mm is formed, the application of the continuity conditions simply reduces to equating the forward and backward components of

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the plane wave fields on the matching interface. Thus,     F  F   n    n     c  c   n   n F  F   n   n 



  F   m   c ,   m F   m 

or

Fn c n  Fm c m

(2.127)

T T where cm  [cT m 0 ] . Solving these equations, we deduce that the generalized Fresnel matrices Rnm and Tnm, respectively defined by cn  Rnmc n and  c  T c , are given by m nm n

R nm  (Fn− )1 (I − R m R n )1 (R m  R n )Fn

(2.128)

Tnm  (Fm )1 (I  R n R m )1 (I  R n2 )Fn

(2.129)

1 denotes the reflection scattering where in (2.128) and (2.129) Rj  Fj (F j) matrix of a semi-infinite crystal of material Mj from free space [75,81]. With the modes normalized according to (2.121) and (2.122), it can be shown [81] that the Fresnel scattering matrices satisfy reciprocity relations that are elegantly expressed in terms of scattering matrix symmetry, i.e., RTnm  Rnm and TTnm  Tmn, and energy conservation relations that follow the form of the plane wave scattering matrix conservation relations, e.g., H I H H R nm bm,n R nm  Tnm I bm,m Tnm  I bm,n  iR nm I bm,n  iI bm,n R nm (2.130) H I H H R mn bm,m Tnm  Tmn I bm,n R nm  iTmn I bm,n  iI bm,m Tnm

(2.131)

The Fresnel matrices given in (2.128) and (2.129) exploit the complete basis of modes, the dimensions of which are determined by the dimension of the plane wave basis used in computing the plane wave scattering matrices for each grating layer. While it is necessary to have a common dimension for all plane wave quantities, it can be advantageous to be able to restrict the basis of modes that is used in field calculations, particularly in those situations in which comparatively few modes dominate the propagation problem. The system of equations (2.127) then needs to be considered in a least squares sense, the solution of which relies on the modal orthogonality relations (2.122). To derive this, we begin with the equation Fncn  Fmcm, which expresses field continuity at the matching interface, and seek to minimize the norm of the difference of the left- and righthand sides. In a manner analogous to Galerkin’s method, we project the total field equation onto one of the modal bases and the field derivative equation onto the other basis. We commence by premultiplying both sides of the equation by FH n Ipw to derive FnHIpwFn c n  Ibm,n c n  FnHIpwFm c m

© 2006 by Taylor & Francis Group, LLC

(2.132)

From Multipole Methods to Photonic Crystal Device Modeling

107

projecting the system of equations onto the modal basis for Mn. Premultiplying through (2.132) by [I I] then allows us to extract the field derivative equation cn  c +n  J nH K nm cm

(2.133)

    with Knm  GH n JpG m, G n  F n  F n , and Jq  Iq  iIq–. In the definition of Jq, the quantities Iq and Iq– refer to the relevant components of either of the constituent matrices of the matrix Ipw (2.119) for plane waves or its Bloch mode equivalent Ibm,n (2.122) applicable to the region Mn. Correspondingly, projecting the field matching equations onto the modal basis for Mm by premultiplying through Fncn  Fmcm by [I  I]F H m Ipw, we derive the total field equation H   c  J K (c  c ), which is the pair to (2.133). Finally, combining these m n m nm n leads to expressions for the Fresnel reflection and transmission matrices

R nm  (I  Bnm )1 (I  Bnm ),

H (I  B )1 Tnm  2 J m K nm nm

(2.134)

   H H according to c n  Rnmc n and c m  Tnmc n . In (2.134), Bnm  Jn KnmJmKnm, takes the role of an impedance matrix, the dimension of which is bn  bn where bn is the dimension of the truncated modal basis for region Mn. Thus, Rnm and Tnm respectively have dimensions of bn  bn and bm  bn. Both the symmetry and energy conservation relations continue to hold analytically, a consequence of the normalizations of the modes in (2.121) and (2.122).When the modal bases are not truncated and the reflection and transmission matrices are square and have order identical to that of the cardinality of the set of plane waves used in the calculation, the expressions in (2.134) reduce readily to those in (2.128) and (2.129), again due to the mode energy normalization (2.122) and the completeness of the basis. In conclusion, we observe that all of these results hold analytically, independent of field truncation errors, when the individual layer scattering matrices are computed using the multipole method. It is shown in [38] that the multipole method preserves analytically the corresponding reciprocity and energy properties in the plane wave basis, and it is demonstrated in [81] that this scattering matrix formulation carries these results through into the Bloch mode basis.

2.5.7 RECURSIVELY COMBINING PC STACK SEGMENTS Having characterized the single layer propagation problem (Section 2.5.5) and shown how the interfacing of adjacent layers may be characterized with Fresnel matrices (Section 2.5.6), we now formulate the solution of the propagation problem in Figure 2.20. We consider an incident field with Bloch mode amplitudes d incident from M1 onto the M1  M2 interface giving rise to a reflected Bloch mode field r in M1 and a transmitted Bloch mode field t in M3. In region M2 comprising L  L2 layers, the field is expanded in terms of forward and backward Bloch modes, with respective amplitudes c 2 (with phase origin on the M1  M2

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Electromagnetic Theory and Applications for Photonic Crystals

boundary), and c2 (with phase origin on the M2  M3 boundary). The field matching equations respectively on the M1  M2 and M2  M3 boundaries are then written as r  R12 D  T21L2 c2 ,

c2  T12 D  R 21L2 c2

(2.135)

c2  R 23L2 c2 ,

t  T23L2 c2

(2.136)

derived from a consideration of the reflection and transmission of Bloch fields at each boundary and the propagation of the field within M2. After solving (2.135) and (2.136), we may then infer the reflection (R) and transmission (T) scattering matrices for the three-segment structure from r  RD and t  TD. We thus derive R  R13  R12  T21L2 R 23L2 (I  R 21L2 R 23L2 )1 T12

(2.137)

T  T13  T23L2 (I  R 21L2 R 23L2 )1 T12

(2.138)

forms that are clearly generalizations of the Airy Formulae for a Fabry–Perot interferometer. Naturally, this result is easily generalized for multisegment PC devices comprising more than three layers [81].

2.6 MODELING OF PHOTONIC CRYSTAL DEVICES 2.6.1 BACKGROUND With the Bloch mode “toolkit,” relying on multipole methods for computing scattering matrices, now in place, we go on to discuss their application to the study of two-dimensional photonic crystal devices. Such devices rely on a 2D photonic bandgap for lateral confinement and guiding of light within a dielectric slab. Devices based on these structures have generated a great deal of interest recently because of their compact size and their ability to control the propagation of light on the wavelength scale. The Bloch mode calculation methods that use the multipole scattering matrix methods described in this chapter have proven to be very efficient for studying certain types of PC devices. In particular, devices incorporating multiple sections of identical waveguides are dealt with in an efficient manner using the transfer matrix methods described in Section 2.5 because the scattering matrices need be calculated only once for each distinct grating type. A number of numerical examples of the Bloch mode method applied to PC devices have been published elsewhere, so rather than simply reproducing the results here, we investigate in more detail some specific properties of three PC devices that have already been studied. The first two of these devices, the “folded directional coupler” (FDC) [84,85] and the coupled Y-junction [85] operate in an identical manner, and are treated together in Section 2.6.3. In Section 2.6.4 we

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discuss the recirculating Mach–Zehnder interferometer [86] and present an analogous design using conventional optics. Although most of the discussions in this section do not rely explicitly on the Bloch mode or multipole methods, the Bloch mode method is a powerful tool that aids in the development of such simple models. By treating structures as grating stacks, we can use many of the tools developed for thin film optics to develop a better understanding of the underlying physics.

2.6.2 STRAIGHT PC WAVEGUIDES Many PC-based devices can be modeled as a sequence of waveguides of varying widths and lengths coupled together. Each of the devices discussed in this section is of this type. With the tools already developed earlier in this chapter, it is a straightforward procedure to calculate the transmission properties of simple waveguide structures. To model a straight photonic crystal waveguide, we choose a grating layer consisting of multiple cylinders per unit cell, with each supercell typically consisting of 11–21 cylinders. Removing one or more cylinders from each supercell results in a periodic array of straight waveguides when the grating layers are stacked to form a 2D crystal. The supercell period D is chosen to be large enough that the waveguide(s) in one supercell do not affect the waveguides in the neighboring supercells. The multicylinder supercell is treated as a single element of a grating with period D, and the techniques used in Section 2.5 can be applied. Recall that eigenvalues m with magnitude | m|  1 correspond to evanescent (nonpropagating) modes, whereas propagating modes have |m|  1. Waveguide modes appear as propagating modes at frequencies within the bandgap of the perfect photonic crystal. Mode dispersion curves are then easily calculated by extracting the longitudinal propagation constant (in the direction of the waveguide, i.e., k0  a2) from m  exp(ik0  a2) for a range of wavelengths. When studying structures formed from finite lengths of waveguide, it is often sufficient to include only the propagating modes when calculating transmission or reflection properties. In this situation, the Bloch mode transmission and reflection matrices are greatly simplified, and it is often possible to derive scalar analytic expressions to describe the properties of the structure [47]. To exemplify this, we consider a finite single mode waveguide through an L layer 2D PC, bounded on either side by free space. The Airy transmission matrix (2.138) then reduces to T  F2 (I  R 22 )L2 (I  R 2 L2 R 2 L2 )1 (F2 )1

(2.139)

after deriving the following forms for the relevant Fresnel matrices: 2  1  1  T23  F 2 (I  R2), T12  (F2 ) , and R23  R21  R2  (F2 ) F 2 . Then, provided that the single mode guide is sufficiently long (typically L 

5), the term 2L  mLwwH, approaching this form in the limit as L → . Here, wT  [1 0 0 … 0], and the limiting form of L2 is of unit rank. In such circumstances, only the propagating mode needs to be considered, and Equation (2.139) may be simplified using the Sherman-Morrison formula [88] to invert the rank one

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perturbation. Using energy conservation relations (2.131), a simplified form of the transmitted flux may be derived [79] eF , y 

1  |r|2 |N_D|2 |1  r 2 m 2 |

(2.140)

In (2.140) m is the eigenvalue of the single propagating mode and r is the (1, 1) element of R2, its complex reflection coefficient. The denominator of (2.140) characterizes the resonance, while the numerator (1  | r |2) expresses the net energy propagation of a unit flux forward mode and a backward mode of flux | r |2 generated by reflection at the rear surface. Coupling of the incident field into the primary mode is given by an overlap integral N_D, where N_ denotes row 1 of (F2 )1. As is evident, such an analysis can provide accurate asymptotic results as well as real physical insight into the propagation characteristics of the structure being studied. In this and subsequent examples, the semianalytic expressions often require numerical input of the propagation constants for various waveguide modes, which are simply calculated using the procedure described above.

2.6.3 RESONANT FILTERS AND JUNCTIONS A large number of photonic crystal based optical devices have been demonstrated both theoretically and experimentally in recent years. Many of these devices are simply photonic crystal analogues of conventional optical devices, and as such they do not fully exploit the unique properties of the photonic crystal. One such property is the total reflection of light from a photonic crystal slab regardless of incident angle when operating at frequencies within a bandgap. In the case of a realistic 2D photonic crystal slab, this is only an approximation as light can be coupled into radiation modes above and below the slab. Although we do not explicitly treat out-of-plane losses here, the properties discussed are generic, and realistic device designs could be proposed to minimize these effects. Figure 2.21 shows two basic PC devices that can be studied efficiently using the techniques of this chapter. The device in Figures 2.21(a) and 2.21(b) is a folded directional coupler (FDC) — a notch-rejection filter with a very high-Q response [84]. A similar device is the symmetric coupler, or coupled Y-junction, shown in Figures 2.21(c) and 2.21(d). This exhibits very high transmission over a large bandwidth [85], a property that is used to design a PC-based Mach–Zehnder interferometer in Section 2.6.4 [86]. Both devices in Figures 2.21 are modeled as three distinct waveguide sections, M1–M3, where M1 and M3 are semi-infinite stacks of identical gratings and M2 is a stack of identical gratings. In the case of the FDC, the input waveguide and output waveguides in M1 and M3 support single modes denoted by |cL and |cR, respectively. Section M2 consists of two waveguides and thus supports two modes — one symmetric (|c), and one anti–symmetric (|c) — across much of the bandgap. The coupled Y-junction consists of a single mode input waveguide in M1, triple waveguides in M2, and a pair of output waveguides in M3. Although the triple and double waveguide sections in the structure of Figures 2.21(c)

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From Multipole Methods to Photonic Crystal Device Modeling

M1

111

L

L

M2

M3 R

(a)

(b)

c

(c)

(d)

FIGURE 2.21 (a) Geometry of the FDC and (b) schematics of the propagating modes in each of the three sections. All of the cylinders are identical, and the shading is to emphasize the three distinct waveguide sections. (c) Geometry of the coupled Y-junction and (d) schematics of the propagating modes of even symmetry. The photonic crystal is a square lattice of cylinders with refractive index n  3, period d, and radius 0.3d in a background of air.

and 2.21(d) can both support an antisymmetric propagating mode, the symmetry of the device relative to the input waveguide means that only modes of even symmetry can be coupled. Thus, the junction can also be considered to support a single input mode |cc; two symmetric modes, |c1 and |c3, in M2; and a single output mode, |c in M3. There is thus a direct mapping between the modes in each section of the FDC and those of the equivalent sections of the junction. In a full Bloch mode analysis of the two structures, both the propagating and evanescent Bloch modes are included in the calculations, with the coupling between the Bloch modes at an interface given by the generalized Fresnel reflection and transmission matrices Rnm and Tnm defined in Equations (2.128) and (2.129). In a typical calculation using the complete basis, the Bloch mode basis and the plane wave basis have the same dimension, and thus the Bloch mode calculation includes a large number of evanescent modes. Often, however, it is sufficient to include only the propagating modes by truncating Rmn and Tmn to the dimensions of the propagating modes in each region.

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The devices in Figure 2.21 each have two interfaces, so we require six Fresnel matrices, R12, R21, R23, T12, T21, and T23, to describe fully propagation from M1 to M3 using Equations (2.128) and (2.129). The truncated orders of each of these matrices are determined by the number of propagating modes on either side of the relevant interface. For instance, R12 describes modes incident and reflected in section M1 (which only supports a single mode) and hence R12 is characterized by a scalar. For the same reason, R21 and R23 are 2  2 matrices. Since T12 describes transmission of the single mode in M1 into the two modes of M2, it must be column vector with two rows (i.e., a 2  1 matrix). Similarly, T21 and T23 are 1  2 matrices (i.e., row vectors). While the transmission properties of the FDC and coupled Y-junction are described accurately by the propagating mode treatment, the computing requirements are only moderately reduced because it is still necessary to include the full basis of plane waves to represent the propagating Bloch modes. A further approximation can be used to derive a semianalytic expression for both the FDC and the Y-junction transmission [85]. To do this, we use approximate relationships between the modes of each region to write down simple forms for the Fresnel matrices. For instance, the input mode of the Y-junction, |cc, can be approximated by a superposition of the modes in the triple guide region, as – |cc  (|c1  |c3)/2, and similar expressions can be found for the modes of the output waveguide. Ignoring any back reflection that may occur when the single input waveguide interfaces to region M2, we can then write down expressions – – for the matrices R12  [0] and T12  [1/2  1/2]T. The remaining Fresnel matrices are found using similar arguments and substituted into Equation (2.129) for the three-layer structure where propagation through the central layer is represented by the matrix , which is similarly truncated to a 2  2 diagonal matrix of eigenvalues mj  exp(ibjd), where j  1, 2. Finally we express the propagation constants of the two modes in M2 in terms of the average propagation constant – and define b  (b1  b2)/2 and b  (b1  b2)/2 for the Y-junction and similar expressions for the FDC by replacing the subscripts (1, 2) with (, ). This last approximation is valid when the mode splitting in M2 is relatively small. After some manipulation, the complex reflection and transmission amplitudes r and t can be written as r

cos2 (bL ) exp (2ibL ) , 1  sin 2 (bL ) exp (2ib) L

t

i exp (ibL ) sin(bL )(1  exp (2ibL )) 1  sin 2 (bL ) exp (2ibL ) (2.141)

Thus, the transmitted and reflected intensity can be expressed as eR r2 

1 B , eT t2  , 1 B 1 B

where

B

4sin 2 (bL ) cos 2 ( bL )

cos 4 (bL )

(2.142) so that eR  eT  1. Although the two devices presented here have identical expressions for their transmittance, they were designed to have very different properties. As shown in © 2006 by Taylor & Francis Group, LLC

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113 1

109

0.8

103

0.6

10–3

0.4

10–9

0.2

T

1015

10–15 3.015

3.02

3.025 /d (a)

3.03

3.035

1

109

0.8

103

0.6 T

1015

10−3 10−9 10−15

0.4

cos2(
L) sin2(∆
L) cos4(∆
L) B = 4cos2(
L)sin2(∆
L)/cos4(∆
L) Transmission = B/(1+B)

3

3.05

3.1

3.15

3.2

0.2

3.25

/d (b)

FIGURE 2.22 (a) Transmission of the FDC shown in Figure 2.21(a) (right axis) plotted as a function of normalized wavelength and values of the various terms in Equation (2.142). (b) Transmission of the coupled Y-junction shown in Figure 2.21(b) (right axis). The other lines show the same functions as are plotted in Figure 2.21(a) for the different waveguide parameters.

Figure 2.22(a), the FDC operates as a high-Q notch rejection filter, whereas the junction is designed for a maximum high-transmission bandwidth (Figure 2.22[b]). The transmission properties of these devices can vary widely with changes in the waveguide separation or the length L. To identify each of the features in a typical transmission spectrum, we need to consider the effects of each of the terms in Equation (2.142). In Figure 2.22, the trigonometric functions are plotted as a function of wavelength along with the value of B and the device transmittance, calculated from Equation (2.142) for both the FDC and the Y-junction of Figure 2.21. From this figure and Equation (2.142), it can be seen that eT → 0 when B → 0, and eT → 1 when B → , where each of these cases corresponds to one of the trigonometric functions vanishing. The relative positions of these zeros determine the sharpness, or otherwise, of the resonances of B, and, in turn, the overall response of the –L) and cos4(bL) occur close device. In the case of the FDC, the zeros of cos2(b together, resulting in B changing from 0 to over a very short wavelength range © 2006 by Taylor & Francis Group, LLC

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and leading to the high-Q resonance observed in the transmission function. Similarly, the parameters of the Y-junction design are such that B is large across a considerable wavelength range resulting in a wide bandwidth of hightransmission. From these examples we see that the Bloch mode analysis from Section 2.5 is very useful for investigating photonic crystal based devices. The key insight from this analysis is that, by working in the basis of Bloch functions, photonic crystal based devices can be analyzed in a manner similar to that for conventional guidedwave devices. It is then easily seen that the different responses of these two device classes are associated with, for example, a perfectly reflecting cladding. Apart from this, however, their responses can be understood in the same way.

2.6.4 MACH–ZEHNDER INTERFEROMETER A third photonic crystal device to which the multipole and Bloch methods have been applied is a Mach–Zehnder interferometer (MZI). In its simplest form, an MZI consists of an input beam, or waveguide, and two beam splitters, or Y-junctions — the first to divide the input into two beams and the second to recombine the beams after each has propagated along a different path. A basic free-space MZI is illustrated in Figure 2.23(a), where the input beam passes through a beam splitter (Y1) and then each propagates along a different path and is recombined by the second beam splitter (Y2). If the two paths are identical, the beams interfere constructively at the output port and a strong signal is detected. If, however, the path lengths of the two beams are different, the beams no longer combine constructively at the output, and some of the light is lost as radiation perpendicular to the measured output beam. In [86], we studied the MZI of Figure 2.23(b) consisting of single mode input and output waveguides, and efficient coupled Y-junctions. Using a simple modal model and full Bloch mode calculations, it was shown that such a device exhibits unusual recirculating behavior caused by reflection of light that would normally be radiated and lost in a conventional MZI. These reflections occur because the PC cladding surrounding the MZI does not allow light to radiate, and hence any light that is not coupled into the output port must be reflected back into the arms of the interferometer. With particular choices of path length L and/or wavelength, it is possible to design a recirculating MZI of this type with a considerably sharper phase response than a conventional MZI. The semianalytic expression derived for the recirculating MZI transmission is expressed as a function of the phase difference between the two arms w and the variable x  bL  w/2 where b is the propagation constant of the single waveguide mode. In this form, the transmitted intensity is given by eT 

4sin 2 (x) cos2 (w /2) /sin 4 (w /2) 1  4sin 2 (x) cos2 (w /2) /sin 4 (w /2)

(2.143)

Observe that this expression can also be written in the form of Equation (2.142) where B is replaced by BMZ  4sin2(x) cos2 (w/2)/sin4(w/2). This observation that the FDC, the Y-junction, and the recirculating MZI all can be represented by an

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In

50/50 beamsplitters Out

(a)

L

(b) m1 In

h

50/50 beamsplitters Out h

(c)

m2

FIGURE 2.23 (a) Conventional Mach–Zehnder interferometer using mirrors and beam splitters to separate and recombine the light. (b) Geometry of a photonic crystal based recirculating MZI using coupled Y-junctions to separate and recombine the light. (c) Conventional MZI modified to exhibit recirculating transmission characteristics.

expression with the same form may at first be a little surprising. However, this occurs because at some level each of the devices can be considered as a generalization of a Fabry–Perot resonator consisting of two interfaces that behave as selective mirrors separated by a cavity in which the light bounces back and forth. In the cases of the FDC and the Y-junction, the interfaces occur where one or more waveguide sections terminate. For the MZI, the Y-junction design behaves as an almost perfect symmetric junction over the operating bandwidth, and so each junction can be thought of as a distributed mirror that reflects modes with odd symmetry and transmits modes of even symmetry. Thus the MZI also displays Fabry–Perot-like resonant behavior. One feature of the recirculating MZI that is of particular interest is the phase response when bL  p/2, 3p/2, 5p/2 …, for which eT 

4cot 4 (w /2) 1  1  4cot 4 (w /2) 1  tan 4 (w /2) /4

(2.144)

This quartic response in tan(w/2) is considerably sharper than the quadratic response of a conventional MZI [86], and hence could be used to improve the switching characteristics of Mach–Zehnder interferometers in both linear and nonlinear devices.

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The recirculating properties of the PC MZI described here occur because light that is not coupled into the output waveguide is reflected back into the interferometer. The same result can be obtained in the free-space MZI illustrated in Figure 2.23(a) by placing reflectors in the path of the beam that is usually radiated. Such a design is illustrated in Figure 2.23(c), where the additional mirrors are labeled m1 and m2. When the beam splitters are characterized by complex reflection and transmission coefficients r  |r|exp(ifr) and t  |t|exp(i(fr  p/2)) where |r|2  |t|2  1/2 and arg(t)  arg(r)  p/2, the transmission intensity is given by Equation (2.143) where x  k0(L  2h)  fr  w/2, k0  2p/l, w is the phase difference and L is the length of the arms. With a choice of parameters such that k0(L  2h)  fr  p/2, 3p/2, 5p/2, …, the quartic phase response of Equation (2.144) is obtained. The examples discussed here illustrate that the multipole and Bloch mode methods not only provide an efficient numerical tool for studying the properties of a many 2D photonic crystal devices, but they also give insight into the underlying physics of the devices allowing simple models to be developed. In turn, these simple semianalytic expressions are an important tool for efficient device optimization and design.

2.7 DISCUSSION AND CONCLUSIONS This chapter has been built around three major themes in photonic modeling: (a) the systematic and integrated development of the multipole method (Section 2.2) as a key technique for modeling finite and infinite periodic structures (based on cylindrical scatterers); (b) the direct application of the multipole method in the study of two paradigm applications, namely the emission properties of sources embedded in photonic crystal clusters and propagation in microstructured optical fibers; and (c) the development of design tools based on the Bloch mode scattering matrix techniques and their application in the design of photonic crystal devices, our implementation of which has relied on multipole methods. In the investigations described in Sections 2.3 and 2.4, the multipole method has worked outstandingly well, at various times leading to results either not possible with other methods or not achievable with comparable accuracy or efficiency. Its strengths, of course, lie in the application of a basis that is well tailored to the geometry and symmetry of the structure and that provides rapidly convergent field expansions that underpin its use in accurate and efficient computational methods. Its weakness lies in the limitations of the structures that can be modeled because only for particular highly symmetric geometries is it possible to construct the necessary basis of multipole functions. In some circumstances, it is possible to adapt the method in a hybrid manner to handle arbitrary geometries. One particular example is the adoption of a numerical technique [35] that enables cylinders of arbitrary cross section to be handled. This technique integrates the wave equation (using a differential method) between the circles that inscribe and circumscribe the boundary and proceeds to encapsulate the electromagnetic properties of the scatterer in terms of an exterior

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multipole reflection matrix — which is dense, in contrast to the diagonal matrices that arise when dealing with purely cylindrical geometries. In the case of microstructured optical fibres comprising a large number of identical cylinders, the significant computational time required for this initial step can be amortized over the multipole calculation for the entire structure with little overall reduction in efficiency. Despite retrofits of this type, however, multipole methods can never tackle the range of problems that are accessible to general purpose techniques. They are clearly special purpose methods and should be used in this context, exploiting the analytic flexibility and the computational advantages they afford for those structures for which they are applicable. In future work, we envisage extending our use of these methods to handle various 3D systems and anisotropic media. The Bloch mode toolkit, elaborated in Sections 2.5 and 2.6, has proven to be physically intuitive and analytically tractable. Of critical importance is the implicit concept of a photonic crystal as a stack of gratings (1D or 2D). From this emerges the key quantity R , the scattering matrix that characterizes the reflection from a semi-infinite photonic crystal. This quantity, which is the photonic crystal analogue of the Fresnel reflection coefficient in thin film optics, encapsulates the mode structure of the bulk PC and underpins the modeling of PC devices in a manner analogous to that adopted for their conventional guided wave counterparts. The quantity R is also important in underpinning the modeling of genuinely infinite structures that are often important in dealing with fundamental issues (e.g., existence of cutoffs, symmetry questions) that, in general, are not otherwise addressable. Typifying the style of questions that can be dealt with is that of the symmetry of modes in coupled photonic crystal waveguides — for which it can be shown that the fundamental may be either odd or even, in contrast to conventional directional couplers in which the fundamental mode is always even. The result [89] follows naturally from a simple model of coupled resonators (with infinite cladding) and an asymptotic analysis that is applicable to the long wavelength regime in which the properties are dominated by the sole propagating (specular) grating order. This physically intuitive analysis [89] reveals that PC directional couplers indeed have quite different properties than conventional couplers — with the fundamental mode being either odd or even depending on the separation of the guides, the characteristics of the bandgap, and the phase on reflection off the bulk photonic crystal — a result that cannot be derived using alternative approaches. In addition to its intuitive and tractable qualities, the Bloch mode method is also computationally easy to implement — with our particular implementation comprising a combination of Mathematica [90] functions and Fortran — with the applications suite implemented in the former, thus exploiting the excellent programming language and numerical linear algebra library, and with the multipole based grating scattering matrices (that define the transfer matrix) being implemented in the latter, with the two linked together using the MathLink toolkit [90]. While scattering matrices can be computed using a variety of techniques, the multipole method has been central to most of the applications that we have studied. The sole proviso for its use is that adjacent layers do not interpenetrate, thus allowing the use of plane wave expansions of fields at matching interfaces. This requirement, strictly speaking, is a

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sufficient but not a necessary condition. Provided that the extent of the interpenetration is not too severe, plane wave expansions are still valid on the matching interfaces — a situation that is directly related to the Rayleigh controversy of diffraction grating theory [91] concerning the validity of plane wave representations for outgoing fields within the grooves of diffraction gratings. For noninterpenetrating cylinder gratings, however, the multipole technique is arguably the preferred method as the formulation is analytically elegant and structurally embodies key properties such as reciprocity and energy conservation analytically within the formulation. These properties may be verified effectively to within machine precision and are analytically inherited by the Bloch mode method making it both analytically tractable and easy to validate numerically. Nevertheless, in those circumstances when the multipole methods can no longer be used, a range of alternative techniques including differential, integral, and finite element methods may be employed. The Bloch mode scattering matrix method for modeling PC devices displays real computational advantages over general purpose techniques (e.g., FDTD methods) when handling extended devices that comprise many identical gratings because the propagation problem in a long segment can be handled with only a single set of modes. That said, however, there is no performance advantage to be gained when dealing with structures such as waveguide tapers that vary from layer to layer. The choice of the diffraction grating paradigm naturally imposes a supercell on the structure, thus modeling the behavior not of a single device but of an infinite parallel array of devices. The utility of the method is thus limited to gap frequencies, which, of course, is where photonic crystals are at their most useful, and further necessitates that the supercell be chosen sufficiently large to avoid cross-talk between adjacent structures. The most important feature of the Bloch mode techniques, however, is their capacity to provide physical insight into coupling and propagation problems in photonic crystal devices. This is achieved through the use of the natural Bloch mode basis and the introduction of Fresnel reflection and transmission matrices that characterize the interaction of fields at an interface between two PC devices, thereby enabling the formulation of the propagation problem in a manner that is essentially a generalization of the theory of thin film optics. Its structure also enables various asymptotic simplifications. In particular, the orthogonality properties enable the modal basis to be truncated to include only the most significant modes, while simultaneously preserving key properties such as energy conservation analytically within the formulation. This feature is quite significant in that it leads to the identification of an impedance matrix that may be of potential use in the characterization and optimization of devices based on photonic crystals. The combination of these features suggests that it may be possible to incorporate the successful design paradigms of thin film optics into the new context of photonic crystal devices. This is certainly an exciting possibility that merits further investigation. We conclude by noting that while the Bloch mode modeling theory in Section 2.5 has been presented in the context of cylindrical photonic crystals, it is readily extensible to general 3D systems, thus allowing the analysis of PC devices in 3D photonic crystals and photonic crystal slabs.

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ACKNOWLEDGMENT This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program.

REFERENCES [1] J.W.S. Rayleigh, The Theory of Sound, Dover, New York, 1945. [2] J.W.S. Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium, Philos. Mag., 34, 481, 1892. [3] W. von Ignatowsky, Zur Theorie der Gitter, Ann. Physik (Leipzig), 44, 369, 1914. [4] J.R. Wait, Scattering of a plane wave from a circular dielectric cylinder at oblique incidence, Can. J. Phys., 33, 189, 1955. [5] J. Korringa, On the calculation of the energy of a Bloch wave in a metal, Physica, 13, 392, 1947. [6] W. Kohn and N. Rostoker, Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium, Phys. Rev., 94, 1111, 1954. [7] N. Stefanou, V. Yannopapas, and A. Modinos, MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals, Comput. Phys. Commun., 132, 189, 2000. [8] V. Twersky, Elementary function representations of Schlömilch series, Arch. Rational Mech. Anal., 8, 323, 1961. [9] P.P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentiale, Ann. Phys. (Leipzig), 64, 253, 1921. [10] S.K. Chin, N.A. Nicorovici, and R.C. McPhedran, Green’s function and lattice sums for electromagnetic scattering by a square array of cylinders, Phys. Rev. E, 49, 4590, 1994. [11] R.C. McPhedran, et al., Lattice sums for gratings and arrays, J. Math. Phys., 41, 7808, 2000. [12] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059, 1987. [13] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett., 58, 2486, 1987. [14] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [15] J.C. Knight, et al., All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett., 21, 1547, 1996; Errata, Opt. Lett., 22, 484, 1997. [16] T.A. Birks, et al., Full 2-D photonic bandgaps in silica/air structures, Electron. Lett., 31, 1941, 1995. [17] A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibre, Kluwer, Boston, 2003. [18] R. Sprik, B.A. van Tiggelen, and A. Lagendijk, Optical emission in periodic dielectrics, Europhys. Lett., 35, 265, 1996. [19] C.M. Soukoulis, Photonic Crystals and Light Localization in the 21st Century, Kluwer, Dordrecht, 2001. [20] K.M. Leung and Y.F. Liu, Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media, Phys. Rev. Lett., 65, 2646, 1990. [21] C.T. Chan, Q.L. Yu, and K.M. Ho, Order-N spectral method for electromagnetic waves, Phys. Rev. B, 51, 16635, 1995.

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[22] M. Neviere and E. Popov, Light Propagation in Periodic Media, Marcel Dekker, New York, 2003. [23] M. Koshiba, Full-vector analysis of photonic crystal fibers using the finite element method, IEICE Trans. on Electron. C, 85, 881, 2002. [24] N.A. Nicorovici, R.C. McPhedran, and L.C. Botten, Photonic band gaps for arrays of perfectly conducting cylinders, Phys. Rev. E, 52, 1135, 1995. [25] D. Felbacq, G. Tayeb, and D. Maystre, Scattering by a random set of parallel cylinders, J. Opt. Soc. Am. A, 11, 2526, 1994. [26] L.-M. Li and Z.-Q. Zhang, Multiple-scattering approach to finite-sized photonic band-gap materials, Phys. Rev. B, 58, 9587, 1998. [27] E. Hoskinson and Z. Ye, Phase transition in acoustic propagation in 2D random liquid media, Phys. Rev. Lett., 83, 2734, 1999. [28] A.W. Snyder and J.D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983. [29] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [30] W. Wijngaard, Guided normal modes of two parallel circular dielectric rods, J. Opt. Soc. Am., 63, 944, 1973. [31] T.P. White, et al., Multipole method for microstructured optical fibers. I. Formulation, J. Opt. Soc. Am. B, 19, 2322, 2002. [32] T.P. White, et al., Multipole method for microstructured optical fibers. I. Formulation: errata, J. Opt. Soc. Am. B, 20, 1581, 2003. [33] B.T. Kuhlmey, et al., Multipole method for microstructured optical fibers. II. Implementation and results, J. Opt. Soc. Am. B, 19, 2331, 2002. [34] R.C. McPhedran, N.A. Nicorovici, and L.C. Botten, The TEM mode and homogenisation of doubly periodic structures, J. Electromagn. Waves Appl., 11, 981, 1997. [35] S. Campbell, et al., Differential multipole method for microstructured optical fibers, J. Opt. Soc. Am. B, 21, 1919, 2004. [36] L.C. Botten, et al., Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method, J. Opt. Soc. Am. A, 17, 2165, 2000. [37] R.C. McPhedran, et al., Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem, in IMA Volumes in Mathematics and its Applications, G. Papanicolau, ed., vol. 96, pp. 155–186, Springer-Verlag, New York, 1997. [38] L.C. Botten, et al., Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation, J. Opt. Soc. Am. A, 17, 2177, 2000. [39] G.H. Smith, et al., Cylinder gratings in conical incidence with applications to modes of air-cored photonic crystal fibers, Phys. Rev. E, 66, 056604, 2002. [40] G.H. Smith, et al., Cylinder gratings in conical incidence with applications to woodpile structures, Phys. Rev. E, 67, 056620, 2003. [41] T.M. Monro, et al., Holey fibers with random cladding distributions, Opt. Lett., 25, 206, 2000. [42] M.A. Duguay, et al., Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures, Appl. Phys. Lett., 49, 13, 1986. [43] N.M. Litchinitser, et al., Antiresonant reflecting photonic crystal optical waveguides, Opt. Lett., 27, 1592, 2002. [44] T.A. Birks, J.C. Knight, and P. St. J. Russell, Endlessly single-mode photonic crystal fiber, Opt. Lett., 22, 961, 1997.

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121

[45] C.G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Computation, 19, 577, 1965. [46] P.R. McIsaac, Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results, IEEE Trans. Microwave Theory Tech., MTT-23, 421, 1975. [47] L.C. Botten, et al., Semianalytic treatment for propagation in finite photonic crystal waveguides, Opt. Lett., 28, 854, 2003. [48] N.A. Mortensen, Effective area of photonic crystal fibers, Opt. Exp., 10, 341, 2002. [49] B.T. Kuhlmey, R.C. McPhedran, and C.M. de Sterke, Modal cutoff in microstructured optical fibers, Opt. Lett., 27, 1684, 2002. [50] S. Wilcox, et al., Modeling of defect modes in photonic crystals using the fictitious source superposition method, Phys. Rev. E, 71, 056606, 2005. [51] B.T. Kuhlmey, et al., Microstructured optical fibers: wheres the edge?, Opt. Exp., 10, 1285, 2002. [52] J.C. Knight, et al., Anomalous dispersion in photonic crystal fiber, IEEE Photonic Tech. Lett., 12, 807, 2000. [53] A. Ferrando, et al., Nearly zero ultraflattened dispersion in photonic crystal fibers, Opt. Lett., 25, 790, 2000. [54] W.H. Reeves, J.C. Knight, and P.St.J. Russell, Demonstration of ultra-flattened dispersion in photonic crystal fibers, Opt. Exp., 10, 609, 2002. [55] G. Renversez, B. Kuhlmey, and R. McPhedran, Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses, Opt. Lett., 28, 989, 2003. [56] B. Kuhlmey, G. Renversez, and D. Maystre, Chromatic dispersion and losses of microstructured optical fibers, Appl. Optics, 42, 634, 2003. [57] J.A. West, et al., Sufrace modes in air-core photonic band-gap fibers, Opt. Exp., 12, 1485, 2004. [58] A. Einstein, Zur Quantentheorie der Strahlung, Z. Physik, 18, 121, 1917. [59] H. Morawitz, Superradiant level shift and its possible detection in a transient optical experiment, Phys. Rev. A, 7, 1148, 1973. [60] E.M. Purcell, H.C. Torrey, and R.V. Pound, Resonance absorption by nuclear magnetic moments in a solid, Phys. Rev., 69, 681, 1946. [61] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059, 1987. [62] S. John and T. Quang, Localization of superradiance near a photonic band gap, Phys. Rev. Lett., 74, 3419, 1995. [63] T. Quang, et al., Coherent control of spontaneous emission near a photonic band edge: a single-atom optical memory device, Phys. Rev. Lett., 79, 5238, 1997. [64] A.G. Kofman, G. Kurizki, and B. Sherman, Spontaneous and Induced atomic decay in photonic band structures, J. Mod. Opt., 41, 353, 1994. [65] P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, San Diego, 1995. [66] O.J.F. Martin and N.B. Piller, Electromagnetic scattering in polarizable backgrounds, Phys. Rev. E, 58, 3909, 1998. [67] A.A. Asatryan, et al., Two-dimensional Green tensor and local density of states in finite-sized two-dimensional photonic crystals, Waves in Random Media, 13, 9, 2003. [68] D.P. Fussell, et al., Three-dimensional local density of states in a finite two-dimensional photonic crystal composed of cylinders, Phys. Rev. E, 67, 045601(R), 2003; D.P. Fussell, R.C. McPhedran, and C.M. de Sterke, Three-dimensional Green’s ten-

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122

[69] [70]

[71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82]

[83] [84] [85] [86] [87] [88] [89] [90] [91]

Electromagnetic Theory and Applications for Photonic Crystals sor, local density of states, and spontaneous emission in finite two-dimensional photonic crystals composed of cylinders, Phys. Rev. E, 70, 066608, 2004. W.C. Chew, Waves and Fields in Inhomogeneous Media, IEEE Press, New York, 1995. U. Grüning, et al., Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 m, Appl. Phys. Lett., 68, 747, 1996; J. Schilling, et al., A model system for two-dimensional and three-dimensional photonic crystals: macroporous silion, J. Opt. A: Pure Appl. Opt., 3, S121, 2001. M. Wubs and A. Lagendijk, Local optical density of states in finite crystals of plane scatterers, Phys. Rev. E, 65, 046612, 2002. F. Fogli, et al., Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers, Opt. Exp., 10, 54, 2002. J.B. Pendry, Photonic band structures, J. Mod. Opt., 41, 209, 1994. K. Busch, et al., The Wannier function approach to photonic crystal circuits, J. Phys. Condens. Matter, 15, R1233, 2003. L.C. Botten, et al., Photonic band structure calculations using scattering matrices, Phys. Rev. E, 64, 046603, 2001. B. Gralak, S. Enoch, and G. Tayeb, Anomalous refractive properties of photonic crystals, J. Opt. Soc. Am. A, 17, 1012, 2000. S.F. Mingaleev and K. Busch, Scattering matrix approach to large-scale photonic crystal circuits, Opt. Lett., 28, 619, 2003. Z.Y. Li and K.M. Ho, Light propagation in semi-infinite photonic crystals and related waveguide structures, Phys. Rev. B, 68, 155101, 2003. L.C. Botten, et al., Semianalytic treatment for propagation in finite photonic crystal waveguides, Opt. Lett., 28, 854, 2003. L.C. Botten, et al., Photonic crystal devices modelled as grating stacks: matrix generalizations of thin film optics, Opt. Exp., 12, 1592, 2004. L.C. Botten, et al., Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory, Phys. Rev. E, 70, 056606, 2004. E.G. McRae, Electron diffraction at crystal surfaces: I. Generalization of Darwin’s dynamical theory, Surface Science, 11, 479, 1968; Electron diffraction at crystal surfaces: II. The double-diffraction picture, Surface Science, 11, 492, 1968. S.Y. Lin, et al., A three-dimensional photonic crystal operating at infrared wavelengths, Nature (London), 394, 251, 1998. T.P. White, et al., Ultracompact resonant filters in photonic crystals, Opt. Lett., 28, 2452, 2003. T.P. White, et al., Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications, Phys. Rev. E, 70, 056607, 2004. T.P. White, et al., Recirculation-enhanced switching in photonic crystal MachZehnder interferometers, Opt. Exp., 12, 3035, 2004. M. Hamermesh, Group Theory and Its Application to Physical Problems, Dover, New York, 1989. W.H. Press, et al., Numerical Recipes in Fortran, 2nd ed., Cambridge University Press, London; New York, 1992. C.M. de Sterke, et al., Modes of coupled photonic crystal waveguides, Opt. Lett., 29, 1384, 2004. S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media/Cambridge University Press, London; New York, 2000. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, vol. 22, Topics in Current Physics, pp. 1–52, R. Petit, ed., Springer-Verlag, Berlin, 1980.

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of Photonic 3 Modeling Crystals by Multilayered Periodic Arrays of Circular Cylinders Kiyotoshi Yasumoto and Hongting Jia

CONTENTS 3.1 Introduction ............................................................................................124 3.2 Scattering by a Single Cylinder ..............................................................126 3.2.1 Two-Dimensional Scattering ......................................................126 3.2.2 Two-Dimensional T-Matrix of a Circular Cylinder ....................128 3.2.3 Three-Dimensional Scattering ....................................................129 3.2.4 Three-Dimensional T-Matrix of a Circular Cylinder ................130 3.3 Scattering by a Periodic Array of Cylindrical Objects ..........................131 3.3.1 Two-Dimensional Scattering ......................................................131 3.3.2 Lattice Sums and T-Matrix ........................................................136 3.3.3 Three-Dimensional Scattering ....................................................139 3.4 Two-Dimensional Scattering from Layered Periodic Arrays ..................142 3.4.1 Reflection and Transmission Matrices ......................................142 3.4.2 Generalized Reflection and Transmission Matrices ..................145 3.4.3 Floquet-Mode Approach for Layered Identical Arrays ..............145 3.4.4 Layered Arrays Embedded in a Slab ..........................................149 3.5 Three-Dimensional Scattering from Layered Crossed-Arrays ................151 3.5.1 Scattering from a Parallel Array of Circular Cylinders ............152 3.5.2 Unit Cell of a Crossed-Array of Circular Cylinders ..................157 3.5.3 Layered Crossed-Arrays of Circular Cylinders ..........................160 3.6 Modal Analysis of Two-Dimensional Photonic Crystal Waveguides ....161 3.6.1 Dispersion Equation ....................................................................161 3.6.2 Mode Field Analysis ..................................................................164 3.7 Numerical Examples ..............................................................................167 3.7.1 Two-Dimensional Scattering from Layered Parallel Arrays ......167 3.7.2 Three-Dimensional Scattering from Layered Crossed-Arrays ......174 123

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3.7.3

Guided Modes of Two-Dimensional Photonic Crystal Waveguides ....................................................................181 3.8 Conclusions ............................................................................................187 References ......................................................................................................187

3.1 INTRODUCTION Photonic crystals are periodic dielectric or metallic structures in which any electromagnetic wave propagation is forbidden within a fairly large frequency range. This frequency range is termed the photonic bandgap, which is analogous to the electron bandgap in natural crystals. Since the pioneering work by Yablonovitch [1] and John [2], photonic crystals have received growing attention because of promising applications to the bandgap-based devices [3,4] in the optical to microwave wave region, including narrow-band filters, laser mirrors, high-quality resonant cavities, and optical waveguides and substrates for microwave antennas. A periodic array of infinitely long parallel cylindrical objects is typical of discrete periodic structures. The frequency response of the array is characterized [5–7] by the scattering properties of individual cylinders and the multiple scattering due to the periodic arrangement of scatterers. When the array is layered, it constitutes a two-dimensional photonic crystal. In the layered system, the multiple interaction of space-harmonics scattered from each of the array layers suppresses the propagation of electromagnetic waves within a particular frequency range, and a photonic bandgap is formed. The electromagnetic scattering and guidance by two-dimensional photonic crystals consisting of layered periodic arrays of parallel circular cylinders has been extensively investigated during the past decade. The frequency response in reflectance, the frequency range of bandgaps, and the mode propagation have been reported using the cylindrical-harmonic expansion method [8–10], the Fourier modal method [11,12], the finite element method [13], the differential method [14], the time domain techniques [15,16], and the finite-difference frequencydomain method [17]. Among these approaches, the cylindrical harmonic expansion method [8–10] is rigorously analytical, especially when the lattice elements are infinitely long circular cylinders. But it requires an intricate formulation in order to deal with the periodic Green’s function. When the layered arrays are embedded in a dielectric slab, the frequency response is significantly modified due to the presence of a multiple scattering of fields between the array plane and the slab boundaries. The bandgap nature of the embedded arrays may be controllable by varying the slab thickness and material combination. Although a photonic crystal consisting of layered arrays of parallel cylinders exhibits various interesting frequency responses, it is essentially a twodimensional structure. When the array is illuminated by electromagnetic waves propagating obliquely to the cylinder axis, it is rather difficult to fully control the frequency response by the structural parameters alone. This deficiency of the two-dimensional structure may be resolved by introducing a third dimension through the layering process. A fully three-dimensional structure is realized when the axes

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of parallel cylinders are rotated by a constant angle in each successive layer [18,19]. Such a three-dimensional structure requires much complicated analysis even for numerical approaches. In this chapter, we reexamine the cylindrical harmonic expansion method and bring it to a simple and tractable formulation for modeling the electromagnetic scattering and guidance by photonic crystals consisting of layered parallel or crossed arrays of cylindrical objects standing in free space or embedded in a dielectric slab. In the method, the reflection and transmission matrices based on the space harmonic fields are defined for a periodic array in isolation. The matrices are expressed in terms of the one-dimensional lattice sums [20,21] and the transition matrix (T-matrix) [22], [23–25] of cylinders located within a unit cell. The lattice sums uniquely characterize a periodic arrangement of scatterers and are independent of the polarization of the incident field and the individual configuration of scatterers. The details of scattering from each array element are included in the T-matrix, which is obtained in closed form for circular cylinders. The array elements per unit cell can contain two or more cylinders, which may be dielectric, conductor, gyrotropic medium, or mixtures thereof with different dimensions. When the arrays are layered, the reflection and transmission matrices are concatenated [26,27] to obtain the generalized reflection and transmission matrices for the entire layered system. This yields a recurrence formula for the generalized reflection and transmission matrices. The dimensions and configurations of cylindrical elements in different layers may be different so long as the array periods are identical over all layers. For a N-layered arrays, the recursion process requires N-1 times computations of inversion of matrices. When the layered system consists of identical arrays separated by an equal distance, a concept of Floquet modes propagating in the layered direction [28,29] is incorporated into the concatenation to calculate the generalized reflection and transmission matrices without using the recursion process. Most of the photonic crystals consisting of layered periodic arrays of cylinders can be treated by combining the recursion process with the concept of Floquet modes. The generalized reflection and transmission matrices are used to calculate the frequency responses in the reflectance and transmittance of two-dimensional or three-dimensional photonic crystals and to characterize the modal properties of guided waves supported by two-dimensional photonic crystal waveguides. Particular attention is turned to the calculation of field distributions inside the photonic crystals. The field distributions inside the homogeneous strip regions of the background medium can be calculated by superposing the space-harmonic fields. However this space-harmonic expansion does not converge in the inhomogeneous grating regions periodically occupied by the array elements. Then the fields in the grating regions are calculated in terms of a cylindrical-wave expansion of the scattered field with the local origin at the center of the individual cylinders. To confirm the applicability and accuracy of the proposed method, various numerical examples are demonstrated for the frequency response in reflectance of free-standing or embedded photonic crystals as well as for the mode dispersion and field distributions in two-dimensional photonic crystal waveguides.

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This chapter is organized as follows. In Section 3.2, the two-dimensional or three-dimensional scattering of a plane wave by an infinitely long circular cylinder isolated in free space is formulated using the T-matrix. The results are used in Section 3.3 to analyze the reflection and transmission by a single layer of periodic array of circular cylinders. The closed form expressions for the reflection and transmission coefficients are given in terms of the lattice sums and T-matrix. In Section 3.4, the derivation of the generalized reflection and transmission matrices for the multilayered parallel arrays is described in detail. The extension of the formulation to the three-dimensional scattering by multilayered crossed-arrays is discussed in Section 3.5. The modal analysis of two-dimensional photonic crystal waveguides using the T-matrix and lattice sums is presented in Section 3.6. Numerical examples are demonstrated and discussed in Section 3.7.

3.2 SCATTERING BY A SINGLE CYLINDER In this section, the scattering of an electromagnetic plane wave from a cylindrical object is formulated using the cylindrical wave expansion and the transition matrix (T-matrix). The closed form expressions for the two-dimensional T-matrix and three-dimensional T-matrix of a circular cylinder are discussed.

3.2.1 TWO-DIMENSIONAL SCATTERING A cylindrical object is situated in a background medium with a permittivity e0 and a permeability 0 as shown in Figure 3.1. The cylinder is uniform and infinitely long in the z direction. We consider the scattering of an electromagnetic plane wave whose direction of incidence is normal to the cylinder axis and forms an angle i with respect to the x axis. The scattering problem is reduced to a twodimensional one. The Ez field in TM-wave problem and the Hz field in TE-wave problem is denoted by (x, y). Let i(x, y) be the incident field with unit amplitude. Then i(x, y) is given in the x-y coordinate system as follows: i ( x , y)  ei ( kx , 0 xk y , 0 y )

(3.1)

k x ,0 k0 cos  i ,

(3.2)

k y ,0 k0 sin  i

where k0   e 00 is the wavenumber in the background medium and  is the angular frequency. Using a complete orthogonal set of cylindrical wave functions, Equation (3.1) is expressed in the cylindrical coordinate (, ) system as follows: i ( , r)  eik0r cos(f−f )  i



∑ (i)m eimf Jm (k0 r)eimf

m

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i

(3.3)

Modeling of Photonic Crystals

127 y


i x

z

FIGURE 3.1 Two-dimensional scattering by a cylindrical object uniform in the axial direction.

where Jm is the Bessel function of the m-th order. Introducing a vector notation, Equation (3.3) is rewritten as i (r, f)  JT  p0

(3.4)

with J  [ Jm (k0 r)eimf ]

(m  0, 1, 2,…)

p0  [(i)m eimf ]

(3.5)

i

(3.6)

where x  [xm] defines a column vector whose elements are xm and the superscript T denotes the transpose of the indicated vector. The scattered field exterior to the cylinder is expressed using another orthogonal set of cylindrical wave functions representing an outgoing wave as follows: s (r, f)  HT  a0s

(3.7)

H  [ H m(1) (k0 r)eimf ]

(3.8)

with

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(1) where H m is the m-th order Hankel function of the first kind and a0s is a column vector whose components represent unknown scattering amplitudes. Solving the boundary value problem when the configuration of the cylinder is specified, a characteristic matrix T that relates a0s to p0 is obtained as follows:

a0s  T  p0

(3.9)

The matrix T is usually referred to as the transition matrix (T-matrix) [22], which characterizes the scattering properties of the cylindrical object concerned.

3.2.2 TWO-DIMENSIONAL T-MATRIX

OF A

CIRCULAR CYLINDER

The T-matrix defined by Equation (3.9) is obtained in close form especially when the cylindrical object has a circular cross section. For a circular dielectric cylinder with radius r, permittivity e, and permeability , we have T  [tm dmn ]

(3.10)

with    z 0 Jm (k0 r ) Jm′ (kr )   Jm (kr ) Jm′ (k0 r )  z 0 Jm′ (kr ) H m(1) (k0 r )  zJm (kr ) H m′(1) (k0 r ) tm    zJm (k0 r ) Jm′ (kr )  z 0 Jm (kr ) Jm′ (k0 r )   zJm′ (kr ) H m(1) (k0 r )  z 0 Jm (kr ) H m′(1) (k0 r )  0 

m0 , e0



m e

(TM wave) (3.11) (TE wave)

(3.12)

where J m and H m

(1) denote the derivatives with respect to the indicated arguments and mn is the Kronecker’s delta. If the circular cylinder is of perfect conductor, Equation (3.11) is reduced to  Jm (k0 r )  (1)  H m (k0 r ) tm     Jm′ (k0 r )  H ′(1) (k r ) 0 m 

(TM wave) (3.13) (TE wave)

The closed-form expressions of the T-matrix are also obtained [27,30] for a dielectric or gyrotropic circular cylinder located in a background of gyrotropic or dielectric medium.

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3.2.3 THREE-DIMENSIONAL SCATTERING A dielectric cylinder with permittivity e and permeability  is situated in free space as shown in Figure 3.2. The cylinder is uniform and infinitely long in the z direction. A plane wave with a unit amplitude and a wavevector ki is indcident on the cylinder where ki  (k0, i, i) in the spherical coordinate system with respect to the origin O. The z components of the incident fields are expressed in the rectangular coordinate system as follows: Ezi  ez ,0 ei (kx ,0 xk y ,0 ykz ,0 z ) ,

H zi  hz ,0 ei (kx ,0 xk y ,0 ykz ,0 z )

(3.14)

with hz ,0  sin ci cos fi

ez ,0  cos ci cos fi , k x ,0 k0 cos fi sin ui ,

k y ,0 k0 sin fi ,

(3.15)

kz ,0  k0 cos fi cos ui

(3.16)

~

0 Hzi denotes the normalized magnetic field and  i is the polarwhere Hzi   0 /e ization angle of the incident electric field. The incident fields (3.14) may be rewritten in the cylindrical coordinate system (p, , z) as follows: Ezi  eikz ,0 z JT  pz ,0 ez ,0 ,

E

i

i

H zi  eikz ,0 z JT  pz ,0 hz , 0

(3.17)

p^ y

s^

x


i

O i

z

FIGURE 3.2 Three-dimensional scattering by a cylindrical object uniform in the axial direction.

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where J  [Jm (g0 r)eimf ],

g0  k02  kz2,0

pz ,0  [(i)m eimax ,0 ],

cos a x ,0 

k x ,0 g0

(3.18)

,

Im{sin a x ,0 } 0

(3.19)

The z components of scattered fields outside the cylinder are expressed in terms of cylindrical wave functions as follows: Ezs  eikz ,0 z HT  a0s ,

H zs  eikz ,0 z HT  b0s

(3.20)

with H  [H m(1) (g0 r0 )eimf0 ]

(3.21)

where a0s and b0s are column vectors whose components represent unknown scattering amplitudes in the cylindrical coordinate system. Solving the boundary value problem when the configuration of the cylinder is specified, a characteristic matrix Tz that relates (a0s, b0s) to the incident field is expressed as follows:  as   0T  s z  b0 

 e p   Tee  z ,0 z ,0    z     hz ,0 pz ,0   Tzhe

Tzeh   ez ,0 pz ,0    Tzhh   hz ,0 pz ,0 

(3.22)

The matrix Tz is a three-dimensional T-matrix of the cylindrical object for the excitation of a plane wave with an arbitrary angle of incidence. The submatrices Tzee and Tzhh are the T-matrices into the copolarized scattered fields, whereas the submatrices Tzeh and Tzhe describe the cross-polarization effects in the scattering process. Note that the subscript z on the T-matrix is used to specify the direction of the cylinder in a three-dimensional configuration.

3.2.4 THREE-DIMENSIONAL T-MATRIX

OF A

CIRCULAR CYLINDER

For a circular dielectric cylinder with radius r, permittivity e, and permeability  located in free space, the four submatrices of Equation (3.22) that define the three-dimensional T-matrix are obtained in closed form as [31]  J (w)  K 2  2,m 3,m m  Tzee   m(1) d mn  2 H ( w )  K  m  1,m 2,m m

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(3.23)

Modeling of Photonic Crystals

Tzeh Tzhe 

131

2k0  Km dmn  2 (1) 2 pw k  {H m (w)} 1,m 2,m  K m2 

   

 J (w)  K 2  m 4,m 1,m Tzhh   m(1) dmn  2  H m (w) 1,m 2,m  K m 

(3.24)

(3.25)

with 1,m 

Jm′ (u) e H m′(1) (w)  0 , uJm (u) e wH m(1) (w)

3,m 

Jm′ (u) e J ′ (w)  0 m , uJm (u) e wJm (w)

Km 

2,m 

4,m 

Jm′ (u) m H m′(1) (w)  0 uJm (u) m wH m(1) (w)

m J ′ (w) Jm′ (u)  0 m m wJm (w) uJm (u)

mkz ,0  1 1   2  2  k u w 

w  rg 0 ,

u  r k 2  kz2, 0 ,

(3.26)

(3.27)

(3.28)

g0  k02  kz2, 0

(3.29)

When the direction of incidence of plane wave is normal ( i  90°) to the cylinder axis, we have kz,0  0, w  k0r, u  kr, Km  0, and Tzeh  The z  0. For this case hh Tee , and T are reduced to the two-dimensional T-matrices given by Equation z z (3.11) for the separated TM and TE waves.

3.3 SCATTERING BY A PERIODIC ARRAY OF CYLINDRICAL OBJECTS Let us consider an array of cylindrical objects characterized by a T-matrix T when isolated in free space. The cylinders are infinitely long in the z direction, parallel to each other, and periodically spaced with a distance h along the x axis on the plane y  0 as shown in Figure 3.3. We shall formulate first the scattering problem in a cylindrical coordinate system and then transform the result into an x–y coordinate system. This yields the reflected and transmitted field relative to the array plane, being expressed in terms of the space-harmonic waves for a periodic system.

3.3.1 TWO-DIMENSIONAL SCATTERING The scattering problem is reduced to two dimensions when i  90° in Figure 3.3. The cross-sectional view of the periodic array for the two-dimensional scattering

© 2006 by Taylor & Francis Group, LLC

132

Electromagnetic Theory and Applications for Photonic Crystals i Ei 

^

p y

s^

x


i

h

z i

FIGURE 3.3 A periodic array of parallel cylinders illuminated by a plane wave of general incidence.

y

i x h

h

FIGURE 3.4 Cross section of a periodic array of parallel cylinders illuminated by a plane wave of incidence normal to the cylinder axis.

is shown in Figure 3.4. The incident plane wave is given by Equation (3.1). The scattered field uniform in the z direction is formulated in the same way for the TM wave and TE wave by taking the wave function s(x, y) to be the Ez field and Hz field, respectively. The scattered field is expressed as follows: s (x , y) 



∑

l

HT ,l  als

(3.30)

with H ,l   H m(1) (k0 r l )eimf l    rl  (x  lh)2  y 2 ,

© 2006 by Taylor & Francis Group, LLC

(3.31) cos f l 

x  lh rl

(3.32)

Modeling of Photonic Crystals

133

where (l, l) is the local polar coordinate system whose origin is located at the center of the l-th cylinder and a sl denotes the amplitude vector of the scattered field from the l-th cylinder. Note that (0, 0) corresponds to the reference coordinate (, ) system used in Section 3.2.1. The scattered field satisfies the Floquet theorem: s ( x  h, y)  eikx ,0h s ( x ,y)

(3.33)

where kx,0  k0 cos i. Using Equation (3.33) in Equation (3.30), s(x, y) is rewritten as  s ( x , y) 



∑

l

eikx , 0lh HT,l  a0s

(3.34)

Thus, the problem of two-dimensional scattering from a periodic array of cylindrical objects is reduced to a problem to determine the scattering amplitudes a0s. We will discuss in detail the procedure to calculate a0s. From Equations (3.4) and (3.34), the total fields exterior to the arrayed cylinders are expressed as follows: ( x , y)  JT  p0 



∑

l

eikx , 0lh HT,l  a0s

(3.35)

Using the addition theorem [32] for the Hankel function Hm(1)(k0 l), which can be applied when the observation point is located in the region 0   x2   y2  h, we have a relation HT,l  JT  l (l ≠ 0),

HT, 0  HT

(3.36)

where  1 l  [l, nm ]  [ H n(1) m (k0 |l|h)]  (1)nm 

(l  0) (l  0)

(3.37)

With the substitution of Equation (3.36), Equation (3.35) is rewritten as follows: ( x , y)  JT  ( p0  L  a0s )  HT  a0s

(3.38)

where L  [ Lnm ],

Lnm  Snm (k0 h, k x ,0 h)

(3.39)



Snm (k0 h, k x ,0 h)  ∑ H n(1) m (lk0 h)  eilkx , 0h  (1)nm eilkx , 0h  l1

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(3.40)

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Electromagnetic Theory and Applications for Photonic Crystals

Referring to Section 3.2.1, the first term in Equation (3.38) may be viewed as the incident field impinging on the zeroth cylinder, whereas the second term is the scattered field from the same cylinder. This argument leads to a relationship between a0s and p0: a0s  T  ( p0  L  a0s )

(3.41)

because the cylinder is assumed to have a T-matrix T. Consequently, we have [33] a0s  T  p0

(3.42)

T  ( I  T  L)1  T

(3.43)

with

= where T may be regarded as an aggregate T-matrix for the periodic array system. Using a0s obtained above in Equation (3.34), the scattered field in the exterior of the periodic array can be determined. However, this expression based on the local cylindrical harmonic waves is not convenient for analyzing the reflected and transmitted fields relative to the array plane. Then we transform Equation (3.34) into an expression based on the space-harmonic fields for plane waves. This transformation is performed using the following formula: H n(1) (k0 r)einf

 (1)n

H n(1) (k0 r)einf 

H 0(1) (k0 r)  

∑

1 p

n ∂ ∂  (1)    ∂x + i ∂y  H 0 (k0  )

|n| 1  ∂ ∂  (1)   i   H 0 (k0 r) |n|  ∂x ∂ y   k0 

(n 0)

(n  0)

1

∫ k(j) ei[jxk(j)|y|]dj

eikx , 0lh eijlh 

l

1 k0n

(3.44)

(3.45)

(3.46)

2p  ∑ d(kx ,0  j  2lp/h) h l

(3.47)

with r  x 2  y2 ,

k(j)  k02  j 2 ,

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x r

(3.48)

Im{k(j)} 0

(3.49)

cos f 

Modeling of Photonic Crystals

135

After several manipulations, we have a relationship between the cylindrical harmonic fields and the space-harmonic fields as follows [33,34]:



∑ eik

l

x ,0lh

H m(1) (k0 rl )eimfl

 2(i)m   hk0m =   m  2(i)  hk m 0 

∞

(k x ,l  ik y ,l )m

l

k y, l



(k x ,l  ik y ,l )m

l

k y ,l

∑ ∑

e l ( x, y) ( y  0) (3.50) e l ( x,

y) ( y  0 )

with i ( k x ,l x k y ,l y ) e l ( x , y)  e

k x ,l  k x ,0 

(3.51)

2lp 2lp k0 cos f i + , h h

k y ,l  k02  k x2,l

(3.52)

where kx,l and ky,l denote the propagation constants of the l-th space harmonic wave in the x and y directions respectively. Substituting Equation (3.50) into Equation (3.34), the scattered field s(x, y) in the upper and lower regions separated by the array plane y  0 is expressed as a superposition of the space-harmonic waves. This yields a representation for the reflected and transmitted fields when the down-going plane wave impinges on the periodic array shown in Figure 3.4. Taking into account the orthogonality of the space-harmonic waves, the reflected and transmitted fields can be treated separately for each of the space-harmonic components. The reflected field lr(x, y) in the l-th space harmonic in the upper region y  0 and the corresponding transmitted field lt(x, y) in the lower region y  0 are given respectively as follows [33,35]: clr ( x , y)  rl0 e l ( x , y),

clt ( x , y)  fl0 e l ( x , y)

(3.53)

with T rl0  u l  T  p0,

T fl0  dl 0  u l  T  p0

 2(−i)m ima   l  u e l    k0 h sin al  cos al 

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kx ,l k0

,

Im{sin al } 0

(3.54)

(3.55)

(3.56)

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Electromagnetic Theory and Applications for Photonic Crystals

where p0 is given by Equation (3.6). Note that the incident field i(x, y) must be included in the transmitted field component of the zeroth order harmonic. This  corresponds to the term of l0 in the right-hand side of Equation (3.54). r l0 and f l0 give the reflection and transmission coefficients that relate the reflected and transmitted l-th space harmonics to the incident plane wave corresponding to the downgoing zeroth space harmonic.

3.3.2 LATTICE SUMS AND T-MATRIX The function Snm(k0h, kx,0h) defined by Equation (3.40) is usually referred to as the lattice sum of the (n  m)-th order [20,21]. The lattice sum and associated matrix L given by Equation (3.39) characterize uniquely the periodic arrangement of cylindrical scatterers and are independent of the polarization of the incident field and the individual configuration of scatterers. The matrix L calculated once can be commonly used for both TM and TE waves and for any arrays of cylindrical objects whenever the periods are same. The details of scattering from each array element within a unit cell are included in the T-matrix T defined by Equation (3.9). This is a main advantage of using the lattice sums in the scattering problems for discrete periodic structures. The efficient calculation of lattice sums is crucial in the present formulation. Since the direct sum of Hankel functions in Equation (3.40) converges very slowly, various techniques [36–40] to accelerate the convergence have been developed. We calculate here the lattice sums using an integral form [21] for a semi-infinite sum of the Hankel functions: 

∑ Hn(1) (lk0 h)eilk

x ,0h

l1



j0 (1)n i ( p /4 kx , 0h ) e  ∫ F (t; k0 h, k x , 0 h) 0 p   Gn (t)  Gn (t)  dj

(3.57)

with F (t; k0 h, k x ,0 h) 

eik0h

1t2

 1  t2 1  eikx , 0h eik0h

Gn (t)  ( t  i 1  t2 )

1t2

 

(3.58)

n

(3.59)

where  (1  i) / 2 and 0 is a positive real number chosen so that the integration satisfies a required convergence. The integration in Equation (3.57) is calculated using a simple trapezoidal formula of numerical integration for elementary functions. The accuracy for Equation (3.57) and its numerical integration was confirmed [21] by a substantial number of numerical tests. The T-matrix of cylindrical objects located in a unit cell plays another important role in the present formulation. Any analytical or numerical techniques can be

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Modeling of Photonic Crystals

137

(a)

(b)

(c)

(d)

FIGURE 3.5 Examples of composite cylindrical elements with multiple circular cylinders per unit cell.

employed to calculate the T-matrix. When a unit cell contains one circular cylinder in particular, the T-matrix is obtained in closed form as discussed in Sections 3.2.2 and 3.2.4. If two or more circular cylinders are contained within a unit cell, a recursive algorithm [23] may be used to obtain the aggregate T-matrix for the composite cylindrical system in terms of the reduced T-matrix for each cylinder in isolation. However, the recursive algorithm assumes that the cylinders should be arranged with the ordered distances from the origin. This introduces some restriction on the relative locations of the cylindrical elements. Several examples in which the distances of the cylinders cannot be ordered are shown in Figure 3.5. For such configurations, a numerical technique is employed to obtain the individual T-matrix of each cylinder in the composite cylindrical system. Let us consider N cylinders whose centers are located at Oi(xi, yi) (i  1, 2, … , N) in a unit cell, respectively. The self-coordinate with the origin at Oi is attached to the i-th cylinder. The field quantities associated with the i-th self-coordinate are assigned the subscript i. The N-cylinders may be dielectric, conductor, or a mixture thereof with different dimensions. Following the procedure discussed in Section 3.3.1, the scattered and total fields exterior to the N cylinders may be expressed as follows: N

s ( x ,y)  ∑ (HT ,i  JT,i  L)  ais i1

(3.60)

N

( x ,y)  JT,0  p0  ∑ (HT ,i  JT,i  L)  ais i1

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(3.61)

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Electromagnetic Theory and Applications for Photonic Crystals

T where J,0  p0 indicates the incident field expressed in terms of cylindrical waves with respect to the reference coordinate attached to the zeroth cell, H,i and J,i are the basis vectors representing cylindrical waves in the i-th self-coordinate, and a si is the unknown amplitude vector of scattered field from the i-th cylinder. When the observation point is chosen near the i-th cylinder, Equation (3.61) can be rewritten in the form N   ( x , y)  JT,i   Ai 0  p0  L  ais  ∑ Jij  asj   HT ,i  ais   j ≠i

(3.62)

J ij  B ij  Aij  L

(3.63)

with

where Aij and Bij are the translation matrices [23] for the singular and regular parts of cylindrical harmonics between the i-th and j-th self-coordinate systems. Denoting the T-matrix of the i-th cylinder in isolation by Ti, from Equation (3.62) we have the following relationship: N

ais  Ti  ( L  ais  ∑ J ij  asj  Ai 0  p0 )

(i  1, 2, … , N)

j ≠i

(3.64)

The solution to Equation (3.64) is obtained in a matrix form as follows [25]: ais  Ti (N )  p0

(3.65)

with N

Ti (N )  ∑ Vij Tj A j 0

(3.66)

j1

V  11 V12   V21 V22        VN 1 VN 2

… V1N   I  T1 L T1J12   … V2N   T2 J 21 I  T2 L          … VNN  TN J N 1 TN J N 2

1 … T1J1N   … T2 J 2 N        … I  TN L 

(3.67)

where Ti(N) represents the T-matrix of the i-th cylinder in the presence of N cylinders per unit cell. Substituting Equation (3.65) into Equation (3.60), we have the scattered field as a sum of cylindrical waves expressed in the N-self-coordinate system. The results are easily transformed into the expressions based on the spaceharmonic fields using the procedure described in Section 3.3.1.

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Modeling of Photonic Crystals

139

Let us consider another example of a composite structure of N circular cylinders. The unit cell contains one circular cylinder in which N  1 cylindrical inclusions are eccentrically located as shown in Figure 3.5(d). Provided that each inclusion is a circular cylinder, the T-matrix of the host cylinder viewed from its exterior is obtained as follows [24]: [z H ′  z G ′ G1 H ]1[z J ′  z G ′ G1 J ] (TM wave) 0 N N 0 0 N N 0 N 0 N 0   (3.68) T N   1  1 ′ ′ ′ (TE wave) −[z 0 H0  z N GN GN H0 ] [z 0 J0  z N GN′ G1 N J0 ] with GN  J N  H N TN 1 ,

GN′  J N′  HN′ TN 1

(3.69)

N 1

T N 1  ∑ B 0 ,i T i ( N 1) B i ,0

(3.70)

i1

Ji  [Jm (ki rN )dmm′ ],

Ji′  [Jm′ (ki rN )dmm′ ]

Hi′  [H m(1)′ (ki rN )dmm′ ]

Hi  [H m(1) (ki rN )dmm′ ],

ki  v ei mi ,

zi 

(i  0, N )

mi ei

(3.71) (3.72)

(3.73)

where the subscript N and 0 are used to indicate the quantities associated with the host cylinder and the background medium, and Ti(N1) represents the T-matrix of the i-th inclusion in the presence of the N  1 inclusions inside the host medium of infinite extent. Ti(N1) may be calculated by Equation (3.66) or a recursive algorithm [23].

3.3.3 THREE-DIMENSIONAL SCATTERING Consider a periodic array of cylinders illuminated by a plane wave obliquely impinging on the cylinder axis as shown in Figure 3.3. The incident wave is given by Equation (3.14). Using the Floquet’s theorem, the scattered fields may be expressed in the form Ezs  eikz ,0 z H zs  eikz ,0 z

© 2006 by Taylor & Francis Group, LLC



∑

l 

∑

l

eikx ,0lh HT, l  a0s

(3.74)

eikx ,0lh HT, l  b0s

(3.75)

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Electromagnetic Theory and Applications for Photonic Crystals

with H, l   H m(1) (g0 rl )eimfl 

(3.76)

where 0 is defined by Equation (3.18), and a0s and b0s are unknown amplitude vectors representing the scattered fields from the zeroth cylinder. Following the same procedure as discussed in Section 3.3.1, the total fields outside the zeroth cylinder can be written as Ez  eikz , 0 z  JT  ( pz ,0 ez ,0  Lx  a0s )  HT  a0s   

(3.77)

H z  eikz , 0 z  JT  ( pz ,0 hz ,0  Lx  b0s )  HT  b0s   

(3.78)

with Lx  [Lx , nm ],

Lx , nm  Snm (g0 h, k x ,0 h)

(3.79)



 ilkx ,0h  (1)nm eilkx ,0h  Snm (g0 h, k x ,0 h)  ∑ H n(1)  m (g 0 h)  e l1

(3.80)

where pz,0 is defined by Equation (3.19). The matrix Lx is a square matrix whose elements are given by Lx,nm  Snm(0h, kx,0h) where Snm represents the (n  m)-th order lattice sum [20,21]. The first term in the right-hand side of Equations (3.77) and (3.78) may be viewed as an incident field impinging on the zeroth cylinder, whereas the second term is the scattered field from the same cylinder in isolation. Using the relations of Equations (3.22) and (3.24), the scattering amplitudes a0s and b0s satisfy the relation  as   0 =  s  b0 

 Tee  z  eh Tz

Tzeh   ez ,0 pz ,0  Lx  a0s    Tzhh   hz ,0 pz ,0  Lx  b0s   

(3.81)

Solving Equation (3.81), we have a relationship between (a0s, b0s ) and the incident field as follows [31]:   ee e  as   0    z ,0 p z ,0    Tz    s  Tz    hz ,0 p z ,0   Tzhe  b0 

© 2006 by Taylor & Francis Group, LLC

Tzeh   ez ,0 p z ,0    Tzhh   hz ,00 p z,0   

(3.82)

Modeling of Photonic Crystals

141

with Tzee  (I  Sz1 Lx Sz 2 Lx )1 (Sz 3  Sz1 Lx Sz 2 )

(3.83)

Tzeh  (I  Sz1 Lx Sz 2 Lx )1 Sz1 (I  Lx Sz 4 )

(3.84)

Tzhe (I  Sz 2 Lx Sz1 Lx )1 Sz 2 (I  Lx Sz 3 )

(3.85)

Tzhh  (I  Sz 2 Lx Sz1 Lx )1 (Sz 4  Sz 2 Lx Sz1 )

(3.86)

Sz1  (I  Tzee Lx )1Tzeh , Sz 2  (I  Tzhh Lx )1Tzeh

(3.87)

Sz 3  (I  Tzee Lx )1Tzee , Sz 4  (I  Tzhh Lx )1Tzhh

(3.88)

In Equation (3.82), Tzee to Tzhh define the T-matrices for the three-dimensional scattering from the periodic array of circular cylinders. They are expressed in terms of the matrix Lx characterized by the lattice sums along the x direction and the reduced T-matrices Tzee, Tzeh, and Tzhh of the isolated single cylinder. To obtain a plane-wave representation of the scattered field in the rectangular coordinate system, we use the Fourier integral formula for Hankel functions. After several manipulations using Equations (3.74), (3.75), and (3.82), the amplitudes ~r ~ r ez,l and hz,l of the l-th space harmonics for the reflected fields Ezr and Hrz in the upper ~t t region y  0 and the corresponding amplitudes ez,l and h z,l for the transmitted ~ fields Ezt and H zt in the lower region y  0 are given as T ee eh  ezr ,l  u l  (Tz ez , 0  Tz hz , 0 )  pz , 0

(3.89)

r he hh  T h z ,l  ul  (Tz ez , 0  Tz hz , 0 )  pz , 0

(3.90)

T ee eh  ezt ,l  e z , 0 dl 0  u l  (Tz ez , 0  Tz hz , 0 )  pz,0

(3.91)

T he hh  hzt ,l  hz ,0 dl 0  u l  (Tz ez ,0  Tz hz ,0 )  pz ,0

(3.92)

 2(i)m  ima x ,l   u  e l  g h sin a   0  x ,l

(3.93)

where

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(l  0, 1, …)

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Electromagnetic Theory and Applications for Photonic Crystals

cos a x ,l 

k x ,l g0

,

Im{sin a x ,l } 0

k x ,l k0 cos fi sin ui 

2 lp , h

g0  k02  kz2,0

(3.94)

(3.95)

~ Note that the incident fields Ezi and H zi must be included in the transmitted field components. This corresponds to the term of l 0 in the right-hand side of Equations (3.91) and (3.92).

3.4 TWO-DIMENSIONAL SCATTERING FROM LAYERED PERIODIC ARRAYS The reflection and transmission by a single layer of periodic array standing in free space is described by the reflection and transmission coefficients given by (Equation (3.54)), which relates the reflected and transmitted l-th space harmonics to the zeroth space harmonic denoting the incident plane wave. In a layered array system, the incident wave on each of the arrays consists of a set of space harmonics, and one particular space harmonic wave is reflected and transmitted into another set of space harmonics. This means that the scattering from each of the arrays in the layered system is described using the reflection and transmission matrices whose elements relate the reflected and transmitted m-th space harmonics to the incident l-th space harmonic. In this section, first we derive the reflection and transmission matrices by  extending the expressions rl0 and f l0 in Equation (3.54). Then we present two schemes of concatenating these matrices to obtain the generalized reflection and transmission matrices describing the scattering from an N-layered periodic array. One scheme uses the recursive relations applied to the scattering matrix in a layered system and the other uses the Floquet mode approach applied to a periodically layered identical arrays system. Finally, the treatment of layered arrays embedded in a dielectric slab is discussed.

3.4.1 REFLECTION AND TRANSMISSION MATRICES An N-layered periodic array of cylindrical objects is situated in a background medium of free space as shown in Figure 3.6. The j-th array is located on the plane y  yj ( j  1, 2, 3, … , N), and the array elements are displace by wj in the x direction. The periodic spacing h of cylinders along the x axis should be same over all layers, but the separation distance dj  yj1  yj between two adjacent arrays may be arbitrary. The cylinders in different layers need not be identical in material properties, dimensions, or shape. Various photonic crystals with a rectangular lattice or a triangular lattice can be formed by properly choosing the array parameters h, wj, and dj in the layered system.

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Modeling of Photonic Crystals

143 y i y = yN

y = yN−1 wj

y = yj

h

y = y2 y = y1 z

x

FIGURE 3.6 Cross section of N-layered periodic arrays of circular cylinders standing in free space.

A periodic array scatters an incident plane wave given by Equation (3.1) into a set of space-harmonics as discussed in Section 3.3.1. When the array is multilayered as shown in Figure 3.6, the scattered space-harmonics impinge on the neighbor arrays as new incident waves and are scattered into another set of space harmonics that impinges back on the original array. A series of those processes explains a multiple scattering of wave fields in the layered arrays. To describe the multiple scattering process, we introduce the reflection and transmission matrices for each of the arrays that relate a set of incident space harmonics to a set of reflected and transmitted ones. The approach that led to, the reflection and transmission coefficients (3.53) is easily extended to derive the reflection and transmission matrices. For the incidence of space harmonics down-going from the  region j  1, the reflection matrix R j and the transmission matrix F j of the array located on y  yj are obtained as follow [41,42]:

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Rj  WjUTj PWj

(3.96)

Fj  I  WjUTj PWj

(3.97)

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Electromagnetic Theory and Applications for Photonic Crystals

with   2(i)m  U  [u e imal  lm ]     k0 h sin al

(3.98)

Tj  (I  Tj L)1Tj

(3.99)

m iman ] P  [p mn ]  [(i ) e

(3.100)

Wj  [e ikx ,l w j du ]

(3.101)

where P is a matrix that transforms the down-going n-th space-harmonic wave = to the m-th cylindrical harmonic wave; Tj is the T-matrix of the j-th array defined by Equation (3.43); U and U are the matrices that transform the m-th cylindrical harmonic wave back to the up-going and down-going l-th space-harmonic waves; and Wj are the diagonal matrices denoting the phase correction to each of the space harmonics due to the relative displacement wj of the array elements.  Similarly, the reflection matrix R j and the transmission matrix F j for the incidence of space harmonics up-going from region j are deduced as follows: Rj  WjUTj PWj

(3.102)

Fj  I  WjUTj PWj

(3.103)

m iman ] P  [p mn ]  [(i ) e

(3.104)

where

If the cylinders consist of an isotropic medium, the reciprocity theorem leads to the identity Fj  Fj  Fj

(3.105)

Furthermore, if the cross section of cylinders is up-down symmetric relative to the array plane y  yj, we have the relation Rj  Rj  R j

(3.106)

The relations (3.105) and (3.106) are effectively used when a multilayered array consisting of isotropic and up-down symmetric cylinders is treated.

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145

3.4.2 GENERALIZED REFLECTION AND TRANSMISSION MATRICES The generalized reflection and transmission matrices of the N-layered arrays shown in Figure 3.6 are obtained by successively concatenating the reflection and transmission matrices given by Equations (3.96), (3.97), (3.102), and (3.103). Let = = R j and Fj be the generalized reflection and transmission matrices for an entire system consisting of j-layered arrays, where the superscripts are used in the same way as in Equations (3.96), (3.97), (3.102), and (3.103). When the ( j  1)-th array is stacked at y  yj1 above the j-th array located at y  yj, the generalized reflection = = and transmission matrices R j1 and F j1 for this ( j  1)-layered system are calculated using the recursive relations [27]: Rj1  Rj1  Fj1 Yj1 Aj1 Rj Yj1 Fj1

(3.107)

Fj1  Fj B j Yj1 Fj1

(3.108)

Rj1  Rj  Fj B j Yj1 Rj1 Yj1 Fj

(3.109)

Fj1  Fj1 Yj1 Aj1 Fj

(3.110)

where Aj1  (I  Rj Yj1 Rj1 Yj1 )1

(3.111)

B j  (I  Yj1 Rj1 Yj1 Rj )1

(3.112)

Yj1  [eik y , md j1 dmm ],

(3.113)

k y , m  k02  k x2, m

= where dj1  yj1  yj. The generalized reflection and transmission matrices RN = and FN for the N-layered arrays are calculated from Equations (3.107)–(3.110) = through the (N  1) times recursion process starting with R 1  R1 and =  F1  F1 . The power reflection and transmission coefficients of the N-layered = arrays into the l-th space harmonic component can be easily calculated using RN = and FN .

3.4.3 FLOQUET-MODE APPROACH

FOR

LAYERED IDENTICAL ARRAYS

The generalized reflection and transmission matrices are expressed using only decaying space harmonic components Yj1 defined by Equation (3.113). Therefore,

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no numerical instability occurs in the calculation. However, an N layered system requires an N  1 times recursion process including the matrix inversion. This consumes much computation time, especially when the number of layers increases. If, however, the multilayered system consists of periodically stacked identical arrays with an equal separation distance, a concept of Floquet modes [28], [29] propagating in the layered direction, can be used to calculate the generalized reflection and transmission matrices without the recursion process. A multilayered system shown in Figure 3.6 is assumed to consist of identical arrays stacked periodically in the y direction with an equal distance d. The cylindrical objects in each layer are up-down symmetric relative to the array plane y  yj  jd ( j  1, 2, 3, …, N). Then all the arrays are characterized by a single reflection matrix R and a single transmission matrix F, which are given by Equations (3.96) and (3.97). Let a(y) and a(y) denote the amplitude vectors of up-going and down-going space harmonics as the function of y, respectively. Then we have the following relations for the j-th array plane located at y  jd: a ( jd  0)  R ⋅ a ( jd  0)  F ⋅ a ( jd  0)

(3.114)

a ( jd  0)  R ⋅ a ( jd  0)  F ⋅ a ( jd  0)

(3.115)

where a ( jd  0) are the amplitude vectors just above the j-th array plane and a ( jd  0) are those defined just below the j-th array plane. The relations (3.114) and (3.115) can be rearranged in a matrix form as  a ( jd  0)        C  a ( jd  0)         a ( jd  0)   a ( jd  0) 

(3.116)

 F  RF1 R RF1   C     1  1 F   F R

(3.117)

where

We introduce two additional reference planes y  jd d/2 that are located an equal distance d/2 away from the array plane y  jd. These reference planes define a unit period for the arrays periodically stacked in the y direction. It is easy to show that the amplitude vectors of space harmonics in two reference planes are related as follows:  a ( jd  d /2)   C d     a ( jd  d /2) 

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 a ( jd  d /2)        a ( jd  d/2) 

(3.118)

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147

with D Cd    0

0   D C  D1   0

0   D1 

D  [eik y ,l d / 2 dlm ]

(3.119)

(3.120)

where Cd represents the transfer matrix over a symmetric one period d in the layered system. An important point concerns the matrix Cd, its eigenvalues, and its eigenvectors. When we divide Cd into four submatrices as C  d ,1 Cd ,2  Cd     Cd ,3 Cd ,4 

(3.121)

the analysis of Equation (3.119) together with Equation (3.117) reveals that the inverse of Cd takes the form C 1   d ,4 C  d  Cd ,2

Cd ,3   Cd ,1  

(3.122)

Bearing in mind this special relation between Cd and C1 d , it is shown [28] that if v is an eigenvalue of Cd, then 1/ v is also an eigenvalue, and if (x1,v, x2,v )T is the eigenvector to the eigenvalue v, then the eigenvector to the eigenvalue 1/ v has the form (x2,v, x1,v)T, where v is a positive integer denoting a number of eigenvalues. Using these properties of eigenvalues and eigenvectors of the matrix Cd and defining a new set of amplitude vectors g (y) by the relation  g (y)   a (y)     1    X     g (y)   a (y) 

(3.123)

Equation (3.118) can be rewritten in the form  g (jd  d /2)   C g     g (jd  d /2) 

 g (jd  d /2)        g (jd  d /2) 

(3.124)

X2   X1 

(3.125)

with X 1 X   X  2

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 0   Cg     0 1 

(3.126)

  [lv dvl ]

(3.127)

where  and 1 are the diagonal matrices whose elements are the eigenvalues v and 1/ v of the matrix Cd respectively and X is the matrix that contains in its columns the corresponding eigenvectors (x1,v, x2,v)T and (x2,v, x1,v)T. Equation (3.124) describes the propagation of space harmonics over one period of the periodically multilayered system using the Floquet-mode amplitude g (y), and Equation (3.123) yields the relationship between the Floquet-mode amplitudes and the conventional space-harmonics amplitudes. The pair of eigenvalues v and 1/ v with Re[log( v)]  0 are equal to ei v d and ei v d respectively, where v of complex values represents the propagation constants of upward and downward propagating v-th Floquet mode. Taking into account the property of Floquet modes, the concatenation over N periods in the y direction gives the following relations: g (Nd  d /2)  N ⋅ g (d /2)

(3.128)

g (d /2)  N ⋅ g (Nd  d /2)

(3.129)

N  [lvN dvl ]  [eiNkvd dvl ]

(3.130)

with

It should be noted that the matrix N does not contain exponentially growing elements because | v|  1 is always satisfied. Substituting Equations (3.128) and (3.129) into Equation (3.123), we have  a (Nd  d /2)   N  X1      a (Nd  d /2)   0  a (d /2)   I 0    X1      a (d /2)   0 N 

0   I 

 g (d /2)        g (Nd  d /2) 

 g (d /2)        g (Nd  d /2) 

(3.131)

(3.132)

Eliminating g(Nd  d/2) and g(d/2) from Equations (3.131) and (3.132), we finally obtain a linear system that determines the link between the incident and the scattered space-harmonic amplitudes for the identical N-layered structure as follows [29]:  a (Nd  d /2)   R   N      a (d /2)   FN

© 2006 by Taylor & Francis Group, LLC

FN   a (Nd  d /2)     RN   a (d /2) 

(3.133)

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149

with RN Z2 Q2  Z1N Q1

(3.134)

FN  Z1Q2  Z2 N Q1

(3.135)

Z1 

1 N 1 ( Q2  Q1 )1  (N Q2  Q1 )1 2 2

(3.136)

Z2 

1 N 1 ( Q2  Q1 )1  (N Q2  Q1 )1 2 2

(3.137)

Q1 

1 1 (X  X2 )1  (X1  X2 )1 2 1 2

(3.138)

Q2 

1 1 (X1  X2 )1  (X1  X2 )1 2 2

(3.139)

= = where RN and FN give the generalized reflection and transmission matrices for the N-layered entire system, which are viewed from the reference planes y  Nd  d/2 = = and y  d/2, respectively. The calculation of RN and FN is always stable because they are expressed in terms of only the propagating or decaying Floquet modes. The matrices Q1 and Q2 are independent of the number of layers. The calculation of the diagonal matrix N is straightforward. The logarithm of the eigenvalue v with | v|  1 is first calculated and then eN log( v) is calculated to obtain Nv . = = Thus, RN and FN for arbitrary numbers of layers can be obtained by fewer matrix operations, and the computation times are independent of the number N of layers. The frequency dependence of eigenvalues v of the transfer matrix Cd is closely related to the presence of stopbands in a periodically multilayered array. If all of the eigenvalues are | v|  1 (1/| v|  1) for some frequency interval, N tends to zero as the number N of layers is increased. For this case, it follows from = = 1 Equations (3.134)–(3.139) that Z1  0, Z2  Q1 1 , RN  Q1 Q2, and FN  0 for a larger N. Such a frequency interval corresponds to a stopband of the multilayered arrays in which there is no transmittance. When a unit period of a layered system contains several nonperiodic layers, we may first calculate the generalized reflection and transmission matrices using the recursion formula (3.107)–(3.110) discussed in Section 3.4.2 and then utilize the Floquet-mode approach mentioned above. When several identical layered systems are stacked, on the other hand, the generalized reflection and transmission matrices calculated from Equations (3.134) and (3.135) are linked using the recursion formula (3.107)–(3.110) to obtain the entire reflection and transmission matrices.

3.4.4 LAYERED ARRAYS EMBEDDED

IN A

SLAB

When the arrays are embedded in a slab of transparent medium as shown in Figure 3.7, additional effects are incorporated into the multiple scattering process

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Electromagnetic Theory and Applications for Photonic Crystals y i

0, 0

y=d

2r

h

dN+1 y = yN

y = yj+1 s, s

dj+1 y = yj , 

y = y1 d1 z

0, 0

x

FIGURE 3.7 Cross section of N-layered periodic arrays of circular cylinders embedded in a dielectric slab.

of the space-harmonic fields. These are the change in an effective propagation length of fields inside the slab medium and the multiple scattering of fields between the array planes and the slab boundaries. The combined effects in scattering lend variety to the frequency response of the arrays that may be controllable by varying the slab thickness and material combination. Let the permittivity and permeability of the slab medium be s and s. For the array located in the slab, the reflection and transmission matrices are still given in the form of Equations (3.96) and (3.97), but the expressions (3.98) and (3.100) must be slightly modified due to the Fresnel reflection of space-harmonic waves at the boundaries between the slab medium and the exterior free space as follows [43]:  2(i)m  ( )  U   ulm   e imal      k h sin a  l  s

(3.140)

m imal ] P  [p mn ]  [(i ) e

(3.141)

cos al 

kx ,l ks

,

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k x ,l k0 cos fi +

2lp , h

Im{sin al} 0

(3.142)

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151

= where ks   ess. The T-matrix Tj for the embedded j-th array is calculated using the lattice sums given by ∞

 ilkx ,0h  (1)nm eilkx ,0h  Snm (ks h, k x , 0 h)  ∑ H n(1)  m (lks h)  e i1

(3.143)

and the T-matrix Tj of the isolated circular cylinder located in the slab medium of infinite extent. Since the x component of the wavevector is conserved through the Fresnel reflection, Wj is the same as Equation (3.101). For the N-layered arrays embedded in the slab, the lower boundary of the slab is treated as the zeroth layer and the upper boundary as the (N  1)-th layer. The conventional Fresnel reflection and transmission matrices for the spaceharmonic waves, which are diagonal matrices, are substituted into R , F , = N1 = N1  R 0, and F 0. The generalized reflection and transmission matrices RN and FN for the embedded N-layered arrays are obtained from Equations (3.107)–(3.110) through an (N  1) times recursion process. If the arrays consist of periodically layered identical arrays with an equal separation, we first may calculate the generalized reflection and transmission matrices for the N-layered arrays freestanding in the slab medium of infinite extent by using Equations (3.134) and (3.135) and then concatenate them to the Fresnel reflection and transmission matrices at the upper and lower slab boundaries.

3.5 THREE-DIMENSIONAL SCATTERING FROM LAYERED CROSSED-ARRAYS The geometry considered here is shown in Figure 3.8. Periodic arrays of circular cylinders are stacked in free space with a separation distance d along the y direction. The cylinders in each layer are infinitely long and parallel to each other, while the cylinder axes are rotated 90° in each successive layer [19]. The stacking sequence repeats every two layers. The array consisting of z-directed cylinders is referred to as the Z-array, and the array of x-directed cylinders as the X-array. The cylindrical elements and their periodic spacing may be different in the Z-array and X-array. We assume an incident plane wave of unit amplitude varying as ei(kx,0 xkz,0 z), where kx,0  k0 cos i sin  i, kz,0  k0 cos i cos i, and (i,  i ) denotes the angle of incidence as depicted in Figure 3.8. The z components of the incident fields are given by Equation (3.17) in Section 3.2.3. Since the structure is periodic in both x and z directions, the scattered fields consist of a set of space harmonics varying as ei(kx,l xkz,mz), where kx,l  kx,0  2l/hx, kz,m  kz,0  2m/hz, hx and hz are the periodic spacing in the Z-array and X-array respectively, and l and m are integers denoting the order of space harmonics. We first formulate the reflection and transmission matrices for the Z-array and X-array separately. Then we use the results to derive the reflection and transmission matrices for the crossed-array, which doubles the Z-array and X-array and constitutes a unit cell in the stacking sequence. The generalized reflection and transmission

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Electromagnetic Theory and Applications for Photonic Crystals i Ei 

p^ y

s^

x
i

hx

z i d

hz

FIGURE 3.8 Schematic view of layered crossed-arrays of circular cylinders. (kx,ᐉ, ky,ᐉ,m, kz,m)

y x

hx z

FIGURE 3.9 A periodic array of z-directed circular cylinders illuminated by the (l, m)-th space-harmonic wave varying as ei(kx,l xk z,mz).

matrices for the layered system in Figure 3.8 are obtained by concatenating those of the crossed-array over the repeating number of cells in the y direction.

3.5.1 SCATTERING

FROM A

PARALLEL ARRAY

OF

CIRCULAR CYLINDERS

Assume a Z-array of circular cylinders with radius r1, permittivity e1, and permeability 1 situated in free space as shown in Figure 3.9. The scattering problem is ~ formulated by employing the Ez and Hz   0/eH 0 z fields as the leading fields.

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153

~ The fields can be characterized in terms of the amplitudes {ez,l,m} and {h z,l,m} for ~ each of the space harmonics. Let ez and hz be the column vectors of (2L  1)  (2M  1) dimensions defined as T T ez   ez (M )  ez (0)  ez (M )  , ez (m)   ez ,L ,m  ez ,0,m  ez ,L ,m   

(3.144)

T T  hz    hz (M )   hz (0)   hz (M )  ,  hz (m)   h z ,L ,m  h z ,0,m  h z ,L ,m  (3.145)    

where m  M and l  L denote the orders of truncation of space harmonics. The analytical procedure described in Section 3.3.3 is easily extended to the general case for the incidence of the space-harmonic waves varying as ei(kx,l xkz,mz). The ampli~ ~ tude vectors (ezr, hzr ) and (ezt , hzt ) for the reflected and transmitted waves are related i ~i to the incident vectors (ez , hz ) as follows:  er   z   R    r      hz 

 ei   R ee  z   i   he  hz   R  

R eh   R hh  

 ei   z  i   hz 

(3.146)

 et   z   F    t      hz 

 ei   F ee  z   i   he  hz   F  

F eh   F hh  

 ei   z  i   hz 

(3.147)

      0       R m (M )  

(3.148)

      0       F m (M )  

(3.149)

with  R m (M )        0 R m       0  



0









 F m (M )        0 F m       0  

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 R m (0)   







0





0









 F m (0)   







0



0

0

154

Electromagnetic Theory and Applications for Photonic Crystals 



where R || and F || (  ee,eh,he,hh) represent the reflection and transmission   matrices of the Z-array, and R || (m) and F || (m) (m  M, …, 0, …, M) denote their submatrices of (2L  1)  (2L  1) dimensions defined for each of m-th spacehe eh he harmonic components relative to the z-coordinate. Note that Reh || , R || , F || , and F|| describe the reflection and transmission into the cross-polarized space harmonic components. From the analogy with Equations (3.89)–(3.92), the submatrices   R || (m) and F || (m) are derived as follows [44]:   m  R (m)  U z (m)T z (m)Pz (m)

( m  ee, eh, he, hh)

(3.150)

  m  F  (m)  I  U z (m)T z (m)P z (m)

( m  ee, hh)

(3.151)

 m  F (m)  U z (m )T z (m )P z (m )

( m  eh, he)

(3.152)

  2(i)n (m )]   U ( m )  [ u e inax,l ( m )  z z, ln   h sin a (m)  x, l  m x 

(3.153)

with



n P z (m)  [ p z, nl (m )]  [(i ) e

cos a x, l (m) 

k x, l gm

2l , hx

k x, l  k x, 0 

,

ina x, l ( m )

]

Im{sin a x, l (m)} 0 gm  k02  kz2, m

(3.154) (3.155)

(3.156)

where Pz(m) is the (2N  1)  (2L  1) matrix whose (n, l)-element pz,nl(m) represents the n-th coefficient in the cylindrical wave expansion of the downgoing l-th space-harmonic varying as ei[kx,l xky,l (m)y] for each of the indicated order m, Uz (m) are the (2L  1)  (2N  1) matrices whose (l, n)-element represents the transformation of the n-th cylindrical harmonic wave into and the up-going and down-going space-harmonics varying as ei[kx,l x ky,l (m)y] respectively, and n  N is the order of truncation in the cylindrical wave expansion. In Equations (3.150)–(3.152), the (2N  1)  (2N  1) matrices Tz(m) (  ee, eh, he, hh) are the T-matrices [31] of the Z-array for the incidence of (l, m)-th space-harmonics with the indicated order m. Tz(m) are obtained in the same form as Equations (3.83)–(3.88) by replacing L, Sz,1 to Sz,4, and Tz by Lx(m), Sz,1(m) to Sz,4(m), and Tz (m), respectively. Lx(m) is a square matrix whose (n, n )-element is given by the (n–n )-th order lattice sum for the Z-array defined as 

S x, n – n′ (m)  ∑ H n(1–)n′ (vgm hx )[eivkx, 0hx  (1)n – n′ eivkx, 0hx ] v1

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(3.157)

Modeling of Photonic Crystals

155

On the other hand, Tz (m) (  ee,eh.he,hh) are obtained using the following substitutions into Equations (3.23)–(3.29): kz ,0 → kz ,m ,

w → wm  r1gm ,

u → um  r1 k12  kz2,m

e → e1 ,  → 1 , k → k1 Next, we consider an X-array of circular cylinders with radius a2, permittivity e2, and permeability 2 situated in free space as shown in Figure 3.10. Employing ~ the Ex and Hx   0 /eH 0 x fields as the leading fields, the scattering problem can be formulated in the same way as for the Z-array. The fields are characterized ~ in terms of the amplitudes {ex,m,l} and {h x,m,l} of each of the space harmonics. Let ~ ex and h x be the column vectors of (2M  1)  (2L  1) dimensions defined as e x  [ex (L )  ex (0)  ex (L )]T ,

ex (l )  [ex ,M,l  ex ,0,l  ex , M,l ]T

(3.158)

hx  [hx (L )  hx (0)  hx (L )]T ,

hx (l )  [hx ,M,l  hx ,0,l  hx ,M,l ]T

(3.159)

~ ~ Then the amplitude vectors (exr, hxr ) and (ext , h xt) for the reflected and trans~ mitted waves are related to the incident vectors (exi, h xi) as follows:  i   ee  er   x    R   e x    Rx   r   x    i   he  hx   Rx  hx 

(kx,ᐉ, ky,ᐉ,m, kz,m)

Rxeh   e xi    Rxhh   hxi 

(3.160)

y x

z

hz

FIGURE 3.10 A periodic array of x-directed circular cylinders illuminated by the (l, m)-th space-harmonic wave varying as ei(kx,l xkz,mz).

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Electromagnetic Theory and Applications for Photonic Crystals

 et   i   ee  x    F   e x    Fx   t   x    i   he  hx   hx   Fx

Fxeh   e xi    Fxhh   hxi 

(3.161)

with  R m (L )  x     Rxm =  0      0 



0









 F m (L )  x      m Fx =  0      0 

 Rxm (0)  







0





0











Fxm (0)











0



      0      Rxm ( L ) 

(3.162)

      0      Fxm ( L ) 

(3.163)

0

0

  the reflection and transmission where R – x and –Fx (  ee, eh, he, hh) represent matrices of the X-array, and R x (l) and Fx (l) (l  L, …, 0, …, L) denote their submatrices of (2M  1)  (2M  1) dimensions. Note that the arrangements of ~ ~ elements in ex and hx are different from those in ez and hz defined by Equations (3.144) and (3.145) for the Z-array. From the analogy with Equations (3.150)–(3.155), the submatrices Rx(l) and Fx(l) are derived as [44] m  Rxm (l )  U x (l )Tx (l ) Px (l )

( m  ee, eh, he, hh)

(3.164)

m  Fxm (l )  I  U x (l )Tx (l ) Px (l )

( m  ee, hh)

(3.165)

m  Fxm (l )  U x (l )Tx (l ) Px (l )

( m  eh, he)

(3.166)

with   2(i)n inaz , m ( l )   U ( l )  [ u ( l )]  e x x , mn  h h sin  (l )   l z  z,m

(3.167)

n ina z , m ( l ) ] Px (l )  [ p x , nm (l )]  [i e

(3.168)

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Modeling of Photonic Crystals

157

cos  z , m (l )  kz , m  kz , 0 

kz , m hl

,

2 mp , hz

Im{sin a z , m (l )} 0

(3.169)

hl  k02  k x2,l

(3.170)

In Equations (3.164)–(3.166), the (2N  1)  (2N  1) matrices Tx(l) (  ee, eh, he, hh) are the T-matrices of the X-array for the incidence of (m, l)-th space harmonic with the indicated order l. Tx(l) are obtained in the same form as Equations (3.83)–(3.88) by replacing L, Sz,1 to Sz,4, and Tz with Lz(l), Sx,m(l) to Sx,4(l), and Tx (l), respectively. Lz(l) is a square matrix whose (n, n )-element is given by the (n–n )-th order lattice sums for the X-array defined as 

Sz , n –n (l )  ∑ H n(1)–n (hl hz )[eivkz , 0hz  (1)n –n eivkz , 0hz ]

(3.171)

v1

However, Tx (l) (  ee, eh, he, hh) are obtained using the following substitutions into Equations (3.23)–(3.29) kz ,0 → k x ,l , e → e2 ,

3.5.2 UNIT CELL

w → wl  r2 hl ,

 → 2 , OF A

u → ul  r2 k22  kx2,l

k → k2

CROSSED-ARRAY

OF

CIRCULAR CYLINDERS

Assume a crossed-array in which the Z-array and X-array are doubled with a center to center separation of d in the y direction as shown in Figure 3.11. The Z-array and X-array in the preceding sections have been treated separately by taking the electric and magnetic fields parallel to the cylinder axis in each configuration as the leading fields. When they are layered, however, the use of common leading fields is required to describe the multiple scattering process between the two orthogonal ~ arrays in a unified manner. Then we choose the Ez and Hz fields as the leading fields common to the crossed-array and rewrite the reflection and transmission matrices of the X-array in terms of these field components. We introduce the matrix  which transforms the order of elements of amplitude vectors defined by Equations (3.144) and (3.145) into those of Equations (3.158) and (3.159) as follows: ex  ex ,

hx  hx

(3.172)

~ We note also that the amplitudes of the (l, m)-th space harmonic in the (Ex, Hx) ~ field and (Ez, Hz ) field are related as ex ,l ,m 

© 2006 by Taylor & Francis Group, LLC

k x ,l kz ,m gm2

ez , l , m 

k0 ( )k y ,l ,m gm2

h z ,l ,m

(3.173)

158

Electromagnetic Theory and Applications for Photonic Crystals (kx,ᐉ, ky,ᐉ,m, kz,m)

y x

hx

z

d hz

FIGURE 3.11 A unit cell of crossed-arrays illuminated by the (l, m)-th space-harmonic wave varying as ei(kx,lxkz,mz).

h x ,l ,m 

k0 ( )k y,l ,m gm2

ez , l , m 

k y ,l ,m  k02  k x2,l  kz2,m ,

k x ,l kz ,m gm2

h z ,l ,m

Im{k y ,l ,m } 0

(3.174)

(3.175)

where the upper and lower signs on ky,l,m correspond to the up-going and downgoing waves in the y direction, respectively. Substituting Equations (3.172)– (3.174) into Equations (3.160) and (3.161), the relations between the amplitude vectors ~ in the X-array are rewritten in terms of the Ez and Hz fields as follows [44]:  er   i   ee  z   [ ]  ez    R R   i   he  r   hz   hz   R

eh   i  R   ez    hh  i R   hz 

(3.176)

 et   i   ee  z   [ ]  ez    F F   i   he  t   hz   hz   F

eh   i  F   ez    hh  i F   hz 

(3.177)

Rxeh   K1 K2    K1  Rxhh   K2

(3.178)

with  Ree    he  R

Reh   K3 K4   Rxee   K3   Rxhe Rhh   K4

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Modeling of Photonic Crystals

 Fee    he  F

Feh   K3  Fhh  K4 m

Rxm  1 R x ,

159

K4   Fxee  K3   Fxhe

Fxeh   K1 K2    K1  Fxhh   K2

Fxm  1 Fxm 

(3.179)

(3.180)

where Rx and Fx are defined by Equations (3.162)–(3.166). In Equations (3.178) and (3.179), K1, K2, K3, and K4 are the (2L  1)(2M  1)  (2L  1)(2M  1) matrices whose submatrices are defined in the same form as Equations (3.148) and (3.149) and are given by k k  K1 (m)   x ,l 2z ,m dll  ,  gm 

k k  0 y ,l ,m  d K2 (m) =   ll

2 g   m

(3.181)

k k  K3 (m)   x ,l 2z ,m dll  ,  hl 

k k  0 y ,l ,m  d K4 (m)    ll

2  hl 

(3.182)

The calculation of Equations (3.178)–(3.180) can be easily carried out by changing the sequence of the array elements defined in the computer program. Now the reflection and transmission matrices (R||, F||) and (R, F) of the Z-array and X-array for the incidence of down-going space harmonics have been expressed as Equations (3.146), (3.147), (3.176), and (3.177) using the common ~ leading fields Ez and H z. When the cylinders are isotropic and symmetric with respect to each array plane, it follows that the reflection and transmission matrices for the incidence of up-going space-harmonics are also given by (R||, F||) and (R, F). Then the reflection and transmission matrices for the crossed-array shown in Figure 3.11 are derived by linking (R||, F||) and (R, F) over the Z-array and X-array stacked with a separation distance d. For the incidence of down-going space-harmonics from the upper region of the Z-array, after several manipulations  the reflection and transmission matrices R  and F of the crossed-array are obtained as follows: R  R  FY  (d ) A R⊥Y (d ) F

(3.183)

F  F⊥ BY (d )F

(3.184)

A  [I  R⊥Y (d )RY (d )]1

(3.185)

with

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Electromagnetic Theory and Applications for Photonic Crystals

B  [I  Y (d )RY (d )R⊥]1

(3.186)

Y ( y)  [eik y ,l ,m y dll ′ ]

(3.187)

 Similarly, the reflection and transmission matrices R  and F  for the incidence of up-going space harmonics from the lower region of the X-array are obtained as follows:

R  R⊥  F⊥ BY (d )RY (d )F⊥

(3.188)

F  FY  (d )A F⊥

(3.189)

3.5.3 LAYERED CROSSED-ARRAYS

OF

CIRCULAR CYLINDERS

Let us consider a 2N-layered array in which the Z-array and X-array are stacked one after the other in the y direction as shown in Figure 3.8. This structure may be treated as an N-stacking sequence of the crossed-array whose reflection and transmission matrices R  and F are defined by Equations (3.183), (3.184), (3.188), and (3.189). The generalized reflection and transmission matrices for the N-layered crossed-arrays are obtained [25] by concatenating R  and F successively over N layers. The separation distance between two center planes of adjacent crossedarrays is 2d when the X-arrays and Z-arrays are periodically stacked in the y direction. Let the ( j  1)-th crossed-array be stacked above the j-th crossed-array. = = Then the generalized reflection and transmission matrices R, j1 and F, j1 for this ( j  1)-layered crossed-array are calculated using the recursive relations: R, j1  R, j1  F, j1Y (2d)A, j1 R, jY (2d )F, j1

(3.190)

F, j1  F, j B, jY (2d )F, j1

(3.191)

R, j1  R, j  F, j B, jY (2d )R, j1Y (2d )F, j

(3.192)

F, j1  F, j1Y (2d )A, j1 F, j

(3.193)

where 1 A, j1   I  R, jY (2d )R, j1Y (2d )   

(3.194)

1 B, j   I  Y (2d )R, j1Y (2d )R, j   

(3.195)

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161

= = The generalized reflection and transmission matrices R,N and F,N for the N-layered crossed arrays are calculated from Equations (3.190)–(3.193) through = = the (N  1) times recursion process starting with R,1  R and F,1  F. Although the notations treating the layered crossed-arrays are a little intricate because of three-dimensional problems, the mathematics and analytical procedure are straightforward and quite similar to those of the two-dimensional scattering problem discussed in Section 3.4. The generalized reflection and transmission matrices of layered crossed-arrays are calculated using the three-dimensional T-matrix of an isolated circular cylinder and the lattice sums for a one-dimensional periodic array.

3.6 MODAL ANALYSIS OF TWO-DIMENSIONAL PHOTONIC CRYSTAL WAVEGUIDES A photonic crystal waveguide is formed by either introducing a defect layer in a photonic crystal or bounding a dielectric space by a photonic crystal. The guided fields are strongly confined because any electromagnetic energy cannot escape through the surrounding medium. Such waveguides have received growing attention in view of their promising applications to new integrated optical devices [3]. The mode propagation in two-dimensional photonic crystal waveguides consisting of a lattice of circular cylinders has been extensively investigated using various computational approaches such as the plane wave expansion method [12], the finite-difference time-domain technique [15], the beam-propagation method [16], and the finite-difference frequency-domain technique [17]. These methods are versatile and can be applied to the analysis of photonic crystal waveguides with irregularities along the guide such as sharp bends, branches, and discontinuities. However, they require a rather intricate discretization technique to get better solutions with fast convergence. In this section, we discuss a rigorous semianalytical approach to modal analysis of two-dimensional photonic crystal waveguides consisting of layered periodic arrays of circular cylinders. The method is an extension of the lattice sums technique combined with the T-matrix approach presented in Section 3.4. The waveguide is assumed to be uniform in the direction of wave propagation. Since the electromagnetic boundary conditions at the interfaces between the lattice elements and the background medium are fully satisfied through the T-matrix of circular cylinders, it is possible to accurately analyze the modal properties even near the cutoffs.

3.6.1 DISPERSION EQUATION The side view of a two-dimensional waveguide is shown in Figure 3.12. The guiding region (t  y  t) is bounded by two photonic crystals. The upper region 1 and the lower region 2 consist of N1-layered and N2-layered arrays of circular cylinders, respectively, which are infinitely long in the z direction and periodically spaced with a common distance h in the x direction. The cylindrical elements should be identical along each layer of the arrays, but those in different

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162

Electromagnetic Theory and Applications for Photonic Crystals y h r1, 1

N1 d1

1

y=t

s x

w 2

y = −t d2

N2 r2, 2 h

FIGURE 3.12 Cross section of a two-dimensional photonic crystal waveguide.

layers need not be necessarily identical in material properties and dimensions. Figure 3.12 shows a typical configuration in which the identical arrays of circular cylinders with the same radius r1(r2) and permittivity e1(e2) are periodically layered with an equal spacing d1(d2) in the upper (lower) region. The parameter w indicates the displacement of the elements along the x direction. An arbitrary lattice configuration of the photonic crystal can be attained by choosing the parameters d1/h, d2 /h, and w/h in the proper way. The background medium is a homogeneous dielectric with permittivity es and permeability s. The guided waves are assumed to be uniform in the z direction and vary in the form eix in the x direction where  is a real propagation constant. The scattering from each layer of the arrays is characterized by the reflection and transmission matrices for the space harmonics with the x-dependence as eilx, where l    2l/h and l are integers. Let us consider an isolated j-th array parallel to the x-z plane and employ the array plane y  yj as the reference plane for the phase of the scattered space harmonics. Following the same analytical procedure discussed in Section 3.4.1 the reflection matrix Rj and the transmission matrix Fj are derived as follows [45], [46]: R j ()  Wj ()U ()Tj ()P ()Wj ()

(3.196)

Fj ()  Wj ()[I  U ()Tj ()P ()]Wj ()

(3.197)

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163

with   2(i)m  U ()  [u e imal ( )  lm ()]     ks h sin l () T j (b)   I  Tj L(b)   

1

(3.198)

(3.199)

Tj

Wj (b)   e ibl w j dll   

(3.200)

m iman (b ) ] P (b)  [p mn (b)]  [(i ) e

(3.201)

∞

ilbh  (1) nm eilbh ] Lnm (b)  ∑ H n(1) m (ks lh)[e

L(b)  [ Lnm (b)],

cos l (b) 

bl  b 

l1

bl , ks

(3.202)

Im{sin l (b)} 0 (3.203)

2pl h

(3.204)

where ks   ess is the wavenumber of the background medium and Tj is the T-matrix of the isolated circular cylinder given by Equations (3.10) and (3.11) for TE and TM waves respectively. Note that the matrices U , L, Wj, and P are given as functions of the propagation constant , whereas the T-matrix Tj does not depend on  and takes the same form as the scattering problem. Since the circular cylinders are up-down symmetric with respect to the array plane, the reflection and transmission matrices are the same for both the down-going and up-going space harmonic waves. The generalized reflection and transmission matrices of an N-layered array can be obtained by using Equations (3.196) and (3.197) in the recursive formula (3.107)–(3.110) or the Floquet mode approach (3.134)–(3.139). For the waveguide shown in Figure 3.12, the upper and lower boundaries viewed from the guiding region t  y  t are characterized by the generalized reflection matrices RN∩1() at y  t  0 and RN∪2 () at y  t  0. Using the method mentioned above, these matrices are calculated as functions of . Denoting the amplitude vectors of space harmonics incoming and outgoing on the    plane y  t  0 by a N1 and bN1 and those on the plane y  t  0 by a N2 and b N2, respectively, the following relations are obtained: ∩  b N  RN (b)  aN , 1

1

1

a  D(b)  b , N N 1

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2

b  RN∪ (b)  a N N 2

2

2

 a N  D(b)  bN 2

1

(3.205) (3.206)

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Electromagnetic Theory and Applications for Photonic Crystals

where D(b)   ei 2k y ,l (b)t dll  ,  

k y ,l  ks2  bl2

(3.207)

Using Equations (3.205) and (3.206), the following relation is derived:   I D(b)RN∪ (b)   a  N  2   10    a    D(b)RN∩1 (b)   N 2   

(3.208)

Equation (3.208) has nontrivial solutions only for discrete values v of  that satisfy the dispersion equation det  I  D(bv )RN∩ (bv ) D(bv )RN∪ (bv )   0 1 2  

(3.209)

The value of v gives the propagation constant of the v-th guided mode propagating along the x direction. The result is substituted into Equations (3.208) and    (3.205) to determine the amplitude vectors a N1, a N2, bN1, and b N2 for the v-th mode. The mode field distribution in the plane transverse to the x axis can be calculated  using a recursion formula starting from a N1 and a N2 as discussed in the following section.

3.6.2 MODE FIELD ANALYSIS The succeeding two layers of the array are schematically shown in Figure 3.13. Consistent with Equations (3.205) and (3.206), the amplitudes of the spaceharmonics incoming on the j-th array are denoted by a j and those outgoing from the same array are by b j . Then the guided mode field within a homogeneous strip region yj  rj  y  yj1rj1 between two layers of arrays is expressed using a superposition of the space-harmonic fields as follows:  j (x , y)  ej1 (x , y)  bj1  ej (x , y)  bj

(3.210)

e j (x , y)   ei[bl x k y ,l (yy j )]   

(3.211)

where

Taking the ray tracing for the space harmonics across the two layered arrays, on the other hand, we have the following relations: bj1  Rj1  aj1

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(3.212)

Modeling of Photonic Crystals

165

=

a −j +1

R +j+1

b +j+1 y = yj +1

2rj +1

=

R +j

a +j +1

b −j+1

a −j

b +j y = yj

2rj a +j

b −j

FIGURE 3.13 Incoming and outgoing space harmonics on a stack of two array elements.

b j1  R j1  aj1  Fj1  a j1

(3.213)

aj  Y (d j1 )  bj1

(3.214)

bj  Rj  aj

(3.215)

bj  Rj  Y (d j1 )  bj1

(3.216)

Y(y)   eik y ,ly dll  ,  

k y ,l  ks2  bl2 ,

d j1  y j1  y j

(3.217)

= where Rj denotes the generalized reflection matrix of the j-layered arrays viewed from the down-going incident space harmonics with aj. Solving Equations (3.212)   and (3.213), a j1 and bj1 are related to a j1 as follows: aj1  Fj11 (Rj  R j )  aj1

(3.218)

bj1   Fj1  R j1 Fj11 (Rj  R j )   aj1  

(3.219)

 From Equations (3.216) and (3.219), b j1 and b j in Equation (3.210) are  related to a j1. Thus the mode field in the homogeneous strip region yj  rj  y  yj1  rj1 can be calculated when a j1 is specified. However, this field expression based on the space-harmonic expansion does not converge in the

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inhomogeneous grating region |y  yj|  rj that contains the periodic arrays of circular cylinders of radius rj. For the grating region, we must turn to the original expression of the scattered field using the cylindrical wave expansion. Let us consider the grating region within |y  yj1|  rj1. The incident waves on the ( j  1)-th array are the down-going and up-going space harmonics with amplitudes a j1. Referring to Equations (3.38), (3.42), and (3.43), the scattered field outside the zeroth cylinder and within the unit cell is expressed as follows:  (r0 , 0 )   JT  (I  LTj1 )  HT  Tj1   (P  aj1  P  aj1 )  

(3.220)

where J   Jm (ks r0 )eim0   

(3.221)

H   H m(1) (ks r0 )eimf0   

(3.222)

m imnn ] P  [ p mn ]  [(i ) e

(3.223)

Applying the boundary condition on the cylindrical surface 0  rj1, from Equation (3.220) the scattered field inside the zeroth cylinder is obtained as follows:  T    ( I  LT )   T   ( P  a  P  a )  (r0 , f0 )   J  J j1 H j1  j1 j1

(3.224)

with   [ J (kr )eimf0 ]  0 J m

(3.225)

 J (k r )  m s j1  J   dmm   Jm (krj1 ) 

(3.226)

 H (1) (k r )  m s j1  H   dmm ′   Jm (krj1 ) 

(3.227)

where k   e is the wavenumber inside the circular cylinder. Thus the scattered fields (0, 0) and (0, 0) within a unit cell including the zeroth cylinder can be calculated using the amplitudes a j1 of incoming space harmonics,  where a is related to a through (3.218). Equations (3.212) to (3.218) are used j1 j1  recursively to obtain a j for all layers of arrays by starting from a N1 and a N2, which are obtained as the solutions to (3.208). This recursion process is performed by a

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167

straightforward matrix calculus. The mode field distribution as a function of y varies within a unit cell along the x axis but the same pattern is repeated with the period h.

3.7 NUMERICAL EXAMPLES In this section, we discuss numerical examples for scattering and guiding problems in several fundamental geometries of photonic crystals using the model of layered periodic arrays of circular cylinders. Before demonstrating the results, we briefly mention the accuracy of solutions obtained by the present analysis. The model uses a T-matrix of an isolated circular cylinder or an aggregate T-matrix of multiple circular cylinders, the lattice sums for a one-dimensional periodic system, and the matrix equations for concatenating the layered structure. Firstly, note that that the T-matrix is given in closed form by Equation (3.11) or Equations (3.23)–(3.25), and the aggregate T-matrix is given in semi-analytical form by Equations (3.66) and (3.67) or Equation (3.68). Thus the T-matrix and aggregate T-matrix are well defined if the dimensions of the circular cylinders are moderate compared with the wavelength. We consider here such a situation. Secondly, the lattice sums calculated using the integral form (3.57) are very accurate. It has been shown [21] that the calculated values of the lattice sums enable one to obtain the free-space periodic Green’s function with accuracy to fourteen decimal places. Finally, the matrix equations used for the concatenations are always stable because all the elements in Equations (3.107)–(3.110) and Equations (3.190)–(3.193) are expressed in terms of only propagating or decaying space harmonic waves. Thus, the results of the present analysis are almost rigorous except that the cylindrical harmonic expansions to calculate the T-matrix and the order of space harmonics are truncated by a finite number. After confirming the convergence of solutions, the numerical results were obtained by truncating the cylindrical wave expansion at m  10 and taking account of the lowest 7 and 49 space harmonics for parallel and crossed arrays respectively. Throughout this section, the permeability of the medium is assumed to be 0 of free space, and the dielectric constant er  e/e0 is used as a parameter of dielectric medium.

3.7.1 TWO-DIMENSIONAL SCATTERING ARRAYS

FROM

LAYERED PARALLEL

The wavelength response of a single layer of the array is controllable only through the parameters pertinent to the array element and spacing. If the array is layered, on the other hand, the number of layers and the separation distance between the adjacent layers are available as additional freedoms to control the wavelength response. In order to get a physical sketch for a multilayered system, we have calculated the power reflectance R0 of the fundamental space-harmonic for six-layered identical arrays of dielectric cylinders, using the recursive algorithm (3.107) for the generalized reflection matrix. The results are shown in Figure 3.14 by the solid lines, where r  0.3h, er  2.5, d  0.7h, and i  90° in Figure 3.6. The power

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Electromagnetic Theory and Applications for Photonic Crystals

transmittance can be viewed as 1  R0, because the energy conservation relation is accurately satisfied. We have confirmed that the same numerical results are reproduced using the Floquet mode approach defined by Equation (3.134). Several resonance peaks appear for both polarizations. As the number of array layers increases, the number of peaks increases and the resonance profiles become sharp. The multiple scattering of evanescent space harmonics between adjacent arrays plays an important role [47] in the wavelength response of the layered arrays. For the sake of comparison, the dotted lines in Figure 3.14 show the results obtained

1

TM wave 6 Layers

0.8 0.6 R0

d 0.4 0.2

h

r = 2.5 r = 0.3h, d = 0.7h

0 0.7

0.8

(a)

0.9

1

h/0

1

TE wave

0.8

6 Layers

R0

0.6 d 0.4

h

r = 2.5

0.2 r = 0.3h, d = 0.7h 0 0.7 (b)

0.8

0.9

1

h/0

FIGURE 3.14 Power reflectance R0 of the fundamental space harmonic vs. normalized wavelength h/ 0 for the normal incidence of (a) TM wave and (b) TE wave on 6-layered identical arrays of circular dielectric cylinders, where r  0.3h, er  2.5, and d  0.7h. Solid lines: with evanescent space harmonics, dotted lines: without evanescent space hamonics.

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by an approximate treatment that ignores the effects of evanescent space harmonics. We note that the wavelength response in the approximate treatment is quite different from the rigorous ones given by solid lines. When the evanescent space harmonics are ignored, the reflection and transmission matrices at each array plane are replaced by the simple reflection and transmission coefficients for the fundamental space harmonic with l  0. This reduces the layered periodic arrays problem to a conventional layered slab problem. However, such an approximation leads to erroneous results in the reflection and transmission characteristics of the layered arrays. If the number of layers is a few tens, the recursive algorithm works very efficiently. However, the computation time in the recursion process increases with the number of layers due to the repetitious calculation of matrix inversions. The Floquet-mode approach (3.134) is more effective if the number of layers is more than around 100. The power reflectances R0 calculated using the Floquet-mode approach are shown in Figure 3.15 for the normal incidence of (a) TM wave and (b) TE wave on the 100 layered identical arrays of circular dielectric cylinders in free space, where r  0.3h, d  h, er  5.0, and i  90°. The curves in Figures 3.15(a) and (b) are plotted for 1600 points. The computation time for each point is about 0.6 sec on the 533 MHz Alpha21164A workstation using the data sets of the lattice sums stored in the memory. We can see that when the number of layers is increased, several sharp stopbands with complete reflection are formed. The locations and widths of the stopbands are different in the TM wave and TE wave. The next examples are layered arrays embedded in a dielectric slab. Figure 3.16 shows the power reflectance R0 of the fundamental space harmonic as a function of the normalized wavelength h/ 0 for the normal incidence of (a) TM wave and (b) TE wave on 9-layered arrays of dielectric circular cylinders embedded in a dielectric slab. The parameters shown in Figure 3.7 are rj  0.2h, ers  1.3, d  9h, d1  d10  0.5h, and d2  d9  h. The dotted lines are for dielectric cylinders with er 4  er 6  1.3, er2  er3  er7  er 8  1.2, and er1  er 9  1.1; the dashed lines are for air holes with er1  er 9  1.0; and the solid lines are for the dielectric slab without arrays. The dielectric slab is almost completely transparent for the incidence of both TM and TE waves when the cylindrical arrays are not contained. However, several sharp resonance peaks appear in the reflectance when the arrays are embedded. The number of peaks and the locations of peaks change sensitively depending on the slab thickness, the material constants of cylinders and slab, the radii of cylinders, and the number of layers and the separation distance of the embedded arrays. Figure 3.17 shows similar plots of power reflectance R0 for the TM wave normally incident on 9-layered arrays of perfect conductor embedded in a dielectric slab of thickness d  8.2h, where d2  d9  h, d1  d10  0.1h, and other parameters are the same as those in Figure 3.16. The dashed line indicates the result for the dielectric slab without arrays, and the dotted line is for the embedded arrays when all cylinders have a same radius r1  r9  0.1h. For the solid line, the radii of cylinders in succeeding layers are varied such that r5  0.1h, r4  r6  0.086h, r3  r7  0.054h, r2  r8  0.024h, and r1  r9  0.008h.

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1

R0

0.8

0.6

0.4

0.2

0 0.2

0.4

(a)

0.6 h/0

0.8

1

0.8

1

TM wave

1

R0

0.8

0.6

0.4

0.2

0 0.2 (b)

0.4

0.6 h/0 TE wave

FIGURE 3.15 Power reflectance R0 of the fundamental space harmonic for the normal incidence of (a) TM wave and (b) TE wave on the 100 identical layers of periodic arrays of circular dielectric cylinders standing in free space, where r  0.3h, er  5.0, d  h, and i  90° in Figure 3.6.

The embedded arrays of perfect conductor exhibit a characteristic of a low-pass filter. Several small ripples appear in the reflectance of the pass-band when the layered arrays are formed by an identical array. These ripples can be suppressed by properly apodizing the radii of cylinders in the succeeding array elements, and very fine characteristics of passband and stopband are achieved.

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Modeling of Photonic Crystals

171 1

R0

R0

1

0.5

0.5

0

0 0.94

0.96 h/0

(a)

0.98

TM wave

1

0.94 (b)

0.96 h/0

0.98

1

TE wave

FIGURE 3.16 Power reflectance R0 of the fundamental space harmonic vs. normalized wavelength h/ 0 for the normal incidence of (a) TM wave and (b) TE wave on 9-layered arrays of parallel circular cylinders embedded in a dielectric slab. The parameters shown in Figure 3.7 are rj  0.2h, ers  1.3, d  9h, d1  d10  0.5h, and d2  d9  h. Dotted lines: dielectric cylinders with er4  er6  1.30, er2  er3  er7  er8  1.2, and er1  er9  1.1; dash-dotted lines: air holes with er1  er 9  1.0; and solid lines: dielectric slab without arrays.

R0

1

0.5

0 0.8

0.9

1

h/0

FIGURE 3.17 Power reflectance R0 of the fundamental space harmonic for the normal incidence of TM wave on 9-layered arrays of conducting circular cylinders embedded in a dielectric slab. The parameters shown in Figure 3.7 are d  8.2h, d2  d9  h, d1  d10  0.1h, ers  1.3. Dashed lines: dielectric slab without arrays; dotted lines: embedded arrays with r1  r9  0.1h; and solid lines: embedded arrays with r5  0.1h, r4  r6  0.086h, r3  r7  0.054h, r2  r8  0.024h, and r1  r9  0.008h.

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Figure 3.18 shows the power reflectance R0 obtained when the layered array treated in Figure 3.15 is embedded in a dielectric slab with ers  2.0 and d  101h. From the comparison with Figure 3.15, it follows that the wavelength response of the reflectance is quite different from that of the freestanding arrays. This difference is caused by the change in the effective propagation length of waves inside

1

R0

0.8

0.6

0.4

0.2

0 0.2

0.4

0.6

0.8

1

0.8

1

h/0 (a)

TM wave

1

R0

0.8

0.6

0.4

0.2

0 0.2

0.4

0.6 h/0

(b)

TE wave

FIGURE 3.18 Power reflectance R0 of the fundamental space harmonic for the normal incidence of (a) TM wave and (b) TE wave on the 100 identical layers of periodic arrays of circular dielectric cylinders embedded in a dielectric slab, where ers  2.0, d  101h, and other parameters are the same as those in Figure 3.15.

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the slab medium and the multiple scattering of waves between the layered arrays and slab boundaries. Finally, we discuss the reflection characteristics of arrays consisting of multiple circular cylinders per unit cell. Figure 3.19(a) shows the power reflectance

1 TM TE 0.8

h

R0

0.6

0.4

0.2

0 0.92

0.96

(a)

1

h/0 1 TM TE

0.8

h

R0

0.6

0.4

0.2

0 0.92 (b)

0.96

1

h/0

FIGURE 3.19 Power reflectance R0 of the fundamental space harmonic vs. normalized wavelength h/ 0 for the normal incidence of TM and TE waves. (a) Array of composite cylinders with one eccentric inclusion and (b) array of homogeneous cylinders without inclusion, where r1  0.3h and er1  1.5 for the host cylinder, and r2  0.027h and er2  4.0 for an inner cylinder located with an eccentricity of 0.5 relative to r1.

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R0 of the array of two circular cylinders for the normal incidence of plane wave. The host cylinder with r1  0.3h and er1  1.5 contains a smaller dielectric cylinder with r2  0.027h and er2  4.0 as an eccentric inclusion. The eccentricity is 0.5 relative to r1, and the line connecting two centers of the outer and inner cylinders is perpendicular to the array plane. For comparison, similar results obtained for the array consisting of only host cylinders are plotted in Figure 3.19(b). The wavelength response of the array of single circular cylinder per unit cell is generally different in two orthogonal polarizations. In contrast, the array of composite circular cylinders with one eccentric inclusion in Figure 3.19(a) has almost the same resonance wavelengths at around h/ 0  0.961 for two polarizations. Figure 3.20 shows the dependence of R0 on the incident angle i at the wavelength h/ 0  0.961 for the same array as shown in Figure 3.19(a). The close view of (a) around i  90° is depicted in (b). Note that the response of the array of composite cylinders is strongly sensitive to the incident angle [48]. Only the incident wave impinging at around i  90° is selectively reflected over the incident angle 45°  i  135°. Figures 3.19 and 3.20 show the results for the case of a single layer of array. These characteristics discussed above will be more pronounced when the arrays are multilayered. Figure 3.21 shows another example of an array consisting of multiple cylinders per unit cell. Four small dielectric cylinders with radii ri (i  2, 3, 4, 5) and dielectric constants eri are placed in free space around a host dielectric cylinder with r1 and er1 as the parasitic lattice elements. The locations of the four small cylinders are symmetric with respect to the center of the host cylinder. The power reflectance R0 of 10-layered arrays shown in Figure 3.21(a) are plotted in Figure 3.21(b) for three different incident angles, where ri  0.05h (i  2, 3, 4, 5), eri  10, x2  y2  0.4h, r1  0.3h, and er1  10. For comparison, similar results obtained for the arrays without four parasitic elements are shown in Figure 3.22. Note that by introducing the parasitic elements, a new transmission band is formed around h/ 0  0.5 in the stopband of the square lattice of host cylinders without parasitic elements.

3.7.2 THREE-DIMENSIONAL SCATTERING CROSSED-ARRAYS

FROM

LAYERED

The power reflectance R00 of the fundamental space harmonic with l  m  0 vs. the normalized wavelength h/ 0 in 32-layered crossed-arrays of dielectric circular cylinders is plotted in Figure 3.23(a) for two typical polarizations of incident wave. The parameters are assumed to be hx  hz  h, r1  r2  0.25h, d  h, er1  er2  5.0. The polarization angles  i  90° and  i  0° correspond to the TE wave and TM wave with respect to the z axis, respectively. For this symmetrically layered crossed-array, the polarization dependency of reflection is suppressed within the indicated frequency range h/ 0  1.0, and there is no significant difference in the power reflectance for both polarizations. This is because the multiple scattering process between the orthogonal arrays is governed only by the fundamental space harmonic. For the frequency range h/ 0  1.0, the higher order space

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1 TMR 0

0.8

i

TER 0

Rl

0.6

0.4

0.2

TM

R−1

TMR

TE

R−1

TE

1

R1

0 0°

45°

90°

135°

180°

i

(a) 1

TM TE

0.8

R0

0.6

0.4

0.2

i

0 86° (b)

88°

90°

92°

94°

i

FIGURE 3.20 Power reflectance Rl of the array shown in Figure 3.19(a) vs. the incident angle i at the resonance wavelength h/ 0  0.961. Figure 3.20(b) is a close view of (a) around i  90°.

harmonics become also propagating waves and take part in the scattering process. In this case, there appears a polarization dependency of reflection even for the symmetric configuration. For comparison, the power reflectance obtained for 64-layers of only Z-array is shown in Figure 3.23(b). The frequency response

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Electromagnetic Theory and Applications for Photonic Crystals

h (x2, y2)

h 3

2 1 0

4

5

r1, 1

r2, 2

(a)

i = 90°

i = 60°

i = 30°

1

0.8

R0

0.6

0.4

0.2

0 0.2 (b)

0.3

0.4

0.5

0.6

h/0

FIGURE 3.21 Geometry of layered arrays with five circular cylinders per unit cell in free space and power reflectance R0 vs. normalized wavelength h/ 0. Small cylinders are placed around the host cylinder at four symmetrical positions, where r1  0.3h, er1  10, ri  0.05h (i  2, 3, 4, 5), eri  10, and x2  y2  0.4h.

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h h

r1, 1

(a) i = 90°

i = 60°

i = 30°

1

0.8

R0

0.6

0.4

0.2

0 0.2 (b)

0.3

0.4 h/λ0

0.5

0.6

FIGURE 3.22 Geometry of layered arrays with one circular cylinder per unit cell in free space and power reflectance R0 vs. normalized wavelength h/ 0, where r1  0.3h and er1  10.0.

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Electromagnetic Theory and Applications for Photonic Crystals i = 90° (or 0°) 1

R00

0.8

0.6

0.4

0.2

0 0.2

0.4

(a)

0.6 h/0

i = 90°

1

0.8

1

i = 0°

0.8

R00

0.6

0.4

0.2

0 0.2 (b)

0.4

0.6 h/0

0.8

1

FIGURE 3.23 Power reflectance of fundamental space harmonic vs. normalized wavelength h/ 0 for the normal incidence of plane wave with a polarization angle i  90° or i  0° on layered arrays of circular dielectric cylinders, where hx  hz  h, r1  r2  0.25h, d  h, er1  er2  5.0. (a) Power reflectance R00 of 32-layered crossed-arrays. (b) Power reflectance R0 of 64-layered Z-arrays.

© 2006 by Taylor & Francis Group, LLC

Modeling of Photonic Crystals

179 i = 90°

i = 0°

1

R0 0

0.8

0.6

0.4

0.2

0 0.2

0.4

0.6 h/0

0.8

1

FIGURE 3.24 Power reflectance R00 of fundamental space harmonic vs. normalized wavelength h/ 0 for the normal incidence of plane wave with a polarization angle i  90° or  i  0° on 32-layered crossed-arrays of asymmetric configuration, where r1  0.25h in the Z-arrays, r2  0.45h in the X-arrays, and other parameters are the same as those in Figure 2.23.

of the reflectance is quite different in both polarizations. These two numerical examples suggest that the layered crossed-arrays realize the polarization independent reflection characteristics with wider stopbands, compared with the layered parallel arrays. Figure 3.24 shows the power reflectance R00 of the fundamental space harmonic versus normalized wavelength h/ 0 for the normal incidence (i  90°) of plane wave with a polarization angle i  90° or i  0° on 32-layered asymmetric crossed-arrays, where hx  hz  h, r1  0.25h in the Z-arrays, r2  0.45h in the X-arrays, and er1  er2  5.0. The radius of cylinders in the X-array is larger than that in the Z-array. There appears a difference in the frequency response for the TM and TE polarizations. We consider another configuration of crossed-arrays in which a crossed-array of perfect conducting cylinders is periodically stacked over a crossed-array of dielectric cylinders. The two crossed-arrays, corresponding to 4-layered arrays, constitute a unit cell of the stacking sequence in the y direction. The power reflectance R00 of the crossed-arrays stacked over 32-unit cells is plotted in Figure 3.25 as functions of the normalized wavelength h/ 0 for the normal (i  90°) incidence

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Electromagnetic Theory and Applications for Photonic Crystals 1

0.8

R00

0.6

0.4

0.2

0.2 (a)

0.4 h/0

0.6

1

0.8

R00

0.6

0.4

0.2

0.2 (b)

0.4 h/0

0.6

FIGURE 3.25 Power reflectance R00 of the crossed-arrays stacked over 32-units vs. normalized wavelength h/ 0 for the normal (i  90°) incidence of a plane wave with polarization angle  i  45°. The one unit of arrays is formed by a crossed-array of conducting cylinders stacked over a crossed-array of dielectric cylinders, where hx  hz  h and dj  h commonly for all 128-layered arrays. r1  0.3h and er1  5.0 for the dielectric cylinders. The radius of the conducting cylinders is (a) r2  0.1h and (b) r2  0.02h. The dotted lines indicate the results for the crossed-arrays formed by a unit cell of 4-layered arrays.

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181

of a plane wave with polarization angle i  45°. The geometrical parameters are chosen such that hx  hz  h, dj  h commonly for all 128-layered arrays, er1  5.0, and r1  0.3h for the dielectric cylinders. The radius of the conducting cylinders is r2  0.1h in Figure 25(a) and r2  0.02h in Figure 25(b). Note that the power reflectance includes the reflected powers in both TM and TE waves. For comparison, the results obtained for the crossed-arrays formed by a unit cell of 4-layered arrays are also shown by the dotted lines. It is seen that the multilayered crossedarrays have several stopbands in which both TM and TE wave components are completely reflected. The locations and widths of the stopbands can be controllable by adjusting the radii of the cylindrical elements. The last example demonstrates a perfect stopband of layered crossed-arrays. One unit of crossed-arrays is formed by the Z-array and X-array consisting of identical cylinders with r1  r2  0.45h, er1  er2  10.0, hx  hz  h, and d  h. The power reflectance R00 for a plane wave with a polarization angle i  45° was calculated by varying the incident angle over 0°  i  45° and 15°  i  90°. The results of one unit of the crossed-arrays are shown in Figure 3.26(a). Only the fundamental space harmonic with l  m  0 becomes a propagating wave within the indicated wavelength range. The power reflectance in this case exhibits moderate wavelength responses that depend on the angle of incidence. Figure 3.26(b) shows the similar plots for 32-layered crossed-arrays. All of the plane waves with the indicated range of incident angles are perfectly reflected within 0.361  h/ 0  0.371. This wavelength band corresponds to a full bandgap of the layered crossed-arrays. The incident plane wave with a polarization angle i  45° contains equally both TM and TE wave components. In the case of layered parallel arrays, the bandgaps for TM and TE waves are formed at different wavelength bands. When the crossed-arrays are layered, on the other hand, the bandgaps in each of the parallel arrays are overlapped due to the multiple scattering between two orthogonal array elements, and a full bandgap is formed.

3.7.3 GUIDED MODES WAVEGUIDES

OF

TWO-DIMENSIONAL PHOTONIC CRYSTAL

We will now discuss the results of a modal analysis of two-parallel photonic crystal waveguides and a coupled cavity photonic crystal waveguide. When two identical photonic crystal waveguides are brought in close proximity, they form a directional coupler that can be used for various applications in integrated optics such as power division, switching, and wavelength or polarization selection. The characteristics of coupling are determined by the propagation constants and field distributions of two eigenmodes, an even mode and an odd mode, supported by the two-waveguides system. Their precise analysis is significantly important to designing the coupler. However, a coupled cavity waveguide is formed by removing particular lattice elements periodically along one entire row of a complete lattice. The defect acts as a resonant cavity, and the guided waves propagate through the coupling between adjacent cavities along the defect layer. The resonance

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Electromagnetic Theory and Applications for Photonic Crystals 1

0.8

R00

0.6

0.4

i = 45° 0°< i < 45° 15°< i < 90°

0.2

0.35

N=2

0.36

0.37

0.38

0.39

0.4

0.39

0.4

h/0

(a)

1

0.8

R00

0.6

0.4 i = 45° 0°< i < 45° 15°< i < 90°

0.2

0.35 (b)

N = 64

0.36

0.37

0.38 h/0

FIGURE 3.26 Power reflectance R00 of (a) one unit of crossed-arrays and (b) 32 units of crossed-arrays vs. normalized wavelength h/ 0 for plane waves with various different angles of incidence, where hx  hz  h, d  h, r1  r2  0.45h, er1  er2  10.0, and i  45°.

© 2006 by Taylor & Francis Group, LLC

Modeling of Photonic Crystals

183 y h

N

s

2t r, 

M

x

s

2t

d N

FIGURE 3.27 Coupled two-parallel photonic crystal waveguides separated by M-layered arrays of circular cylinders.

coupling makes the guided waves more dispersive than conventional photonic crystal waveguides in which one layer of lattice is entirely removed. The coupled waveguide is schematically shown in Figure 3.27. Two parallel identical waveguides with a width 2t  1.5h, formed by photonic crystals with a square lattice of circular air holes on a dielectric substrate, are separated by M rows of the lattice where h is the lattice constant. The number of rows of the lattice in the upper and lower regions is assumed to be N. The dielectric constant of the substrate is ers  12.25, and the radius of air holes is r  0.475h . Figure 3.28 shows the dispersion curves and field distributions of the lowest even and odd TE modes for three different numbers M of the separating layers, where the guided modes with (Ez, Hx, Hy) fields are termed TE modes with respect to the x direction of wave propagation. The field distribution is plotted for a wavelength h/ 0  0.255 as functions of y in the plane x  0.5h crossing the homogeneous background free space. The mode field patterns are rather complicated being compared with those of the conventional dielectric waveguides. The fields of both even and odd modes extend over five rows of the lattice outside of the guiding region and exhibit

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184

Electromagnetic Theory and Applications for Photonic Crystals 10

0.9 Even mode

5

x /h = 0.5

Even mode

h/0 = 0.255

0.8 Ez

h/2

Odd mode

−5

0.7

0.23

0

0.24

(a)

0.25 0.26 h/0

0.27

h/2

Even

h/2

Odd

= 0.775686 Odd mode

−10

= 0.767546

−5

0 y/h

(b)

5

10

0.9 Even mode

5

x/h = 0.5

0

−5

0.7

Even =

h/2

0.24

(c)

0.25 h/0

0.26

0.27

−10

= 0.772632

Odd

−5

0 y/h

(d)

0.9

0.771357 Odd mode

h/2

0.23

Even mode

h/0 = 0.255

0.8 Ez

h/2

Odd mode

5

10 Even mode

5

0.8

Even mode

0

−5

0.7

(e)

x/h = 0.5 h/0 = 0.255

Ez

h/2

Odd mode

0.23

0.24

0.25 h/0

0.26

0.27

−10 (f)

h/2

Even

h/2

Odd

= 0.772360 Odd mode

−5

= 0.771757

0 y/h

5

FIGURE 3.28 Dispersion curves and mode field distributions of the lowest even and odd TE modes in the two-parallel photonic crystal waveguides shown in Figure 3.12 for three different numbers M of the separating layers. The lattice elements are air holes of a radius r  0.475h; the dielectric constant of the substrate is ers  12.25; and the waveguide width is 2h  1.5h where h is the lattice constant.

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Modeling of Photonic Crystals

185

an oscillatory behavior as a consequence of confinement due to the cumulative Bragg reflections from the periodic lattice in the y direction. The coupling length of two waveguides is described in terms of the difference Even  Odd between the propagation constants of the even and odd modes. Since the mode field pattern changes as a function of x with period h as a nature of a periodic system, an effective power transfer from one waveguide to the other is obtained when the coupling length takes an integral multiple of h. Let the coupling length be lc  M1h where M1 is a positive integer. Then the phase-matching condition for a coupler is given by bEven (l0 )  bOdd (l0 ) 

(2 M 2  1)p M1h

(3.228)

where M2 is a nonnegative integer. A very accurate calculation of Even and Odd as functions of the wavelength 0 is required to realize the phase-matching condition (3.228). The rate of power transfer also depends on the field profiles of the even and odd modes. The transferred power takes a maximum when the sum and difference of field profiles in the even and odd modes coincide with those of the fundamental mode in each of two waveguides in isolation. We can seen that the mode field distributions obtained for M  3 well satisfy such a requirement. Figure 3.29 shows the cross section of a two-dimensional coupled-cavity photonic crystal waveguide. The lattice elements are alternately removed along

h

60°

rc, c

s

r, 

FIGURE 3.29 Cross section of a two-dimensional coupled-cavity photonic crystal waveguide with a triangular lattice.

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Electromagnetic Theory and Applications for Photonic Crystals

0.25

0.25

0.2

0.2

0.15

0.15

h/2

h/2

the center layer of a triangular lattice of circular cylinders. The train of a missing lattice periodically spaced with the distance 2h forms a coupled-cavity photonic crystal waveguide. The radius rc and dielectric constant erc of circular cylinders on the center layer may be different from those r and er of the arrays in the upper and lower layers. The dispersion curves of the lowest even and odd TM modes with (Hz, Ex, Ey) fields are plotted in Figure 3.30 for a triangular lattice of air holes on a background dielectric with ers  11.4, where r  0.25h, er  erc  1.0, and (a) rc  0.2h, (b) rc  0.25h, and (c) rc  0.3h. The guided modes of the coupled-cavity waveguide become more dispersive that those of the conventional photonic crystal waveguide [46]. The dispersion characteristics are controllable by adjusting the parameters of the lattice elements on the center layer.

r/h = 0.2

0.1

r/h = 0.25

0.1

Even mode

(a)

0.05

Odd mode

0.05 0

Even mode

0.22

0.23 h /0

0.24

0.25 (b)

Odd mode

0 0.22

0.23

0.24

0.25

h/0

0.25

h/2

0.2 0.15 r/h = 0.3

0.1 Even mode Odd mode

0.05 0 0.22 (c)

0.23

0.24

0.25

h/0

FIGURE 3.30 Dispersion curves of the lowest even and odd TM modes for the twodimensional coupled-cavity waveguide formed on a photonic crystal with a triangular lattice of circular air holes on a dielectric background with ers  11.40, where er  1.0 and r  0.25h for the cylinders in the upper and lower layers and erc  1.0 and (a) rc  0.2h, (b) rc  0.25h, and (c) r  0.3h for the cylinders in the center layer.

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3.8 CONCLUSIONS A semianalytical approach for analyzing the electromagnetic scattering and mode guidance by photonic crystals has been formulated using a model of layered periodic arrays of circular cylinders. The method is based on the T-matrix of circular cylinders located within a unit cell, the one-dimensional lattice sums for a periodic arrangement of cylindrical scatterers, and the generalized reflection and transmission matrices for a layered system. The lattice sums technique assumes an infinite periodic system for each of the layered arrays and a real wavevector parallel to the array plane. Due to this limitation, the method is not applicable to inhomogeneous photonic crystal waveguides that include bends, branches, and discontinuities along the wave propagation. Otherwise, the formulation is quite general and can be applied to various configurations of photonic crystals. The unit cell may include two or more cylinders that may be of dielectrics, conductors, gyrotropic materials, or a mixture thereof with different dimensions. The array elements in different layers may be different. The distance between each of the layered arrays may be arbitrary. The accuracy and efficiency of the approach has been tested through a substantial number of numerical examples that are presented in the referenced literature.

REFERENCES [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059–2062, 1987. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett., 58, 2486–2489, 1987. [3] A. Scherer, T. Doll, E. Yablonovitch, H.O. Everitt, and J.A. Higgins, eds., Special section on electromagnetic crystal structures, design, synthesis, and applications, J. Lightwave Technol., 17, 1928–2207, 1999. [4] A. Scherer, T. Doll, E. Yablonovitch, H.O. Everitt, and J.A. Higgins, eds., Minispecial issue on electromagnetic crystal structures, design, synthesis, and applications, IEEE Trans. Microwave Theory Tech., 47, 2057–2150, 1999. [5] V. Twersky, On scattering of waves by the infinite grating of circular cylinders, IRE Trans. Antennas Propagation, 10, 737–765, 1962. [6] K. Ohtaka and N. Numata, Multiple scattering effects in photon diffraction for an array of cylindrical dielectric, Phys. Lett., 73a, 411–413, 1979. [7] R. Petit, ed., Electromagnetic Theory of Grating, Springer-Verlag, Berlin, 1980. [8] G. Tayeb and D. Maystre, Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities, J. Opt. Soc. Am. A, 14, 3323–3332, 1997. [9] K. Ohtaka, T. Ueta, and K. Amemiya, Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods, Phys. Rev. B, 57, 2550–2568, 1998. [10] R.C. McPhedran, L.C. Botten, A.A. Asatryan, N.A. Nicorovici, C. Martijn de Sterke, and P.A. Robinson, Ordered and disordered photonic band gap materials, Aust. J. Phys., 52, 791–809, 1999.

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[11] M. Plihal and A.A. Maradudin, Photonic band structure of two-dimensional systems: the triangular lattice, Phys. Rev. B, 44, 8565–8571, 1991. [12] H. Benisty, Modal analysis of optical guides with two-dimensional photonic bandgap boundaries, J. Appl. Phys., 79, 7483–7492, 1996. [13] G. Pelosi, A. Cocchi, and A. Monorchio, A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam, IEEE. Trans. Antennas Propagation, 48, 973–980, 2000. [14] E. Popov and B. Bozhkov, Differential method applied for photonic crystals, Appl. Opt., 39, 4926–4932, 2000. [15] Y. Naka and H. Ikuno, Guided mode analysis of two-dimensional air-hole type photonic crystal optical waveguides, IEICE Tech. Rep., EMT-00-78, pp. 75–80, 2000. [16] M. Koshiba, Y. Tsuji, and M. Hikari, Time-domain beam propagation method and its application to photonic crystal circuits, J. Lightwave Technol., 18, 102–110, 2000. [17] C.P. Yu and H.C. Chang, Compact finite difference frequency-domain method for the analysis of two-dimensional photonic crystals, Opt. Express, 12, 1397–1408, 2004. [18] E. Özbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K.-M. Ho, Micromachined millimeter-wave photonic band-gap crystals, Appl. Phys. Lett., 64, 2059–2061, 1994. [19] R. Gonzalo, B. Martinez, C.M. Mann, H. Pellemans, P.H. Bolivar, and P. de Maagt, A low-cost fabrication technique for symmetrical and asymmetrical layer-by-layer photonic crystals at submillimeter-wave frequencies, IEEE Trans. Microwave Theory Tech., 50, 2384–2392, 2002. [20] A.N. Nicorovici and C.R. McPhedran, Lattice sums for off-axis electromagnetic scattering by grating, Phys. Rev. E, 50, 3143–3160, 1994. [21] K. Yasumoto and K. Yoshitomi, Efficient calculation of lattice sums for free-space periodic Green’s function, IEEE Trans. Antennas Propagation, 47, 1050–1055, 1999. [22] P.C. Waterman, New formulation of acoustic scattering, J. Acoust. Soc. Am., 45, 1417–1427, 1969. [23] WC. Chew, Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York, 1990. [24] H. Toyama, T. Iwasaki, and K. Yasumoto, Electromagnetic scattering from a dielectric cylinder with multiple eccentric cylindrical inclusions, Prog. Electromagnetics Res., PIER 40, 113–129, 2003. [25] H. Jia and K. Yasumoto, A new analysis of electromagnetic scattering from twodimensional electromagnetic band-gap structures, Proceedings of 2004 International Conference on Computational Electromagnetics and Its Applications, pp. PS-21–PS-24, Beijing, China, Nov., 2004. [26] L. Li, Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings, J. Opt. Soc. Am. A, 13, 1024–1035, 1996. [27] H. Jia, K. Yasumoto, and H. Toyama, Reflection and transmission properties of layered periodic arrays of circular cylinders embedded in magnetized ferrite slab, IEEE Trans. Antennas Propagation, 53, 1145–1153, 2005. [28] S.F. Helfert and R. Pregla, Efficient analysis of periodic structures, J. Lightwave Technol., 16, 1694–1702, 1998. [29] K. Yasumoto, H. Toyama, and T. Kushta, Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders

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[30]

[31]

[32] [33]

[34]

[35] [36]

[37] [38]

[39]

[40] [41]

[42]

[43]

[44]

[45]

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using lattice sums technique, IEEE Trans. Antennas Propagation, 52, 2603–2611, 2004. K. Yasumoto, T. Kushta, and H. Toyama, Reflection and transmission from periodic composite structures of circular cylinders: refinement of reflectance and transmittance, Proceedings of the Fourth International Kharkov Symposium on Physics and Engineering of Millimeter and Sum-Millimeter Waves, pp. 96–101, Kharkov, Ukraine June, 2001. K. Yasumoto and H. Jia, Three-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders, Proceedings of 2002 China-Japan Joint Meeting on Microwaves, pp. 301–304, Xi’an, China, April, 2002. G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 2 ed., 1995. K. Yasumoto and T. Ueno, Rigorous analysis of scattering by a periodic array of cylindrical objects using lattice sums, Proceedings of 1998 China-Japan Joint Meeting on Microwaves, pp. 247–250, Beijing, China, August, 1998. H. Roussel, W.C. Chew, F. Jouvie, and W. Tabbara, Electromagnetic scattering from dielectric and magnetic gratings of fibers: a T-matrix solution, J. Electromagnetic Waves and Appl., 10, pp. 109–127, 1996. T. Kushta and K. Yasumoto, Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell, Prog. Electromagnetics Res., PIER 29, 69–85, 2000. R. Lampe, P. Klock, and P. Mayes, Integral transforms useful for the accelerated summation of periodic, free-space Green’s functions, IEEE Trans. Microwave Theory Tech., MTT-33, 734–736, 1985. R.E. Jorgenson and R. Mittra, Efficient calculations of the free-space periodic Green’s function, IEEE Trans. Antennas Propagation, AP-38, 633–642, 1990. S. Singh and R. Singh, Application of transforms to accelerate the summation of periodic free-space Green’s functions, IEEE Trans. Microwave Theory Tech., MTT-38, 1746–1748, 1990. N.A. Nicorovici, R.C. McPhedran, and R. Petit, Efficient calculation of the Green’s function for electromagnetic scattering by gratings, Phys. Rev. E, 49, 4563–4577, 1994. A. Moroz, Exponentially convergent lattice sums, Optics Lett., 26, 1119–1121, 2001. K. Yasumoto, Generalized method for electromagnetic scattering by twodimensional periodic discrete composites using lattice sums, Proceedings of 2000 International Conference on Microwave and Millimeter Wave Technology, pp. P-29–P-34, Beijing, China, Sept., 2000. H. Toyama, K. Yasumoto, and H. Jia, Electromagnetic scattering and guidance by two-dimensional photonic bandgap structures, Proceedings of 27th General Assembly of URSI, Paper No. P0172, Maastricht, Netherlands, August, 2002. K. Yasumoto, T. Kushta, and K. Yoshitomi, Electromagnetic scattering by a periodic array of cylindrical objects embedded in a dielectric slab, Proceedings of 2000 International Symposium on Antennas and Propagation, pp. 1371–1374, Fukuoka, Japan, August, 2000. K. Yasumoto and H. Jia, Electromagnetic scattering from multilayered crossedarrays of circular cylinders, Proceedings of SPIE: Microwave and Optical Technology 2003, 5445, pp. 200–205, The International Society for Optical Engineering, August, 2003. K. Yasumoto and H. Jia, Modal analysis of two-dimensional photonic crystal waveguides using lattice sums technique, Proceedings of 2002 International Conference

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on Microwave and Millimeter Wave Technology, pp. 1063–1066, Beijing, China, August, 2002. [46] K. Yasumoto, H. Jia, and K. Sun, Rigorous analysis of guided mode, Proceedings of 2004 URSI International Symposium on Electromagnetic Theory, 2, pp. 739–741, Pisa, italy, May, 2004. [47] K. Yasumoto, T. Kushta, and H. Toyama, Reflection and transmission of electromagnetic waves by multilayered periodic arrays of cylindrical objects: role of evanescent space-harmonics, Proceedings of 2001 International Symposium on Microwave and Optical Technology, pp. 457–460, Montreal, Canada, June, 2001. [48] T. Toyama and K. Yasumoto, Electromagnetic scattering from periodic arrays of composite circular cylinder with internal cylindrical scatterers, Prog. Electromagnetics Res., PIER 52, 321–333, 2005.

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and 4 Simulation Optimization of Photonic Crystals Using the Multiple Multipole Program Christian Hafner, Jasmin Smajic, and Daniel Erni CONTENTS 4.1 4.2 4.3 4.4

Introduction and Overview ......................................................................191 Introduction to Photonic Crystal Simulation ..........................................193 Basics of the Multiple Multipole Program ..............................................194 Handling Periodic Symmetries while Using Periodic Boundary Conditions ..............................................................................198 4.5 Advanced MMP and MAS Eigenvalue Solvers ......................................200 4.6 Computation of Waveguide Modes in Photonic Crystals ......................206 4.7 Computation of Waveguide Discontinuities ............................................210 4.8 Sensitivity Analysis of Photonic Crystal Devices ..................................212 4.9 Optimization Based on the Sensitivity Analysis ....................................216 4.10 Achromatic 90° Bend ..............................................................................217 4.11 Filtering T-Junction ................................................................................218 4.12 Conclusions and Outlook ........................................................................222 Acknowledgment ..............................................................................................223 References ........................................................................................................223

4.1 INTRODUCTION AND OVERVIEW The multiple multipole program (MMP) is a powerful semianalytical technique that was developed during the last 25 years. Because of its accuracy and reliability it is especially well suited for numerical model optimization. This is highly 191

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important for the design of structures within photonic crystals because currently almost no useful design rules are available for devices embedded in photonic crystals. For the efficient analysis of photonic crystals several techniques were added to the original MMP codes. These include: 1. The modeling of periodic structures using the concepts of fictitious boundaries and periodic boundary conditions 2. Novel eigenvalue solvers based on fictitious excitations and weighted residual minimization techniques 3. So-called connections, a unique macro feature of MMP that allows one to accurately handle waveguide discontinuities without any absorbing boundaries 4. Eigenvalue estimation techniques and parameter estimation techniques for a drastic reduction of the computation time of frequency domain methods These techniques also are valuable in conjunction with other numerical methods. Beside the MMP features mentioned above, we present a promising alternative to the well-known supercell approach for the analysis of waveguides in photonic crystals. Our new approach is considerably faster than the well-known supercell approach, and it is more realistic at the same time. It allows us to compute not only the propagation constants of all modes but also the radiation losses of the waveguide modes in photonic crystals of finite size. Finally, we embed the MMP approximation of each mode of a waveguide into a connection that can be used exactly as an analytic expansion that describes the electromagnetic field of the mode everywhere. This combination of MMP with the mode matching technique allows us to model waveguide discontinuities efficiently and accurately. Furthermore, it avoids all problems associated with reflected waves at the output ports caused by absorbing boundary conditions. As a consequence, we can drastically reduce the model size and computation time. We then address the problem of fabrication tolerances, which has a strong impact on the quality of photonic crystal devices. With an extensive sensitivity analysis we explore the impact of the individual cell geometry (for example, radii and locations of the rods of a 2D photonic crystal) on device characteristics such as reflection and transmission coefficients. We finally take advantage of the sensitivity analysis in order to improve the device characteristics. The entire procedure is demonstrated using two relatively simple but interesting examples, the 90° bend and the filtering T-junction. We demonstrate that a 90° bend can be optimized so that almost zero reflection is obtained over a very broad frequency range that covers almost the entire bandgap. Furthermore, we outline a design procedure for filtering T-junctions that is based on the analysis of waveguide modes while avoiding time-consuming modeling optimization of the entire device. In both examples, we apply our sensitivity analysis for the optimization of these structures.

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4.2 INTRODUCTION TO PHOTONIC CRYSTAL SIMULATION In 1987 Yablonovich [1] proposed photonic crystals (PhCs) as an optical equivalent to semiconductors (i.e., in PhCs photons play essentially the role that electrons play in semiconductors). Although PhCs are rarely observed in nature, nanotechnology allows us to fabricate both 2D and 3D PhCs [2–4]. Pure PhCs are strictly periodic structures that exhibit photonic bandgaps [3,4] equivalent to the bandgap in semiconductors. Therefore, the bandgap computation is closely related to the computation of periodic structures, such as gratings. Since the waves in perfect PhCs are solutions of an eigenvalue problem, codes for computing band diagrams that describe a perfect PhC must be able to efficiently handle both periodic structures and eigenvalue problems. Since no photon can penetrate a PhC within the bandgap, a PhC of finite size totally reflects light within a limited frequency range that corresponds to the bandgap. Although this is an interesting effect, only the introduction of defects or cell modifications — corresponding to the doping of semiconductors — explains the high attention that is currently paid to PhCs in integrated optics. In the design of finite PhCs with modified cells, one has much more freedom than in the doping of semiconductors because nanotechnology allows one easily to modify the geometry of any particular cell of a PhC. For example, when the initial PhC consists of a regular array of circular rods, one can modify the radius and location and radius (or even shape) of each rod. Furthermore, tunable materials [5,6] allow one also to modify the material properties within some range. When identical cell modifications are introduced along a line in a PhC, one obtains simple waveguides. We will show that the characteristic properties of these waveguides may be varied in a wide range. The composition of different waveguides of finite length can then be used for more advanced structures such as filters. Although PhC waveguides are similar to standard waveguides, the proper computation of PhC waveguides is more demanding because the simple cylindrical symmetry of standard waveguides is replaced by periodic symmetry. As a consequence, the computation of PhC waveguide modes requires the proper handling of periodic symmetry in addition to the eigenvalue problem that defines the modes. Since these ingredients also are required for the computation of band diagrams, essentially the same codes may be used for both the computation of the band diagrams of perfect PhCs and for the computation of PhC waveguide modes. A more severe problem in the PhC waveguide computation results because the symmetry perpendicular to the direction of the waveguide is broken by the defect or cell modification that generates the waveguide. As a consequence, the PhC waveguide model is periodic along the waveguide but not periodic in the transverse direction. Thus, the theoretical waveguide model is infinite and must be either truncated or modified in such a way that the numerical effort remains finite. In the following, we will present two different techniques for solving this problem. Once waveguides have been introduced in a PhC, it is natural also to introduce waveguide discontinuities. Already the most simple waveguide discontinuities (i.e., waveguide bends) make the PhC concept very attractive for integrated optics for the

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following reason. Conventional waveguide bends in integrated optics either exhibit significant radiation losses or are large in size, which is the most important reason for the large extent of traditional optical chips. Within PhCs one can easily obtain sharp waveguide bends without any radiation loss and even with zero reflection coefficients [7]. In the following we will analyze and optimize a sharp 90° bend in a PhC. From the numerical point of view, the analysis of waveguide discontinuities is most demanding because one typically must model structures that are large compared to the wavelength and consist of many cells. First of all, the symmetry of the initial crystal is completely broken. Furthermore, the model must be appropriately truncated. Finally, a high accuracy is required because small variations of some model parameters may have unexpectedly big effects. This especially holds when structures are being optimized. We will show that optimization may render some parts of the PhC very critical or sensitive, which causes problems for the fabrication and may be attractive at the same time when one is interested in tuning or switching. Finally, the critical parts are attractive for the design of sensors. Because no design rules are currently known for the design of PhC waveguide devices, the only way to develop useful structures without overly stressing intuition is to combine numerical simulation with optimization. When this is done, it is most important to know that optimizers are heavily disturbed by inaccuracies of the forward solver that is used to analyze the PhC structure. We therefore prefer accurate numerical techniques that are close to analytic solutions but flexible enough for computing all kind of PhC structures. In the following, we outline the basics of the MMP, which may be considered a semianalytic method. We show how MMP can handle all important problems associated with PhC simulations, namely periodic symmetries, eigenvalue problems, and waveguide discontinuity problems. Finally, we use MMP for the detailed analysis of PhC structures and for the optimization of such structures. For reasons of simplicity, we focus on 2D PhCs. Beside much higher memory requirements, much longer computation time, and difficulties in the graphic representation, the handling of 3D PhC models provides no essential new problems (i.e., the fundamental procedures remain the same for 3D).

4.3 BASICS OF THE MULTIPLE MULTIPOLE PROGRAM Two-dimensional PhCs consist of infinite 2D arrays of either dielectric rods in free space (or some other material) or of holes (air or some other material) within a dielectric background material as illustrated in Figure 4.1. In order to obtain a photonic bandgap, the dielectric contrast (i.e., the ratio of the higher refraction index divided by the lower refraction index) must be sufficiently big. Typically, this ratio is near 3. For a numerical method, it does not matter whether the refractive index in the rod is bigger or smaller than in the surrounding medium, that is, if one has an array of rods or of holes. In the following, we will focus on the rod type PhCs.

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195 r

r

a

a
1


2

Hexagonal lattice

Square lattice

FIGURE 4.1 The perfect photonic crystal structure. On the left-hand side one can see the rods (holes) arranged in a hexagonal lattice; the right-hand side represents the crystal structure with a square lattice. The selected materials should be with a large permittivity contrast, i.e., 1/2 (or 2/1)  10.

First we consider a single rod. For a general shape, no analytic solution is available, but for a circular rod, we easily obtain an analytic solution from the following procedure: 1. Introduction of an appropriate coordinate system (i.e., a cylindrical coordinate system (r, w, z)) with origin r  0 in the center of the rod. In this coordinate system the boundary of the rod is simply defined as r  R, which will be essential for fulfilling the boundary conditions. 2. Formulation of the wave equations for the electromagnetic field inside and outside the rod using the cylindrical coordinate system. Because of dependences of the field components caused by the Maxwell equations, it is sufficient to explicitly solve the equations for the longitudinal components Ez and Hz only. That is, we have only two scalar field equations for both the domain D1 inside the rod and D2 outside. 3. Time separation or Fourier analysis leads to the standard frequency domain formulation with complex notation and time dependence described by exp(ivt). 4. Separation of the z dependence leads to the well known z dependence described by exp(igz), where g  b  ia is the propagation constant in z direction. 5. Separation of the w dependence leads to the angular dependence exp(inw). 6. Finally, one obtains the radial dependences Hn(i)(kj r), where Hn(i) denotes Hankel function of order n and kind i  1, 2. kj   k2j g 2 is the transverse wavenumber, whereas kj  v mj j denotes the wavenumber in the domain Dj that is described by the permittivity j and the permeability mj.

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We easily can assume that the parameters v, g, n, j, mj are complex because we use complex notation anyway. However, the time dependence is usually given by some light source that is most frequently characterized by a real angular frequency v. Furthermore, the material properties j, mj are complex only when material losses are present. Since these losses are usually avoided in optical structures, j, and mj are often assumed to be real. As a consequence, kj becomes real as well, and kj becomes either real or imaginary. Since the field must be 2p periodic in w direction, n must be an integer number. As a result, we obtain  (1, 2 ) (k r )exp(iw)ex p (i (gz  vt ))) E zj (r , w, z , t )  Re( AEj H n j

(4.1)

for the longitudinal component of the electric field and a similar formula for the magnetic field. Because of the singularity of the Hankel function H at the origin r  0, we call this an electric or magnetic multipole of order n with amplitude A. Note that two different kinds of Hankel functions (upper index 1 or 2) represent outgoing and incoming waves, respectively. When we consider a scattering problem, where the dielectric rod is illuminated by some incident plane wave, the scattered field outside the rod is simply a superposition of multipoles of various orders n of the outgoing wave type only. Inside the rod, we have standing waves (i.e., a superposition of incoming and outgoing waves). Since the total field may not be singular in the origin, the singularities of the incoming and outgoing waves must cancel each other at r  0. This leads to the introduction of Bessel functions Jn  (Hn(1)  Hn(2))/2

(4.2)

as radial dependence. We call this Bessel type of expansion rather than multipole expansion. Thus, the field inside the rod is a superposition of Bessel type expansions of order n  0, 1, 2, …, whereas the field outside the rod is a superposition of (outgoing) multipole expansions of order n  0, 1, 2, … . Note that the longitudinal component of the wave vector of the incident plane wave determines the propagation constant, that is, one has g  kzincident

(4.3)

Thus, only the amplitudes A are unknown parameters. When guided waves propagating along the cylindrical structure (i.e., in z direction) are considered, g plays the role of the eigenvalue and must be computed together with the amplitudes. Finally, the amplitudes of the electric and magnetic field Bessel expansions and of the electric and magnetic field multipole expansions must be computed so that the boundary conditions (continuity of tangential components of E and H) on the surface r  R of the rod are fulfilled for all angles w. This can be done analytically for the circular rod by an angular Fourier decomposition of the incident wave, which leads to small matrix equations for all orders (2 by 2 for n  0 and

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4 by 4 for n  0). For noncircular rods, no analytic solution is available (except for special cases). Instead of doing this with a pure numerical method, we can still approximate the field inside the rod by a truncated sum of Bessel expansions of orders 0, 1, 2, …, N and outside the rod by a truncated sum of multipoles of orders 0, 1, 2, …, N. We call this simply Bessel expansion and multipole of maximum order N. Since we have two angular dependences for each order n  0 and only one for n  0, we have 1  2N unknowns for both the electric and magnetic expansions (i.e., a total number of 2  4N unknowns). Imposing at least 2  4N continuity equations on the surface of the rod, we obtain a matrix equation that is characterized by a dense matrix. This procedure corresponds to the standard boundary method that can easily be generalized so that more complicated problems may also be solved in an approximate way. The multipole-Bessel approach is very close to the analytic approach and may therefore be called semianalytic. Despite this, there are several difficulties that are not obvious at all. First, it is quite difficult to prove the completeness of this approach. A rigorous proof has been given by Vekua [8]. From this, we know that we have a complete approximation basis when the boundary is Hoelder-continuous, but we do not know how well it converges and where we optimally place the origin of the cylindrical coordinate system. It seems that one can formulate theorems on the convergence similar to the much more simple Fourier series of periodic functions. Essentially, the convergence depends on the number of derivatives of the boundary that are continuous. When infinitely many derivatives are continuous, exponential convergence is obtained. Although this looks nice at first sight, it does not guarantee that the maximum order N required for obtaining a reasonable accuracy is small enough for numerical treatment. When analyzing, for example, an elliptical rod with large aspect ratio, this approach turns out to be inefficient despite the exponential convergence. The most fundamental idea of the MMP [9] is to use more than a single multipole expansion of maximum order N for modeling the field in any domain as illustrated in Figure 4.2. Because of the summation theorems for Bessel and Hankel functions, neighbor multipole expansions may cause numerical dependencies that lead to ill-conditioned matrices. At first sight, it seems reasonable to avoid ill-conditioned matrices, but when reasonable matrix solvers are applied, much more accurate results may be obtained as illustrated in [9]. Furthermore, reasonable strategies for the distribution of multipoles have been developed [9]. However, to obtain a good MMP code, the derivation of the matrix equation from the boundary conditions is crucial. In our MaX-1 implementation [10] some sort of generalized point matching technique with properly weighted residuals is used. This leads to an overdetermined system of equations. The direct solution of this system is essential. An interesting way to reduce problems with proper multipole settings is to work with zero order multipoles only. This is done in several methods, namely the method of auxiliary sources (MAS) [11], method of fictitious sources [12], and charge simulation techniques [13,14]. When one smoothly distributes monopoles along the boundaries, one can easily construct appropriate matching points along

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Ω2

FIGURE 4.2 The basic idea of the MMP field computation in the case of the domain with an arbitrary geometry. The multipolar expansions denoted by  and by O are used for the field modeling in the domains 1 and 2, respectively.

the boundary so that standard point matching leads to a square matrix system that may be solved with standard methods after some simple regularization. MAS codes usually distribute the monopoles along auxiliary lines that are obtained from conformal mapping procedures. This leads to an optimal monopole placement for sufficiently simple geometries. As long as PhCs consisting of circular rods are considered, the multipole-Bessel approach seems sufficient because it is optimal to place one polar coordinate system in the center of each rod for expanding the field outside the rods with a multipole expansion and the field inside with a Bessel expansion. Nonetheless, we will introduce noncircular fictitious boundaries in the following because we must truncate the infinite PhC model to a finite size. This is shown in the following sections.

4.4 HANDLING PERIODIC SYMMETRIES WHILE USING PERIODIC BOUNDARY CONDITIONS When we consider a perfect 2D PhC (see Figure 4.1), we have an infinite structure consisting of infinitely many cells. No numerical method can handle such a structure directly. A simple but inefficient way is to consider a finite PhC; that is, to truncate the model and to increase the model size until convergence is obtained. For an efficient treatment, the periodic symmetry should be considered. One way is to use Floquet theory and to replace the multipole expansions by appropriate periodic sets of multipoles. This approach is rather difficult because the convergence of multipole arrays is rather bad. Furthermore, this approach is not very

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2E−6

2E−6

1E−6

1E−6

0

0

0

1E−6

2E−6

2E−6

0

1E−6

2E−6

2E−6 Periodic boundary

1E−6 Original cell

Periodic boundary

1E−6 Original cell

0

Fictitious excitation 0

1E−6

Bessel, and multipole expansions

2E−6

0

Fictitious excitation 0

1E−6

2E−6

Bessel, and multipole expansions

FIGURE 4.3 The MMP application to the PhC periodic structures. The infinite PhC structure is truncated by introducing fictitious periodic boundaries, and the original cell is defined. The crystal structure with a square lattice is presented on the left and hexagonal lattice on the right-hand side of the figure. The cell size and shape are determined by the lattice constant a and the type of lattice, respectively.

flexible. The following approach is much simpler and very flexible. It can be applied easily to arbitrary periodic structures (PhCs as well as gratings, antenna arrays, etc.). First we truncate a single cell of the PhC with arbitrary fictitious boundaries as shown in Figure 4.3. We call this the original cell in conjunction to the PhC lattice’s primitive cell. In most cases, the fictitious boundaries are straight lines. When one knows that the resulting field is complicated in some region, it is reasonable to define the fictitious boundaries in such a way that they do not cut the problematic region. Otherwise, the numerical effort would be increased. For example, the fictitious boundaries might cut a rod of the PhC. Since we know that the field is most complicated on the surface of the rod, it is better to avoid this. In 2D applications that are periodic in two different directions, the original cell essentially has four different fictitious boundaries: left, right, upper, and lower. The shapes of the left and right boundaries must be identical because the right boundary is simply obtained from the left by a translation with a displacement vector that describes the periodicity in one direction. Without any loss of generality, we can assume that this is the x direction. Similarly, the upper boundary is obtained from the lower boundary with a translation in a direction different

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from x but not necessarily perpendicular to x. Therefore, we need to explicitly define only two of the four fictitious boundaries together with the two displacement vectors (primitive vectors) that describe the periodicity. For example, we define only the left and upper boundaries of the original cell. For each point Pleft(x, y) on the left boundary we have a corresponding point Pright(xa, y) on the left boundary and for each point Ptop(x, y) on the top boundary we have a corresponding point Pbottom(xbx, yby) on the bottom boundary. For the orthogonal case, we have bx  0 and by  b. From Floquet theory or from simple Fourier analysis, we know that the field satisfies the following conditions: Field(Pright)  Field(Pleft)exp(iaCa)

(4.4)

Field(Ptop)  Field(Pbottom)exp(ibCb)

(4.5)

where Ca and Cb are characteristic numbers that become complex when losses are present either inside or outside the rods. b is the length of the second displacement vector with the Cartesian components (bx, by). Note that the periodicity conditions hold for all coordinates of the electromagnetic field; that is, we obtain six conditions for each point on the fictitious boundaries. When we impose such periodicity conditions on a fictitious boundary, we call this a periodic boundary. Since MMP works with overdetermined matrix equations, the number of periodicity conditions is not important. For codes with square matrices, it would be necessary to use only four periodic boundary conditions. Once the original cell is defined, one needs to know only the field inside this cell. The field in all other cells can be easily computed from the periodicity conditions. The most important thing now is that the domain outside the rod no longer extends to infinity because the original cell is finite. Obviously, we can model the field inside the rod as before with a Bessel expansion. The domain outside the rod is no longer unbounded. Therefore, we do not have only waves propagating away from the rod that can be modeled with a multipole located in the rod. We also have incoming waves from the rods outside the original cell entering it through the fictitious boundaries. To simulate these waves, it is not necessary to define a multipole expansion for each rod. It is sufficient to distribute either a few multipoles around the fictitious boundary or to add another Bessel expansion that takes into account the waves coming from outside. The reliability of this approach can be deduced from the summation theorems of Bessel functions [15]. Therefore, we can model the field inside the original cell with only three sets of expansions: a multipole and a Bessel expansion for the field outside and a Bessel expansion for the field inside the rod.

4.5 ADVANCED MMP AND MAS EIGENVALUE SOLVERS Perfect PhCs are characterized by band diagrams that essentially show the dispersion relations; that is, the modes that can propagate in a PhC depending on the

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frequency (or energy or wavelength) and on the characteristic constants Ca and Cb that describe the periodicity of the field. In fact, we can consider the frequency as the eigenvalue that depends on the propagating modes as functions of Ca and Cb. This means that we have an eigenvalue problem very similar to the eigenvalue problem describing a resonator. However, the dependence on the characteristic constants Ca and Cb makes the problem more delicate. These constants span the so-called reciprocal lattice space that is real when no losses are present, as illustrated in Figure 4.4. Because of the periodic symmetry, it is sufficient to consider only a small section of the reciprocal lattice space, the so-called first irreducible Brillouin zone (IBZ). This considerably reduces the computational effort, but the computation and visualization of eigenvalues that depend on two variables remains quite demanding. Fortunately, the maximum and minimum values of each eigenvalue are always found on the border of the IBZ, which usually forms a triangle connecting the three high symmetry points of the IBZ. Usual band diagrams therefore only show the eigenfrequencies along the border of the IBZ, as shown in Figure 4.5, where , M, and X are the aforementioned corners of the IBZ in Figure 4.4.

Original lattice

Reciprocal lattice f1 = 2

e2 e1

e1 = e2 = a

f2 = 2

f3 = 2

e2 × e3 . e1 (e2 × e3)

f2

e3 × e1

f1

e1. (e2 × e3) e1 × e2 e1. (e2 × e3)

f1 = f2 =

2 a

Reciprocal lattice space

M Irreducible 1st BZ Γ

X

1st Brillouin zone (1st BZ)

FIGURE 4.4 The reciprocal lattice space. The vector relations for the calculation of the basis vector of the reciprocal lattice space are given. The first Brillouin zone is marked, and the irreducible part of the first BZ is separated by considering the high symmetry points.

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Electromagnetic Theory and Applications for Photonic Crystals 0.99 0.90 0.81 Bandgap 2

0.72 a /2c

0.63 0.54 0.45 0.36

Bandgap 1

0.27 0.18 9E−2 0

Γ

X

M

Γ

FIGURE 4.5 The band structure of the perfect crystal is presented. The crystal structure is formed by using dielectric rods with relative permittivity   11.56 and radius r  0.18 arranged in a square lattice with lattice constant a  1m. The first six modes are calculated, and the bandgaps (more than one) are present.

Note that the eigenvalue problem associated with guided waves along cylindrical waveguides lies somehow between the simple resonator problem (where the eigenfrequency does not depend on any characteristic constant) and the 2D PhC problem (where one has two characteristic constants). Here one can either consider the frequency as a function of the propagation constant or the propagation constant as a function of the frequency — which is more convenient for engineers. However, we assume that the reader is familiar with band diagrams and guided waves and focus on the methods for solving eigenvalue problems using matrix methods. When one is working with square matrices S, an eigenvalue problem with eigenvalue e typically leads to a homogeneous matrix equation of the form S(e)X(e)  0

(4.6)

whereas a scattering problem leads to SX  B

(4.7)

where S is the system matrix, X the eigen vector, and B the vector that describes the excitation. Beside the uninteresting trivial solutions X  0, the eigenvalue problem has one or several nontrivial solutions if the determinant of the system matrix S in (4.6) vanishes. From the condition Det(S(e))  0

© 2006 by Taylor & Francis Group, LLC

(4.8)

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one can therefore iteratively find the eigenvalues e of the nontrivial solutions and after this the eigenvectors X(e). That is, one can consider f1(x)  Det(S(x))

(4.9)

as a nonlinear function of x and search for its zeros. As soon as a zero is found, it defines at least one eigenvalue. It is well known that this procedure implies strong numerical problems. Furthermore, it cannot be used when one is working with overdetermined systems characterized by rectangular matrices. In this case, we obtain matrix equations of the form R(e)X(e)  E(e)

(4.10)

where R is the rectangular matrix and E is the error vector to be minimized. It is well known that the multiplication of (4.10) with the adjoint matrix R* leads to a symmetric square matrix system of the form S(e)X(e)  R*(e)R(e)X(e)  R*(e)E(e)

(4.11)

that minimizes the square norm of the residual vector E, when we set S(e)X(e)  R*(e)R(e)X(e)  0

(4.12)

However, because we are aware of the numerical problems of the determinant technique and because we want to avoid squaring the condition number of R by computing R*R for obtaining (4.11) and (4.12), we look for a method to solve (4.10) directly in such a way that the square norm of E is minimized. It is natural to hope that the condition Norm(E(e))  minimum

(4.13)

might replace (4.8). Therefore, we define a new function f2(x)  Norm(E(x))

(4.14)

and search for its minima. In fact, this technique works only in simple cases. In general, it is disturbed by almost trivial solutions that may appear. Such solutions are characterized by spurious field values. In order to avoid this, one can define some field amplitude and search for the minima of f3(x)  Norm(E(x))/Amplitude(x)

(4.15)

Proper but numerically expensive definitions of the amplitude are integrals over the energy density (in the original cell of a PhC or in a resonator) over the time average of the Poynting flux along the z axis of a waveguide. In most cases, an

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Electromagnetic Theory and Applications for Photonic Crystals 1 0.1

n=5

1E−2 1E−3

1

1E−6

fn (e) = 1n f2 1 f1(e) = A(e)

1E−7

f2(e) = E(e)

1E−4 1E−5

n=2

f1(e) =

1E−8

1 A(e)

1E−9 1E−10 1E−11 2.136E14

2.139E14 2.142E14 Frequency (Hz)

2.145E14

FIGURE 4.6 Different definitions of the eigenvalue tracing function. A perfect PhC made by dielectric rods arranged in a square lattice was treated. The twin-minima phenomenon can be suppressed by an appropriate selection of the tracing function.

extremely rough approximation of these integrals turns out to be sufficient (i.e., one usually can use a single or a few test points) [9]. This is because the amplitude definition shall only avoid almost trivial solutions. Figure 4.6 shows the different definitions of the tracing function f(x) for a perfect PhC consisting of dielectric rods arranged in a square lattice. Up to now, we have described a procedure that is tantamount to the standard MMP eigenvalue solver. In the MAS, one usually works with a square matrix. Therefore one could work with the determinant. However, in order to avoid numerical problems associated with the determinant, an alternative method is used. This method essentially mimics the way measurements of resonators are done. A perfect resonator is energetically closed. For the measurement, the cavity must be accessed. That is, one has to introduce a port that connects some generator and a port that connects some voltage or power meter. In the MAS, these two ports are two points within the resonator: a point Pin where a fictitious field source (monopole) is introduced and a test point Pout where the field is measured. Thus, one essentially defines an amplitude in the test point that is similar to the most rough amplitude definition in the MMP eigenvalue solver. Now, the amplitude in the test point is a spectral function associated to the eigenvalue (see Figure 4.6) where each peak defines the location of an eigenvalue. The MAS eigenvalue solver easily can be used also within other codes, namely within MMP. The numerical detection of the peaks of g4(x)  Amplitude(Pin, Pout, x)

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(4.16)

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can essentially use the same numerical techniques as the search for the minima of the MMP eigenvalue search function (4.15). Obviously, one can also search for the minima of f4(x)  1/g4(x)  1/Amplitude(Pin, Pout, x)

(4.17)

A detailed analysis [16] shows that g4 exhibits a strange double-peak phenomenon when the resolution of the function representation is sufficiently fine. As a consequence, f4 exhibits a twin minimum instead of a single minimum for each eigenvalue. This phenomenon strongly disturbs the automatic numerical search routines as soon as the search step is fine enough. Since f3 does not exhibit twin minima, it seems to be better than f4. When comparing the two definitions, we see that f4 depends on Pin (i.e., on the location of the excitation), whereas f3 does not use any excitation. A detailed analysis of the procedure shows that one of the MMP expansion functions implicitly plays the role of the excitation. In the current MMP eigenvalue solver, the last MMP expansion plays this role. It may happen that such an excitation cannot excite all modes when it is placed at an inappropriate position. This mainly is observed when a mode exhibits special symmetries that are different from the symmetry properties of the fictitious excitation in the MAS solver or different from the symmetry properties of the last expansion in the MMP solver. By a reasonable selection of the point Pin, one easily can avoid this in the MAS case. Typically, it is sufficient to place Pin at some random position. For f3, this is more difficult because of the excitation’s implicit definition. The main question here is whether the double-peak phenomenon or the eigenvalue suppression by the inappropriate excitation is disturbing more. For the band diagram computation of PhCs, we would like to obtain a method that is fully automatic, accurate, efficient, and robust at the same time. Therefore we cannot accept unintended mode suppression, nor can we accept the problems caused by the double-peak phenomenon. Thus, we have implemented a sophisticated eigenvalue search procedure that combines f2, f3, and f4 and simultaneously traces these three functions for finding the full set of eigenvalues [16]. While computing the band diagram for a PhC, each of the eigenvalues turns out to be a function of the characteristic constants Ca, Cb; that is, we have e(Ca, Cb). For each point (Ca, Cb) in the reciprocal lattice space, we must iteratively find e from the analysis of the eigenvalue search functions f2, f3, and f4. This is numerically demanding. When knowing e in some point (Ca, Cb), one can use this information as a starting value for the eigenvalue search in a neighbor point (Ca  da, Cb  db) and if (da, db) is sufficiently small, only a few iterations will be needed to obtain e(Ca  da, Cb  db). As soon as we know e in several points, we can estimate e in a neighbor point even more precisely by an appropriate higher order extrapolation. We call this procedure the eigenvalue estimation technique (EET), or parameter estimation technique (PET) [9]. The PET makes the MMP computation of band diagrams highly efficient. The overall procedure is illustrated in Figure 4.7.

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Electromagnetic Theory and Applications for Photonic Crystals 0.8 0.7 Rough search 0.6

fa/c

0.5

EET

EET

0.4 Rough search

0.3

Rough search

0.2 0.1 0

EET Γ

X

EET

EET M

EET Γ

FIGURE 4.7 The perfect PhC band structure computation using MMP is depicted. The position along the x-axis of the band diagram is determined by periodic boundary conditions in the X and Y directions (square lattice) defined by (Ca, Cb). At this position a rough eigenvalue search is performed over a wide frequency range (of our interest) and those computed eigenfrequencies are the initial data for the eigenvalue estimation technique (EET), or advanced parameter estimation technique (PET), for the calculation of the rest of band diagram.

4.6 COMPUTATION OF WAVEGUIDE MODES IN PHOTONIC CRYSTALS As shown in Figure 4.8, a waveguide in a PhC is periodic in the direction of the wave propagation, but the periodic symmetries in all other directions are broken by the waveguide channel. Therefore, waveguides in 2D and 3D PhCs are periodic in only one direction and extend to infinity in all other directions. The latter causes severe numerical problems. The most frequently used “supercell” approach [3,4] removes these problems by replacing the single waveguide with an infinite set of identical parallel waveguides as also illustrated in Figure 4.8. As a consequence, one again obtains periodic symmetry in the same directions as the original perfect PhC, but now the size of the original cell increases along with the distances between the neighboring waveguides. One may assume that the modes of the single waveguide and of the corresponding array of parallel waveguides become identical when the distances between neighboring waveguides are sufficiently large. As soon as this is guaranteed, the supercell computation proceeds exactly as the computation of the band diagram of the perfect PhC. The numerical costs for the supercell evaluation rise with increasing distances between neighboring waveguides not only because the size of the supercell increases but also because the number of modes in the band diagram increases, as

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Periodic boundary conditions y

x

9a Propagation direction

a

Supercell

FIGURE 4.8 The 2D PhC waveguide. The waveguide is realised by introducing a vacancy line defect (substitutional line defect is also possible). The supercell method is depicted here.

illustrated in Figure 4.9. This renders the supercell approach rather inefficient. From a device perspective, the assumption of arrays of parallel waveguides is not realistic. In fact, the assumption of any infinite PhC is not realistic as well, and hence, a truncated PhC model is mandatory; however, such a model also includes coupling problems at the ports of the waveguides. To study the properties of the waves propagating along the waveguide, we therefore assume better that the waveguide is infinite in the propagation direction and we only truncate the model perpendicular to this direction, as shown in Figure 4.10. In this model, we have PhC “walls” of finite size on both sides of the waveguide. Such walls do not perfectly confine the light. As a consequence, the resulting finite size PhC waveguide will exhibit some radiation loss, which makes the computation more complicated. Note that this radiation is responsible for the coupling between neighboring waveguides in the supercell approach. When we consider, for example, a waveguide consisting of a single defect line (i.e., a W1 waveguide) within a 2D PhC with a square lattice with lattice constant d, the distance D between neighbor waveguides required for reasonable supercell results is D  (n  1)d

(4.18)

where n is an integer number. The original supercell then is a rectangle with the side lengths D and d. While truncating a waveguide model to n/2 PhC layers on each side of the waveguide, we essentially obtain the same model size for both models.

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Electromagnetic Theory and Applications for Photonic Crystals y

0.48 0.46

Bandgap

0.44 0.42 0.40 fa/c

x

9a

0.38 0.36 0.34 0.32 0.3

Bandgap

0.28 0.26 Γ

a

X

ky kx

2 Γ

9a

X

2/a

FIGURE 4.9 The original cell, the irreducible Brillouin zone, and the band structure of the 2D PhC waveguide. We have a defect mode covering almost the entire bandgap and a bunch of modes above and below the bandgap. Radiation

Periodic boundary conditions y 9a

x

Propagation direction

Supercell

Radiation a

FIGURE 4.10 The direct approach for PhC waveguide eigenvalue analysis. The fictitious periodicity in the Y-direction is removed, and radiation losses are introduced. Consequently, the eigenvalue problem becomes complex.

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Arbitrary fictitious boundaries have to be introduced in order to laterally truncate the PhC waveguide model. Therefore, one can use the upper and lower periodic boundaries of the supercell as truncation lines. Thus, the supercell model and the truncated model may look very similar, but there is an important difference: multipoles usually are placed above the upper and below the lower boundary of the supercell. These multipoles approximate the field incident from the upper and lower parallel waveguides. In the truncated model, no such multipoles are present because no energy is incident from top and bottom. But now energy is leaving the original cell through the top and bottom boundaries, and the field outside the truncation lines therefore is not zero. This radiating field is modeled with multipoles inside the original cell or with Rayleigh expansions that are well known from grating computations. As a result, the MMP matrix for a supercell computation has the same size as the MMP matrix for a truncated model computation. Although the supercell model and the truncated model are similar, a fundamental difference in the formulation of the eigenvalue problem makes the truncated model much more efficient and more practical: the supercell model analyzes the band diagram and introduces the waveguide modes as new modes in this diagram within the bandgap of the original perfect PhC. The truncated model does not compute any band structure. It considers a periodic waveguide structure with radiation losses. When we assume that x is the waveguide direction and d accounts for the periodicity in this direction, we know from Floquet theory that the complex amplitudes of the electromagnetic field satisfy the condition Field(x, y, z)  Field(x  nd, y, z) exp(ind C)

(4.19)

where C is a complex constant that plays the role of the propagation constant g of a cylindrical waveguide structure, where one usually has Field(x, y, z)  Field(x  x0, y, z) exp(ix0g)

(4.20)

Regarding the dispersion in cylindrical waveguide structures, one usually considers g as an eigenvalue and plots its frequency dependence g(v). Within the same picture, we can compute C(v) or the inverse function v(C). When neglecting the imaginary part of C that describes the damping due to the radiation loss, i.e., considering v(Real(C)), we essentially obtain the function that describes the mode in the band diagram from the X to the point (along the border of the irreducible Brillouin zone) in the supercell model as depicted in Figure 4.9. Using the complex eigenvalue search procedure, one easily obtains the complex values of C leading to the frequency dependence of the radiation loss (Figure 4.11) as a function of the number of confining PhC layers on both sides of the waveguide channel (Figure 4.12). Because a complex eigenvalue search is considerably more demanding than the real one, it is important to take advantage of sophisticated techniques such as the PET mentioned earlier. For small radiation loss, one can easily start with a real eigenvalue search that neglects losses. From the resulting field, one can estimate

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Electromagnetic Theory and Applications for Photonic Crystals 2E6

fa/c=0.340 Radiation

1.8E6 1.6E6

Periodic boundary conditions

9a

1.2E6

Cx im

y

1.4E6

x

1E6 8E5 6E5 4E5 2E5

a

Radiation

f = 1.125E14

0 0

Time-average poynting vector

2E5

4E5

6E5

8E5

1E6 1.2E6 1.4E6 1.6E6 1.8E6 2E6

Cx re

FIGURE 4.11 The complex eigenvalue analysis of the 2D PhC waveguide. Left: direct approach and radiation losses; center: time-average Poynting vector field at the given frequency; right: eigenvalue tracing function in the periodic constant (C) complex plane at the given frequency. The eigenvalue is characterized by the black spot near the bottom. When the frequency is modified, it moves along the gray line.

the radiation loss from the integral of the Poynting vector field over the truncation boundary and over the periodic boundaries [17].

4.7 COMPUTATION OF WAVEGUIDE DISCONTINUITIES Waveguide discontinuities break all periodic symmetries of the perfect PhC. Such structures can only be approximated in very simple cases by the supercell method (i.e., by a periodic continuation in all periodic directions of the original PhC). Since supercell models of waveguide discontinuities become even much less efficient than supercell models of PhC waveguides, it seems natural to use models that are truncated in all corresponding directions. This approach is frequently applied, namely in FDTD and other domain methods [18]. The truncation of a waveguide port is quite difficult and cannot be done accurately using standard absorbing boundary conditions [19]. Typically, truncated models must contain sufficiently long accessing waveguide sections to cope with the mismatch at the waveguide ports. As a consequence, even simple waveguide discontinuity problems become rather bulky containing a huge number of cells of the original PhC. Thus, the system matrices and memory requirement also become huge, and the computational costs become nearly unbearable especially within an optimization environment. The waveguide model presented in the previous section used a special truncation with a fictitious boundary. Outside this boundary, the field was modeled with an analytic set of expansions (multipoles or Rayleigh expansions). Similarly, we may introduce fictitious boundaries to truncate a waveguide discontinuity model and model the field outside these boundaries using the field of the waveguide

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Simulation and Optimization of Photonic Crystals 0.44

1 Layer 2 Layers

0.42

3 Layers 4 Layers

0.40

fa/c

211

0.38 0.36 0.34

4 3 2

1

0.32 0

3E5

6E5

9E5 1.2E6 1.5E6 1.8E6 2.1E6 2.4E6 2.7E6 3E6

CX RE 0.44

1 Layer 2 Layers

0.42

3 Layers 4 Layers

fa /c

0.40 0.38 0.36 0.34

3

2

1

4

0.32 0

6E4

1.2E5

1.8E5

2.4E5

3E5

3.6E5

4.2E5

4.8E5

CX IM fa /c =0.33

1

fa /c =0.33

2

fa /c =0.33

3

fa /c =0.33

4

FIGURE 4.12 The complex periodic constant with respect to the frequency and the number of surrounding crystal layers as a parameter. Top: real part of the periodic constant for different numbers of crystal layers; center: imaginary part of the periodic constant for different numbers of crystal layers; bottom: time-average Poynting vector of 2D photonic crystal waveguides for different numbers of crystal layers at the given frequency.

modes in the different ports as illustrated in Figure 4.13. For doing this, the MMP code contains a unique feature called connections [9]. Connections allow one to use solutions retrieved from another model as a new MMP expansion that is inserted into the new MMP model. This is possible because MMP solutions are linear superpositions of known solutions of Maxwell equations within homogeneous domains. Because of the linearity of Maxwell’s equations, the MMP solutions are

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Electromagnetic Theory and Applications for Photonic Crystals Fictitious boundary

E

Fictitious boundary

Waveguide

Discontinuity

Waveguide

T

R

D

D

FIGURE 4.13 The waveguide discontinuity treatment using MMP. The model is truncated using fictitious boundaries in order to perform a field matching between the structure and eigenfields of the modes of the connected waveguides. Distance D should be large enough to prevent an evanescent field produced by the defect from reaching a fictitious boundary.

again solutions that may be used as expansions for MMP. Obviously, connections may be nested (i.e., a connection may contain another connection). However, MMP allows us to pack any guided and any evanescent mode of an infinite waveguide into a connection and to use such connections for modeling the field outside the truncation boundaries. This technique is closely related to the mode matching technique [20] that is frequently used for standard waveguide discontinuities. It has been used successfully for MMP computation of standard waveguide discontinuities [9,10] and of PhC waveguide discontinuities [21,22]. In the following, we present some simple but illustrative examples. The approximation of the field outside the truncation boundary with connections allows one to minimize the computational domain. Therefore, the MMP matrices describing waveguide discontinuities are very small compared to the matrices obtained from other methods. A simple and intuitive alternative technique has been proposed within the framework of MAS. As in the MAS eigenvalue solver, the MAS model essentially mimics a measurement situation, where a reference plane is defined in each waveguide port and standing waves are observed along the accessing waveguides. For the observation of the standing waves, some section of each port must be modeled explicitly. Therefore, the MAS model always becomes larger than the MMP model. However, excellent agreement of the results is obtained [23], which is important for the validation of our results.

4.8 SENSITIVITY ANALYSIS OF PHOTONIC CRYSTAL DEVICES When we consider any waveguide discontinuity in a PhC, it is intuitively clear that the cells near the discontinuity have a stronger influence on the characteristic

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1.2

fa /c = 0.42

R +T

1.0

T 0.8 0.6 0.4 0.2

R 0 0.34

0.36

0.38

0.40

0.42

0.44

a /2c

FIGURE 4.14 The 90° PhC waveguide bend. Left: time-average pointing vector at the given frequency; right: frequency response within the first photonic bandgap of the power reflection R and power transmission T. At certain frequencies, one obtains ideal performance; that is, we have no power reflection at all (R  0%), and power transmission is maximum (T  100%).

properties of the discontinuity than do the cells far away, but it is not clear how great the influence of some cells is or how much fabrication tolerances affect the characteristic properties. For example, having a 90° bend in a defect waveguide within a PhC consisting of circular posts on a square lattice (see Figure 4.14), we know [7] that zero reflection may be obtained at some frequency, but when we modify the radius or the location of any of the rods, we do not know if the spectral response of the reflection coefficient is affected. Therefore, it might happen that we obtain much higher reflections for a fabricated a 90° bend than predicted by the numerical model. To get an impression of the influence of the fabrication tolerances, we can perform a numerical sensitivity analysis. During this analysis, we slightly modify the parameters of each cell and compute the output characteristics for each configuration. Assuming each cell having m parameters and a truncated model with n cells, we must analyze at least m  n  1 slightly different models (i.e., an unperturbed and m  n perturbed models). For example, given a truncated model that consists of 100 circular posts, we should study at least the influence of the radius and location of each post, which leads to at least 301 different models to be computed. For 3D PhCs, noncircular posts, etc., the number of models obviously increases. When we consider the frequency dependence of the reflection coefficient or of another characteristic property, the computation becomes even more demanding. Therefore, efficient computations of waveguide discontinuities are extremely important for the sensitivity analysis. Within the MMP environment, one may take advantage of the PET both for the efficient computation of the frequency dependence and for the efficient

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Electromagnetic Theory and Applications for Photonic Crystals

computation of similar numerical models. Assume that the MMP matrix equation of the form R1X1  A1

(4.21)

has been solved for a first model M1. We then get a solution vector X1. When any of the model parameters is slightly varied, the second model M2 has a solution X2 that should be close to X1. Thus, we have a good starting vector X1 for iteratively finding X2. This procedure may be repeated for a third model and so on. If more than a single model is known, we can use extrapolation techniques for finding an even better, higher-order estimation of the solution vector. In our case, the model parameter may be the frequency, the radius of any rod, the coordinates of any rod, or the material properties. Although the PET is very flexible and can reduce the computation time significantly, it has an important drawback: it requires an iterative matrix solver. Up to now, we have not found an iterative matrix solver for the efficient computation of rectangular matrices that may be ill-conditioned. Therefore, we can apply the PET only when the MMP matrix is sufficiently well conditioned. Although the discontinuities break the periodic symmetries of the original PhC, the system matrices obtained exhibit a special block structure that is different from the structure of a random set of rods because most of the rods are still of the same shape and located at the original positions of the original PhC. The system matrix therefore typically consists of several sets of identical blocks. Although it should be possible to drastically reduce memory and computation time with an appropriate block matrix solver, we did not find an appropriate solver that is flexible enough for the sensitivity analysis. Therefore, we currently can consider only relatively simple configurations with only a few cells as shown in Figure 4.15.

+

−

+

−

+ − −

−

−

−

FIGURE 4.15 The most important rods (variables) of the 90° PhC waveguide bend. This selection of rods at the waveguide corner is based on the sensitivity analysis.

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The sensitivity analysis shows that only the first two rows of cells on each side of a waveguides have a significant impact on characteristic constants, such as the reflection and transmission constants. Therefore, we can drastically restrict the observation area for the sensitivity analysis. We have mentioned that we need at least one numerical model for each model parameter. One model per parameter is sufficient only when the dependence of the characteristic constants on the parameter is almost linear. Since we were not sure about the linearity, we usually used two models per parameter. This allows us to use central differences approximations for the gradient approximation and allows us to test the linearity. According to our experience, the dependences of the characteristic constants on the various model parameters are smooth in a rather wide area. Therefore, the linear approximation is usually quite good. Another problem is associated with the graphic representation of the results of the sensitivity analysis. To obtain a quick overview of the results of hundreds or even thousands of models, we need such a graphic representation. Since we currently focus on PhCs consisting of circular rods with identical material properties, we have only three geometric parameters per rod: the radius r and the coordinates x, y. It is most natural to illustrate the dependence on the coordinates by a 2D vector and the dependence on the radius by coloring or shading as depicted in Figure 4.16. ∆R (%) ∆r = +10%

4

3

−

2

− − −

−

−

1

− 0

−1

FIGURE 4.16 Results of the sensitivity analysis for the 90° bend: the power reflection is computed for several models with a single rod increased by 10%. The gray-scale value of a rod indicates its influence on the power reflection caused by an increased radius. A decrease of the power reflection is observed when the radius of a dark rod is increased or when the radius of a bright rod is decreased. Note that there seem to be only two categories of rods with equal gray-scale values. In fact, the precise values of all rods are different.

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Electromagnetic Theory and Applications for Photonic Crystals

4.9 OPTIMIZATION BASED ON THE SENSITIVITY ANALYSIS The most promising result of the sensitivity analysis is that the variation of a parameter may also allow one to improve the model. For example, when observing some nonzero reflection coefficient for some frequency of a given model, we can try to reduce the reflection coefficient by appropriate modifications of the model parameters. This means that the sensitivity analysis opens the door to the optimization of any waveguide discontinuity. The first step of such an optimization is the proper definition of the optimization goal; that is, the fitness function that depends on the model parameters. The fitness definition can be as simple as Fitness(p1, p2, …)  1/R(p1, p2, …)

(4.22)

where R is the reflection coefficient at a specific frequency, but it often is much more complicated and often not even unique. As soon as the fitness is defined, one can search for its maxims. Unfortunately, one often observes a huge number of local maxims, and it becomes extremely difficult and time consuming to find the global optimum. It is well known that deterministic, classical optimizers rapidly converge to a local optimum, where they become trapped. For the reliable search of the global optimum, many stochastic optimizers recently have been developed based on various concepts. In a highdimensional multimodal search space, these optimizers usually require thousands or millions of fitness evaluations. Since each fitness evaluation requires the computation of at least one (depending on the fitness definition) rather large PhC waveguide discontinuity problem, we currently prefer deterministic optimizers and hope to intuitively find a reasonable starting point that allows us to find a good optimum rather than the global one within relatively few iteration steps. From the sensitivity analysis we usually see that several or even all model parameters allow us to improve the fitness. To reduce the dimension of the search space, it is reasonable to consider only those model parameters that have the strongest influence on the fitness function. For this preselection, the sensitivity analysis is very helpful. When we have selected M parameters, we search for the maximum fitness in an M dimensional space. The sensitivity analysis directly gives us an approximation of the gradient in this space in a single point that corresponds to the initial intuitively defined model. We can expect that the fitness will be increased when we proceed in the direction given by the gradient. However, how big our step size should be is not clear. Standard optimizers based on gradient or conjugate gradient information are known to be quick, but they usually assume that the cost for the computation of the gradient is not high. When M is large, our gradient computation becomes extremely expensive. Therefore, we should avoid gradient computations whenever possible. We therefore perform several steps in the direction of the gradient until we have found a local optimum in this direction. This leads us to a new model, where we can restart the sensitivity analysis and gradient computation, and so on.

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In the following sections, we show two simple examples of optimized PhC structures.

4.10 ACHROMATIC 90° BEND It is well known that a 90° bend in a PhC defect waveguide as shown in Figure 4.14 can be designed so that zero power reflection is obtained at some frequency [7]. This is important in integrated optics because bends in conventional optical waveguides exhibit either strong radiation losses or strong reflections or become large in extent compared to the wavelength. In [22] it was shown that different geometries of the 90° bend allow one to obtain zero reflection at different frequencies. In practice, an achromatic 90° bend that has zero reflection, or at least low reflection, over the entire bandgap would be more desirable. When we perform our sensitivity analysis at a given frequency where the power reflection coefficient is relatively high (e.g., fa /c  0.42; Figures 4.14 and 4.16) we notice that we can reduce the reflection coefficient at this frequency by increasing the radius of a few rods and by decreasing the radius of a few other rods (Figure 4.16). Note that we only consider the dependence on the rod radii and neglect the locations of the rods here. This considerably reduces the search space. As one can see from Figure 4.16, only 3  7  10 rods have a relatively large influence on the reflection coefficient. According to the optimization procedure outlined in the previous section, we should now proceed in the direction of the gradient of our 10-dimensional work space. To simplify the fabrication, we go in a slightly different direction by admitting only three rod radii for all rods: roriginal, rdecreased, rincreased. Now we have to find the optimal values for rdecreased and rincreased. This might be considered as a one-dimensional optimization problem. We reduce this to a one-dimensional optimization problem because we set rdecreased  (1  a)roriginal and

rincreased  (1  a)roriginal

(4.23)

With an initial value a  0.1, we find that the power reflection coefficient is reduced from R  8.6% to R  2.6%. Assuming that R linearly depends on a, we can easily extrapolate a solution with R  0.8%. As a result, we find the frequency dependence shown in Figure 4.17. Obviously, we can repeat this procedure for different frequencies. Initially we expected that we might obtain zero reflection for each frequency with another set of rod radii. Surprisingly, it turned out that we find a single configuration with almost zero reflection over a very wide frequency range that covers almost the entire bandgap, as shown in Figure 4.17. When we restart the sensitivity analysis after some optimization steps, we see that one of the rods has become extremely sensitive [22]. While modifying its radius from 0.156a to 0.109a, we see that the reflection coefficient at one frequency ( fa/c  0.335) is increased from 0 to 100 percent. From the fabrication perspective, such pronounced sensitivity may have a severe impact on the device characteristics. However, it becomes very interesting for tuning, switching, and

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Electromagnetic Theory and Applications for Photonic Crystals 1.2 fa/c =0.42

1.0

R+T T

0.8 0.6 0.4 0.2 R

0 0.34 0.36 0.38 0.4 a/2c

0.42 0.44

FIGURE 4.17 Optimal solution for the 90° bend. We now have an almost perfect waveguide bend (reflection by means of power below 1%) over the entire bandgap.

for sensor applications. When we are able to externally tune the material properties of this single rod, we will have essentially the same effect as when we modify its radius. Thus, it is possible to obtain a switch that can be set on or off by tuning one single rod of the entire structure. Furthermore, when we replace the sensitive rod with an appropriate tube through which some particles are flowing, we might obtain an excellent sensor. Similarly, we might obtain a sensor when the sensitive rod is allowed to move.

4.11 FILTERING T-JUNCTION A filtering T-junction is a small structure with one input and two output ports (Figure 4.18). By appropriately modifying the cells in the two output ports, we can make the structure asymmetric so that it transmits the energy mainly to the first output for a frequency v1 and to the second output at another frequency v2. Although this structure is not much different from the 90° bend structure, its optimization is now much more demanding because the goal is much more complicated: 1. We want to have small power reflection coefficients R for both frequencies. 2. We want to maximize the power transmission coefficient T1 to the output port 1 while having a minimal power transmission coefficient T2 at frequency v1. 3. We want to have minimum power transmission T1 to the output port 1 and a maximum power transmission T2 at frequency v2. This is a typical multiobjective optimization problem. To define the fitness, we must weight the different objectives, and we must compute the reflection and

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Output port 2

Output port 1

Input port

FIGURE 4.18 PhC filtering T-junction. It consists of the input port and two output ports.

transmission coefficients at two frequencies. A possible fitness definition is: fitness  (1/R(v1))2  (1/R(v2))2  (1/T2(v1))2  (1/T1(v2))2  (T1(v1))2  (T2(v2))2 Note that such a fitness definition is never unique and always has some influence on the solution that will be found. Since we currently work on personal computers, our hardware does not allow us to solve this problem using the sensitivity analysis outlined above. Therefore, we replace computer power by brain power (i.e., by some theoretical considerations). First, we need filtering structures in the two output ports that allow transmitting power at the desired frequencies. We can obtain such a filtering when we introduce short sections of waveguides with different dispersion characteristics in the output ports. In the first port, we should insert a waveguide that matches well with the waveguide of the input port and contains a propagating mode at v1. This means that the propagating mode at v1 should have a propagation constant and group velocity similar to the exciting mode at the input port. Similarly, we should insert a waveguide in the second port that matches well with the waveguide of the input port and contains a propagating mode at v2. Furthermore, the waveguide in the first port should not have any propagating mode at v2, and the waveguide in the second port should not have any propagating mode at v1. Figure 4.19 shows that such waveguides exist and are obtained when a substitutional line defect consisting of larger rods is introduced. To keep the structure small, we introduce very short waveguide sections consisting of only three cells. Obviously, the filtering characteristics might be improved by inserting longer sections. Figure 4.20 shows the resulting solution at the frequencies v1 and v2. We have thus obtained a filtering T-junction, but its quality is not very high. To improve the T-junction, we now might start a sensitivity analysis. Of course, we expect that the six rods in the output ports are most important. Since we already have optimized their radii in the design of the waveguide sections, we

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Electromagnetic Theory and Applications for Photonic Crystals Vertical waveguide 0.48

Left propagation range

0.46

Right propagation range Bandgap

0.42

Right waveguide

0.44 0.410

Left waveguide

fa/c

0.40 0.38 0.36 r = 0.35a

0.346

0.34

r = 0.25a

0.32 0.3 0.28

Bandgap

Γ

X

FIGURE 4.19 Waveguides of the PhC filtering T-junction. Top: photonic crystal waveguides geometry; center: corresponding dispersion curves. The basic idea is to find a two-line defect waveguide with corresponding dispersion characteristics regarding the guided modes.

fa/c = 0.346

fa/c =0.346 R =35.37%

fa/c = 0.410

fa/c =0.410 R =36.51%

FIGURE 4.20 Initial solution of the PhC filtering T-junction. The quality of this design is not high (reflection is significant).

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Simulation and Optimization of Photonic Crystals R (x100%) at fa/c =0.346

R (x100%) at fa/c = 0.410

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0 4 x10−7

221

0 4

2

0 d-left (m) −2

−4 −4

−2

4

2 0 d-right (m)

x10−7

x10−7

2 0 d-left (m) −2

−4 −4

−2

2 0 d-right (m)

4 x10−7

FIGURE 4.21 Sensitivity analysis of the T-junction initial solution. The power reflection is mapped against the positions of the defect rods (with a larger radius) for two different operating frequencies. A set of three rods in the left channel is considered, and the simultaneous movement of this set defines the first variable (d-left). The same situation is given for the right channel, where we define the second variable (d-right). T-left (x100%) at fa/c = 0.346

T-right (x100%) at fa/c = 0.346

1.0 0.8 0.6 0.4 0.2 4

2 0 x10−7 −2 −4 −4 d-left (m)

4 2 0 −2 x10−7 d-right (m)

0.012 0.010 0.008 0.006 0.004 0.002 0 4

2 0 x10−7 −2 −4 −4 d-left (m)

T-left (x100%) at fa/c = 0.410 3.0 2.5 2.0 1.5 1.0 0.5 4

−2

0

2

4

x10−7 d-right (m)

T-right (x100%) at fa/c = 0.410 1.0 0.8 0.6 0.4

2

0 x10−7 −2 d-left (m) −4 −4

−2

0

2

d-right (m)

4 x10−7

0.2 4

2 0 x10−7 −2 d-left (m) −4 −4

4 2 0 x10−7 −2 d-right (m)

FIGURE 4.22 Power transmission coefficients for the left and right channels of the example presented in Figure 4.21. These coefficients are computed for two operating frequencies with respect to two selected variables (positions of defects in the left and right channel).

now should analyze the influence of their locations in the junction. Even when we consider only the locations, we obtain a 12-dimensional search space. Because of our limited computer resources, we consider only the influence of the simultaneous movement of the three rods in the first channel as a first variable and the simultaneous movement of the three rods in the second channel as a second variable. Figure 4.21 shows the corresponding dependence of the reflection coefficients at v1 and v2 as functions of these two variables. Figure 4.22 shows the corresponding power transmission coefficients. We have different optimum locations for the six

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fa/c = 0.346

fa/c = 0.346 R = 9.72% T-left = 88.64% T-right = 0.33%

fa/c = 0.410

fa/c = 0.410 R = 11.89% T-left =0.17% T-right = 87.75%

FIGURE 4.23 Optimized filtering T-junction. This design performs better than the initial design in Figure 4.20. We have reduced the reflection by a factor of more than three at both operating frequencies.

contributions defining our fitness; therefore, our optimum is some compromise, as shown in Figure 4.23. Despite this simple optimization, we have obtained a reasonable filtering T-junction with an extremely small size. As far as we know, this is the smallest T-junction that has ever been proposed for integrated optics. Since we did a very rough optimization, it is obvious that even much better filtering T-junctions may be designed in the future.

4.12 CONCLUSIONS AND OUTLOOK PhCs are rich, often surprising, and counterintuitive in their behavior and therefore also interesting and promising for future applications in integrated optics. Because few useful design rules are available, the analysis and optimization of PhC structures requires efficient, accurate, and reliable simulations. We have presented several techniques based on MMP for the simulation and optimization of such structures. We also have embedded these techniques into numerical optimizers that allow us to find PhC structures with interesting and promising properties. We are convinced that

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more powerful computers and further improvements of the numerical methods will allow us to find various promising structures for ultradense integrated optics based on the concept of PhCs. Provided that the technological problems can be solved in such a way that PhCs can be fabricated efficiently and with sufficient precision, we assume that these structures will be widely used in the near future.

ACKNOWLEDGMENT This work was sponsored by the Swiss National Science Foundation.

REFERENCES [1] E. Yablanovich, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059–2062, 1987. [2] J. Mills, Photonic crystals head toward the marketplace, Nov. 2002, http://optics.org/ articles/ole/7/11/1/1. [3] K. Sakoda, Optical Properties of Photonic Crystals, Springer, Berlin, 2001. [4] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Molding the Flow of Light, Princeton University Press, Princeton, N.J., 1995. [5] K. Busch and S. John, Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum, Phys. Rev. Lett., 83, 967–970, 1990. [6] A. Figotin and Y.A. Godin, Two-dimensional tunable photonic crystals, Phys. Rev. B, 57, 2841–2848, 1998. [7] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, High transmission through sharp bends in photonic crystal waveguides, Phys. Rev. Lett., 77, 3787–3790, 1996. [8] I.N. Vekua, New Methods for Solving Elliptic Equations, North-Holland, Amsterdam, 1967. [9] Ch. Hafner, Post-Modern Electromagnetics Using Intelligent MaXwell Solvers, John Wiley & Sons, New York, 1999. [10] Ch. Hafner, MaX-1: A Visual Electromagnetics Platform, John Wiley & Sons, New York, 1998. [11] F.G. Bogdanov, D.D. Karkashadze, and R.S. Zaridze, The Method of Auxiliary Sources in Electromagnetic Scattering Problems in Generalized Multipole Techniques for Electromagnetic and Light Scattering, T. Wriedt, ed., 143–172, Elsevier, Amsterdam, 1999. [12] G. Tayeb, The Method of Fictitious Sources Applied to Diffraction Gratings (Appl. Computational Electromagn. Soc. J., 9, No. 3, 1994). [13] H. Singer, H. Steinbigler, and P. Weiss, A Charge Simulation Method for the Calculation of High Voltage Fields, IEEE Trans. PAS, 88, 1802–1814, 1969. [14] N.H. Malik, A review of the charge simulation method and its applications, IEEE Trans. on Electrical Insulations, 24, 3–20, 1989. [15] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1965. [16] J. Smajic, Ch. Hafner, and D. Erni, Automatic calculation of band diagrams of photonic crystals using the multiple multipole method, Appl. Computational Electromagn. Soc. J. (ACES), 18, 172–180, 2003.

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[17] J. Smajic, Ch. Hafner, K. Rauscher, and D. Erni, Computation of Radiation Leakage in Photonic Crystal Waveguides, Proceedings of Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, 21–24, 2004. [18] M. Qui and S. He, A non-orthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions, J. Appl. Phys., 87, 8268–8275, 2000. [19] A. Mekis, S. Fan, and J.D. Yoannopoulos, Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides, IEEE Microwave Guided Wave Lett., 9, 502–504, 1999. [20] B.Z. Katsenelenbaum, Theory of Non-Regular Waveguides with Slowly Varying Parameters (Russian). Academy Sci. USSR, Moscow, 1961. [21] J. Smajic, C. Hafner, and D. Erni, On the design of photonic crystal multiplexers, Opt. Express, 11, 566–571, 2003, http://www.opticsexpress.org/abstract.cfm? URL=OPEX-11-6-566. [22] J. Smajic, C. Hafner, and D. Erni, Design and optimization of an achromatic photonic crystal bend, Opt. Express, 11, 1378–1384, 2003, http://www.opticsexpress.org/abstract.cfm? URL=OPEX-11-12-1378. [23] D. Karkashadze, R. Zaridze, A. Bijamov, Ch. Hafner, J. Smajic, and D. Erni, Reflection compensation scheme for the efficient and accurate computation of waveguide discontinuities in photonic crystals, Appl. Computational Electromagn. Soc. J., (ACES), 19, 10–21, 2004.

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5 Mode-Matching Technique Applied to Metallic Photonic Crystals Hongting Jia and Kiyotoshi Yasumoto

CONTENTS 5.1 Introduction ............................................................................................226 5.2 Analysis of a Metallic Array Composed of Rectangular Cylinders ......227 5.2.1 TM Polarized Case ....................................................................227 5.2.2 TE Polarized Case ......................................................................230 5.2.3 Numerical Examples and Discussion ........................................231 5.3 Analysis of Photonic Crystals Consisting of Metallic Cylinders with Arbitrary Cross Section ..................................................................233 5.3.1 A Scattering Matrix of One Sliced Layer with TM Polarized Incidence ....................................................................................234 5.3.2 A Global Matrix of the Original Array with Arbitrary Cross Section ..............................................................................236 5.3.3 Adjustment of the Reference Phase Plane ................................237 5.3.4 S-Matrix Solution for TE Polarized Incidence ..........................237 5.3.5 A Multilayered Structure Problem ............................................238 5.3.6 2D Photonic Crystal Waveguides Consisting of Metallic Cylinders with Arbitrary Cross Section ......................239 5.3.7 Numerical Examples ..................................................................242 5.4 Analysis of Metallic Photonic Crystals for a General Incidence ..........255 5.4.1 Formulation ................................................................................255 5.4.2 Numerical Examples ..................................................................260 5.5 Scattering Analysis of Crossed Photonic Crystals Consisting of Arbitrarily Shaped Cylinders ..............................................................260 5.5.1 Reflection and Transmission Matrices of One Z-Array Layer ..262 5.5.2 S-Matrix of One X-Array Layer ................................................265 5.5.3 Numerical Examples ..................................................................269 225

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5.6 Diffraction from a Conductive Slab Cut Periodically by Rectangular Holes ..................................................................................272 5.6.1 Formulation ................................................................................273 5.6.2 Numerical Examples ..................................................................279 5.7 Scattering Analysis of a Conductive Slab Cut Periodically by Rectangular Holes in an Arbitrary Direction ....................................283 5.8 Conclusion ..............................................................................................291 Acknowledgment ............................................................................................292 References ........................................................................................................292

5.1 INTRODUCTION Metallic periodic structures have been widely used in microwave, millimeter wave, and optical systems as filters [1,2], frequency selective surfaces [3,4], substrate of antennas [5,6], beam splitters, and mirrors [7], etc. [8,9]. The periodic problems have been studied from the early 1950s as gratings problems [10]. Recently, a periodic system with a particular arrangement of scatterers including conductors and dielectric has received growing attention [11], because such a system may behave like negative refractive index materials [12] within a certain frequency range. The characteristics of a negative refractive index are realized, in general, by arranging multielement scatterers per unit cell. On the other hand, these structures may also be applied as mimicking surface plasmons [13] and high impedance surfaces [3]. Many approaches [14–25] have been proposed to analyze them, such as the mode matching method, Finite difference time domain (FDTD) technique, finite elements method, and Fourier series method. Although most numerical methods are very powerful when applied to dielectric photonic crystals, the computing costs become very high when using them to analyze metallic photonic crystals. The reason for this is that the singularity around metal edges is very strong. Many terms of expansion functions are needed in order to approximate the original fields without using special functions. Although the mode matching method is well known as an effective technique to deal with metallic gratings, the number of unknowns becomes very large for multilayered structure problems. Another problem of a classical mode matching method is that the relative convergence phenomenon [26] may occur when we match directly the electromagnetic fields in two layers that are expressed by the modal expansions. Hence, this classical mode method is very difficult to apply to metallic photonic crystals composed of arbitrary shapes or multilayered structures. The aim of this chapter is to propose a new process [27–29] to deal with multilayered periodic structures consisting of conductive objects in order to study surface plasmon problems, frequency selective surfaces problems, and artificial negative index refraction material. In this chapter, we shall discuss an accurate method for analyzing electromagnetic properties scattering from periodic metallic structures. In this approach, the original structure is sliced into many layers of rectangular rods or rectangular apertures. This reduces the original periodic array to a stacking sequence of

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lamellar gratings with metallic rectangular rods or rectangular apertures arranged periodically. First, the reflection and transmission matrices for each layer of lamellar gratings are derived using the mode matching technique. Then the results are concatenated through the entire layered system to obtain a global reflection and transmission matrices for the original periodic structure. The elements of the reflection and transmission matrices are expressed in terms of the modal series with fast convergence, which converges with O(n3), where n is the number of modal series, and the scattered fields are calculated by a simpler matrix operation. Although the present formula is very simple without requiring any special treatment for metal edges, the computing precision is also very high, and the relative convergence phenomenon has been overcome perfectly. The reason is that in this method the electromagnetic fields have been correctly calculated by infinite sums. In this chapter, two-dimensional problems and three-dimensional problems are discussed. Mode properties of two-dimensional waveguides consisting of metallic photonic crystals are also discussed.

5.2 ANALYSIS OF A METALLIC ARRAY COMPOSED OF RECTANGULAR CYLINDERS 5.2.1 TM POLARIZED CASE Let us consider a simple problem whose cross section is illustrated in Figure 5.1. The rectangular metallic cylinders are infinitely long in the z direction and periodically spaced with a distance 2h in the x direction. The background medium is free space. If a plane wave illuminates this array at f0 incident angle, the scattering fields may be expressed by the superposition of space-harmonic waves. If we assume that the incident wave is a TM polarized plane wave, then the incident wave, reflected, and transmitted waves are expressed as: Ezi ( x , y)  ei ( kx 0 xk y 0 y ) Ezs ( x , y)  Ezt ( x , y) 



∑

m

∑

m

Am ei ( kxm xk ym y ) , Bm ei[ kxm xk ym ( yt )] ,

(5.1) y>0 y < t

(5.2) (5.3)

2  k — k02  where kxm  k0 cos f0  pm xm, and Im(kyn)  0. h (m  0, 1, 2 …), kym  Am and Bm are unknown coefficients. Considering periodic properties, the electromagnetic fields in the strip region (t  y  0) may be expanded in terms of waveguide modes as follows:

Ez ( x , y)  ei 2 hpkx 0 ∑ sin j ( x1  a)[Ceig ( yt )  Ceig y ] n1

x1 ∈ {x1: a < x1 < a and x1  x  2hp}

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(5.4)

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np where jn  — , gn   k20   j2n , Im(gn)  0, and p is an integer number. C n are 2a unknown coefficients. Since the z-component of the electric fields crosses the boundary planes y  0 and y  t must be continuous, we have

eikx 0 x 



∑



m

∑

m

Am eikxm x  Q( x )ei 2 hpkx 0 ∑ sin j ( x1  a)[Ceig t  C ] (5.5) n1



Bm eikxm x  Q( x )ei 2 hpkx 0 ∑ sin j ( x1  a)[C  Ceig t ]

(5.6)

n1

where  1 Q( x )    0 

x ∈Ξ

(5.7)

otherwise

 Ξ   x : x ∈  

 [a  2hp, a  2hp]  p 

∪

(5.8)

Multiplying both sides of (5.5) and (5.6) by an exponential function eikxnx and integrating them on the interval [h, h], then the following equations are obtained 2h [dn 0  An ]  ∑ [Ceig t  C ]Sn a n1 2h Bn  ∑ [C  Ceig t ]Sn , a n1

(5.9)

n  0,  1,  2,

(5.10)

where Sn =

1 a

a

∫a sin j ( x  a)eik

xn x

dx 

j 2 a(k xn

 j2 )

[(1) eikxna  eikxna ]

(5.11)

The x-component of the magnetic fields may be derived from (5.1)–(5.4) by Hx  (i/vm0)( Ez / y). Applying the continuous condition of the magnetic fields across the boundary planes y  0 and y  t, we have k y 0 Sl0  



∑

m



∑

m

  g [Ce igl t  C ] Am k ym Slm l l l

  g [C  Ce igl t ], Bm k ym Slm l l l

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l  1, 2, 

(5.12) (5.13)

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The solution of unknown coefficients Am, Bm, and C n may be obtained by solving the set of linear equations (5.9), (5.10), (5.12), and (5.13) after truncating the expansion up to m, n  M and n, l  L, but this method is not the best one. The first reason is that the number of unknowns is very large on its own. The second, more important reason is that there is a relative convergence phenomenon [26]. In order to account for these shortcomings, we shall modify the linear equations. From (5.12) and (5.13), we derive the following relative equation. 



Cl 

  1    k S e igl t  ( Am eiglt  Bm )k ym Slm   ∑ 0 0 y l 2 igl t   gl (1  e )  m 

Cl 

  1    k S  ( Am  Bm eiglt )k ym Slm   ∑ 2 igl t  y 0 l 0  gl (1  e )  m 

(5.14)





(5.15)

Substituting (5.14) and (5.15) into (5.9) and (5.10), a new set of linear equations for unknowns Am and Bm may be derived.

∑

dn 0  An  0 n  Bn  0 n 



∑

m

m

[ Am mn  Bm mn ]

[ Am −mn  Bm mn ]

(5.16) (5.17)

where

ak ym (1  e 2ig t ) Sm Sn

n1

2hg (1  e 2ig t )

mn  ∑

ak ym eig t Sm Sn

n1

hg (1  e 2ig t )

mn  ∑

(5.18)

(5.19)

If we substitute (5.9) and (5.10) into (5.12) and (5.13), another set of linear equations for unknowns C n may be derived as follows:

2k y 0 Sl0  ∑ [Ceig t  C ] n1



∑ [C  Ceig t ] n1

a 2h



∑

m

  g [Ce igl t  C ] k ym Sm Slm l l l

a ∑ k S S  gl [Cl  Cleiglt ] 2h m ym  m lm

(5.20) (5.21)

Solving this set of linear equations and substituting the results into (5.9) and (5.10), the amplitudes of the reflected waves and the transmitted waves are obtained.

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5.2.2 TE POLARIZED CASE When a TE polarized plane wave illuminates a periodic structure as shown in Figure 5.1, the incident wave and scattering waves are expressed as: H zi ( x , y)  ei ( kxm xk ym y ) H zs ( x , y)  H zt ( x , y) 



∑

m

∑

m

(5.22)

Am ei ( kxm xk ym y ) ,

y>0

Bm ei[ kxm xk ym ( yt )] ,

(5.23)

y < t

(5.24)



H z ( x , y)  ei 2 hpkx 0 ∑ cos jn ( x1  a) [ Dneign ( yt )  Dneign y ] n0

 t < y < 0,

x1 ∈ {x1 : a < x1 < a and x1  x  2hp}

(5.25)

In a similar manner, two sets of linear equations may be derived. The first set, corresponding to (5.16) and (5.17), is An  dn 0  Bn 



∑

m



∑

m

[Bm mn  Am mn ]

(5.26)

[Am mn  Bm mn ]

(5.27)

where mn 

a 2hk yn

mn 

a hk yn

Cnn 

1 a



gn (1  e 2ignt )CnmCnn

n0

(1  dn 0 )(1  e 2ignt )

∑



g eignt CnmCnn

∑ (1 nd

n0

(5.28)

n 0 )(1 − e

a

2 ignt

ik xn x

∫a cos jn ( x  a)e

(5.29)

)

dx 

ik xn 2  j2 ) a(k xn n

[(1)n eikxna  eikxna ] (5.30)

y


0

x 2h

−t

2a

FIGURE 5.1 Geometry of an infinite periodic array of rectangular conductive rods.

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Mode-Matching Technique Applied to Metallic Photonic Crystals

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The second set, corresponding to (5.20) and (5.21), is



2Cl0  ∑ [ Dneignt  Dn ] n0





∑ [Dneig t  Dn ] ∑ n

n0

m

∑

m

 agnCnmClm  (1  dl 0 ) [ Dleiglt  Dl ] 2hk ym

 agnCnmClm  (1  dl 0 ) [Dl  Dleiglt ] 2hk ym

(5.31)

(5.32)

5.2.3 NUMERICAL EXAMPLES AND DISCUSSION We will now investigate the convergent properties of Equations (5.16) and (5.17) and Equations (5.20) and (5.21). Figure 5.2 shows convergence behaviors of amplitudes and phases of 1-th order, zero-th order, and first order space harmonics with TM polarized incidence as functions of the truncated number M for a square rods array with 2h/l0  0.6, a/h  0.5, t/h  0.3, and an incident angle f0  60°. The dotted lines are the results of Equations (5.16) and (5.17), and the solid lines are obtained by Equations (5.20) and (5.21). The convergent speed is very fast for all the space harmonics. For the TE polarization case, the convergent behaviors of Equations (5.26) and (5.27) and Equations (5.31) and (5.32) are plotted in Figure 5.3. In this figure, the dotted lines and solid lines denote the results of Equations (5.26) and (5.27) and Equations (5.31) and (5.32) respectively. It is found that the convergent property is the same speed as in the TM polarized case. In the following sections, we shall apply Equations (5.16) and (5.17) and Equations (5.26) and (5.27) to derive an S-matrix formula, because they may allow the S-matrix to be expressed in a very simple form. If using Equations (5.20) and (5.21) and Equations (5.31) and (5.32), the expression of the S-matrix formula will become

0.28

294

0.92 0–order

0.9 0.88 0.5

–1–order

218 216 214 212 210 300

300

0.21

306 0–order

0.45

300 –1–order

0.4

(a)

5

10

15 20 M (N)

25

304 302

296 0.4

304 302

0.4

298 294

305

0.22

Amplitude

Amplitude

0.29

296

1–order

0.23

298

Phase (°)

1–order

0.3

Phase (°)

0.31

0.35

30

5

(b)

10

15 20 M (N)

25

300 30

FIGURE 5.2 Convergence behaviors of amplitudes and phases of the 1-th order, zero-th order, and first order space harmonics scattered from a square rods’ array with TM polarized incidence as functions of the truncated number M, where 2h/l0  0.6, a/h  0.5, t/h  0.3, and the incident angle is chosen to be f0  60°. (a) Reflected waves (b) Transmitted waves.

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complicated, moreover, the dimensions of some matrices that must be included in the S-matrix formula must become very large. In order to check our computer codes, Figure 5.4 shows a comparison of the transmission efficiency calculated by a conventional mode matching method [24] and the present method, where h  0.5 cm, a  0.38 cm, t  2.06 cm, and the

133

0.22

330

0.6

0–order

0.58

328 326

0.56

1–order

322 320 318 316 58

0.82 0–order 0.8

56

Phase (°)

0.2 0.62

0.24

Amplitude

0.21

134

Phase (°)

1–order

Amplitude

0.26

135

0.22

54

324

0.31 0.3 0.29

1–order

5

10

(a)

15 20 M (N)

25

95

0.55

90

0.5

1–order

337 5

30

338

(b)

10

15 20 M (N)

25

30

FIGURE 5.3 Convergence behaviors of amplitudes and phases of the 1-th order, zero-th order, and first order space harmonics scattered from a square rods’ array with TE polarized incidence as functions of the truncated number M, where 2h/l0  0.6, a/h  0.5, t/h  0.3, and the incident angle is chosen to be f0  60°. (a) Reflected waves. (b) Transmitted waves. 0

Pt−1/PINC (dB)

−1

−2

h = 0.5 cm a = 0.38 cm t = 2.06 cm
0 = 58.5°

−3 25

30 Frequency (GHz)

35

FIGURE 5.4 Comparison of the transmission efficiency calculated by a mode matching method [24] (dotted line) and the present results (solid line).

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Mode-Matching Technique Applied to Metallic Photonic Crystals

233

angle of incident TE polarized wave is f0  58.5°. The dotted curve represents the power transmission efficiency into the 1-th order harmonic, which is obtained by using a mode matching method with 96 basis functions [24]. The solid curve represents the results obtained by the present formula with M  6. The agreement between the two results is very good.

5.3 ANALYSIS OF PHOTONIC CRYSTALS CONSISTING OF METALLIC CYLINDERS WITH ARBITRARY CROSS SECTION A cross section of 2D metallic cylinders embedded in a dielectric slab is illustrated in Figure 5.5. The cylinders are parallel with each other, infinitely long in the z direction, and periodically spaced with a distance 2h in the x direction. The slab including metallic arrays is in the background medium with a relative permittivity erb and relative permeability mrb. To formulate this problem, we slice it into many layers. In each layer, the original shape is approximated by several rectangular rods. Using the same manner discussed in the preceding section, the electromagnetic fields may be expanded in terms of waveguide modes between any two adjacent metal rods, and then a set of linear equations is derived by applying the continuous condition of tangential electromagnetic fields across the boundary planes. If we directly solve the problem shown in Figure 5.5 using a conventional mode matching method, the number of unknowns becomes very large, and a relative convergence phenomenon [26] will occur. In order to overcome these difficulties, we shall introduce a scattering matrix based on space harmonics to characterize the electrical properties of a metal array. By concatenating sequentially these scattering matrices of each sliced layer, a global S-matrix may be derived. All the electrical properties of the original problem are characterized by this global S-matrix. Since all the scattering matrices are defined in the same harmonics’ system, the relative convergence problem can be overcome easily. Moreover this process has very high precision. In this method, the electromagnetic fields at any position are expressed in terms of space harmonics, for all the layers have the same period.

y rb rb


0

2h

x rb rb

FIGURE 5.5 Geometry of periodic metallic cylinders with arbitrary cross section.

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Electromagnetic Theory and Applications for Photonic Crystals

5.3.1 A SCATTERING MATRIX OF ONE SLICED LAYER WITH TM POLARIZED INCIDENCE Figure 5.6 shows a geometry of an infinite periodic structure in which a unit cell includes several rectangular conducting rods. As stated above, the incident waves and scattering waves are expressed as follows: Eziu ( x , y) 



m

Ezid ( x , y)  Ezr ( x , y)  Ezt ( x , y) 

∑

∑

m

∑

m

∑

m

Amei ( kxm xk ym y ) , y > 0

(5.33)

Amei[ kxm xk ym ( yt )] , y < t

(5.34)

Bmei ( kxm xk ym y ) , y < 0

(5.35)

i[ k xm xk ym ( yt )] B , y > t me

(5.36)

2 k

rbmrb. where kxm  kb cos f0  pm — , kym   kb2  xm, Im(kym)  0, and kb  k0  h   A m and B m are the amplitudes of incident waves and unknown coefficients, respectively. The scattering fields in the region bounded by two adjacent metal rods located in the zero-th unit cell may be expanded in terms of waveguide modes as follows:

Ezl ( x , y)  ∑ [Cln eigln y  Cln eigln ( yt ) ] sin jln ( x  al  wl ) n1

for l  1, 2,  , L

(5.37)

np 2  where jln  — kl2  j erlmrl, wl is the orientation of ln, Im(gln)  0, kl  k0 2al , gln  th the x coordinate of the l slot in the zero-th unit cell region, C ln are unknown coefficients, and L denotes the total number of discrete rectangular rods per unit

Y

t

2a 1

2a 2

 r 1, r1

r 2 r 2

Unit cell rb, rb rb , rb

X

w1 w2

2h

FIGURE 5.6 Geometry of an infinite periodic structure in which a unit cell includes several rectangular conductive rods.

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Mode-Matching Technique Applied to Metallic Photonic Crystals

235

cell. The tangential components of electric and magnetic fields derived from Equations (5.33)–(5.37) must be continuous across the boundary planes at y  0 and y  t. Taking into account a periodic condition, these boundary conditions may be expressed as follows: Eziu ( x , 0)  Ezr ( x , 0)  e 2ikx 0hp Ezl ( x  2hp, 0)Q( x )

(5.38)

Ezid ( x , t )  Ezt ( x , t )  e 2ikx 0hp Ezl ( x  2hp, t )Q( x )

(5.39)

H xiu ( x , 0)  H xr ( x , 0)  e 2ikx 0hp H xl ( x  2hp, 0), x  Ξ

(5.40)

H xid ( x , t )  H xt ( x , t )  e 2ikx 0hp H xl ( x  2hp, t ), x  Ξ

(5.41)

where  1 Q(x )    0 

otherwise

 Ξ   x : x   

 [al  2hp  wl , al  2hp  wl ]  p 

x Ξ

(5.42)



∪

(5.43)

Applying boundary conditions (5.40) and (5.41) and considering the orthogonality of a set of trigonometric functions {sin jln(x  al  wl)} in the interval [al  wl, al  wl], we have Cln  Cln 



mrl

∑

mrb gln (1  e 2iglnt ) m

mrl mrb gln (1  e

Gl,nm 

2 iglnt

∑

) m

   iglnt ] k ym jlnGl,nm [ A m  Bm  ( Bm  Am )e

(5.44)

iglnt  k ym jlnGl,nm [( Am  B  B m )e m  Am ]

(5.45)

eikxm wl [(1)n eikxmal  eikxmal ] 2  j2 ) al (k xm lv

(5.46)

Substituting (5.44) and (5.45) into boundary conditions (5.38) and (5.39) and applying the orthogonality of a set of exponential functions {eikxmx} in the interval [0, 2h], a system of linear equations for the unknown coefficients of space harmonics B m is derived. If the number of space harmonics is truncated up to m  M, the system of linear equations may be rewritten in matrix form as follows: a  b   (a  b )   ( b  a )

(5.47)

a  b   (a  b )   ( b  a )

(5.48)

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Electromagnetic Theory and Applications for Photonic Crystals

where t

a  [ AM  A0  A M]

(5.49)

t

 b  [ BM  B 0  BM ]

 k yn   [ mn ]    2hmrb  k yn   [ mn ]   h m  rb

L

(5.50) 2   1  e 2iglnt jlvGl ,nnGl ,nm lnt gln n1

∑ al mrl ∑ 1  e2ig l1

L

eiglnt jl2nGl,nnGl,nm 2 ig t gln n1 1  e ln

∑ al mrl ∑ l1

   

   

(5.51)

(5.52)

Solving the linear Equations (5.47) and (5.48), we have 1  b     I      I    a             I      I   a   b   

(5.53)

Thus, a scattering matrix of one single layer of periodic arrays may be defined as: 1  R T     I      I      S        I      I   T R   

(5.54)

Matrices R and T are called reflection matrices and transmission matrices respectively.

5.3.2 A GLOBAL MATRIX OF THE ORIGINAL ARRAY WITH ARBITRARY CROSS SECTION The reflection and transmission matrices characterizing the original structure shown in Figure 5.5 may be calculated by concatenating all of the sliced layers.  Assuming R  n and T n to be the reflection and transmission matrices of the grating  stacked by the top n-layers and R  n1 and T n1 to be those of the (n  1)-th array, which is located behind the n-th array, the total scattering matrix of the (n  1)layered grating may be obtained.  R  n1 Sn1    Tn1

   Tn1   R  n  Z Rn1Tn    ZTn R   n1 

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     R Z R  T n n1 n1  ZTn1

(5.55)

Mode-Matching Technique Applied to Metallic Photonic Crystals

237

where   1 Z  Tn[ I  R n1 Rn ]

(5.56)

 1 Z  Tn1[ I  R  n Rn1 ]

(5.57)

If we cascade sequentially all the sliced layers starting at the top layer, the global reflection and transmission matrices may be derived by this recursive formula (5.55).

5.3.3 ADJUSTMENT

REFERENCE PHASE PLANE

OF THE

From the above statement, it is found that the reference phase plane is set by the values of wp, so we can freely shift it. However it is necessary to recalculate the reflection and transmission matrices for each change. The following formula gives a simple procedure for any shift without repeated calculations. We assume R′ and T′ to be the reflection matrix and the transmission matrix of a periodic array in the x′o′y′ coordinate, whose origin is (s, 0) in the xoy system. From a simple calculation, the reflection matrix and transmission matrix viewed in the xoy coordinate are given as follows: R  X R′ X ,

T  XT ′X

(5.58)

where ik xm s X   [ x dmn ] mn ]  [e

(5.59)

This formula is very convenient and will be widely applied in the following discussion to analyze photonic crystals arranged in a nonrectangular lattice.

5.3.4 S-MATRIX SOLUTION

FOR

TE POLARIZED INCIDENCE

The reflection and transmission matrices with TE polarized incidence may be derived in a similar way by employing Hz as the leading field. We assume that the  vectors a h and b h are the amplitude vectors of incident waves and scattering waves defined respectively as follows:    t a h  [ Ah (M )  Ah 0  AhM ]

(5.60)

   t b h  [ Bh (M )  Bh 0  BhM ]

(5.61)

where H ziu ( x , y) 

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M

∑

mM

 e i ( k xm xk ym y ) , y > 0 Ahm

(5.62)

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Electromagnetic Theory and Applications for Photonic Crystals

H zid ( x , y)  H zr ( x , y)  H zt ( x , y) 

M

∑

 e i[ k xm xk ym ( yt )] , y < t Ahm

(5.63)

M

 e i ( k xm xk ym y ) , y > 0 Bhm

(5.64)

i[ k xm xk ym ( yt )] B , y < t hm e

(5.65)

mM

∑

mM M

∑

mM

Thus we have    b   h  S  ah       ah   bh 

(5.66)

The scattering matrix S of TE polarized case is in the same form as the TM case defined by (5.54). The matrices  are defined as:  L k k e a 1  e 2iglnt g G G ln l , nn l , nm   [ mn ]   ∑ xm xn rb l ∑ 2 iglnt 1  dn 0 2 hk e 1  e  l1 ym rl n0

   

(5.67)

   L k k e a eiglnt glnGl ,nnGl ,nm   [ mn ]   ∑ xm xn rb l ∑ 2 ig t 1  dn 0  l1 hk ym erl n0 1  e ln

   

(5.68)

5.3.5 A MULTILAYERED STRUCTURE PROBLEM Figure 5.7 shows a cross section of a multilayered structure. For the grating slabs (j  1, 3, 5, …), the S-matrix may be calculated using a recursive formula discussed in Section 5.3.2. For the interval region between two adjacent grating slabs (j  2, 4, 6, …), the S-matrices may be easily written as follows: 0 Y  S     Y 0 

(5.69)

Y  [ ymn ]  [eik ymd dmn ]

(5.70)

Thus, the total S-matrix of a multilayered structure may be derived by using the same recursive formula (5.55). This process seems inefficient because we must run it many times. However, if all the slabs and their intervals are the same, the number of running times may be sharply reduced by using a binary number process (1  1 → 22 → 44 → 88 → 1616 → …); for example, a 100-layered structure requires only 7 calculation times.

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239

rb, rb 2h

 j=1

X j=2

d

j=3 j=4 j=5 j=6 j=7 rb, rb

FIGURE 5.7 Cross section of multilayered arrays.

5.3.6 2D PHOTONIC CRYSTAL WAVEGUIDES CONSISTING OF METALLIC CYLINDERS WITH ARBITRARY CROSS SECTION Figure 5.8 shows the side view of a two-dimensional photonic crystal waveguide composed of metallic cylinders with an arbitrary cross section. The cylindrical elements should be the same along each layer of the arrays, but those in different layers do not need to be the same. The background medium is a homogeneous dielectric with relative permittivity erb and relative permeability mrb. The guided waves are assumed to be uniform in the z direction and to vary in the form eibx in the x direction, where b is a propagation constant. Let us consider the electromagnetic fields in the 0-th region shown in Figure 5.8. According to the above discussion, the reflection matrices R∪0 and R∩0 may be easily obtained, in which R∪0 and R∩0 are the reflection matrices of upper photonic crystals viewed from the 0-th region to the y direction and to the y direction respectively. A column vector a 0 is assumed to be a normalized amplitude vector of downgoing space harmonics. Then we can derive the following relation:  Y0 R0∪Y0 R0∩ a 0  a0

(5.71)

where Y0 is a diagonal matrix whose diagonal elements are eikymd0. For nontrivial a 0, the discrete values bm of b must be a root of the following equation. det[Y0 R0∪Y0 R0∩  I ]  0

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(5.72)

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Electromagnetic Theory and Applications for Photonic Crystals

j +1 j

dj

R

∩ j

R

a −j = j a −j

∩ j

a +j

b +j

b −j

j−1

3 2 1

d0 0

∩

R0

R

∩ 0

a +0 = 0 a+0

b −0 = 0 b −0

a −0 = 0 a −0

b +0 = 0 b +0

Waveguide region

−1 −2 −3

FIGURE 5.8 Schematic of a two-dimensional inhomogeneous photonic crystal waveguide consisting of metallic cylinders with arbitrary cross section.

The value of bm is the propagation constant of the m-th guided mode. The electromagnetic fields may be expressed with the amplitude vectors (aj, b j ), (j  0, 2, 4, …). Substituting the root bm into (5.71), the normalized mode  amplitude vector a 0 may be obtained. Then the amplitude vectors (a 0, b 0 ) may be derived by setting a0  1 as follows: a0  a 0,

∩  b 0  R0 a 0 ,

a0  Y0 b 0,

∪  b 0  R0 a 0

(5.73)

Assuming (R j, T j ) to be the reflection and transmission matrices of j-th array, the amplitude vectors (aj, b j ) in j-th region may be calculated from a recursion

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Mode-Matching Technique Applied to Metallic Photonic Crystals

241

process by starting at j  2 or j  2:   aj  (Tj1 )1[ b j2  R j1 a j2 ], for j  2, 4, 6

(5.74)

  aj  (Tj1 )1[ b j2  R j1 a j2 ], for j 2, 4, 6 

(5.75)

However, the accuracy of this formula is not high, when we run it in finite-digit numbers, because the recursion formula easily accumulates the computing error for each step. In fact, the solution bm of (5.71) must be a root of the following equation, when and only when the electromagnetic field does exist in the j-th local region. Yj R∪j Yj R∩j aj  aj

(5.76)

where R∪j, R∩j are the reflection matrices to characterize the upper and lower boundaries, respectively, viewing from the j-th region. Hence, the normalized mode amplitude aj may be derived by solving (5.76); the other vectors (aj , bj ) may also calculated using the same process to be similar to Equation (5.73). Since the primary values of the space-harmonic vectors aj  aj aj and bj  aj bj must satisfy the following relative equations, where aj and aj2 are complex numbers a j Tj1 aj  a j2 [ bj2  R j1 aj2 ], for j  2, 4, 6, 

(5.77)

a j Tj1 aj  a j2 [ bj2  R j1 aj2 ], for j 2, 4, 6, 

(5.78)

then we have aj a j2 aj a j2



( bj2  R j1 aj2 )t (Tj1 aj )* , for j  2, 4, 6,  (T  a )t (T  a )*

(5.79)

( bj2  R j1 aj2 )t (Tj1 aj )* , for j 2, 4, 6,  (T  a )t (T  a )*

(5.80)

j1 j



j1 j

j1 j

j1 j

The coefficients aj can uniquely be determined by setting a0  1. Comparing (5.74), (5.75) and (5.79), (5.80), it is obvious that the computing accuracy of (5.79) and (5.80) is higher than (5.74) and (5.75). The unknown in (5.79) and (5.80) is only a scalar number (complex number), whereas the unknown in (5.74) and (5.75) is a vector number. The mode-field distributions inside the homogeneous strip region (j  0, 2, 4, 6 …) of the background dielectric are calculated by superposing the upgoing waves (bj ) and the downgoing waves (bj ). The mode fields in the grating regions ( j  1, 3, 5 …) are easily calculated by using

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Electromagnetic Theory and Applications for Photonic Crystals

the original expressions. For example, substituting (aj , bj ) into (5.44) and (5.45), the expanding coefficients C ln are derived, and using these results and (5.37), the z-component of electric fields may be obtained.

5.3.7 NUMERICAL EXAMPLES Since the structure of each layer shown in Figure 5.6 is up-down symmetric, the reflection and transmission matrices for downgoing incidence and upgoing incidence should be the same, i.e., R  R and T  T. This may be easily confirmed from Equation (5.54). It is worth mentioning that in the proposed method, the mode matching of the fields between two adjacent layers of sliced metal rods has been taken through a buffer layer of free space whose thickness is finally reduced to zero. The relative convergence phenomenon [26], may occur if the fields expressed by the modal expansions are directly matched across the two sliced layers like the conventional mode-matching technique. To avoid this, the fields are matched to those in the buffer layer satisfying the periodic condition. By projecting the modal fields in the grating regions into the space-harmonic fields that constitute an orthogonal complete basis for the specified periodic system, the terms of the original modal expansions are uniquely transformed into the semiinfinite sums denoted by the index n in (5.51), (5.52), (5.67), and (5.68), respectively. These semi-infinite sums can be calculated almost rigorously because they converge very rapidly with a rate O(n3) as n increases. The similar semi-infinite sums have been used in the analysis of waveguide discontinuities [30], and the accurate and stable solutions have been obtained without using the edge conditions at the corners of waveguide walls. We first consider a problem of periodic structures composed of circular conductive cylinders. Since the structure is symmetrical about the x-axis, only the upper half structure must be sliced into lamellar layers to calculate the reflection and transmission matrices. This solution is also available to the lower half structure. Figure 5.9 shows a comparison of the results obtained by an analytical method [21], which has been explained in Chapter 3, and those obtained by the present method for a periodic array of circular cylinders whose radius is 0.6h and angle of incident plane wave is 45°. The solid lines denote the results of the present method, in which the truncated number is M  6 and the upper half-region of circular cylinders was divided into 30-layers. The dotted lines are the results calculated by an analytical method [21] with 31 cylindrical harmonic expansions and 31 space harmonics. Very good agreement was found between the two methods for all of the space harmonics in the whole frequency band. The example shows that the computing precision of the present method is very high and the convergence is very fast. Next, let us investigate the properties of metallic photonic crystals arranged in a square lattice as shown in Figure 5.10(a). In the following examples the truncation number is chosen to be M  15 except as noted. Figure 5.11 shows the reflection efficiency scattered by a metallic photonic crystal in a square lattice as functions of normalized wavelength 2h/l0 for three TM incident plane waves.

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Mode-Matching Technique Applied to Metallic Photonic Crystals 300

0.4

200

0.2

100

First-order

100

100

Zero-order 0 -1-order

0.2

100

0

300 200 100 Zero–order

0 –1-order

300 200

0

0.1 0

0

0.4

0.2

300 200

0.1

100

0

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 h/0

(a)

Amplitude

200

200

0 0.2 Phase (°)

Amplitude

300

300

0.05

0

1

0.6

First–order

Phase (°)

0.6

243

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 h/0

(b) 1.2 200

0.8

1

300 200 100

Amplitude

Zero-order

Phase (°)

Amplitude

100 0

0

0.4

0.8

Zero-order

300 200

0.4

100 0

0 300

0 1.2

0.8

200

0.8

200

0.4

100

0.4

100

0 1.2

0 (c)

200

0.4

100 0 0.8

300

First-order

–1-order

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 h/0

0 (d)

Phase (°)

First-order

–1-order

300

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 h /0

FIGURE 5.9 Comparison of the reflected and transmitted waves calculated by an analytical method (dotted lines) and the present method (solid lines) for a periodic array of circular cylinders, where r  0.6h and f0  45°. (a) Reflected waves in TM incidence. (b) Transmitted waves in TM incidence. (c) Reflected waves in TE incidence. (d) Transmitted waves in TE incidence.

A perfect bandgap appears in the lower frequency band, when the dimension of square cylinders is very small. The width of the perfect bandgap will extend with an increase of the cylinders’ dimension. However, there is not any perfect bandgap for TE polarized incidence in Figure 5.12, which shows the computing results for TE polarized waves. But there are some part bandgaps when a/(2h)  0.3. A part bandgap means that the total reflection may be observed only for a part of the incident waves; for example, the plane waves at incident angle f0  60° will be reflected perfectly around 2h/l0  0.5, or a perfectly reflected wave may be observed around 2h/l0  0.7 for TE incident waves at f0  60°. From Figure 5.12(c), a square photonic crystal may be applied as a narrow stop-band filter for TE polarized incidence. The reflection efficiency scattered from a metallic photonic crystal arranged with a regular triangular lattice as functions of normalized wavelength 2h/l0 is

© 2006 by Taylor & Francis Group, LLC

244

Electromagnetic Theory and Applications for Photonic Crystals
0

a a

2h

2h

(a)
0

a a

60° 60°

2h 60° 60° (b)

FIGURE 5.10 Two types of metallic photonic crystals. (a) In square lattice. (b) In regular triangular lattice. 1

0.8


0 = 90°
= 30° 0

0.6 0.4 0.2 0 0

(a)


0 = 60°

Reflection efficiency

Reflection efficiency

1

0.2

0.4 0.6 2h/0

0.8

0.8 0.6

0.2 0 0

1 (b)


0 = 30°
0 = 60°
0 = 90°

0.4

0.2

0.4 0.6 2h /0

0.8

1

FIGURE 5.11 Reflection efficiency from a metallic photonic crystal in square lattice as functions of normalized wavelength 2h/l0 with three TM incidences. (a) a/(2h)  0.01 (b) a/2(h)  0.3.

shown in Figures 5.13 and 5.14 for TM and TE polarized incidence respectively. Although the characteristics shown in Figures 5.11(a) and 5.13(a) are very similar, there is a big difference between Figures 5.11(b) and 5.13(b). There are two bandgaps in Figure 5.11(b), whereas there is only one bandgap in Figure 5.13(b). However, the bandgap in Figure 5.13(b) is wider than that in

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals 0.002

1
0 = 30°


0 = 30° Reflection efficiency


0 = 60°

Reflection efficiency

245


0 = 90° 0.001

0.8


0 = 60°
0 = 90°

0.6 0.4 0.2

0 0

0.2

0.4

(a)

0.6

0.8

0 0

1

2h/0

0.2

(b)

0.4 0.6 2h/0

0.8

1

1

Reflection efficiency


0 = 30°
0 = 60°

0.8


0 = 90°

0.6 0.4 0.2 0

0

0.2

0.4

(c)

0.6

0.8

1

2h/0

FIGURE 5.12 Reflection efficiency from a metallic photonic crystal in square lattice as functions of normalized wavelength 2h/l0 with three TE incidences. (a) a/(2h)  0.01 (b) a/(2h)  0.3 (c) a/(2h)  0.1. 1

1

0.6

Reflection efficiency

Reflection efficiency


0 = 30°

0.8
0 = 60°

0.4
0 = 90°

(a)

0.6
0 = 30°

0.4


0 = 60°
0 = 90°

0.2

0.2 0 0

0.8

0.2

0.4

0.6

2h/0

0.8

0 0

1

(b)

0.2

0.4

0.6

0.8

1

2h/0

FIGURE 5.13 Reflection efficiency from a metallic photonic crystal in regular triangular lattice as functions of normalized wavelength 2h/l0 with three TM incidences. (a) a/(2h)  0.01 (b) a/(2h)  0.3.

© 2006 by Taylor & Francis Group, LLC

246

Electromagnetic Theory and Applications for Photonic Crystals 1
0 = 30°

0.8

Reflection efficiency

Reflection efficiency

1


0 = 60°
0 = 90°

0.6 0.4

0.6


0 = 30°
0 = 60°
0 = 90°

0.4 0.2

0.2 0 0

0.8

0.2

0.4

0.6

0.8

2h/0

(a)

0 0

1

0.2

0.4

0.6

0.8

1

2h/0

(b)

FIGURE 5.14 Reflection efficiency from a metallic photonic crystal in regular triangular lattice as functions of normalized wavelength 2h/l0 for three TE incidences. (a) a/(2h)  0.1 (b) a/(2h)  0.3.
0

a r 1

a

r 1 2h 2h r 1 (a)


0 60°

a 60°

r 1

a

r 1 2h 60° 60°

r 1

(b)

FIGURE 5.15 Schematic of two types of metallic photonic crystals in which the metallic arrays are embedded in dielectric slabs. (a) In square lattice. (b) In regular triangular lattice.

Figure 5.11(b). Let us consider a scattering problem of the metallic arrays embedded in dielectric slabs as shown in Figure 5.15. Figures 5.16 and 5.17 show the reflection efficiency scattered from metallic photonic crystals embedded in slabs with square and regular triangular lattice arrangements, respectively, as functions

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals (a)

1

247


0 = 30°
0 = 60°

Reflection efficiency

0.8
0 = 90° 0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

2h /0

(b)

1

Reflection efficiency

0.8

0.6

0.4
0 = 30°
0 = 60°

0.2


0 = 90° 0

0

0.2

0.4 2h/0

FIGURE 5.16 Reflection efficiency from a photonic crystal composed of metallic arrays embedded in dielectric slabs and arranged with a square lattice as functions of normalized wavelength 2h/l0 for three TM incidences. (a) a/(2h)  0.01 (b) a/(2h)  0.3.

of normalized wavelength 2h/l0 for three TM incident waves, where the relative permittivity of the dielectric slab is er1  10. There is a narrow pass band within the perfect bandgap region as a/(2h)  0.3. This characteristic may be applied as a narrow bandpass filter, and the passband width and the working

© 2006 by Taylor & Francis Group, LLC

248

Electromagnetic Theory and Applications for Photonic Crystals (a)

1
0 = 30°

Reflection efficiency

0.8


0 = 60°

0.6


0 = 90°

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

2h /0

(b)

1

Reflection efficiency

0.8

0.6


0 = 30°

0.4


0 = 60° 0.2

0


0 = 90°

0

0.2

0.4

2h/0

FIGURE 5.17 Reflection efficiency from a photonic crystal composed of metallic arrays embedded in dielectric slabs and arranged with a regular triangular lattice as functions of normalized wavelength 2h/l0 for three TM incidences. (a) a/(2h)  0.01 (b) a/(2h)  0.3.

frequency band may be designed by changing their dimensions and the electrical parameters. Figures 5.18 and 5.19 show the reflection efficiency from a photonic crystal composed of metallic arrays embedded in a dielectric slab as functions of

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals (a)

1


0 = 30°

0.8 Reflection efficiency

249


0 = 60°


0 = 90°

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

2h/0

(b)


0 = 30°


0 = 60°


0 = 90°

1

Reflection efficiency

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

2h/0

FIGURE 5.18 Reflection efficiency from a photonic crystal composed of metallic arrays embedded in dielectric slabs and arranged with a square lattice as functions of normalized wavelength 2h/l0 for three TE incidences. (a) a/(2h)  0.1 (b) a /(2h)  0.3.

normalized wavelength 2h/l0 with TE polarized incidences. Comparing Figure 5.12(b) with Figure 5.18(b) and Figure 5.14(b) with Figure 5.19(b), we may find that the widths of perfect bandgaps in Figures 5.18(b) and 5.19(b) are broader than those in Figures 5.12(b) and 5.14(b). These results show that metallic

© 2006 by Taylor & Francis Group, LLC

250

Electromagnetic Theory and Applications for Photonic Crystals 1 0.8


0 = 90°

Reflection efficiency

Reflection efficiency


0 = 30°

1


0 = 30°


0 = 60°

0.6 0.4


0 = 60°


0 = 90°

0.8 0.6 0.4

0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

2h /0

(a)

1

0

0.2

(b)

0.4

0.6 2h / 0

0.8

1

FIGURE 5.19 Reflection efficiency from a photonic crystal composed of metallic arrays embedded in dielectric slabs and arranged with a regular triangular lattice as functions of normalized wavelength 2h/l0 for three TE incidences. (a) a/(2h)  0.1 (b) a/(2h)  0.3.

0.5

0.4 Mode 1

h/

0.3

0.2 Mode 3

0.1 Mode 2

0

0.4

0.6 2h /0

FIGURE 5.20 Dispersion curves of a 2D photonic crystal waveguide consisting of rectangular conductive cylinders in a square lattice arrangement with TM polarization. The solid lines denote the even modes and the dotted line denotes an odd mode.

photonic crystals embedded in slabs are more suitable as photonic crystal waveguides than those that are not embedded in slabs. Next, we shall discuss the problem of metallic photonic crystal waveguides. Figure 5.20 shows the dispersion curves of a 2D photonic crystal waveguide

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

251

FIGURE 5.21 Cross section of metallic photonic crystal waveguides. (a) In rectangular lattice. (b) In regular triangular lattice.

consisting of rectangular conductive cylinders in a square lattice arrangement whose geometry is shown in Figure 5.21(a) in a TM polarization case, where a  0.4h, d  3h, and the truncated number M is 15 including 31 space harmonics. The solid lines denote the even modes; the dotted line denotes an odd mode; and the square mark denotes cutoff frequency. In Figure 5.11(b), we may find that only one mode (mode 1) locates in the perfect bandgap region, whereas two modes (mode 2 and mode 3) are in a part bandgap region. The distribution of electric fields is plotted in Figure 5.22 for three modes at 2h/l0  0.4 and 0.7, in which the corresponding normalized propagation constants are bh/p  0.288769, 0.353135, and 0.427878, respectively. The electric fields are concentrated around the defect region for two even modes, whereas they are distributed in a larger region for the odd mode. This means that the odd mode is a weak mode. Thus, the method can calculate all the guided modes, even if they are not in perfect bandgap ranges. Figure 5.23 shows the dispersion curves of a 2D photonic crystal waveguide consisting of rectangular conductive cylinders in a square lattice arrangement for TE polarized modes. The values of the parameters are the same as those given in Figure 5.20 except for the polarization of an incident wave. This photonic crystal waveguide has two modes. One is an even mode and the other is an odd mode. Their magnetic field distributions are plotted in Figure 5.24, at 2h/l0  0.8 and

© 2006 by Taylor & Francis Group, LLC

252

Electromagnetic Theory and Applications for Photonic Crystals 0.2

1

Ez

0.6

0

Mode 1

−0.2

h/ = 0.288769

Ez

0.8

2h /0 = 0.4

0.4

−0.4 −0.6

0.2

−0.8 2h/0 = 0.7 −1

0 −10

0 y/(2h )

(a)

−10

10

Mode 2

h/ = 0.353135

0 y/(2h )

(b)

10

1

Ez

0.5 0 −0.5 Mode 3

2h/0 = 0.7

−1 h/ = 0.427878 −10 (c)

0 y/(2h )

10

FIGURE 5.22 Distributions of electrical fields for the three lowest TM modes where solid lines are at x/(2h)  0 and dotted lines are at x/(2h)  0.5. (a) Field distributions of an even mode. (b) Field distributions of an even mode. (c) Field distributions of an odd mode. 0.5

0.4

h/

0.3 Even mode 0.2

Odd mode

0.1

0

0.7

0.8 2h /0

0.9

FIGURE 5.23 Dispersion curves of a 2D photonic crystal waveguide consisting of rectangular conductive cylinders in a square lattice arrangement for TE polarization modes. © 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

253

1

1

0.5

0.5

0

0

Hz

Hz

bh/l0  0.207316 and bh/l0  0.192617. Because there is no perfect bandgap in the TE polarization case, both guided modes are in part bandgaps. Next, we shall discuss photonic crystal waveguides consisting of metallic cylinders embedded in dielectric slabs whose cross section is shown in Figure 5.22(b). The dispersion curves are plotted in Figure 5.25 for TM polarized modes

−0.5

−0.5 2h/0 = 0.8 h/ = 0.192617

2h/0 = 0.8

−1

h/ = 0.207316

−10

−1 0

(a)

−10

10

0

(b)

y/(2h)

10

y/(2h)

FIGURE 5.24 Distributions of magnetic fields for the three lowest TM modes. Solid lines are at x/(2h)  0, and dotted lines are at x/(2h)  0.5. (a) Field distributions of an even mode. (b) Field distributions of an odd mode.

0.5

0.4 Mode 3 0.3 h/

Mode 2 Mode 4

0.2

0.1

0

Mode 1

0.4

0.5

0.6 2h/0

0.7

0.8

FIGURE 5.25 Dispersion curves of a 2D photonic crystal waveguide consisting of rectangular conductive cylinders embedded in dielectric slabs and arranged with a square lattice for a TM polarization case. © 2006 by Taylor & Francis Group, LLC

254

Electromagnetic Theory and Applications for Photonic Crystals 1

1

Mode 2

Mode 1

Ez

Ez

0.5 0.5 2h/0 = 0.5

0

h/ = 0.394016

2h/0 = 0.55 h/ = 0.431360

0 −4 (a)

0

4

-8

-4

(b)

y/(2h)

0 y/(2h)

0.5

0

4

8

Mode 4

Ez

Ez

0 −0.5 Mode 3

2h/0 = 0.61 h/ = 0.484840

−0.5

−1

−1 −5

(c)

2h/0 = 0.78 h / = 0.379927

0 y/(2h)

−10

5 (d)

0 y/(2h)

10

FIGURE 5.26 Distributions of electrical fields for the three lowest TM modes. Solid lines are at x/(2h)  0, and dashed lines are at x/(2h)  0.5. (a) Field distributions of an even mode. (b) Field distributions of an even mode. (c) Field distributions of an even mode. (d) Field distributions of an even mode.

where a  0.4h, er1  10.0, and d  3h. There are four even modes and no odd mode. The distributions of electric fields are plotted in Figure 5.26 where normalized wavelengths are 2h/l0  0.5, 0.55, 0.61, 0.78, and the corresponding normalized propagation constants are bh/p  0.394016, 0.431360, 0.484840 and 0.379927, respectively. Figure 5.27 shows the dispersion curve and field distributions of the lowest odd mode of the same waveguide for TE polarized waves. There is only one mode, and it is odd. Finally, we applied the method to analyze a periodic structure that includes two split metallic cylinders of double C-shaped cross section per unit cell as shown in Figure 5.28. The transmission coefficient T0 of the zero-th order space harmonic is shown in Figure 5.29 for a TM polarized incidence, where ao  bo  1.2h, ai  bi  h, co  ci  0.2h, so  si  0.05h, erb  1, ero  eri  35, and M  20. The solid and dotted lines represent the amplitude and phase of T0. This figure shows a decrease in the phase of the transmission coefficient T0 as the frequency increases in the range of 2h/l0  0.546  0.5526 and 2h/l0  0.553  0.565.

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

255

0.5 1 0.4

Hz

h/

0.5 0.3 0

0.2 −0.5 2h/0 = 0.4

0.1 −1 0

0.4

(a)

h/ = 0.217681

−10

0.5

0

(b)

2h/0

10

y/(2h)

FIGURE 5.27 Dispersion curve and field distributions of the lowest odd mode in a 2D metallic photonic crystal waveguide of a slab type. (a) Dispersion curve (b) Field distributions.

2h

Φ0

bo

ai

rb

co ri

si

bi

ci

ro so

ao

FIGURE 5.28 Geometry of a periodic structure with double C-shaped cross section.

These phase characteristics are similar to those observed in so-called negative reflective index materials.

5.4 ANALYSIS OF METALLIC PHOTONIC CRYSTALS FOR A GENERAL INCIDENCE In the above section, we have discussed problems of metallic photonic crystals composed of cylinders with arbitrary cross section, but the formula is only for normal incidence. In this section, we shall investigate the properties at any angle incidence shown in Figure 5.30. It is obvious that it becomes a three-dimensional problem, although the geometrical configuration is still two dimensional.

5.4.1 FORMULATION In the same manner discussed in the preceding section, we first sliced the original cylinders into many layers. The cross-section of one sliced layer has been

© 2006 by Taylor & Francis Group, LLC

256

Electromagnetic Theory and Applications for Photonic Crystals 1

350

0.6

300 0.4 Phase

0.2

Phase of T0 (°)

Amplitude of T0

0.8

Amplitude 250

0

0.55

0.56 2h/0

FIGURE 5.29 Transmission coefficient T0 of double C-shaped periodic array. Ei

i

p^

y

i

^

s

x i z

FIGURE 5.30 Geometry of a dielectric slab embedding conductive cylinders with arbitrary cross section.

drawn in Figure 5.30. If we assume eikz0z to be the variation in z-direction, the incident waves are written as: Eziu ( x , y, z )  H ziu ( x , y, z ) 

© 2006 by Taylor & Francis Group, LLC



∑

m

∑

m

Ameei ( kxm xk ym ykz 0 z )

(5.81)

Amh i ( kxm xk ym ykz 0 z ) e Z0

(5.82)

Mode-Matching Technique Applied to Metallic Photonic Crystals

∑

Ezid ( x , y, z ) 

m

H zid ( x , y, z ) 



∑

m

257

Ameei[ kxm xk ym ( yt )kz 0 z ]

(5.83)

Amh i[ kxm xk ym ( yt )kz 0 z ] e Z0

(5.84)

mp 2 2  , kym   k2b  k m0/e0. The scattering waves where kxm  — xm  kz0 , and Z0   h in the lower and upper half spaces are expressed

∑

Ezr ( x , y, z ) 

m

H zr ( x , y, z ) 



∑

m

∑

Ezt ( x , y, z ) 

m

H zt ( x , y, z ) 



∑

m

Bmeei ( kxm xk ym ykz 0 z )

(5.85)

Bmh i ( kxm xk ym ykz 0 z ) e Z0

(5.86)

Bmeei[ kxm xk ym ( yt )kz 0 z ]

(5.87)

Bmh i[ kxm xk ym ( yt )kz 0 z ] e Z0

(5.88)

The scattering fields in the region bounded by two adjacent rectangular rods located in the zero-th unit cell may be expanded in terms of waveguide modes.

Ezl ( x , y, z )  ∑ [Cleneikln y  Cleneikln ( yt ) ] sin jln ( x  wl  al )eikz 0 z

(5.89)

n1

H zl ( x , y, z ) 

1 Z0



∑ [Clhneik

n0

ln y

 Clhneikln ( yt ) ] cos jln ( x  wl  al )eikz 0 z

(5.90)

np 2  where jln  — k2l   j2ln  k z0 , and Im(kln)  0. The continuous condi2al , kln  tions of the tangential components of electromagnetic fields cross the boundary planes at y  0 and y  t, and may be expressed:

Eziu, x ( x , 0)  Ezr, x ( x , 0)  e 2ikx 0hp Ezl , x ( x  2hp, 0)Q( x )

(5.91)

Ezid, x ( x , t )  Ezt , x ( x , t )  e 2ikx 0hp Ezl , x ( x − 2hp, −t )Q( x )

(5.92)

H ziu, x ( x , 0)  H zr, x ( x , 0)  e 2ikx 0hp H zl , x ( x  2hp, 0),

(5.93)

H zid, x ( x , t )  H zt , x ( x , t )  e 2ikx 0hp H zl , x ( x  2hp, t ),

© 2006 by Taylor & Francis Group, LLC

x ∈Ξ x ∈Ξ

(5.94)

258

Electromagnetic Theory and Applications for Photonic Crystals

where  1 Q( x )    0 

x ∈Ξ

(5.95)

otherwise

 

Ξ   x : x ∈  

 



∪

p

[al  2hp  wl , al  2hp  wl ]  

(5.96)

We first apply the boundary conditions (5.93) and (5.94), and the following relative equations may be derived. Clhn 

Clhn  Clen 

ik xm (1 + n 0 )(1 − e



2 ilnt

∑

[ Amh  Bmh  ( Amh  Bmh )eiklnt ]Gl,nm

(5.97)

[ Amh  Bmh  ( Amh  Bmh )eiklnt ]Gl,nm

(5.98)

) m

ik xm



∑

(1  dn 0 )(1  e 2iklnt ) m

gl

∑

gb k0 kln erl (1  e 2ikl t ) m

jlnGl ,nm

{kz 0 k xm [ Bmh  Amh  ( Bmh  Amh )eiklnt ]  k0 k ym erb [ Bme  Ame  ( Bme  Ame )eiklnt ]} 

Clen 

ijln kz 0 k0 kln erl

Clhn

(5.99)

gl gb k0 kln erl (1  e

2 iklnt

∑

) m

jlnGl ,nm

{kz 0 k xm [( Bmh  Amh )eiklnt  ( Bmh  Amh )]  k0 k ym erb [( Bme  Ame )eiklnt  ( Bme  Ame )]} 

i j ln k z 0 k0 kln erl

Clhn

(5.100)

Applying (5.91) and (5.92), and noting (5.97)–(5.100), we may derive a set of linear equations. If we truncate the number of space harmonics up to m  M, the simultaneous equations may be written in matrix form as follows: e  h   be   bh ( I   1 )b  2 b 1 2 e   e   a h  ( 1  I ) a  2 ah   a 1 2

© 2006 by Taylor & Francis Group, LLC

(5.101)

Mode-Matching Technique Applied to Metallic Photonic Crystals

259

h   be   bh (  4 ) be  ( I   3 )b 4 3  e  h   ae   a h ) a  ( 4   ) a  ( I   3 4 3

(5.102)

e   bh  ( I +  ) be   bh  1b 2 1 2 e   a h  (   I ) ae   a h   1a 2 1 2

(5.103)

h  (   ) be  ( I   ) bh  4 be   3b 4 3  e  h  (   ) ae  ( I   ) a h  4 a   a 3 4 3

(5.104)

The column vectors ae, ah, be, and bh are defined as a

e

e ]  [ AeM  A0e  AM

(5.105)

h ] ah  [ AhM  A0h  AM

(5.106)

e ] be  [ BeM  B0e  BM

(5.107)

h ] bh  [ BhM  B0h  BM

(5.108)

The coefficient matrices (j , j  1, … , 4) and  are defined as  

k yn

  

k0

   1  [ 1mn ]  

 

2  [ 2 mn ]    

 3

    [ 3mn ]    

erb gl2

L

∑e l1

kz 0 k xn k02 k xn k xm k03k ym

2 rl g b

L

l1 L

∑ l1

k xm kz 0 k yn

 

k02 k ym mrb

 

  [ mn ]  

(5.109)

  

   1  gl2   1 w1l , mn  2      rl  g b 

∑e

 

4  [ 4 mn ]  

 

w1l , mn 

k xm kz 0

 k0 k ym mrb 

(5.110) 

 1  2  (gb2 w 2 l , mn  kz 0 w1l , mn )  mrb erl 

(5.111)





 e ∑ erb w1l , mn   l1 rl L

(5.112)



 

dmn 

(5.113)

 

and

ak0 jl2n (1  e 2iklnt )

n1

2hkln (1  e 2iklnt )

w1l , mn  ∑

ak0 jl2n eiklnt

n1

hkln (1  e 2iklnt )

w1l , mn  ∑

© 2006 by Taylor & Francis Group, LLC

Gl,nnGl,nm

Gl,nnGl,nm

(5.114)

(5.115)

260

Electromagnetic Theory and Applications for Photonic Crystals

ak0 (kl2  jl2n )(1  e 2iklnt )

n0

2hkln (1  d 0 )(1  e 2iklnt )

w 2 l , mn  ∑

ak0 (kl2  jl2n )eiklnt

n0

hkln (1  dn 0 )(1  e 2iklnt )

w 2 l , mn  ∑

Gl,nnGl,nm

(5.116)

Gl,nnGl,nm

(5.117)

Solving Equations (5.101)–(5.104), it is very easy to derive the S-matrix solution.    I    2 1  1     4  4   3  I S    2  1 I 1       4  3 ( 4   )     I 2  1  1     4  4   I  3    2  1 I 1       4 3   4 

1 2     3   2     3  I  2     3   2    I   3 

(5.118)

5.4.2 NUMERICAL EXAMPLES To test the accuracy of the present formula, let us consider a scattering problem of a periodic array consisting of circular cylinders located in free space. Figure 5.31 shows the reflected and transmitted electromagnetic fields (Ez and Hz) of three space harmonics, where the radius of the cylinders r  0.6h, the angle of an incident plane wave ui  30° and fi  45°, and the polarized angle ci  45°. The upper half of a circular cross section of the cylinders was sliced into 50 layers of thin rectangular rods, and the order of space harmonics was truncated by M  5. For comparison, the rigorous results obtained by an analytical approach [22] using the cylindrical harmonic expansion are also plotted by the dotted lines. Both results are in very close agreement with respect to the phase as well as the amplitude of the reflected waves. The comparison confirms that the present method yields a highly accurate solution even if the truncation order M of space harmonics is relatively small.

5.5 SCATTERING ANALYSIS OF CROSSED PHOTONIC CRYSTALS CONSISTING OF ARBITRARILY SHAPED CYLINDERS A cross-stacked periodic structure consisting of metallic cylinders embedded in a dielectric slab with an arbitrary cross section is illustrated in Figure 5.32. The cylinders in each slab are infinitely long and parallel to each other, while the cylinder axes are rotated 90° in each successive slab. The stacking sequence repeats every two slabs. The array of z-directed cylinders is referred to as the Z-array and the array of x-directed cylinders as the X-array. This structure has a double period viewed from y-direction, so that it is a bigrating. To formulate the © 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals Phase

1

200 First–order

100

Phase

300

Phase

0.4

200

0.2

100

0 0.6

0 Phase

0.5

1

(a)

1.5 2h/0

2

2.5

0

0.5

1

1.5 2h/ 0

2

200

Phase

0.1

100

Amplitude

0 0.3

Zero–order

0

Phase

300

Phase

200 100

0

0 Amplitude

0.6 0.5 0.4 0.3

Zero–order

300

Phase

200 100 0

1

300

0.2

0 3

Amplitude

0

0.2

2.5

First–order

1

Phase (°)

Amplitude (Ez )

100

0

100

300

300 200

200

0 0.3

0

–1–order

Amplitude

300

Amplitude

–1–order

300

Phase

200

0.1

200 Amplitude

100

–1–order

Amplitude

0

0 0

(c)

100 Phase

0

(b)

First–order

Phase

200

0.5

0 3

0.1

300

Amplitude

0.2

100

0 0

0 Zero–order

Phase

200

0.2 Amplitude

100

0.4

300

–1–order

0.4

Phase (°)

Zero–order

Amplitude (Ez )

Amplitude

200

Amplitude

0

0

0 0.6

Amplitude (Hz)

Amplitude (Hz )

0.2

300

First–order

0.5

1

1.5 2h/0

2

2.5

Phase (°)

Amplitude

Phase (°)

300

0.4

261

0

3

100 0

0

0.5

(d)

1

1.5

2

2.5

3

2h/0

FIGURE 5.31 Comparison of the reflected and transmitted electromagnetic fields Ez and Hz obtained by the present method (solid lines) and the analytical approach [22] (dotted lines) for a periodic array of circular cylinders. (a) Reflected Ez field. (b) Reflected Hz field. (c) Transmitted Ez field. (d) Transmitted Hz field.

problem, we slice each slab into many layers in which several rectangular rods are included per unit cell and arranged periodically. When a plane wave varying as ei(kx0xkz0z) illuminates the structure, from Floquet theorem, the scattered fields are expressed in superposition of space harmonics varying as ei(kxmxkzpz), where kxm  kx0  mp/hx, kzp  kz0  pp/hz, kx0  ks sin ui cos fi, kz0  ks sin ui sin fi, and 2hx and 2hz are the spacing periods in the X-array and Z-array; respectively. (ui, fi) denotes the angle of incident plane wave. Both m and p are integer numbers. The set of space harmonics {ei(kxmxkzpz)} forms a complete orthogonal system. In other words, all the scattered fields can be expressed in their superposition for each incident wave that belongs to the system. Thus, the properties of multilayered crossarrays can be described by a scattering matrix (S-matrix) based on this complete orthogonal system consisting of the space harmonics. The S-matrix can be calculated by two steps. The first step is to calculate the S-matrices of each sliced layer. The second step is to derive the total S-matrix by using a simple recursive process. © 2006 by Taylor & Francis Group, LLC

262

Electromagnetic Theory and Applications for Photonic Crystals

Ei

i

p^

y i

s^

x z

i

FIGURE 5.32 Geometry of multilayered cross-arrays of conductive cylinders embedded in dielectric slabs with arbitrary cross section.

5.5.1 REFLECTION AND TRANSMISSION MATRICES ONE Z-ARRAY LAYER

OF

Figure 5.33 shows the geometry of an infinite periodic structure in which each unit cell includes L rectangular rods. Because the structure is periodic only in the x-direction, there is no coupling between different waves of kzp and kzq (p  q). Thus the complete system {ei(kxmxkzpz)} may be divided into several orthogonal subsystems. In each subsystem, kzp is a constant. It is found that each subsystem also forms a complete orthogonal system. For each subsystem, we may derive a sub S-matrix to describe its property, which has been discussed in the previous section. Using these sub S-matrices together, the global S-matrix may be expressed as follows:  es   ei   z   z   s        hi h R T  z   z z   z       es   T R   ei  z   z   z   z  s   i   hz   hz     

(5.119)

s where the column vectors ezi, ezs, hi z , and hz are defined as s s s t ezs  [ezs ,M ,P , ez ,M1,P , ez ,M2 ,P , … , ez , M , P ]

(5.120)

ezi  [ezi,M ,P , ezi,M1,−P , e1z,M2,P ,…, ezi, M , P ]t

(5.121)

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

i

Ei

263

y

p^ i

^

s

x i z

(a)

2hx Y

t

2a1

2a2

r 1, r 1

r 2 r 2

Unit cell rb , rb rb , rb

X

w1 w2

2hx

(b)

FIGURE 5.33 An array of z-directed parallel cylinders. (a) Three dimensional geometry. (b) Cross-section view at a plane z  0. s s s t hzs  [hzs ,M ,P , hz ,M1,P , hz ,M2 ,−P , … , hz , M , P ]

(5.122)

 i t hzi  [hzi,M ,P , hzi,M1,P , hz1 ,M2 ,P ,…, hz , M , P ]

(5.123)

and  R  z    Tz

1 z 2   z 3  I z 4 z 3   z 2 z1  I z 2   z 3 (z 4   z ) z 3  I     I z 2  z 2 z1  z1    z 4 z 3    z 4   z I  z 3       I       z1 z2 z1 z2          z 4     I   z3 z z4 z3  

   I  z1   Tz   z 4   z     R z    z1   z 4 

© 2006 by Taylor & Francis Group, LLC

z 2

z1

(5.124)

264

Electromagnetic Theory and Applications for Photonic Crystals

z

   P ,       0      0 

  P      z   0      0 



0









 0 ,











0

 0      0       P 



0









 0  







0



–

0      0  ,     P ,  

  1, 2, 3, 4

(5.125)

(5.126)

–

The sub-matrices  p, and p are defined as follows: k ynp p,1  [ p,1, mn ]    k0

erb gl2, p

L

∑e l1

2 rl g b , p

k k  zp xn p, 2  [ p, 2, mn ]   2  k0 

∑e

k k xm p,3  [ p,3, mn ]   xn 3  k0 k ymp

∑m

L

l1 L

l1

k k k xm zp ynp p, 4  [ p, 4 , mn ]   2 k  0 k ymp mrb  

 p  [ p, mn ]  



n1

L

l1

(5.130)

(5.131)

2 ikl ,npt

)

2 ikl ,npt

)

2hx kl ,np (1  e

al k0 jl2,n eikl ,npt

n1

hx kl ,np (1  e 2ikl ,npt )

© 2006 by Taylor & Francis Group, LLC

rl

 w1l , p, mn  

 



w1l , p, mn  ∑

e

∑ erb

2 gb2, p  k02 erbrb  kzp

al k0 jl2,n (1  e

(5.128)

 1  2  (gb2, p w 2 l , p , mn  kzp w1l , p , mn )  (5.129)  rb erl

dmn 

 k0 k ymp mrb 

w1l , p, mn  ∑

 2  1  gl , p     2  1 w1l , p, mn    gb, p  rl  

(5.127)

 

k xm kzp

2 gl2, p  k02 erl mrl  kzp

 w1l , p, mn  

Gl ,nmn

Gl ,nmn

(5.132) (5.133)

(5.134)

Mode-Matching Technique Applied to Metallic Photonic Crystals

w 2 l , p , mn  ∑

n0

2 ikl ,npt

2hx kl ,np (1  dn 0 )(1  e

)

2 ikl ,npt



al k0 (kl2  jl2,n )eikl ,npt

n0

hx kl ,np (1  dn 0 )(1  e 2ikl ,npt )

w 2 l , p , mn  ∑ Gl ,nmn 

al k0 (kl2  jl2,n )(1  e

)

265

Gl ,nmn

(5.135)

Gl ,nmn

(5.136)

ei ( kxn kxm wl ) [(1)n eikxnal  eikxnal ] 2  j 2 )(k 2  j 2 ) (k xn ln xm ln

 [(1)n eikxmal  eikxmal ]

5.5.2 S-MATRIX

OF

(5.137)

ONE X-ARRAY LAYER

Figure 5.34 shows the geometry of an infinite periodic structure in which each unit cell includes several rectangular conducting cylinders. Since it is uniform in the x-direction, we choose Ex and Hx fields as the leading fields. Let us assume that ex and hx are column vectors of (2M  1)(2P  1) dimensions Ei

i

p^

y

i ^

s

x

z

i

2hz

(a) Y Unit cell rb, rb Z

rl, rl

2a 2

2a 2

r 2 r 2

r1, r1

t w1

2hz

w2

(b)

FIGURE 5.34 An array of x-directed parallel cylinders. (a) Three dimensional geometry. (b) Cross-section view at a plane x  0.

© 2006 by Taylor & Francis Group, LLC

266

Electromagnetic Theory and Applications for Photonic Crystals

that are defined as e x  [ex ,M ,P , ex ,M ,P1 , ex ,M ,P2 , … , ex , M , P ]t

(5.138)

h x  [hx ,M ,P , hx ,M ,P1 , hx ,M ,P2 , … , hx , M , P ]t

(5.139)

Then the amplitude vectors (exs, hxs) for the scattered waves are related to the incident waves (exi, hxi) by the S-matrix as follows:  e i   e s   x   x     s     h x   Rx Tx   hix       e s   T R   e i   x   x x   x   i   s   hx   hx     

(5.140)

where

 R  x    Tx

   I x 2 x1 x1   Tx  (x 4   x ) x 3  I x 4    R x1 x 2 x1  I x     x 4 x 3 x 4   x     I x 2 x1   (x 4   x ) I  x 3   x1 x 2    x 4 x 3 

   I  x1    x 4   x    x1    −x 4 

x 2

x1

I  x 3

x 4

x 2

x1  I

x 3

 x  x 4

x1 −x 4 x1  I x 4   x

1 x 2   x 3   x 2   x 3  I 

x 2   x 3   (5.141) x 2   I  x3 

x 2   x 3   x 2   I  x 3 

(5.142)

and

x

   M ,       0      0 

© 2006 by Taylor & Francis Group, LLC



0









 0 ,











0



0      0  ,     M ,  

  1, 2, 3, 4

(5.143)

Mode-Matching Technique Applied to Metallic Photonic Crystals

  M      x   0      0 



0









0      0        M 

 0  







0



k ymq m ,1  [ m ,1, pq ]   k  0

(5.144)

 w1l , m , pq  

erb gl2, m

L

∑e l1

2 rl g b , m

k k zq xm m , 2  [ m , 2, pq ]   2  k0

∑e

k k zq zp m ,3  [ m ,3, pq ]   3 k  0 k ymp

∑m

L

l1

L

l1

k k k xm zp ymq m , 4  [ m , 4 , pq ]   2  k0 k ymp mrb

(5.145)

 2  1  gl , m   w  1  2  1l , m , pq  g    rl b,m  1  2  (gb2, m w 2 l , m , pq  k xm w1l , m , pq )  e  rb rl

L

267

e

∑ erb l1

rl

 w1l , m , pq  

(5.146)

(5.147)

(5.148)

 k k  xm zp  m  [ m , pq ]   d pq   k0 k ymp mrb 

(5.149)

2 gl2, m  k02 e rl mrl  k xm

(5.150)



a l k0 j 2l ,n (1  e 2ik l , mn t )

n1

2hz k l , mn (1  e 2ik l , mnt )

w1l , m , pq  ∑

a l k0 j 2l ,n eik l , mn t

n1

hz k l , mn (1  e 2ik l , mn t )

w1l , m , pq  ∑

w 2 l , m , pq

w 2 l , m , pq

2 g b2, m  k02 erb mrb  k xm

(5.151)

Gl ,npq

(5.152)

Gl ,npq



a l k0 ( k l2  j 2l ,n )(1  e 2ik l , mn t )

n0

2hz k l , mn (1  dn 0 )(1  e 2ik l , mn t )

∑

a l k0 ( k l2  j 2l ,n )eik l , mn t

n0

hz k l , mn (1  dn 0 )(1  e 2ik l , mn t )

∑

© 2006 by Taylor & Francis Group, LLC

Gl ,npq

Gl ,npq

(5.153)

(5.154)

268

Electromagnetic Theory and Applications for Photonic Crystals

Gl ,npq 

ei ( kzqkzp ) wl 2  j 2 ) (k 2  j 2 ) a l2 (kzq ln zp ln  [(1)n eikzq a l  eikzq a l ][(1)n eikzpa l  eikzpa l ]

(5.155)

If we define the column vectors ex and hx as follows ex  [ex ,M ,P , ex ,M1,P , ex ,M2,P , … , ex , M , P ]t

(5.156)

hx  [hx ,M ,P , hx ,M1,P , hx ,M2,P , … , hx , M , P ]t

(5.157)

e x  ϒex , h x  ϒhx

(5.158)

then

where ϒ  [ϒi , j ]  [ϒmp, nq ]  [dmn d pq ] i  1, 2, 3,  , (2 M  1)(2 P  1)  (2 P  1)(m  M )  P  1  p j  1, 2, 3,  , (2 M  1)(2 P  1)  (2 M  1)(q  P )  M  1  n

(5.159)

The column vectors (ex, hx) and (ez, hz) satisfy the following relative equations.  e   A B  z  x  z      hx  Cz Az 

 e   z     hz 

(5.160)

 e   A   z    x Bx       hz  Cx Ax 

 e   x    hx 

(5.161)

where Az  [ Az ,ij ] 

  k k xm zp    k2  k2  b zp 

Cz  [Cz ,ij ] 

 k ke  zmp 0 rb   k2  k2  b zp 

  ij   

  ij   

 

k ymp k0 mrb

  

2 kb2  k xm

Bx  [ Bx ,ij ]  

Bz  [ Bz ,ij ] 

,

,  

ij  ,  

      

k ymp k0 mrb 2 kb2  kzp  

k xm kzp

Ax  [ Ax ,ij ]  

 k2  b 

2  k xm

  dij     

dij 

 

k ymp k0 erb

  

2 kb2  k xm

Cx  [Cx ,ij ]  

(5.162)

(5.163)

    

dij    

(5.164)

The following equation may be easily confirmed. 1  A    x Bx    Az Bz      Cx Ax  Cz Az 

© 2006 by Taylor & Francis Group, LLC

(5.165)

Mode-Matching Technique Applied to Metallic Photonic Crystals

269

Substituting (5.160) into (5.140),  es   ei   z   z   s         hz   R⊥ T⊥   hzi       es   T R   ei   z   ⊥ ⊥  z   s   i   hz   hz     

(5.166)

where

 R      T

 A B 0 x  x  0 T   C x Ax   Ax 0 R   0   0 0 Cx

1 0    0 0 0      0   0  0 0   R Tx     x  Bx   0 0  0   Tx R x    Ax   0 0 0  

  0 0 0   Az Bz 0     Az 0  0  0 0   Cz    0 0  0 0 0 Az     0 Cz  0 0 0    0

0   0  Bz   Az 

(5.167)

The S-matrix expression of a crossed-array may be derived by using the same recursive formula (5.55) with (5.124) and (5.167).

5.5.3 NUMERICAL EXAMPLES Although the notations are a little intricate because of the three-dimensional problems, the mathematics derivation of the proposed method is straightforward. By considering the orthogonality and completeness of the space harmonics, the multilayered system can be characterized by using the independent solutions to each of the layers in isolation. For the same reason, the whole system of space harmonics to crossed-array bigratings can be reduced to orthogonal monoperiodic systems in each layer. Therefore, this method can be applied to the multilayered crossed-arrays at a smaller computer cost. Although the dimensions of the total reflection and transmission matrices of the crossed-arrays are larger, the dimensions of each submatrix to be calculated are very small. This is the main difference from the classical methods [10]. Hence, the number of unknowns does not increase, even if the number of layers is very large. Note that the infinite sums with respect to variable n in Equations (5.133)–(5.136) and Equations (5.151)–(5.154) converge with a faster than O(n3) rate. To test the convergence behavior, consider the crossed-array shown in Figure 5.35(a). The amplitudes and the phases of (0, 1)-th, (0, 0)-th, and (1, 0)-th order

© 2006 by Taylor & Francis Group, LLC

270

Electromagnetic Theory and Applications for Photonic Crystals p^

i

Ei

y i

s^

x i

z

b

2hz d 2hx

b

a

a

(a) p^

i

Ei

y i

s^

One set

x

b

i

a

z

d

2hz

b

a

hz

2hx

hx

(b) Ei

i

p^

y One set

i

s^

b a

x i

z

2hz hz

d a

b

hx

(c)

FIGURE 5.35 Three types of crossed arrays.

© 2006 by Taylor & Francis Group, LLC

2hx

Mode-Matching Technique Applied to Metallic Photonic Crystals

194

225 (0,0) order

220

0.07 0.00084

101

0.00082

Amplitude (Hz )

0.3 0.08

Phase (°) (Ez )

193

30 (0,0) order

0.07

20 247

(1,0) order

246

0.0088 2

4

(a)

6

8

10

0

234 (0,1) order

(0,0) order

215

8

24

230

22 36

0.7035 (0,0) order

0.703

35

0.7025 114

0.14 (1,0) order

(1,0) order

228 2

4

6 M

8

10

(0,1) order

Amplitude (Hz)

0.7035

6 M

0.0018 Phase (°) (Ez)

232 216

0.7025 0.042 0.041 0.04 0.039

4

0.0019

233

0.019

0.703

2

(b)

M 0.02

Amplitude (Hz )

70

0.08

0.009

100

0.00078

(c)

72

0.0092

(1,0) order

0.0008

74 (0,1) order

Phase (°) (Hz )

0.31 Amplitude (Hz )

0.079 0.078 0.077 0.076

195 (0,1) order

0.13 0

226 10

(d)

Phase (°) (Hz)

0.32

271

2

4

6

8

113 112 111 10

M

FIGURE 5.36 Convergent behaviors of the amplitudes and phases of (0, 1)-th, (0, 0)-th, and (1, 0)-th order harmonics as functions of the truncated number M  P, where c i  45°. (a) Reflected waves. (b) Reflected wave. (c) Transmitted waves. (d) Transmitted wave.

harmonics are plotted in Figure 5.36 as functions of the truncated number M  P where hx  hz  0.45  l0, a  0.2hx, b  0.6hx, d  2hx, u i  0°, fi  180°, and c i  45°. The solid lines denote the amplitudes, and the dotted lines denote the phases. In this case, all of the space harmonics except for the (0, 0)-th harmonic are evanescent waves. The numerical results show that the convergent speed is very high. The power reflectance of space harmonics is shown in Figures 5.37 and 5.38, for d  2hz and d  hz, respectively, as functions of normalized wavelength 2hz/l0 from 32 sets of crossed-arrays as shown in Figure 5.35(a). The width of the perfect bandgap in Figure 5.38 is larger than that in Figure 5.37. The bandgaps appear in the lower frequency side. Figures 5.39 and 5.40 show the power reflectance of space harmonics scattered from 32 sets of crossed-arrays as shown in Figures 5.35(b) and 5.35(c), respectively. Although their structures are very similar, with a difference only in the stacking sequence, the scattering properties are quite different. Comparing these two figures, we find that the width of the bandgap in Figure 5.40 is broader than that in Figure 5.39. The results indicate that the sequence is also important.

© 2006 by Taylor & Francis Group, LLC

Electromagnetic Theory and Applications for Photonic Crystals 1

1

0.8

0.8

Reflected efficiency

Reflected efficiency

272

0.6 0.4 0.2

i = 0,
i = 90° i = 30°,
i = 90° i = 45°,
i = 45°

0 0.2

0.4

(a)

0.6

0.6 0.4 i = 60°,
i = 90° i = 75°,
i = 45° i = 75°,
i = 90°

0.2 0

0.8

1

0.2

0.4

(b)

2h/

0.6

0.8

1

2h/

1

1

0.8

0.8

0.6 0.4 0.2

i = 0,
i = 90° i
i = 90° i = 30°, i = 45°,
i = 45°

0 0.2

(a)

Reflected efficiency

Reflected efficiency

FIGURE 5.37 Power reflectance scattered from 32 sets (64 layers) of crossed-arrays shown in Figure 5.35(a) as functions of normalized wavelength 2hz /l0, where a  b  0.4hz, d  2hz and ci  45°.

0.4

0.6 2hz /

0.6 0.4 0.2

i = 90,
i = 90° i = 75°,
i = 45° i = 75°,
i = 90°

0 0.8

1

0.2

(b)

0.4

0.6 2hz /

0.8

1

FIGURE 5.38 Power reflectance of space harmonics as functions of normalized wavelength 2hz /l0 from 32 sets (64 layers) of crossed-arrays where d  hz. The values of other parameters are the same as those given in Figure 5.37.

5.6 DIFFRACTION FROM A CONDUCTIVE SLAB CUT PERIODICALLY BY RECTANGULAR HOLES In the previous section, we have discussed the problems of periodically metallic structures including two-dimensional problems and three-dimensional problems, but the structures are all composed of two-dimensional cylinders. In this section, we shall extend the S-matrix theory combined with a mode matching technique to three-dimensional structures. The aim of this extension is to study threedimensional negative refractive index materials, frequency selective surface problems, and surface plasmon problems.

© 2006 by Taylor & Francis Group, LLC

1

1

0.8

0.8

Reflected efficiency

Reflected efficiency

Mode-Matching Technique Applied to Metallic Photonic Crystals

0.6 0.4 i = 0,
i = 90° i = 30°,
i = 90° i = 45°,
i = 45°

0.2 0 0.2

0.4

(a)

0.6

0.6 0.4 i = 60,
i = 90° i = 75°,
i = 45° i = 75°,
i = 90°

0.2 0

0.8

0.2

1

0.4

(b)

2hz /

273

0.6

0.8

1

2hz /

1

1

0.8

0.8

Reflected efficiency

Reflected efficiency

FIGURE 5.39 Power reflectance scattered from 32 sets (128 layers) of crossed-arrays shown in Figure 5.35(c) of all space harmonics as functions of normalized wavelength 2hz /l0, where hx  hz, a  b  0.4hz, d  hz, and c i  45°.

0.6 0.4 0.2

 i = 0,
i = 90°  i = 30°,
i = 90°  i = 45°,
i = 45°

0 0.2

(a)

0.4

0.6 2hz /

0.6 0.4 0.2

 i = 60,
i = 90°  i = 75°,
i = 45°  i = 75°,
i = 90°

0 0.8

1

0.2

(b)

0.4

0.6

0.8

1

2hz /

FIGURE 5.40 Power reflectance scattered from 32 sets (128 layers) of crossed-arrays shown in Figure 5.35(b) as functions of normalized wavelength 2hz /l0, where hx  hz, a  b  d  hz, and c i  45°.

5.6.1 FORMULATION Let us consider a problem in which several rectangular holes arranged periodically in a given lattice are cut into a perfectly conductive slab as shown in Figure 5.41. The holes are fully filled by dielectric media with relative permittivity erl and relative permeability mrl. All the base lines of the rectangular holes must be parallel to one another. We set the x-axis to be parallel to the base lines and the y-axis to be orthogonal to that. From the periodic properties, the scattering natures may be described by an S-matrix based on space harmonics. Therefore, the incident wave may be expanded in terms of space harmonics.

© 2006 by Taylor & Francis Group, LLC

274

Electromagnetic Theory and Applications for Photonic Crystals

E yi ( x , y, z )  H yi ( x , y, z )  E yi ( x , y, z )  H yi ( x , y, z ) 

i

Ei





∑ ∑

m n



∑ ∑

m n



∑ ∑

m n



∑ ∑

m n

p^

i e i ( k xm xk yn ykzmn z ) Emn

(5.168)

i H mn i k xk yk z e ( xm yn zmn ) Z0

(5.169)

i e i ( k xm xk yn ykzmn ( zt )) Emn

(5.170)

i H mn i k x k y k z t e ( xm  yn  zmn (  )) Z0

(5.171)

z i

^

s

y i t

x

(a)

y

Unit cell (2c1, 2d1) 2b1 (2c 2, 2d2)

0 r1 0r1

2b 2 2w

2a1 0 r 2  0 r1 2a 2

x

0 r 3 0r1

(2c 3, 2d 3)

2b 3

2a3

(b)

2h

FIGURE 5.41 Geometry of three rectangular holes cut into a perfectly conductive slab and arranged periodically in a given lattice. (a) Three dimensional illustration. (b) Top view.

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

275

mp np 2  2  where kxm  kx0  — , kyn  ky0  — kb2  k xm  kyn, and Im(kzmn)  0. w , kzmn  h Then, the scattering fields in the upper and lower half-spaces may be written as:



∑ ∑

E ys ( x , y, z ) 

m n

H ys ( x , y, z ) 





∑ ∑

m n



∑ ∑

E ys ( x , y, z ) 

m n

H ys ( x , y, z ) 





∑ ∑

m n

 e i ( k xm xk yn ykzmn z ) Amn

(5.172)

 Bmn i k x k y k z e ( xm  yn  zmn ) Z0  e i[ k xm xk yn ykzmn ( zt )] Amn  Bmn i k x k y k z t e [ xm  yn  zmn (  )] Z0

(5.173) (5.174) (5.175)

 where A mn, B mn are unknown coefficients. The scattering fields in the l-th rectangular region are expressed by electric and magnetic Hertzian vectors

yel  ∑



∑ sin jlr ( x  al  cl ) cos hln ( y  bl  dl )

r1 n0

(5.176)

leigl , rn z  C le iglrn ( zt ) ]  [Crn rn



yhl  ∑



∑ cos jlr ( x  al  cl ) sin hln ( y  bl  dl )

r0 n1

leigl , rn z  Dle iglrn ( zt ) ] Z  [ Drn rn 0

(5.177)

pp np 2  —  where jlr  — k2l  jlr  hlv2 , kl  k0 erlmrl , and Im(glrv  0). 2al , hlv  2bl , glrv  The tangential components of electric and magnetic fields derived from Equations (5.168)–(5.177) should be continuous across the boundary planes at z  0 and z  t. Taking into account the periodic natures, the boundary conditions may be expressed as follows:

E xi, y ( x , y, 0)  E xs, y ( x , y, 0) e

2 i ( hpk x 0 wqk y 0 )

E xl , y ( x  2hp, y  2 wq, 0)Q( x , y)

(5.178)

E xi, y ( x , y, t )  E xs, y ( x , y, t ) e

2 i ( hpk x 0 wqk y 0 )

E xl , y ( x  2hp, y  2 wq, t )Q( x , y)

(5.179)

H xi, y ( x , y, 0)  H xs , y ( x , y, 0 ) e

2 i ( hpk x 0 wqk y 0 )

H xl , y ( x  2hp, y  2 wq, 0),

( x , y) ∈ Ξ

(5.180)

H xi, y ( x , y, t )  H xs , y ( x , y, t ) e

2 i ( hpk x 0 wqk y 0 )

© 2006 by Taylor & Francis Group, LLC

H xl , y ( x  2hp, y  2 wq, t ),

( x , y) ∈ Ξ (5.181)

276

Electromagnetic Theory and Applications for Photonic Crystals

where  1 ( x , y) ∈ Ξ Q( x , y)    0 otherwise 

(5.182)

   Ξ  ( x , y) :  x ∈ ∪ [al  2hp  cl , al  2hp  cl ]    p    ∩  y ∈ ∪ [bl  2wq  dl , bl  2wq  dl ]   q 

(5.183)

The y-component of magnetic fields may be derived from (5.177)

H yl ( x , y, z )  ∑



∑(kl2  hl2n ) cos jlr ( x  al  cl ) sin hln ( y  bl  dl )

r0 n1

 eiglrn z  D e iglrv ( zt )  Z   Drn  0 rn 

(5.184)

Substituting (5.169) (5.171) (5.173) (5.175) and (5.184) into (5.180) and (5.181) and considering the orthogonality of trigonometric functions {cos jlr(x  al  cl)} and {sin hlv(y  bl  dl)}, we may derive the following relative equations l  Drn

ihln 

l  Drn

 hl2n )(1  dr 0 )(1  e 2iglrnt ) i  B )e iglrnt {( H mn mn m n

(kl2

∑ ∑

i  B )} k G a G b  ( H mn mn xm l , rm l , nn

(5.185)

i  B )}k G a G b  ( H mn mn xm l , rm l , nn

(5.186)

ihln  hl2n )(1  dr 0 )(1  e 2iglrnt ) i  B )e iglrnt {( H mn mn m n

(kl2 

∑ ∑

a b and Gl,nn are defined as where the functions Gl,rm

Gla, rm 

eikxmcl [(1)r eikxmal  eikxmal ] 2  j2 ) al (k xm lr

(5.187)

Glb, nn 

eik yndl [(1)n eik ynbl  eik ynbl ] 2  h2 ) bl (k yn ln

(5.188)

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

277

In the same manner, applying the continuous conditions of the x-component of magnetic fields (5.180) and (5.181), we may obtain two other equations. C rn 

jlr hln erl k0 glrn

  Drn



b ik yn jlrGla, rm Gl , nn

n

kb2  k y2n



1

∑ ∑

erl k0 glrn (1  dn 0 )(11  e 2iglrn ) m

  H i  ( B  H i )e iglrnt ]  {k xm k yn [ Bmn mn mn mn   E i  ( A  E i )e  k0 erb kzmn [ Amn mn mn mn

C rn 

jlr hln erl k0 glrn

D rn 

iglrnt

(5.189)

]}

1



∑ ∑

erl k0 glrn (1  dn 0 )((1  e 2iglrn ) m n i

b ik yn jlrGla, rm Gl , nn 2 kb2  k yn

t

  H i )e glrn  ( B  H i )]  {k xm k yn [( Bmn mn mn mn   E i )e  k0 erb kzmn [( Amn mn

iglrnt

i  ( A mn  Emn )]}

(5.190)

Applying the relative Equations (5.185) (5.186) and (5.189) (5.190) and the boundary conditions (5.178) (5.179), and noting the orthogonality of exponential functions ei(kxmxkyny), it leads to a set of linear equations in terms of unknown  coefficients A mn and B mn. L

  A  E ipq ∑ pq l1

al bl ∑ ∑ ik j (k 2  hl2 )(Crn  Crneiglrnt )Gla,rpGlb,nq (5.191) 4wh r1  =0 yq lr l

1 )  k m k  i [k xp k yq ( Apq  E ipq 0 rb zpq ( B pq  H pq )] 2 kb2  k yq L

∑ l 1

al bl 4wh





∑ ∑ ikxphlnGla,rpGlb,nq  [jlrhl (Crn  Crneig

lr t

)

r0 n1

 iglrnt )]  mrl k0 glrn ( D rn  Drn e L

  A  E ipq ∑ pq l1

(5.192)

al bl ∑ ∑ ik j (k 2  hl2n )(Crneiglrnt  Crn )Gla,rpGlb,nq (5.193) 4wh r1 n0 yq lr l

1 i )  k m k  [k xp k yq ( Apq  E ipq 0 rb zpq ( B pq  H pq )] 2 kb2  k yq L

∑ l 1

al bl 4wh





∑ ∑ ikxphlnGla,rpGlb,nq  [jlrhln (Crneig

r0 n =1

iglrnt  mrl k0 glrn ( D  D rn e rn )]

© 2006 by Taylor & Francis Group, LLC

lrnt

 C rn )

p, q  0,  1,  2,  (5.194)

278

Electromagnetic Theory and Applications for Photonic Crystals

If the number of expansion coefficients is truncated by m  M and n  N, the simultaneous Equations (5.191)–(5.194) may be rewritten in matrix form as follows:  I   2  1 1    ) I     (       4 3 4     1 2 I   1             4 3 4     I  1     4 =   1    4 

2 I   3 2  3

2   a        3  b     2   a      I   3   b   2   ei  1     i  4   h  3      1  I 2   ei      i 4   I   3   h 

(5.195)

where the column vectors a, b, ei, and hi are defined as t a  [ A(M )(N ) , A(M )(N 1) , A(M )(N 2) , … , A MN ]

(5.196)

t b  [ B(M )(N ) , B(M )(N 1) , B(M )(N 2) , … , B MN ]

(5.197)

 i i i t ei±  [ E(i M )(N ) , E(M )(N 1) , E(M )(N 2)) , … , E MN ]

(5.198)

 i i i t hi  [ H(i M )(N ) , H (M )(N 1) , H (M )(N 2)) , … , H MN ]

(5.199)

 and the submatrices ( 1  4 ) are expressed as follows:  e

rb k yn kzmn k yq 2 2  k0 (kb  k yn ) 

   1  [ 1ij ]  

2

  k k2 k xm yn yq   [ 2ij ]   2 2 2  k0 (kb  k yn )    

 

1)  x(pqmn 

1)  (pqmn

  k xm k xp   2  mrb k0 kzpq  k0  

   3  [ 3ij ]  

2 kb2  k yq

(5.200)

 



k xm k yq

3)  x(pqmn

 

2 e k k k kb2  k yq rb yn xp zmn

 

2 ) mrb k0 kzpq k0 (kb2  k yn

4  [ 4ij ]  

i  (p  M)(2N  1)  N  1  q,

© 2006 by Taylor & Francis Group, LLC

k02

  ( )  2 x pqmn    2 k k k yn xm xp

2 ) k02 (kb2  k yn

(5.201)    

2 )   x(pqmn       

(5.202)

 

2 )  x(pqmn 

(5.203)

 

j  (m  M)(2N  1)  N  1  n.

Mode-Matching Technique Applied to Metallic Photonic Crystals

279

In Equations (5.200)–(5.203), the terms xpqmn include double infinity sums, which are defined as 1) x(pqmn

L

∑ l1 L

1)  x(pqmn ∑ l1

2 ) x(pqmn

L

∑ l1 L

2 )  x(pqmn ∑ l1

3) x(pqmn

L

∑ l1 L

3)  x(pqmn ∑ l1

2 ig t k0 al bl jl2r (kl2  hl2n )(1  e lrn ) a a b b G G G G (5.204) ∑∑ 4erl wh r1 n0 g (1  d )(1  e 2iglrnt ) l , rm l , rp l ,nn l ,nq lrn n0

k0 al bl 2erl wh





∑∑

r1 n0

jl2r (kl2  hl2n )eiglrnt glrn (1  dn 0 )(1  e 2 ig

2 iglrnt

)

a b b Gla,  m Gl , rpGl , nn Gl , nq (5.205)

t

k0 al bl jl2r hl2n (1  e lrn ) a a b b G G G G ∑∑ 4erl wh r1 n1 g (1  e 2iglrnt ) l , rm l , rp l ,nn l ,nq

(5.206)

jl2r hl2n eiglrnt k0 al bl G a G a G b G b ∑∑ 2erl wh r1 n1 g (1  e 2iglrnt ) l , rm l , rp l ,nn l ,nq lrn

(5.207)

lrn

2 ig t k0 al bl (kl2  jl2r )hl2n (1  e lrn ) a a b b G G G G (5.208) ∑∑ 4erl wh 0 n1 g (1  d )(1  e 2iglrnt ) l , rm l , rp l ,nn l ,nq lrn

k0 al bl 2erl wh





∑∑

r0 n1

r0

(kl2  jl2r )hl2n eiglrnt glrn (1  dr 0 )(1  e 2

iglrnt

)

a b b Gla, rm Gl , rpGl , nn Gl , nq (5.209)

Solving Equation (5.195), an S-matrix may be obtained.  I   2  1 1     4 (  4 ) I  3 S   2 I   1 1      4 3   4     I  1     4    1     4 

2

 1

I   3

4

2

 1 

 3

I

4  

1 2    3   2    I   3  2     3   2    I   3 

(5.210)

5.6.2 NUMERICAL EXAMPLES The S-matrix is in a form similar to Equations (5.118) and (5.124), but the elements of the matrices  j ( j  1, 2, 3, 4) include double infinite sums. All the infinite sums in Equations (5.204)–(5.209) converge at a rate faster than O(r3) or O(n3).

© 2006 by Taylor & Francis Group, LLC

Electromagnetic Theory and Applications for Photonic Crystals 114 0.06

0.29 (0,1) order

28 27

0.205

Amplitude (Hy )

29 (0,0) order

Phase (°) (Ey)

0.67

27 22

2

4

6

(a)

8

10

(1,0) order

0.085

0.195 109 12

2

4

(b)

M

6

8

21 10

20 12

M 30

0.06

0.19

114

0.18

120 0.23 118

(0,0) order

0.22 0.21

116 119

0.135

(1,0) order

Amplitude (Hy )

115

Phase (°) (Ey )

(0,1) order 0.05 Amplitude (Ey )

28

(1,0) order 110

117 2

4

6

8 M

10

116 12

26 300

0.24 0.23

298

(0,0) order

0.22

296 208

0.21

207

(1,0) order

206 2

(d)

28

(0,1) order

0.07 0.068 0.066 0.064

118

0.13

(c)

(0,0) order

0.09

111

0.2

199 198 30 29

0.675

Phase (°) (Hy)

Amplitude (Ey )

111 30

0.05 0.675

200

0.28

112

0.67

201

113

(0,1) order

Phase (°) (Hy)

280

4

6

8

10

205 12

M

FIGURE 5.42 Convergent behaviors of the amplitudes and phases of (0, 1)-th, (0, 0)-th, and (1, 0)-th order harmonics as functions of the truncated number M  N, where c i  45°. (a) Reflected waves. (b) Reflected wave. (c) Transmitted waves. (d) Transmitted wave.

To show this, we shall investigate their convergence behavior. Figure 5.42 shows the amplitudes and phases of (0, 1)-th, (0, 0)-th, and (1, 0)-th order harmonics scattered from a metallic slab cut periodically by rectangular holes, as functions of the truncated number (M  N), where L  1, h  w  0.3l0, a1  b1  0.6h, c1  d1  0, er1  mr1  1, t  0.1l0, and c i  45°. The solid lines denote the amplitudes, and the dotted lines denote the phases. The relative calculating errors are less than one percent, when the truncated numbers (M, N) are chosen to be 5. Hence, in the following examples, the truncated numbers are chosen to be 5. As our second example, let us discuss the scattering properties of threedimensional arrays. Figure 5.43 shows the reflected efficiency scattered from 32-multilayered slabs cut periodically by rectangular holes, where L  1, w  h, a1  b1  0.8h, c1  d1  0, er1  mr1  1, t  0.4h, c i  45°, and the interval between two adjacent slabs is equal to 1.2h. The structure’s pattern viewed from above is very similar to Figure 5.10(a), which may be obtained by exchanging the rod regions and space regions in Figure 5.10(a), so that the arrangement is a rectangular lattice. In Figure 5.43, there is a perfect bandgap in the lower frequency band. Figure 5.44 shows the power reflection efficiency scattered from 32-multilayered

© 2006 by Taylor & Francis Group, LLC

1

1

0.8

0.8

0.6 i = 0, 0.4

Reflected efficiency

Reflected efficiency

Mode-Matching Technique Applied to Metallic Photonic Crystals


i = 90°

i = 30°,
i = 90°

0.2

0.6

0.4  i = 60°,
i = 90°

0.2 i

= 45°,


i

 i = 75°,
i = 45°

= 45°

0

 i = 75°,
i = 90°

0 0.2

0.4

0.6

0.8

0.2

1

2h/0

(a)

281

0.4

0.6

0.8

1

2h /0

(b)

1

1

0.8

0.8

0.6 0.4 0.2 0

i = 0,


i = 90°

0.4

0.6 0.4 i = 60°,
i = 90° i = 75°,
i = 45°

0.2 i = 45°,
i = 45°

i = 30°,
i = 90°

0.2 (a)

Reflected efficiency

Reflected efficiency

FIGURE 5.43 Reflected efficiency scattered from 32-multilayered slabs cut periodically by rectangular holes, where L  1, w  h, a1  b1  0.8h, c1  d1  0, er1  mr1  1, c i  45°, and the interval between two adjacent slabs is equal to 1.2h.

i = 75°,
i = 90°

0 0.6 2h/0

0.8

0.2

1 (b)

0.4

0.6 2h/0

0.8

1

FIGURE 5.44 Reflected efficiency scattered from 32-multilayered slabs cut periodically by rectangular holes, where L  2, w  2h, a1  b1  a2  b2  0.8h, c1  d1  0, c2  h, d2  2h, er1  mr1  er2  mr2  1, c i  45°, and the interval between two adjacent slabs is equal to 1.2h.

slabs with rectangular holes, arranged in a triangular lattice, where L  2, w  2h, a1  b1  a2  b2  0.8h, c1  d1  0, c2  h, d2  2h, er1  mr1  er2  mr2  1, t  0.4h, c i  45°, and the interval between two adjacent slabs is equal to 1.2h. The width of the perfect bandgap becomes very narrow, in comparison with the rectangular lattice shown in Figure 5.43. As our next example, we consider a scattering problem from overlapped crossed-arrays as shown in Figure 5.45. To solve this problem, we divide the structure into five layers. The five layers are sequentially the upper half space, y-array region, metallic slab region, x-array, and lower half space from top to

© 2006 by Taylor & Francis Group, LLC

282

Electromagnetic Theory and Applications for Photonic Crystals z y x

2w

du b dc a

b 2h a

1

1

0.8

0.8

0.6 0.4 i = 0,
i = 90° i = 30°,
i = 90°

0.2

i = 45°,
i = 45°

0 0.2

(a)

Reflected efficiency

Reflected efficiency

FIGURE 5.45 Schematic of an overlapped crossed-array.

0.4

0.6 2h/0

0.8

0.6 0.4 0.2

i = 60°,
i = 90° i = 75°,
i = 45°

0

i = 75°,
i = 90°

0.2

1

(b)

0.4

0.6

0.8

1

2h/0

FIGURE 5.46 Reflected efficiency scattered from 32 sets of overlapped crossed-arrays, where w  h, a  b  0.4h, dc  du  0.2h, c i  45°, and the interval between two adjacent overlapped crossed-arrays is 1.4h.

bottom. An S-matrix may be derived for each layer by using (5.124), (5.167), and (5.210). Figure 5.46 shows the reflected efficiency scattered from 32 sets of overlapped crossed-arrays, where w  h, a  b  0.4h, dc  du  0.2h, ci  45°, and the interval between two adjacent overlapped crossed-arrays is equal to 1.4h. There is a perfect bandgap around 2h/l0  0.4. Figure 5.47 shows the reflected efficiency scattered from another overlapped crossed-array, where du  0.4h, dc  0, and the interval between two adjacent overlapped crossed-arrays is equal to 1.2h. The values of the other parameters are the same as those in Figure 5.46. The perfect bandgap in Figure 5.46 is narrower than in Figure 5.47. We also find that there is a narrow passband around 2h/l0  0.3 in the bandgap region for a

© 2006 by Taylor & Francis Group, LLC

1

1

0.8

0.8

Reflected efficiency

Reflected efficiency

Mode-Matching Technique Applied to Metallic Photonic Crystals

0.6 0.4 0.2

i = 0,
i = 90° i = 30°,
i = 90° i = 45°,
i = 45°

0 0.2

(a)

0.4

0.6

283

0.6 0.4

i = 60°,
i = 90° i = 75°,
i = 45° i = 75°,
i = 90°

0.2 0

0.8

1

0.2

0.4

(b)

2h/0

0.6

0.8

1

2h/0

FIGURE 5.47 Reflected efficiency scattered from 32 sets of overlapped crossed-arrays, where w  h, a  b  0.4h, dc  0, c i  45°, and the interval between two adjacent overlapped crossed-arrays is 1.2h.

particular incidence (u i  75°, fi  90°) in Figure 5.46. This property may be applied as a directive selector.

5.7 SCATTERING ANALYSIS OF A CONDUCTIVE SLAB CUT PERIODICALLY BY RECTANGULAR HOLES IN AN ARBITRARY DIRECTION In the previous section, we have discussed a three-dimensional problem of a perfectly conductive slab cut periodically by several rectangular holes per unit cell. Because the baselines of all the rectangular holes are parallel, (all the rectangular holes are in a same direction), this is a special case. In this section, we shall discuss a much more common case, in which the baselines of the rectangular holes are not parallel. Figure 5.48 is a top view of a conductive slab cut periodically by rectangular holes in an arbitrary direction, where 2al and 2bl are the width and the length of the l-th rectangular hole, respectively. The center of the l-th rectangular hole is located at (cl, dl), and its rotation angle to the x-axis is al. The l-th rectangular hole is filled by a dielectric medium with relative permittivity erl and relative permeability mrl. The thickness of the conductive slab is assumed to be t. The metallic slab is located in a homogeneous medium with relative permittivity erb and relative permeability mrb. According to the above sections, the incident waves may be expressed as follows: E yi ( x , y, z )  H yi ( x , y, z ) 

© 2006 by Taylor & Francis Group, LLC





∑ ∑

m n



∑ ∑

m n

i e i ( k xm xk yn ykzmn z ) Emn i H mn i k x k y k z e ( xm  yn  zmn ) Z0

(5.211)

(5.212)

284

Electromagnetic Theory and Applications for Photonic Crystals

E yi ( x , y, z ) 





∑ ∑

m n

H yi ( x , y, z ) 



i e i ( k xm xk yn ykzmn ( zt )) Emn

(5.213)

i H mn i k x k y k z t e ( xm  yn  zmn (  )) Z0

(5.214)



∑ ∑

m n

mp np 2  2  2 —  where kxm  kx0  — h , kyn  ky0  w , kzmn  kb  k xm  k yn, and Im(kzmn)  0. Then, the scattering fields in the upper and lower half-spaces may be written as:

E ys ( x , y, z )  H ys ( x , y, z )  E ys ( x , y, z )  H ys ( x , y, z ) 





∑ ∑

m n



∑ ∑

m n



∑ ∑

m n



∑ ∑

m n

 e i ( k xm xk yn ykzmn z ) Amn

(5.215)

 Bmn i k x k y k z e ( xm  yn  zmn ) Z0

(5.216)

 e i[ k xm xk yn ykzmn ( zt )] Amn

(5.217)

 Bmn i k x k y k z t e [ xm  yn  zmn (  )] Z0

(5.218)

y

Unit cell 1

2b

1,

0r 1

d1 )

0r1

)

2

2

,d (c 2

0r2 2w

1

(c

2b

2

2a

0r 2

1

2a

x

2

2a 3

0r 3 0r 3

d 3) (c 3,

2b 3

3

2h

FIGURE 5.48 Top view of a perfectly conductive slab cut periodically by three rectangular holes per unit cell.

© 2006 by Taylor & Francis Group, LLC

Mode-Matching Technique Applied to Metallic Photonic Crystals

285

 where A mn, B mn are unknown coefficients. The scattering fields in the l-th rectangular region are expressed by electric and magnetic Hertzian vectors in the l-th local coordinate system whose origin locates at (cl, dl) in the global coordinate system and whose rotation angle is al.

yel  ∑ l



∑ sin jlr ( xl  al ) cos hl ( yl  bl )[CCrl eig

l , r z

r1 0

yhl  ∑ l



∑ cos jlr ( xl  al ) sin hln ( yl  bl )[Drl eig

l ,rn z

r0 n1

 Crl eiglr ( zt ) ] (5.219) l e iglrn ( zt ) ]/ Z  Drn 0

(5.220) rp np 2  2  2 —  erlmrl, and Im(glrv  0). where jlr  — 2al , hlv  2bl , glrv  kl  jlp  hlv, kl  k0 Assuming ax and ay are x- and y-components in the global coordinate, system respectively, and axl and ayl are xl- and yl components in l-th local coordinate, respectively, then we have  a  x    ay           

   l      yl  

ax a

      



  cos a l     sin al 

 

 

cos al

sin al 

 



     

    yl  

sin al   axl  cos al

   

a 



   ay 

   

sin al   ax 

 cos al 

(5.221)

(5.222)

The x- and y-components of the magnetic fields at the apertures must be continuous across the boundary planes at z  0 and z  t. The continuous conditions are written as follows: [ H xi ( x , y, 0)  H xs ( x , y, 0)] cos al  [ H yi ( x , y, 0)  H ys ( x , y, 0)] sin al e

2 i ( hpk x 0 wqk y 0 )

H xl ( x a , ya , 0), l

( x , y)  Ξl

(5.223)

[ H yi ( x , y, 0)  H ys ( x , y, 0)] cos al  [ H xi ( x , y, 0)  H xs ( x , y, 0)] sin al e

2 i ( hpk x 0 wqk y 0 )

H yl ( x a , ya , 0), l

( x , y)  Ξl

(5.224)

[ H xi ( x , y, t )  H xs ( x , y, t )] cos al  [ H yi ( x , y, t )  H ys ( x , y, t )] sin al e

2 i ( hpk x 0 wqk y 0 )

H xl ( x a , ya , t ), l

( x , y)  Ξl

(5.225)

[ H yi ( x , y, t )  H ys ( x , y, t )] cos al  [ H xi ( x , y, t )  H xs ( x , y, t )] sin al e

2 i ( hpk x 0 wqk y 0 )

H yl ( x a , ya , t ), l

© 2006 by Taylor & Francis Group, LLC

( x , y)  Ξl

(5.226)

286

Electromagnetic Theory and Applications for Photonic Crystals

  Ξl  ( x , y): ∪ [(|x a |  al ) ∩ (|ya |  bl )]   p ,q  

(5.227)

x a  ( x  2hp  cl )cos al  ( y  2 wq  dl )sin al

(5.228)

ya  ( y  2 wq  dl )cos al  ( x  2hp  cl )sin l

(5.229)

From Equations (5.211)–(5.220), we may derive the magnetic components Hi x,y, l Hs and H . Substituting these results into Equations (5.224) and (5.226) and x,y xl,yl using the orthogonality of trigonometric functions cos jlp(xl  al) and sin hln(yl  bl), the following relationships may be obtained. ihln

l  D





∑ ∑ (k yn cos al  kxm sin al )

(kl2  hl2n )(1  dr 0 )(1  e 2igl , rnt ) m n  i  B )cos a  ( H i  B )e iglrnt cos a  Gl,rnmn ( H mn mn l mn mn l   sin al  i i  B )  k e k  2 [k xm k yn ( H mn 0 rb zmn ( Amn  Emn )] mn 2 kb  k yn 

l  D

eigl , rnt sin al 2 kb2  k yn

 i  B )  k e k i  A )] [k xm k yn ( H mn ( E 0 rb zmn mn mn mn   

ihl



(5.230)



∑ ∑

(k yn cos al  k xm sin al ) i t (kl2  hl2n )(1  dr 0 )(1  e 2 gl , rn ) m n   )cos a  (H i  B )e iglrnt cos a  Gl, rnmn ( H min  Bmn l mn mn l   sin al i  i  B )  k e k  2 [k xm k yn ( H mn 0 rb zmn ( Emn  Amn )] mn 2 kb  k yn ig t  e l , rn sin al   E i )] i  B )  k e k (5.231)  k ( H ( A [ k 0 rb zmn xm yn mn mn mn mn  2  kb2  k yn 

where Gl, rnmn  ei ( kxmcl k yndl )

[(1)r eikxm ,l al  eikxm ,l al ][(1)n eik yn ,l bl  eik yn ,l bl ] 2 2 2 2 al bl (k xm , l  j l )(k yn , l  hln ) (5.232)

k xm ,l  k xm cos al  k yn sin al ,

© 2006 by Taylor & Francis Group, LLC

k yn ,l  k yn cos al  k xm sin al

(5.233)

Mode-Matching Technique Applied to Metallic Photonic Crystals

287

In the same manner, applying (5.223) and (5.225), we have

l  Crn

ijlr









∑ ∑

(k xm cos al  k yn sin al ) erl k0 gl , rn (1  dn 0 )(1  e ) m n  i − + B+ )sin  − ( H i + + B− )e i l ,  t sin   Gl, rnmn ( H mn mn l mn mn l   cos al  i i  B )  k e k  2 [k xm k yn ( H mn 0 rb zmn ( Amn  Emn )] mn 2 kb  k yn  eigl , rnt cos al i  B )  k e k i  A )]  [ k k ( H ( E xm yn mn mn 0 rb zmn mn mn  2  kb2  k yn  jlr hln  Dl  (5.234) erl k0 gl , rn rn

l  Crn

2 iglrnt

ijlr

∑ ∑

(k xm cos al  k yn sin l ) erl k0 gl , rn (1  dn 0 )(1  e 2iglrnt ) m n  i  B )e igl ,rnt sin a  ( H i  B )sin a  Gl, rnmn ( H mn mn l mn mn l   eigl , rnt cos al i  i  B )  k e k  [k xm k yn ( H mn mn 0 rb zmn ( Amn  Emn )] 2 kb2  k yn  cos al i  B )  k e k i  A )]  2 [ k k ( H ( E  xm yn mn mn 0 rb zmn mn mn 2  kb  k yn  jlr hln  Dl (5.235) erl k0 gl , rn rn

The continuous conditions for the tangential components of electrical fields may be expressed by E xi ( x , y, 0)  E xs ( x , y, 0)

 e 2i ( hpkx 0wqk y 0 ) [ E xl ( x a , ya , 0)cos al  E yl ( x a , ya , 0)sin al ]Ql ( x , y) l

(5.236)

l

E yi ( x , y, 0)  E ys ( x , y, 0) e

2 i ( hpk x 0 wqk y 0 )

[ E xl ( x a , ya , 0)sin al + E yl ( x a , ya , 0)cos al ]Ql ( x , y) l

(5.237)

l

E xi ( x , y, t )  E xs ( x , y, t ) e

2 i ( hpk x 0 wqk y 0 )

[ E xl ( x a , ya , t )cos al  E yl ( x a , ya , t )sin al ]Ql ( x , y) l

© 2006 by Taylor & Francis Group, LLC

l

(5.238)

288

Electromagnetic Theory and Applications for Photonic Crystals

E yi ( x , y, t )  E ys ( x , y, t ) e

2 i ( hpk x 0 wqk y 0 )

[ E xl ( x a , ya , t )sin al  E yl ( x a , ya , t )cos al ]Ql ( x , y) l

(5.239)

l

where  1 ( x , y) ∈ Ξl Ql ( x , y)    0 otherwise 

(5.240)

s l Substituting Ei x,y, Ex,y, and Exl,yl into (5.236)–(5.239) and noting the orthogonality ikxmxkyny of functions e leads to a set of linear equations.   A ) 4 wh( E ipq pq

L



∑ al bl cos al ∑ ∑ ijl (k xp cos al  k yq sin al )Gl, npq l1

 (kl2

r1 n0

le igl , rnt ]  hl2n )[Crln  Crn

L



 ∑ al bl sin al ∑ ∑ ihln (k yq cos al  k xp sin al )Gl, rnpq l1

r0 n1

l  C le igl , rnt ]  m k g l l igl , rnt ]}  {jlr hln [Crn rn rl 0 l , rn [ Drn  Drn e

(5.241)

4 wh i   A )  k m k  [k xp k yq ( E ipq pq 0 rb zpq ( H pq  B pq )] 2 kb2  k yq

L



 ∑ al bl cos al ∑ ∑ ihln (k yq cos al  k xp sin al )Gl, rnpq r0 n1

l1

l  C le igl , rnt ]  m k g l l igl , rnt ]}  {jlr hln [Crn rn rl 0 l , rn [ Drn  Drn e

L



 ∑ al bl sin al ∑ ∑ ijlr (k xp cos al  k yq sin al )Gl, rnpq l1

 (kl2

r1 n0

l  C le igl , rnt ]  hl2n )[Crn rn

(5.242)

  A ) 4 wh( E ipq pq L





∑ al bl cos al ∑ ∑ ijlr (k xp cos l  k yq sin al )Gl, rnpq l1

r1 n0

l ]  (kl2  hl2n )[Crln eigl , rnt  Crn L





 ∑ al bl sin al ∑ ∑ ihln (k yq cos al  k xp sin al )Gl, rnpq l1

r0 n1

le igl , rnt  {jlr hln [Crn

© 2006 by Taylor & Francis Group, LLC

l ]  m k g l igl , rnt  Dl ]}  Crn (5.243) rl 0 l , rn [ Drn e rn

Mode-Matching Technique Applied to Metallic Photonic Crystals

289

4 wh   A )  k m k  i [k xp k yq ( E ipq pq 0 rb zpq ( B pq  H pq )] 2  k yq

kb2



L



 ∑ al bl cos al ∑ ∑ ihln (k yq cos al  k xp sin al )Gl, rnpq r0 n1

l1

le igl , rnt  {jl hln [Crn

L

l ]  m k g l igl , rnt  Dl ]}  Crn rl 0 l , rn [ Drn e rn

 ∑ al bl sin al ∑ ∑ ijlr (k xp cos al  k yq sin al )Gl, rnpq l1

r=1 n0

 (kl2

le igl , rnt  hl2n )[Crn

l ]  Crn

(5.244)

If the number of unknown coefficients is truncated by m  M and n  N, these linear equations are rewritten in matrix form  i     i ei  a  (  5c  1s )(h b )  ( 2 s  6 c )(a  e )       i i  ( 3s  7c )(h  b )  ( 4 s  8c )(e   a )

(5.245)

i    i    b ( ei  a )  hi  b   a (  1c  5 s )( h  b )   a ( 2 c  5 s )(a  e )   i     a ( 3c  7 s )( h  b ) i    a ( 4 c   (5.246) 8 s )( e  a )

 i     i ei  a  (  5c  1s )(h  b )  ( 2 s − 6 c )(a  e )   i     i   ( 3s  7c )(h  b )  ( 4 s  8c )(e  a )

(5.247)

i    i−    b ( ei  a )  hi  b   a ( 1c  5 s )( h  b )   a ( 2 c  5 s )( a  e )   i     a (3c  7 s )( h  b )  i   a (4 c   (5.248) 8 s )( e  a )

where the column vectors ei, hi, a, and b are defined as: t a  [ A(M )(N ) , A(M )(N 1) , A(M )(N 2) ,  , A MN ]

(5.249)

t b  [ B(M )(N ) , B(M )(N 1) , B(M )(N 2) ,  , B MN ]

(5.250)

 i i i t ei  [ E(i M )(N ) , E(M )(N 1) , E(M )(N 2)) ,  , E MN ]

(5.251)

 i i i t hi  [ H(i M )(N ) , H (M )(N 1) , H (M )(N 2)) ,  , H MN ]

(5.252)

The matrices  are defined  L ab  l l       ϒ(k xm ,l k yq ,l x 1   ∑ 1l , mn w3l , mnpq  k yn , l k yq , l x2 l , mn w1l , mnpq )   l1 4 wh  © 2006 by Taylor & Francis Group, LLC

(5.253)

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Electromagnetic Theory and Applications for Photonic Crystals

 L ab    2   ∑ l l ϒ(k xm ,l k yq ,l x4 l , mn w 3l , mnpq  k yn , l k yq , l x3l , mn w1l , mnpq )   l1 4 wh 

(5.254)

 L ab     ∑ l l ϒ(k k x w     k k x w ) xm , l yq , l 1l , mn 3l , mnpq yn , l yq , l 2 l , mn 1l , mnpq  3   l1 4 wh 

(5.255)

 L ab    4   ∑ l l ϒ(k xm ,l k yq ,l x4 l , mn w 3l , mnpq  k yn , l k yq , l x3l , mn w1l , mnpq )   l1 4 wh 

(5.256)

 L ab     ∑ l l ϒ(k k x w    xm , l xp , l 1l , mn 2 l , mnpq  k yn , l k xp , l x2 l , mn w3l , mnpq )  5   l1 4 wh 

(5.257)

 L ab  l l     ϒ(k xm ,l k xp,l x4 l , mn w 6   ∑ 2 l , mnpq  k yn , l k xp , l x3l , mn w3l , mnpq )   l1 4 wh 

(5.258)

 L ab      7   ∑ l l ϒ(k xm ,l k xp,l x w  k k x w ) yn , l xp , l 2 l , mn 3l ,mnpq  1l , mn 2 l ,mnpq  l1 4 wh 

(5.259)

 L ab  l l     ϒ(k xm ,l k xp,l x4 l , mn w 8   ∑ 2 l ,mnpq  k yn , l k xp , l x3l , mn w3l ,mnpq )   l1 4 wh 

(5.260)

k xm ,l  k xm cos al  k yn sin al ,

k xp,l  k xp cos al  k yq sin al

(5.261)

k yn ,l  k yn cos al  k xm sin al ,

k yq ,l  k yq cos al  k xp sin al

(5.262)

when   c, then ϒ  cos al ,

when   s, then ϒ  sin al

(5.263)

and x 1l , mn  sin al 

cos al kxm k yn

x2l , mn  cos al  x3l , mn  x4 l , mn 

© 2006 by Taylor & Francis Group, LLC

2 kb2  k yn

sin al kxm k yn 2 kb2  k yn

sin al k0 erb kzmn 2 kb2  k yn

cos al k0 erb kzmn 2 kb2  k yn

(5.264)

(5.265)

(5.266)

(5.267)

Mode-Matching Technique Applied to Metallic Photonic Crystals

w1l , mnpq  ∑



∑

r0 n1

w1l , mnpq  ∑



∑

r0 n1

w 2 l , mnpq



∑



∑

r1 n0

w 2 l , mnpq  ∑



∑

r1 n0

w 3l , mnpq

w 3l , mnpq





∑ ∑

r1 n1





∑ ∑

r1 n1

hl2n (kl2  jl2 )(1  e

2 igl , rnt

erl k0 gl , rn (1  dr 0 )(1  e

)

2 igl , rnt

i

Gl, rnpqGl,rnmn

(5.268)

Gl, rnpqGl, rnmn

(5.269)

Gl, rnpqGl,rnmn

(5.270)

Gl, rnpqGl, rnmn

(5.271)

)

t

hl2n (kl2  jl2r )2e gl , rn

erl k0 gl , rn (1  dr 0 )(1  e 2igl , rnt ) i

t

jl2r (kl2  hl2n )(1  e 2 gl , rn ) erl k0 gl , rn (1  dn 0 )(1  e 2igl , rnt ) i

t

jl2 (kl2  hl2n )2e gl , rn erl k0 gl , rn (1  dn 0 )(1  e hl2n jl2r (1  e

2 igl , rnt

erl k0 gl , rn (1  e i

)

2 igl , rnt

)

2 igl , rnt

)

Gl, rnpqGl, rnmn

(5.272)

Gl, rnpqGl, rnmn

(5.273)

t

hl2n jl2r 2e gl , rn erl k0 gl , rn (1  e

291

2 igl , rnt

)

Symbols a and b denote diagonal matrices whose elements are (k2b  k2yq)/ (k0mrbkzpq) and kxpkyq/(k0mrbkzpq), respectively. By solving the equations, a scattering matrix may be derived.    1  4 s   7c   I   6c  2 s 1s  5c 8c 3s        )    )    ) ) I   (   (    (   b   a (2c   5s a 1c 5s a 4c 8s a 3c 7s   S       I  4 s  8c 7c   3s 6c  2 s 1s  5c        ) I   (   )     )    )   (    (     (  a 4 c 8 s a 3 c 7 s a 2 c 5 s a 1 c 5 s b          I   4 s    6c 2s 5c  1s 8c 3s   7 c              a (4 c  8 s )  a (3c  7 s )    b   a (2c  5s ) I   a (1c  5s )          4 s  8c  I  3s   7 c 6c  2 s 5c  1s        ) I   (   )     )   a (2c   )  (     (  a 4 c 8 s a 3 c 7 s 5 s a 1 c 5 s b  

(5.274)

5.8 CONCLUSION We have described a novel approach to analyzing periodic structure problems consisting of metallic objects. This method, which is available to two-dimensional and three-dimensional metallic photonic crystals, combines a mode-matching

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Electromagnetic Theory and Applications for Photonic Crystals

technique and an S-matrix formula. In this method, the original structure is sliced into many layers of rectangular rods or rectangular apertures. This reduces the original periodic array to a stacking sequence of lamellar gratings with metallic rectangular rods or rectangular apertures arranged periodically. First, the reflection and transmission matrices for each layer of lamellar gratings are derived using a mode matching technique. Then the results are concatenated through the entire layered system to obtain the reflection and transmission matrices for the original periodic structure. The elements of the reflection and transmission matrices are expressed in terms of the modal series with fast convergence, which converges with O(n3) where n is the number of modal series, and the scattered fields are calculated from a simpler matrix operation. The novel formula has several advantages. One is that the convergence is very fast and that the computing precision is very high. The second is that the formula is very suitable to multilayered systems. The last, which is very important, is that the relative convergence phenomena, which must appear when treating multilayered problems using a conventional mode matching method, has been prevented perfectly and the singular fields around metal edges may accurately be computed. This is because those singular fields are expressed in infinite sums. We hope that the proposed method will make a contribution to future research in new fields, such as negative refractive index material, frequency selective surfaces, and surface plasmons. We also hope that this method will be applied as a popular technique to study or to design new devices of metallic periodic structures.

ACKNOWLEDGMENT This research was supported in part by the 21st Century COE Program, Reconstruction of Social Infrastructure Related to Information Science and Electrical Engineering.

REFERENCES [1] S. Peng and G.M. Morris, Experimental demonstration of resonant anomalies in diffraction from two-dimensional gratings, Opt. Lett., 21, 549–551, 1996. [2] S. Peng and G.M. Morris, Resonant scattering from two-dimensional gratings, J. Opt. Soc. Am. A, 13, 993–1005, 1996. [3] D. Sievenpiper, L. Zhang, R.F.J. Broas, N.G. Alexópolous, and E. Yablonovitch, High-impedance electromagnetic surfaces with a forbidden Frequency band, IEEE. Trans. Microwave Theory Tech., 47, 2059–2074, 1999. [4] C.M. Horwitz, A new solar selective surface, Opt. Commun., 11, 210–212, 1974. [5] V. Veremey, Superdirective antennas with passive reflectors, IEEE Antennas and Propagation Magazine, 37(2), 16–26, 1995. [6] C. Simovski and S. He, Antennas based on modified metallic photonic bandgap structures consisting of capacitively loaded wires, Microwave Opt. Technol. Lett., 31, 214–221, 2001.

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Mode-Matching Technique Applied to Metallic Photonic Crystals

293

[7] R. Ulrich, Far-infrared properties of metallic mesh and its complementary structure, Infrared Phys., 7, 37–55, 1967. [8] M.M. Sigalas, C.T. Chan, K.M. Ho, and C.M. Soukoulis, Metallic photonic bandgap materials, Physical Review B, 52, 11,744–11,751, 1995. [9] B.K. Minhas, W. Fan, K. Agi, S.R.J. Brueck, and K.J. Malloy, Metallic inductive and capacitive grads: theory and experiment, J. Opt. Soc. Am. A, 19, 1352–1359, 2002. [10] R. Petit, ed., Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980. [11] E. Yablonovitch, Inhibited spontaneous emission in solid state physics and electronics, Phys. Rev. Lett. 58, 2059–2062, 1987. [12] J.B. Pendry, Electromagnetic materials enter the negative age, Physics World, 14, 47–51, 2001. [13] J.B. Pendry, L. Martín-Moreno, and F.J. Garcia-Vidal, Mimicking surface plasmons with structured surfaces, Science, 305, 847–848, 2004. [14] V. Twersky, On Scattering of waves by the infinite grating of circular cylinders, IRE Trans. Antennas Propagat., 10, 737–765, 1962. [15] K. Otaka and N. Numata, Multiple scattering effects in photon diffraction for an array of cylindrical dielectric, Phys. Lett., 73a, 411–413, 1979. [16] K. Ohtaka, T. Ueta, and K. Amemiya, Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods, Phys. Rev., B, 57, 2550–2568, 1998. [17] M. Plihal and A.A. Maradudin, Photonic band structure of two-dimensional systems: the triangular lattice, Phys. Rev., B, 44, 8565–8571, 1991. [18] H. Benisty, Modal analysis of optical guides with two-dimensional photonic bandgap boundaries, J. Appl. Phys., 79, 7483–7492, 1996. [19] G. Pelosi, A. Cocchi, and A. Monorchio, A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam, IEEE. Trans., Antennas Propagat., 45, 185–186, 1997. [20] E. Popov and B. Bozhkov, Differential method applied for photonic crystals, Appl. Opt., 39, 4926–4932, 2000. [21] T. Kushta and K. Yasumoto, Electromagnetic Scattering from periodic arrays of two circular cylinders per unit cell, Progress In Electromagnetics Research, PIER 29, 69–85, 2000. [22] K. Yasumoto and H. Jia, Three-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders, Proc. 2002 China-Japan Joint Meeting on Microwaves, 301–304, 2002. [23] H. Jia and K. Yasumoto, A novel formulation of the Fourier model method in S-matrix form for arbitrary shaped gratings, Int J Infrared Millimeter Waves, 25, 1591–1609, 2004. [24] J. Shmoys and A. Hessel, Analysis and design of frequency scanned transmission gratings, Radio Science, 513–518, 1983. [25] A. Cucini, M. Albani, and S. Maci, Truncated Floquet wave full-wave (T(FW)2) analysis of large periodic arrays of rectangular waveguides, IEEE Trans. Antennas Propagat., 51, 1373–1385, 2003. [26] R. Mittra and S.W. Lee, Analytical Techniques in the Theory of Guided Waves, Macmillan, New York, 1971. [27] H. Jia and K. Yasumoto, S-matrix solution of electromagnetic scattering from periodic arrays of metallic cylinders with arbitrary cross section, IEEE Antennas and Wireless Propagation Lett., 3, 41–44, 2004.

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[28] H. Jia and K. Yasumoto, A rigorous analysis of electromagnetic scattering from multilayered crossed-arrays of metallic cylinders, in Wave Propagation, Scattering and Emission in Complex Media, Y.Q. Jin, ed. Science Press, and World Scientific, 25–262, 2004. [29] H. Jia and K. Yasumoto, Wave scattering from periodic metallic cylinders with arbitrary cross section for a general angle of incidence, Proc. Asia-Pacific Radio Science Conference, 101–104, 2004. [30] H. Jia, K. Yoshitomi, and K. Yasumoto, Rigorous analysis of E-/H-plane junctions in rectangular waveguides using Fourier transform technique, Progress in Electromagnetics Research, PIER 21, 273–292, 1999.

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Method of Lines for 6 The the Analysis of Photonic Bandgap Structures Reinhold Pregla and Stefan F. Helfert CONTENTS 6.1 Introduction ..............................................................................................296 6.2 Basic Theory ............................................................................................298 6.2.1 Material Properties ......................................................................299 6.2.2 Maxwell’s Equations in Matrix Notation ....................................299 6.2.3 Generalized Transmission Line Equations in General Orthogonal Coordinates ..............................................................300 6.2.4 Discretization ..............................................................................301 6.2.5 Absorbing Boundary Conditions ................................................307 6.2.6 Wave Equations ..........................................................................308 6.2.7 Eigenvalue and Modal Matrices ..................................................308 6.3 Impedance/Admittance Transformation ..................................................309 6.3.1 Transformation through Waveguide Sections..............................309 6.3.2 Transformation at Waveguide Concatenations ............................310 6.3.3 Transformation at Metalized Waveguide Interfaces....................310 6.3.4 Steps of the Analysis Procedure..................................................311 6.4 Determination of Floquet Modes ............................................................312 6.4.1 Introduction..................................................................................312 6.4.2 Symmetric Periodic Structures ....................................................313 6.4.3 Propagation Constants of Floquet Modes ..................................314 6.4.4 Wave Impedance/Admittance of Floquet Modes ........................315 6.4.5 Efficient Calculation of the Floquent Mode Parameters ............316 6.4.6 Concatenation of N Periods ........................................................318 6.5 Determining the Band Structures of Photonic Crystals ..........................319 6.5.1 Propagation Algorithm ................................................................319 6.5.2 Eigenmode Algorithm ................................................................323 6.5.3 Lines of Varying Length..............................................................325 6.6 Junctions in Photonic Crystal Waveguides ..............................................328 6.6.1 Port Relations ..............................................................................328 295

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Electromagnetic Theory and Applications for Photonic Crystals

6.6.2 Main Diagonal Submatrices ........................................................330 6.6.3 Off-Diagonal Submatrices: Coupling to Other Ports ..................332 6.7 Numerical Results ....................................................................................336 6.7.1 Band Structure Calculations ........................................................336 6.7.2 PC Waveguide Devices................................................................339 6.7.3 Bragg Gratings ............................................................................342 6.8 Conclusion................................................................................................347 Acknowledgment ..............................................................................................348 References ........................................................................................................348

6.1 INTRODUCTION There is a lot of ongoing research in the area of photonic bandgap structures or photonic crystals (PC) [1]. These PCs are periodic structures that prohibit the propagation of electromagnetic waves of some frequencies. By introducing defects, various circuits may be designed. The research includes the search for materials and structures that are best suited for this purpose on the one hand and the development (or improvement) of numerical tools on the other. The first problem has been addressed in the review paper [2], while some of the modeling requirements can be found, for example, in [3]. In the following we show some examples of PC devices. Waveguides for microwave frequencies are drawn in Figure 6.1. The first shows a microstrip line accompanied with holes on the sides preventing waves from radiating in that direction. The second example is a dielectric film with metal posts. A channel waveguide is obtained by removing one row of these posts. Examples of optical devices are shown in Figure 6.2. A straight waveguide and a 60° bend are presented. Further examples of bandgap structures are shown in Figure 6.3. One is a microstrip meander line and the other is a holey fiber. Generally, bandgap structures can be understood as a concatenation of various waveguide sections. The analysis of single photonic crystals and of more complicated PC devices can be done efficiently and with high accuracy in frequency domain with eigenmode algorithms. For example, material dispersion can be included in the frequency domain without any problems, which is not as easy in the time domain. One of the eigenmode methods is the method of lines (MoL) [4,5]. The MoL is a semianalytic procedure where the cross-section is discretized with finite differences (FD) and analytical expressions are used in the remaining direction. Consequently, the development of suitable algorithms deals with the improvement of the finite differences as well as with the suitable treatment of the analytical expressions. In this chapter we will focus mainly on the analytical part. A possible mathematical analysis of concatenated structures involves transfer matrix formulas. In case of reflections we have waves in two directions described by exponentially decreasing and increasing terms. The increasing terms can lead to numerical problems when using transfer matrix formulas. Therefore, the transfer matrix method is stable only as long as guided modes only are used. However, this

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The Method of Lines for the Analysis of Photonic Bandgap Structures

297

Periodic cell

Microstrip

b

Periodic length a (a)

Dielectric substrate (
r)

w

(b)

FIGURE 6.1 Bandgap structures in microwaves: (a) microstrip line with etched circles in the metallic ground plane; (b) channel guide in a dielectric film with metallic posts on both sides.

(a)

OIBG1520

(b)

OIBG1510

FIGURE 6.2 Photonic crystal waveguide: (a) defect guide; (b) bend.

x z

Air

y

SiO2

FIGURE 6.3 Bandgap structures: (a) microstrip meander line; (b) holey fiber. (Reprinted with permission from A. Barcz, S.F. Helfert, and R. Pregla, ICTON Conf., vol. 5, 126–129, copyright IEEE 2003.)

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Electromagnetic Theory and Applications for Photonic Crystals

is a simple approximation because higher order modes (i.e., evanescent modes, radiation modes, complex modes, etc.) also are required to obtain correct solutions. To avoid these problems the concept of impedance/admittance transformation has been introduced [6–8]. This procedure is analogous to the well-known transmission line theory. By suitable transformation of the analytic expressions, the exponentially increasing terms do not have to be calculated numerically. Hence the numerical problems do not occur. Therefore, this is an essential difference from the transfer matrix method. Instead of impedances, we may also use scattering parameters as shown in [9] to obtain a stable method. As is also known from transmission line theory (and particularly applied in the Smith chart), an impedance/admittance corresponds to a certain reflection coefficient and vice versa. The important extension to the transmission line theory here is that we are dealing with matrices. In this chapter we will show how the impedance/admittance transformation formulas are obtained on the basis of generalized transmission line (GTL) equations. These GTLs can be derived for arbitrary orthogonal [10,11] coordinate systems and for general anisotropic material. For an efficient treatment of periodic structures, Floquet’s theorem ([12] pp. 605–608) was introduced [13,14]. The implementation was done using the impedance/ admittance (or reflection factor) transformation, so that it is numerically stable. This could be shown for Bragg gratings with some 10,000 (ten thousand) periods [13,15]. Also structures with an infinite number of periods can be modeled very easily. As known from transmission line theory, the input impedance of such an infinite structure is just the characteristic impedance; that is, the input reflection coefficient is zero. The concept of crossed discretization lines has been introduced for the analysis of junctions [16]. Waveguide devices realized with photonic crystals allow the design of waveguide junctions and sharp bends with low losses. In the bends of these circuit elements, the incoming and outgoing waves are propagating orthogonal to each other, which the analysis should take into account. This can be done by determining the fields on crossed discretization lines [17]. This chapter is organized in the following way: we start with a general description of the MoL including the impedance transformation. Then we will show how the Floquet mode concept can be introduced. A particular problem for PCs is the calculation of the band structures. Therefore, we will show how band structures are determined with the Floquet modes. After that we give the theory for the analysis of junctions and bends with crossed discretization lines. Numerical results are given from various band structure calculations as well as from circuits. Whenever possible a comparison with other methods is given, showing a very good agreement.

6.2 BASIC THEORY In this section we derive the generalized transmission line (GTL) equations that are analogous to the well-known equations for coupled multiconductor transmission lines in inhomogeneous media [18]: d [U ]  j[vL ][I ] dz © 2006 by Taylor & Francis Group, LLC

d [I ]  j[vC ][U ] dz

(6.1)

The Method of Lines for the Analysis of Photonic Bandgap Structures

299

In the entire chapter we assume a time dependence according to exp(jvt). If we want to examine the general case, we have to add to the right side of each equation a term that is proportional to the quantity on the left side. These transmission line equations are solved by calculating modal matrices [18].

6.2.1 MATERIAL PROPERTIES For the formulation of GTL equations in Cartesian coordinates the material parameters (permeability and permittivity tensors) are assumed to have the following form. (We assume that we will later use analytic expressions in the z-direction.)   xx   r    yx   0 

 xy  yy 0

0   0    zz 

  with r  er

or

   r  mr

(6.2)

When introducing this form of anisotropic material parameter tensors, the generalized transmission line (GTL) equations have the same form as in the isotropic case. The tensor elements can be complex, and no symmetry is assumed. For analyzing complicated circuits the device under study is divided into homogeneous sections in the direction of propagation or the direction of analytical solution. Hence, the material parameters in the cross sections are functions of x and y only. An algorithm for materials with an arbitrary anisotropy is described in [11].

6.2.2 MAXWELL’S EQUATIONS

IN

MATRIX NOTATION

It is very efficient to write the Maxwell’s curl-equations in matrix notation for the derivation of generalized transmission line (GTL) equations. The Maxwell’s curl-equations in Cartesian coordinates can be written in the following form (the magnetic field components H are normalized with the free space wave impedance, ˜ i.e., h0   m0/e: 0 H  h0 H):   j[er ][E ]  [Dˆ c ][H ] j[mr ][H ]  [Dˆ c ][E ] (6.3) [E] and [H˜ ] are vectors with the field components [E ]  [E x , E y , Ez ]t

[ H ]  [H x , H y , H z ]t

(6.4)

[Dˆc] is the matrix operator for the curl-operator  0   ˆ [ Dc ]   Dz  D  y © 2006 by Taylor & Francis Group, LLC

Dz 0 Dx

Dy   Dx   0  

(6.5)

300

Electromagnetic Theory and Applications for Photonic Crystals

We used the abbreviations Du–  ∂/∂u– and u  x, y, z for the derivatives and normalized the coordinates with the free-space wave number k0 (i.e., u–  k0u). With the definition for the matrix divergence operator [Dˆ d ]  [Dx , Dy , Dz ]

(6.6)

we obtain for the Maxwell’s divergence relations  [Dˆ d ][er ][E ]  r /(e0 k0 )

 [Dˆ d ][mr ][H ]  0

(6.7)

Note that [Dˆd][Dˆc]  [0, 0, 0]. Hence, the divergence–relations in charge-free regions are fulfilled by the curl-relations. This is also true after discretization.

6.2.3 GENERALIZED TRANSMISSION LINE EQUATIONS ORTHOGONAL COORDINATES

IN

GENERAL

Generalized transmission line (GTL) equations can be derived in arbitrary orthogonal coordinate systems [11]. Here we will repeat the formulas for the Cartesian case and the anisotropy given in Equation (6.2) [16]. By using the abbreviations Eˆ  [E y , E x ]t

ˆ  [H , H ]t H x y

(6.8)

we obtain from Maxwell’s curl-equations for propagation (or analytical solution) in the z-direction: ∂ ˆ ˆ E  j[RHz ]H ∂z

∂ ˆ H  j[REz ]Eˆ ∂z

(6.9)

The electric and magnetic field components in Equation (6.8) are ordered in such a way that the inner product is proportional to the z-component of the Poynting vector. These equations were obtained from the first two scalar equations in (6.3) where the z-components were introduced according to: ˆ je zz Ez  [Dy• Dx ]H

jm zz H z  [Dx• Dy ]Eˆ

(6.10)

The superscripts at the differential operators correspond to the discretization points z that we will introduce later. The matrices [RE,H ] contain derivatives with respect to x and y as well as the material parameters. We obtain for these matrices  e  D m1D• yy x zz x [ REz ]    e xy  Dy mzz1Dx•  



© 2006 by Taylor & Francis Group, LLC

e yx  Dx mzz1Dy   e xx  Dy mzz1Dy   



(6.11)

The Method of Lines for the Analysis of Photonic Bandgap Structures

 D e1D•  m xx  y zz y  • 1   Dx e zz Dy  m yx  

[ RHz ] 



Dy ezz1Dx  m xy   Dx ezz1Dx  m yy  

301



(6.12)



To obtain the wave equations we need the products of these matrices. Therefore, we split them into two parts according to [REz ]  [REzr ]  [eˆ rt ]

[RHz ]  [RHzr ]  [mˆ rt ]

(6.13)

where  D m1D• D m1D  x zz x x zz y  [ REzr ]     • 1 Dy m zz Dx Dy mzz1Dy    





 D e1D• y zz y [ RHzr ]    Dx ezz1Dy•  





Dy ezz1Dx   Dx ezz1Dx   



e yy [eˆ rt ]   e  xy

e yx   e xx  

(6.14)

 m m xy  xx [mˆ rt ]    m yx m yy 

(6.15)

Therefore, the products can be written as [QEz ]  [RHz ][REz ]  [RHzr ][eˆ rt ]  [mˆ rt ][REzr ]  [mˆ rt ][eˆ rt ]

(6.16)

[QHz ]  [REz ][RHz ]  [REzr ][mˆ rt ]  [eˆ rt ][RHzr ]  [eˆ rt ][mˆ rt ]

(6.17)

This simple result is a consequence of the fact that the product of the matrices [REzr] and [RHzr] is equal to zero. Particularly in case of diagonal tensors in Equation (6.2) or for isotropic materials, the products in Equations (6.16) and (6.17) can be calculated very easily. We can directly transform the GTL equation from Cartesian coordinates into any arbitrary orthogonal coordinate system [11].

6.2.4 DISCRETIZATION The wave equations must now be discretized. Figures 6.4 and 6.5 show cross sections of planar waveguides with the adequate discretization points. In Figure 6.4 we have dielectric regions as well as a part with metal. We will later show how this metal has to be considered in the analysis. Let us begin with the pure dielectric cross section sketched in Figure 6.5. The field components are discretized on different points. Ex lies on the same points as Hy, and Ey lies on the same points as Hx. These are the field components whose product gives the Poynting vector in the z-direction — the assumed direction of wave propagation. Now we collect the discretized field components in column

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Electromagnetic Theory and Applications for Photonic Crystals Electric wall Magnetic wall, ABC Electric wall

x

Magnetic wall, ABC

Ex ,
xx ,
yx Hy, xy , yy Ey ,
xy ,
yy Hx , xx , yx Ez ,
zz Hz , zz

Electric wall

3

2

1

Magnetic wall, ABC

y

HD

1

2

3

FIGURE 6.4 Cross section of a waveguide with metal. (Reprinted with permission from R. Pregla, IEEE Trans. Microwave Theory Tech., MTT-25, 50, 1469–1479, copyright IEEE 2002.)

y

Magnetic wall, ABC

Magnetic wall, ABC

x

Magnetic wall, ABC

z

Ex , Ey ,
xx, yy,
yx, xy Hx , Ey ,
yy , xx,
xy, yx Ez ,
zz Hz , zz

Magnetic wall, ABC

FIGURE 6.5 Cross section of a planar photonic crystal structure with discretization points.

vectors. We order these components by starting with the adequate point in the left upper point and then go downward in the first column. Coming down to the last point in the first column, we continue to the highest point in the second column and so on. Hence, the values in the columns from left to right are placed below each other. Each of the column vectors may be represented by a boldface capital letter of the field component and by a subscript i for the i-th column. The collection of all these vectors in the total column vector is marked by a hat (ˆ).

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If we have Nx° columns of ° discretization points and Ny° points in each column, then the total number of points is Ny°  Nx°. In the case of an ideal metal in the cross section, the total number at the discretization points is reduced by the number of points in the metal area. In the case of a nonideal metal, the metal is modeled by a dielectric with complex permittivity. The number of • columns and rows is Nx• and Ny• respectively. The values of the permittivities and permeabilities are collected in the same order as the field components, not in column vectors but in the main diagonal of a diagonal matrix. Each component of the tensor is discretized on a different permittivity or permeability point. Therefore, we have five permittivity and permeability matrices, respectively. Figure 6.5 shows that, for example, exx is discretized on the same points as Hy. We have three different matrices of each of the parameters even in the case of isotropic materials. Therefore, there is no big difference between the algorithms for isotropic and anisotropic materials. In what follows we assume most of the time that the material parameters are described by diagonal tensors (isotropic or anisotropic). In discretized form we use only one subscript for this case. Summarizing we have for u  x, y: ˆ  [E t , … , E t ]t Eu → E u u1 uN,• x

ˆ  [E t , … , E t ]t Ez → E z z1 zN 

x

ˆ  [H t , … , E t ]t Hu → H u u1 uN• , x

ˆ  [H t , … , E t ]t Hz → H z z1 zN

(6.18)



x

Euu → Eˆ u  Diag(diag(Eu1 ), … , diag(EuN,• )) x

Ezz → Eˆ z  Diag(diag(Ez1 ), … , diag(EzN )) 

x

ˆ u  Diag(diag(Mu1 ), … , diag(M •, )) muu → M uN x

ˆ z  Diag(diag(Mz1 ), … , diag(M )) m zz → M zN 

x

Now the differential operators are replaced by central differences. By combining all differences in a row or a column, we obtain difference operator matrices Dx°,• and Dy°,•, respectively. hx

∂ → Dx,• ∂x

hy

∂ → Dy,• ∂y

(6.19)

The operator matrices Dx°,• and Dy°,• have to fulfill Dirichlet or Neumann boundary conditions respectively in the case of magnetic walls. For electric walls the conditions for the matrices have to be exchanged. Electric and magnetic walls have to be localized in different places (Figure 6.5). For modeling radiation absorbing boundary conditions (ABC) have to be introduced instead of magnetic walls, as

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will be shown later. For magnetic walls as in Figure 6.5, the difference operators in Equation (6.19) have the following form  1     1 1    Dx           1

1 1      1 1    Dy          1 1 

(6.20)

The difference operators for the discretization points on bullets • are given by Dx•  Dx°t and D•y  Dy°t. Particularly, for band structure calculations we also need difference operators with periodic boundary conditions. Let us assume that the structure in Figure 6.5 is periodic in the x-direction. The first column of discretized fields on the points ° and  contains the fields of the previous magnetic wall on the left side. Then we obtain the following difference operators for the x-direction:  1 1            Dx    1 1    N  1  s x

• 1 sN x     1 1  •   Dx           1  1 

(6.21)

where s  ejkxhx describes the propagation in the x-direction according to Floquet’s theorem. In this case, the number of ° columns Nx° is equal to the number of • columns N x•. Furthermore, we have Dx°  (D x•)t. For efficient programming we should normalize the difference operators as described in [4]. The difference operators for 2D discretization can be constructed in the following way. In the case of inhomogeneous cross sections that do not contain metallic subsections, the difference operators can be described by Kronecker products of the above difference operators for the rows (Dx°,•) or columns (Dy°,•) and identity matrices of the order of the number of rows or columns. The collection of all rows and all columns is marked by a hat ( ˆ ). So we have Dˆ y,•  Ix,• ⊗ Dy,•

Dˆ x,•  Dx,• ⊗ I y,•

I x°,• and I y°,• are identity matrices of the order Nx°,• and Ny°,• respectively. In the case of an ideal metallic subsection as in Figure 6.4 (nonideal metal behaves like dielectric with complex permittivity), the cross section is divided with lines in vertical and horizontal directions according to the metal boundaries. Therefore, we obtain in both directions three different subregions in Figure 6.4. First we assume that we have discretization points inside and on the surface of the ideal metal. We do not want to use the discretized components on these points in our

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calculation. Therefore, we reduce the total number of discretization points to those outside the metal. Let us assume that in our example the number N x° consists ° , N ° , and N ° columns in the three subregions 1, 2, 3 in the x-direction. of N x1 x2 x3 ° , N ° , and N ° rows in the three subAnalogously, the number N y° consists of N y1 y2 y3 ° , I ° , I ° , I ° , I ° , and regions in the y-direction. Let us define identity matrices Ix1 x2 x3 y1 y2 ° of the order N ° , N ° , … The complete vector F ° of the field quantity F I y3 x1 x2 c discretized on ° points can now be reduced to the vector F ° with components only outside the metal by F  Jt Fc

(6.22)

The matrix J t is obtained from the matrix °

 ) Ix ⊗ I y  diag(Ix 1 , Ix 2 , Ix 3 ) ⊗ diag(I y1 , I y 2 , I y3

(6.23)

as  I ⊗ I  x1 y    Jt       

 I y1 Ix 2 ⊗   

   I y 3  

           Ix 3 ⊗ I y  

(6.24)

° . In the middle expression we have discarded those rows in Iy° that are also rows in Iy2 t t t The matrices J •, J , and J  can be constructed likewise. Now, the reduced difference operators can be obtained in the following way:   Jt D ˆJ Dˆ xr x   t ˆ ˆ Dyr  J Dy J 



•  Jt D ˆ •J Dˆ xr x • • t ˆ ˆ Dyr  J Dy• J• 

(6.25)



The diagonal matrices for the material parameters have to be reduced according to E xr  Jt E x J

M xr  J•t M x J•

E yr  J•t E y J•

M yr  Jt M y J

Ezr 

Mzr 

JtE 

z J

JtM 

(6.26)

z J

The supervectors of the discretized field components are t ˆ  E ˆt ˆt  E  y , E x 

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t ˆ  H ˆ t ,H ˆt H x y  

(6.27)

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Electromagnetic Theory and Applications for Photonic Crystals

The discretization of the Equations (6.9) according to Figure 6.5 yields the following GTL equations: d ˆ ˆ H  jREz E dz

d ˆ ˆ E  jRHz H dz

(6.28)

For the matrices REz and RHz we obtain Dˆ • t M ˆ zz1 Dˆ x• Dˆ x• t M ˆ zz1 Dˆ y   Eˆ yy x  Rˆ Ez    ˆ • ˆ zz1 Dˆ y   M ˆ zz1 Dˆ x• Dˆ y t M  Dˆ y t M Eˆ    xy xy 

 Dˆ • t Eˆ 1 Dˆ •  y zz y Rˆ Hz    Dˆ x t Eˆ zz1 Dˆ y• 

ˆ  Eˆ  M yx yx   Eˆ xx   

(6.29)

ˆ M ˆ xx ˆ  M Dˆ y• t Eˆ zz1 Dˆ x   M xy xy     ˆ •  ˆ ˆ yy  Dˆ x t Eˆ zz1 Dˆ x  M M yx M yx    

(6.30)



In the case of metal in the cross section (as in Figure 6.4), a subscript r must be added to all matrices. The M-matrices are required for interpolation. We need them if two components from different discretization lines must be combined. In the above equations these matrices are products, according to M xy•  M x M y• 



M yx  M y M x 



M xy  M x M y 



M yx•  M y M x• (6.31) 



The subscript indicates the interpolation direction, and the superscript indicates the component that has to be interpolated. For eigenmode problems with general cross section layers as in Figure 6.6, we assume propagation in the y-direction. The z-direction for analytical solution is now one of the directions in the cross section. To obtain the adequate equations we introduce Dy–  j ereI. ere is the effective dielectric constant for propagation in y-direction. Furthermore,

z z2

B

z1

d A

MMPL1262

Hy Hz Ex
xx
yx xy yy, zz Ey Ez Hx
xy
yy
zz xx yx

FIGURE 6.6 General inhomogeneous anisotropic layer with discretization lines. (Reprinted with permission from R. Pregla, IEEE J. of Sel. Topics in Quantum Electron., 8, 1217–1224, copyright IEEE 2002.)

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all matrices My are identity matrices and can be withdrawn. The matrices D x• and Mx• are replaced by Dxe and M ex respectively. They are used for the quantities on full lines. The matrices Dx° and M x° are replaced by Dxh and Mxh, respectively, and are used for the quantities on dashed lines.

6.2.5 ABSORBING BOUNDARY CONDITIONS To take into account radiation, we introduce absorbing boundary conditions (ABCs). These ABCs can be realized by replacing the Dˆ -matrices in the following way: Dˆ x → Dˆ xa  Dxa ⊗ I y Dˆ x• t → Dˆ x•a  Dxa ⊗ I y• Dˆ y• → Dˆ y•a  Ix• ⊗ Dya Dˆ  t → Dˆ a  I  ⊗ D a y

y

x

(6.32)

y

where we have for example,  a b c  1 1  1    1 1     a   Dx         1 1      c b  a  N N N 

(6.33)

and Dxa  Dx•a

Dy•a  Dya

(6.34)

The coefficients a1,N, b1,N, and c1,N are given in [5]. It can be seen that the following relation holds: Dˆ xa Dˆ ya  Dˆ y•a Dˆ x•a The remaining difference operators in Equations (6.29) and (6.30) have to be chosen as follows: Dˆ x t → Dˆ x t  Dxt ⊗ I y Dˆ y →  Dˆ y t Ix ⊗ Dyt Dˆ y• t → Dˆ y• t  Ix• ⊗ Dyt Dˆ • →  Dˆ • t D t ⊗ I • x

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x

x

y

(6.35)

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The operators Dx and Dy correspond to Dxa and D ya, respectively. We obtain these operators from D xa and D ya by setting the parameters a  1 and b  c  0. Dx and Dy fulfill Dirichlet boundary conditions.

6.2.6 WAVE EQUATIONS Combining Equation (6.28), we obtain for the discretized wave equations: d2 ˆ ˆ 0 H  Qˆ H H dz 2

Qˆ H Rˆ E Rˆ H

d2 ˆ ˆ 0 E  Qˆ E E dz 2

Qˆ E Rˆ H Rˆ E

(6.36)

(6.37)

ˆ xy  M ˆ yx  0, Eˆxy  Eˆyx  0, we have: For M H E ˆ yy Dˆ y• t Eˆ zz1 Dˆ y•  Dˆ x• t M ˆ zz1 Dˆ x• M ˆ xx  Eˆ yy M ˆ xx Q11 H E ˆ yy Dˆ y• t Eˆ zz1 Dˆ x  Dˆ x• t M ˆ yy ˆ zz1 Dˆ y M Q12

(6.38)

H E ˆ xx Dˆ x t Eˆ zz1 Dˆ y•  Dˆ y t M ˆ zz1 Dˆ x• M ˆ xx Q21 H E ˆ xx Dˆ x t Eˆ zz1 Dˆ x  Dˆ y t M ˆ zz1 Dˆ y M ˆ yy  Eˆ xx M ˆ yy Q22 E D ˆ • t Eˆ 1 Dˆ • Eˆ  M ˆ xx ˆ xx Dˆ x• t M ˆ zz1 Dˆ x•  Eˆ yy M Q11 y zz y yy E D ˆ • t Eˆ 1 Dˆ  Eˆ  M ˆ zz1 Dˆ y ˆ xx Dˆ x• t M Q12 y zz x xx

(6.39)

E D ˆ  t Eˆ 1 Dˆ • Eˆ  M ˆ zz1 Dˆ x• ˆ yy Dˆ y t M Q21 x zz y yy E D ˆ  t Eˆ 1 Dˆ  Eˆ  M ˆ zz1 Dˆ y  M ˆ yy Eˆ xx ˆ yy Dˆ y t M Q22 x zz x xx

6.2.7 EIGENVALUE AND MODAL MATRICES By transforming the fields by using the modal matrices TˆH and TˆE according to ˆ ˆ  Tˆ H H H

ˆ ˆ  Tˆ E E 

TˆH1Qˆ H TˆH  ˆ 2

TˆE1Qˆ E TˆE  ˆ 2

(6.40)

the Equations (6.36) and (6.37) reduce to d2 ˆ ˆ H  ˆ 2 H  0 2 dz

d ˆ ˆ2ˆ E E  0 dz 2

(6.41)

We obtain identical diagonal matrices of propagation constants  2 from both eigenvalue problems in Equation (6.40). This relation holds, because both Q-matrices are obtained as products of the R matrices but in different order.

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The general relation between the eigenvector matrices TˆE and TˆH of QE and QH, respectively, is 1 TE  Rˆ H TH Bˆ  1

1 TH  Rˆ E TE Bˆ  2

Bˆ 1Bˆ 2 ˆ 2

Bˆ 1  Bˆ 2  Bˆ

(6.42)

which is known from the theory of matrices. Generally, the amplitudes of the eigenvectors can be chosen arbitrarily. Therefore, we introduced the (diagonal) matrices Bˆ 1 1, 2 for normalization in the method above. By transforming the GTL Equations (6.28) with the two modal matrices, we obtain the following simple transformed expressions: d ˆ ˆ E  jBˆ H dz

d ˆ ˆ H  jBˆ E dz

(6.43)

6.3 IMPEDANCE/ADMITTANCE TRANSFORMATION 6.3.1 TRANSFORMATION THROUGH WAVEGUIDE SECTIONS From the general solution of Equation (6.41) ˆ ˆ ˆ ˆ ˆ F  Ff  Fb  e z A  e z B

(6.44)

– –ˆ –ˆ with Fˆ  E, H, we obtain a relation between the derivatives of the fields and the fields themselves. For that purpose we examine two cross sections A and B (inner – sides) of a longitudinal homogeneous section whose distance is d (d  k0d): d dz

ˆ   ˆ   FA  Gˆ Aˆ   FA  Aˆ  ˆ / sinh(ˆ d )      ˆ  Aˆ Gˆ   ˆ  Gˆ  ˆ / tanh(d ˆ )   FB   FB  

(6.45)

– Using the first part of the general solution (the forward propagating fields Fˆ f), we can –ˆ –ˆ define wave impedance/admittance matrices Z0  I and Y0  I. This simple result is a consequence of the field normalization using the two modal matrices TE and TH. Introducing (6.43) into (6.45) results in  ˆ  ˆ  H A   y1    ˆ   yˆ H B   2

ˆ  yˆ2   EA    ˆ  yˆ1   E  B 

ˆ  ˆ  EA   z1    ˆ   zˆ  E B   2

 ˆ  zˆ2   H A    ˆ  zˆ1  H B  

(6.46)

where y1  Yˆ0 / tanh(ˆ d )

y2 Yˆ0 / sinh(ˆ d )

zˆ1  Zˆ 0 / tanh(ˆ d )

zˆ2  Zˆ 0 / sinh(ˆ d )

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(6.47)

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Electromagnetic Theory and Applications for Photonic Crystals

We obtain open-circuit and short-circuit parameters as known from the circuit theory. Here, these parameters are diagonal matrices. Defining admittances/impedances according to ˆ ˆ H A,B  YˆA,B EA,B

ˆ ˆ EA,B  Zˆ A,B H A,B

(6.48)

results in the admittance/impedance transformation between cross sections A and B

(

YˆA  yˆ1  yˆ2 yˆ1  YˆB

)

1

yˆ2

(

Zˆ A  zˆ1  zˆ2 zˆ1  Zˆ B

)

1

zˆ2

(6.49)

These admittance/impedance transformation formulas are numerically stable.

6.3.2 TRANSFORMATION AT WAVEGUIDE CONCATENATIONS If waveguide sections with different cross sections are concatenated, the tangential fields in the common cross section have to be matched. The matching process must be performed in the original domain. The related impedance transformation is given by [16]: 1T Zˆ1  TE11TE 2 Zˆ 2TH2 H1

(6.50)

6.3.3 TRANSFORMATION AT METALIZED WAVEGUIDE INTERFACES In this subsection we will derive the impedance transformation formulas through metalized interfaces between two layers or two waveguide cross sections with different metalization. Some of the discretization lines in the layers (sections) below (before) and above (behind) the interface end on metal, and the others (in the slots) are common to both layers or sections. This transformation is the generalization of the one described in [16]. We will separate the fields of these two types of discretization lines with the help of matrices J m and J c, respectively. We obtain these matrices from an identity matrix J whose size corresponds to the number of discretization lines in the layer (section). J m is obtained from J by removing the rows of discretization lines that cross the slots from one layer (section) to the other. Analogously, J c contains only the rows of discretization lines in the slots. Matching the tangential electric field results in ˆ 0 JmE (6.51) 1

1

ˆ  J cE ˆ J1c E 1 2 2

(6.52)

ˆ 0  J2m E 2

(6.53)

Therefore, we obtain for the tangential electric field transformation between the two sides  c ct c ˆ ˆ   J2  E ˆ E 1   2  J1 J2 E2 0  

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(6.54)

The Method of Lines for the Analysis of Photonic Bandgap Structures

 c ct c ˆ ˆ   J1  E ˆ E 2   1  J2 J1 E1  0 

311

(6.55)

To explain in more detail, the matrix J 1c reduces the field Eˆ 1 in the original domain ˆ of layer 1 to that part common to the field of layer/section 2. J m 1 E1 is the field part of the metallic front end of layer/section 2. The same is true if we exchange 1 and 2. The matrices with zero rows have to be understood symbolically because these rows are somewhere between the rows in the upper submatrix. Matching the tangential magnetic field results in ˆ  J cH ˆ c ˆ c ˆ ct c ˆ J1c H 1 2 2  J2 Y2 E 2  J2 Y2 J2 J1 E1

(6.56)

The last equation may be rewritten and combined with Equation (6.51) (we did not yet use this equation in the matching process) to:  Jc   c ˆ ct 1 c ˆ   1 E  ( J2 Y2 J2 ) J1 H1  ˆ  m 1   J1    0

(6.57)

ˆ  J ct ( J cYˆ J ct )1 J c H ˆ ˆ ˆ E 1 1 2 2 2 1 1  Z1H1

(6.58)

Zˆ1  J1ct ( J2cYˆ2 J2ct )1 J1c

(6.59)

The final result reads

or

6.3.4 STEPS

OF THE

ANALYSIS PROCEDURE

Now the analysis of complex devices is done as follows. We start at the output of the device and transform the load impedance/admittance through the sections and interfaces to the input. Here we obtain the total fields from a source wave and the input impedance. The fields are then calculated in the whole device, starting at the input and going toward the output. In the case of eigenmode calculations, we transform the impedances/admittances from the upper and lower side of the cross section toward a matching interface. The field matching results in an implicit eigenvalue problem for the effective permittivity ere (or for the frequency in the case of band structure calculations). Then the fields are obtained by transforming the field from the matching interface through the layers toward the upper and lower boundaries [7].

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6.4 DETERMINATION OF FLOQUET MODES 6.4.1 INTRODUCTION In this section we would like to describe the analysis of periodic structures. A 2D Bragg grating example is shown in Figure 6.7. The two vertical dashed lines indicate a possible division of this structure into identical periods. A typical example for periodic structures from the microwave area is a meander line (see Figure 6.3a) of which we show one period in Figure 6.8. In these cases the period is symmetric, which allows the development of an efficient algorithm for its analysis.

n=1 t = 0.5 µm n = 1.53

h = 2.4 µm

n = 1.52

x

C y

Lb La

A

B

D

La = 0.106553 µm

z

Lb = 0.106456 µm

FIGURE 6.7 Bragg grating (COST 240) as an example for a periodic structure. (Reprinted with permission from R. Pregla, ICTON Conf., vol. 6, copyright IEEE 2004.)

Period

A

FIGURE 6.8 Microstrip meander line.

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M

B

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The treatment of asymmetric periodic structures is a little more complicated. Particularly, the numerical effort needed for determining the Floquet modes is higher than for symmetric cases. In this chapter we do not go into details but rather refer to the literature. Algorithms for dealing with asymmetric periodic structures are given in [19] and [20]. The determination of Floquet modes with the MoL is described in [21]. Another way of determining the Floquet modes for symmetric and asymmetric periodic structures is described in [22]. Together with the MoL expressions, the formulas given there could also be used for full 3D devices. However, the formulas are valid for the symmetric and the asymmetric periodic structures. Therefore, the numerical effort for analyzing symmetric structures is as high as for asymmetric periodic structures.

6.4.2 SYMMETRIC PERIODIC STRUCTURES The analysis can be done very efficiently in the case of symmetric periods. A method using reflection coefficients (or, to be more general, scattering parameters) has been presented in [9,13]. The algorithm for determining the Floquet modes for this case was shown in [23]. Here we describe a Floquet mode algorithm with impedances or admittances. First we use the open-circuit or short-circuit matrix parameter description for a generalized two port (Cartesian coordinates: [16]; cylindrical coordinates: [24,25]) which gives: E   z  A    11     E B   z21

z12   H A    z22  H B 

 H  y  A    11    H B   y21

y12   EA    y22   E B 

(6.60)

Now we obtain in the case of symmetrical period sections for the even and odd excitations: 1. Even case: EA  EB, HA  HB → magnetic wall in symmetry plane M EA  ( z11  z12 )H A → Zeven  z11  z12  z11h

(6.61)

1 H A  ( y11  y12 )EA → Yeven  y11  y12  z11h

(6.62)

2. Odd case: EA   EB, HA  HB → electric wall in symmetry plane M 1 EA  ( z11  z12 )H A → Zodd  z11  z12  y11h

(6.63)

H A  ( y11  y12 )EA → Yodd  y11  y12  y11h

(6.64)

The subscript h symbolizes the half of the period. The matrices z11(y11) and z12(y12) can be obtained from z11h and y11h as: 1 ) z11  12 ( z11h  y11 h

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1 ) z12  12 ( z11h  y11 h

(6.65)

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Electromagnetic Theory and Applications for Photonic Crystals 1  y y11  12 ( z11 h 11h )

6.4.3 PROPAGATION CONSTANTS

OF

1  y y12  12 ( z11 h 11h )

(6.66)

FLOQUET MODES

For the determination of Floquet modes, we use the two-port description by transfer matrices. From Equation (6.60) we easily obtain the following relations:  E   z z1  B    22 12    1  H B   z12

1 z   E  z21  z22 z12 11   A     1 z12 z11   H A 

(6.67)

 E   z z1  A    11 21    1  H A   z 21

1  E  z11 z 21 z22  z12   B    1 z   H B  21 z22

(6.68)

1  E   E    y1 y y12 12 11  A  B     1  H  1 y  y  y y  H B   y22 y12 11 211 22 12   A 

(6.69)

These formulas also can be inverted to obtain the transfer relations for the opposite direction. For symmetrical period sections, we may introduce z22  z11, z12  z21 (y22  y11, y12  y21) and obtain  E   z z1 ( z z1 z  z )   E  11 12 11 12   A   B    11 12      1  1 z12 z11   H A   H B   z12

(6.70)

1  E   E    y1 y y12 12 11  A  B       1 1   H B   y11 y12 y11  y122  y11 y12   H A 

(6.71)

Now by using the Floquet-modal matrices SE and SH, we perform a transformation to Floquet modes according to  EA,B  SE E A,B

 H A,B  SH H A,B

(6.72)

With this transformation, we obtain for Equations (6.70) and (6.71) 1 1 1    1 E    B    SE z11 z12 SE SE ( z11 z12 z11  z12 ) SH   EA       1S 1 1 S  H B   SH1 z12   H A  E H z12 z11SH

(6.73)

1 1 1 1      E  E S S E y12 y11SE E y12 SH  B  A     1     1 y  y )S 1 1  H B   SH (y11 y12 11 12 E SH y11 y12 SH   H A 

(6.74)

or

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These equations are equivalent to the following relation for Floquet modes:      E Z 0 sin h F   E A  B    cos h F       cos h F   H  H B  Y0 sin h F A 

(6.75)

This results in the following eigenvalue problems: 1 y S  z z1S  S l  y12 11 E 11 12 E E E 1S  z1 z S  S l  y11 y12 H H H 12 11 H

(6.76)

which gives L E  L H  L  cos h F

(6.77)

The modal matrices SE and SH are not independent of each other. The following relations hold: SE  z11SH D1

1S D SH  z12 E 2

(6.78)

We introduced the diagonal matrices D1,2 because the amplitudes of eigenvectors may be arbitrarily chosen. They will be determined in the next section. For self-consistency they must fulfill the condition D1D2  L1

6.4.4 WAVE IMPEDANCE/ADMITTANCE

OF

(6.79)

FLOQUET MODES

By introducing the expression of Equation (6.78) into the off-diagonal submatrices in Equation (6.73), we obtain for the lower left submatrix 1 1 1 1 S H z12 SE  SH z12 z11SH D1  LD1 1 1 1 1  D 2 SE z12 z12 SE  D 2

(6.80)

The first expression of Equation (6.78) was used in the first term. In the second term, we introduced the second expression of Equation (6.78). Both results must be identical. Therefore, we have again D1D2  L1. For the upper right submatrix in Equation (6.73) we obtain 1 2 1 1 1 2 S E (z11 z12 z11  z12 ) SH  SE ((z11 z12 )  I))SE D 2  (L  I )D 2 1 1 1 1  D SH 1 SH (z12 z11  z11 z12 )S



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1 1 D 1 (L  L )

(6.81)

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Electromagnetic Theory and Applications for Photonic Crystals

Here the second expression of Equation (6.78) was introduced in the first line and the first expression in the second line. Again we obtain D1D2  L1. Furthermore, we took into account the relation 1 1 1 1 1  L1 S H z11 z12 SH  (SH z12 z11SH )

(6.82)

The calculated values in Equations (6.80) and (6.81) should be equal to Y˜0 sinh(F) and Z˜ 0 sinh(F), respectively. Thus, we have LD1  Y0 L2  I

(L2  I )D2  Z 0 L2  I

and

Now, among others, we have two main possibilities in choosing the quantities D1,2 in conjunction with Z˜ 0 and Y˜0: 1. By choosing Z˜ 0  Y˜0  I we obtain D1  L2  I L

D2  I

L2  I

(6.83)

2. Choosing D1  D2  D results in 1

DL

2

1 Z 0  ((L2  I )/L) 2  Y01

(6.84)

The most important problem now is to calculate the eigenvalues L in an efficient way. Z˜ 0 or Y˜0 can be determined easily afterward. These matrices are diagonal but frequency dependent.

6.4.5 EFFICIENT CALCULATION

OF THE

FLOQUET MODE PARAMETERS

The required quantities can be calculated efficiently by using open- and short-circuit matrix parameters of the half periods. Using the relations in Equations (6.65) and (6.66), we obtain, for example, 1  (z 1 1 1 z11 z12 11h  y11h )(z11h  y11h )  (z11h y11h  I )(z11h y11h  I)1

(6.85)

1 y  (y 1 1 1  y12 11 11h  z11h ) (y11h  z11h )  (z11h y11h  I )1 (z11h y11h  I)

(6.86)

1 z  (z 1 1 1 z12 11 11h  y11h ) (z11h  y11h )  (y11h z11h  I )1 (y11h z11h  I)

(6.87)

1  (y 1 1 1  y11 y12 11h  z11h )(y11h  z11h )  (y11h z11h  I )(y11h z11h  I)1

(6.88)

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For the eigenvalues, we obtain from Equations (6.85) and (6.87): 1 1 1 1 1 S E z11 z12 SE  L  SE (z11h y11h  I )SE SE (z11h y11h  I ) SE

 (SE1 z11h y11h SE  I )(SE1 z11h y11h SE  I )1

(6.89)

or 1 1 1 1 1 S H z12 z11SH  L  SH (y11h z11h  I ) SH SH (y11h z11h  I ) SH 1  (SH1 y11h z11h SH  I )1 (S H y11h z11h SH  I )

(6.90)

Thus, we have L  (L h  I )(L h  I )1  (Lh  I )1 (Lh  I )

(6.91)

where we defined L h  SE1 z11h y11h SE  SH1 y11h z11h SH

(6.92)

This is the fundamental equation for determining the Floquet modes. We obtain equivalent equations by using Equations (6.86) and (6.88). The values D1,2 in Equations (6.83) (the alternative with Z0  Y0  I) are now given as D1  2 L h (L h  I )

D2  12 (L h  )

Lh

(6.93)

Due to the relation cosh x  (1  tanh2 –x2 )/(1  tanh2 –x2 ), Lh is related to F by 1  1 tanh  F   L h 2  I  2 

SE1 z11h y11h SE  I

SH1 y11h z11h SH

(6.94)

To obtain a similar expression for F, we have to examine the half of two periods, resulting in tanh F  I

1 S E z11 y11SE  I

SH1 y11 z11SH

(6.95)

This result also can be obtained from tanh F  sinh F /cosh F by introducing the impedance/admittance matrices in Equation (6.76). Another simple way to obtain the Equations (6.95) is the following: From Equation (6.75) we obtain for the matrices z˜11 and y˜11 (see also Equations (6.101) with N  1): z11  Z 0 / tanh F

© 2006 by Taylor & Francis Group, LLC

y11  Y0 / tanh F

(6.96)

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Electromagnetic Theory and Applications for Photonic Crystals

After transformation with SE and SH, these two matrices must be identical to the matrices z11 and y11 of the period in Equation (6.60). So we obtain Z 0 / tanh F  SE1 z11SH

Y0 / tanh F  SH1 y11SE

(6.97)

From the product of these two equations we obtain Equation (6.95). The quotient results in alternative equations for the characteristic impedance or admittance, respectively. The results presented here are analogous to the well-known equations in the so-called image theory of filters. The main difference is that here we have matrices in the equations instead of scalar values. By introducing the result of Equation (6.91) into Equation (6.84) (the alternative with D1  D2), we obtain for the wave impedance Z 0  (L2  I )/L  2 L h  (L2h  I )

6.4.6 CONCATENATION

OF

(6.98)

N PERIODS

In this subsection we give the relevant formulas for N concatenated period sections (Figure 6.7). Because we described each section like a homogeneous waveguide, only F must be replaced by NF if N periods are concatenated. Equation (6.75) now reads:      E   0 sinh(NF )   E C  D    cosh(NF )       cosh(NF )   H C   H D  Y0 sinh(NF )

(6.99)

The ports C and D are the input and output ports of the whole structure. Furthermore, again we can write the field relations between the generalized two-ports C and D with open-circuit impedance or short-circuit admittance (z- and y-) matrices:    E  C    z1     ED   z2

  z2   H C     z1  H D 

    H  C    y1     H D   y 2

  y 2   E C   y1   ED 

(6.100)

with z1  Z 0 / tanh(NF ) y1  Y0 / tanh(NF )

z2  Z 0 / sinh(NF ) y 2 Y0 / sinh(NF )

(6.101)

˜ ˜ ˜ ˜ By using the relation E˜ C,D  Z˜ C,D H C,D and HC,D  YC,D EC,D, impedance/admittance transformation for the whole periodic structure is performed as Z C  z1  z2 (z1  Z D )1 z2

© 2006 by Taylor & Francis Group, LLC

YC  y1  y 2 ( y1  YD )1 y 2

(6.102)

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At the output and input we have the following relations between the impedances/ admittances with the tilde () (Floquet impedances) and bar (–) (mode impedances) Z D  SE1TE ZD TH1SH

YD  SH1THYD TE1TE

(6.103)

Equation (6.103) can be inverted easily to obtain the mode impedances/admittances from the Floquet values.

6.5 DETERMINING THE BAND STRUCTURES OF PHOTONIC CRYSTALS For determining the band structures of photonic crystals, we can apply the method of lines in two ways: a) as a propagation algorithm, and b) as an eigenmode solver. In the first case, we determine the Floquet modes with the algorithms described earlier. In this section we describe both of these methods [26].

6.5.1 PROPAGATION ALGORITHM To use the MoL as propagation algorithm and calculate the band structure, we introduce the Floquet-mode concept into the analysis. The determination of the Floquet modes with impedances and admittances has been described in Section 6.4. Alternatively, we can use the algorithm presented in [23,27] where scattering parameters were used instead. Consider the array of silicon rods in air shown in Figure 6.9. The top and front views of one elementary cell with discretization lines are shown in Figure 6.10.

a z

y

x

FIGURE 6.9 Array of silicon rods in air. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

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320

Electromagnetic Theory and Applications for Photonic Crystals rf, kb

f , b k+1 k+1

Periodic boundary condition

N

Discretization lines

a

z y (a)

1 0

Periodic boundary condition a x Absorbing boundary condition ABC

Periodic boundary condition PBC

PBC

(b)

ABC

FIGURE 6.10 (a) One cell of a photonic crystal with discretization in x-direction; (b) discretization in cross section.

The front view shown in Figure 6.10(b) is more complicated than the structure shown in Figure 6.9; this is to make clear that the devices that can be examined may be more complex and do not have to be symmetric. As indicated in Figure 6.10, we discretize in the x- and y-directions (i.e., in the case of full three-dimensional analysis). On the top and bottom boundaries we apply absorbing boundary conditions. If the structure is symmetric, we may reduce the numerical effort by introducing a symmetry plane (i.e., electric or magnetic walls depending on the polarization).

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Due to the periodicity of the device the fields are periodic, too. With the labels given in Figure 6.10(a) we can write a relation between the fields on the lines numbered 0 and N: F0  exp{jk x a}FN

F  E, H

(6.104)

where the width of the cell is a. This relation is introduced into the discretization scheme. Now, in the z-direction we determine the Floquet modes with the presented algorithm. For the propagation of the Floquet modes from the interface labeled k to the following one labeled k  1 (Figure 6.10[a]) we can write: Fkf1  eF Fkf

Fkb1  eF Fkb

(6.105)

F  jkzF a The following relation between the single Floquet modes and the transformed fields described in Section 4 holds:   Z H  Ff  (E 0 )/2

(6.106)

  Z H  F b  (E 0 )/2

(6.107)

Now the propagation constants of the Floquet modes are a function of the frequency: kzF  kzF (v, k x ) Applying the described method, we obtain a high number of Floquet modes (and therefore various values for kz) for each frequency (i.e., also in the bandgap). (The reader should be reminded that kzF is a matrix containing the propagation constants of all Floquet modes.) However, very few of the Floquet modes are guided (i.e., the ones we are looking for in band-structure calculations). Consequently, the values kz are complex in general, where the imaginary part gives the decrease of the field. For the guided modes, the propagation constant is real (apart from numerical noise). Therefore, we select only those modes, where this condition is fulfilled. Varying v and kx enables us to determine the band structure of our photonic crystal. This algorithm is different from those often used in the literature where the frequency is determined as a function of kx and kz. Unlike the procedure that we describe next (Section 6.5.2) we do not have to solve an implicit eigenvalue problem. Hence, no problems occur with finding suitable start values. It should be mentioned that crystals with a periodicity in all three dimensions can be analyzed with this method as well. Instead of absorbing boundary conditions at the top and bottom of the structure in Figure 6.10, we just have to introduce periodic BCs also at these boundaries. Figure 6.11(a) shows the irreducible Brillouin zone of a square lattice. To determine the –X and X–M bands we take one cell as shown

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322

Electromagnetic Theory and Applications for Photonic Crystals a

kx2 M

a

z,x2

Γ

X

kx1 x,x1 (b)

a

2·

2·

a

(a)

x2 z′ (c)

x1

x′ y

FIGURE 6.11 (a) Irreducible Brillouin zone; (b) one cell for calculating –X, X–M bands; (c) one cell for calculating –M band. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

in Figure 6.11(b) and apply the Floquet algorithm as described. As can be seen, we introduced two coordinate systems. The system with x1 and x2 corresponds to the structure, whereas the system with x and z relates to the algorithm for analysis. This is because we always use analytic expressions for the z-direction and discretize in the other ones. Hence, we have analytic expressions in the x2-direction for the –X direction, whereas for computing the values in X–M direction analytic solution in the x1-direction are used. Particularly, for the –X band we choose exp( jk x1a)  1 and exp( jk x 2 a) 1

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for the X–M direction. We should mention here that we can choose an arbitrary relation between the fields on the opposite side of an elementary cell (besides 1 as we do for the –X or X–M band). Hence, not only values on the outer sides of the triangle in Figure 6.11(a) can be determined but also arbitrary values within the Brillouin zone. This is different to other eigenmode methods where a high symmetry is required, allowing the examination only of special cases. For computing the –M band we have various possibilities. One is shown in Figure 6.11(c). We rotate the coordinate system by 45° and examine the new elementary cell. The Floquet modes are then determined for the new z -direction. In this way we keep a “forward algorithm”; i.e., we do not have to introduce a root solver. However, instead of introducing a rotated coordinate system, we could also use the original coordinate system (Figure 6.11[b]) to compute the band in the –M direction. In this case, kx has to be fixed, and the frequency for which the condition kz  kx is fulfilled must be determined. One can do this either by varying the frequency in an interval with constant frequency steps and thus approach the unknown frequency, or by using a root finder. Obviously, one has to determine the Floquet modes a couple of times before finding the searched value using either of these methods. The number of iterations depends strongly on the start values, particularly for the root finder. Further, if the start values are too far away from the solution, the method might not even converge. Other problems can occur if two (or more) solutions are close to each other. Compared to using a rotated coordinate system, the CPU-time is definitely higher with this approach (the Floquet modes have to be determined a couple of times, maybe even using different start values). Therefore, we usually use the rotated coordinate system for examining the X–M band.

6.5.2 EIGENMODE ALGORITHM When applying the MoL as eigenmode algorithm we divide the structure in homogeneous sections in the vertical direction. Then we position the discretization points in the axis of the rods as shown in Figure 6.12. Here we use periodic boundary conditions on all four boundaries; i.e., we preset kx and ky. For the further analysis consider Figure 6.13. The algorithm is not restricted to the examination of rods in air, but we could have much more different sections in the vertical direction. Therefore, we show more regions in this figure. After determining the analytical solution in each section, we start with the impedances of the top and bottom layers. Usually, these layers are infinitely long so that the impedance is the corresponding wave impedance. These impedances are transformed to a matching plane with the derived transformation formulas (6.49) in homogeneous sections and (6.50) or (6.59) at interfaces. The transformation from the upper sections results in an impedance Zˆu, and Zˆ1 is due to the transformation from the lower boundary. The transverse electric and magnetic fields must be continuous. Hence, we obtain ˆ  Zˆ H ˆ E u

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ˆ  Zˆ H ˆ E 1

(6.108)

324

Electromagnetic Theory and Applications for Photonic Crystals Periodic boundary condition

y z

x

FIGURE 6.12 Top view of an elementary cell with discretization points. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

z Zu E,H x,y

Matching plane

Zl

FIGURE 6.13 Transformation of the impedance from the top and the bottom to a matching plane.

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or ˆ 0 ( Zˆ u  Zˆ1 )H

(6.109)

This equation has only nontrivial solutions if the system determinant is zero: det( Zˆ u  Zˆ1 )  0

(6.110)

Since the impedances depend on the frequency, Equation (6.110) is an implicit eigenvalue problem for v. By variation of kx and ky in the finite differences in Equation (6.21) or Equation (6.104), respectively, we are able to compute the whole band structure. It should be mentioned that instead of giving the propagation constant for both directions we could also give just one of them (e.g., kx) and the frequency. After transforming the impedances to the matching plane, we obtain an implicit eigenvalue problem for the remaining propagation constant in this case. This appears to be similar to the method we described in the previous section. However, the algorithm described here always requires the search for zeros; i.e., an iteration that was not the case in the propagation algorithm. Note that instead of impedances we could also use admittances or scattering parameters. The latter give a relation between the in- and outgoing waves. At the top and bottom of the structure, the corresponding reflection coefficient is equal to zero. Analogously to the described method, we transform this reflection coefficient to a matching plane that again results in an implicit eigenvalue problem for the unknown frequency. The details of this method and the application to band-structure calculations can be found in [9] and [26], respectively.

6.5.3 LINES

OF

VARYING LENGTH

Often photonic crystals are not made of structures with horizontal and vertical boundaries but with round ones like circles. When applying the method of lines, curved boundaries are usually modeled by a staircase approximation. The easiest approximations of circles are squares or crosses (Figure 6.14). The numerical examinations show that this approximation already gives quite good results. However, to improve the accuracy we developed another way of modeling the curved boundaries: lines of varying length [28,29,30,31] (Figure 6.15[b]) were used. We will describe the main principles for the waveguide shown in Figure 6.15, which has a curved shape. On the left we see the usual staircase approximation, and on the right side the modeling with lines of varying length is sketched. For analyzing such structures with curved boundaries not only the transverse (i.e., x-y-) field components have to be determined but also the longitudinal (or vertical) (i.e., z-) ones. Equation (6.10) shows how they are computed from the transverse fields: je zz Ez  [ Dy•

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ˆ Dx ]H

jm zz H z  [Dx•

Dy ]Eˆ

(6.111)

326

Electromagnetic Theory and Applications for Photonic Crystals h

d1 r

d2

b

FIGURE 6.14 Top view of an elementary cell with staircase approximation. III II II I

I

I

FIGURE 6.15 Waveguide modeling: (a) staircase approximation; (b) lines of varying length. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 6, pp. 118–121, copyright IEEE 2004.)

As before, we discretize these equations in the x- and y-directions. Then, by introducing the expressions for the transverse components Equations (6.40)–(6.44), we can write:  ˆ ˆ ˆ ( z )  Tˆ  E E (z )  Eb(z )  z Ez  f 

 ˆ ˆ ˆ ( z )  Tˆ  E H ( z )  E b ( z )  (6.112) z Hz  f 

Note that the signs in front of the backward propagating part are the opposites of the signs of the transverse components. Let us now examine the propagation along a homogeneous section. As an example, we use the transverse electric field. In Figure 6.16(a) the section consists of two horizontal boundaries. In this case the field on the top boundary (z  d) is computed as:  ˆ ˆ ˆ (d )  Tˆ  exp(d )E E E f (0)  exp( d )E b (0)    © 2006 by Taylor & Francis Group, LLC

(6.113)

The Method of Lines for the Analysis of Photonic Bandgap Structures

d (x)

d z

dk

z 0

r

327

x

0

FIGURE 6.16 Homogeneous section with (a) two straight boundaries; (b) one curved boundary. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 6, pp. 118–121, copyright IEEE 2003.)

In contrast we have a curved boundary on the top of the structure shown in Figure 6.16(b). The field on the kth line can be determined as [30]:   ˆ ˆ Ek (d k )  TˆEk  exp(d k )Ef (0)  exp(d k )E b (0)   

(6.114)

T˜ Ek is the kth row of the transformation matrix. Equation (6.114) shows how the field on one line is determined from all eigenmodes. To determine the whole field distribution on the upper boundary, we introduce a column vector d that contains all individual distances and a row vector  with the propagation constants of the eigenmodes. The field on the upper curved boundary is then given as [30]: ˆ ˆ ˆ (d)  Tˆ • exp(d  )E ˆ E E I f (0)  TE • exp( d )E b (0)

(6.115)

Here, “•” indicates the element by element product. The magnetic field and the longitudinal electric field at the curved boundary are determined analogously, but we have to take care with the sign in front of the backward propagating part. Next we have to match the fields at interfaces (Figure 6.17[a]). With the given coordinate system the y components as well as the t components must be continuous. The t components are determined as Ft  Fx cos a  Fz sin a This relation has to be used for the electric field particularly if we have regions with a different permittivity. If the permeability is constant in all sections (as is usually the case in optics), it is sufficient for the magnetic field to match the y and x (instead of the t) components. Now we are in position to fulfill the boundary conditions. After deriving the expressions for the fields at the boundaries, we need to transform the admittances/impedances (or alternatively reflection coefficients) from the output to the input and the field in the opposite direction. This allows us again to determine the Floquet modes, and therefore also the band structures. The details are described in the literature [23,30,31].

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II

Fy  Ft z

I (a)

x

y

zII r II

r BII II d II(x )

r BI

I zI

d I(x )

rI

z (b)

y

x

FIGURE 6.17 (a) Transverse fields at curved boundaries; (b) transformation of the reflection coefficient in sections with curved boundaries. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 6, pp. 118–121, copyright IEEE 2003.)

6.6 JUNCTIONS IN PHOTONIC CRYSTAL WAVEGUIDES 6.6.1 PORT RELATIONS The analysis of waveguide junctions is demonstrated with the photonic crystal waveguide structure given in Figure 6.18(a). For the analysis we use crossed discretization lines. Only a few of the discretization lines in both directions are drawn in the figure. The junction region is bounded by the four ports A to D. In general, all of these four ports have connecting waveguides, in our case WA to WD. The connecting

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329

B WB ABC Bk

B Bk

Ak

Ak D

WD

z1

y1 ns

WC

C

C

D

x2 x1 np

A

z2

y2

WA

A

FIGURE 6.18 (a) Junction of waveguides in photonic crystals; (b) enlarged inner junction. (Reprinted with permission from R. Pregla, ICTON Conf., vol. 5, 116–121, copyright IEEE 2004.)

waveguides may be different than the waveguides inside the junction. In case of sharp bends we have only two ports with connecting waveguides (e.g., the ports A and D with WA and WD). Only the junction region is sketched in Figure 6.18(b). The number of concatenated waveguide sections between the ports A and B is K. The ports of the section k are labeled Ak and Bk. At the sidewalls we assume absorbing boundary conditions (ABC). They are placed on the position of magnetic walls. According to the uniqueness theorem, the field outside the inner region is determined from the tangential field on the outer boundary. The fields at the outer and inner side of the boundary are clearly related. We describe the relation of the tangential fields at the inner side of the four generalized ports A, B, C, and D by open circuit matrix parameters in the form [16]: ˆ   EA   zˆ AB    AB11  ˆ   ˆ AB  E B   zAB21    ˆ   zˆ AB  EC   CD11    ˆ AB  ˆ   zCD21  ED 

AB zˆAB12

CD zˆAB11

AB zˆAB22

CD zˆAB21

AB zˆCD12

CD zˆCD11

AB zˆCD22

CD zˆCD21

 ˆ  CD   H A  zˆAB12    ˆ  CD   H zˆAB22  B   ˆz CD   ˆ  CD12   H C    CD   zˆCD22 ˆ   H D  

(6.116)

We collected the discretized field components in column vectors as described in Section 6.2.4 and should remember that each of the field column vectors –ˆ –t –t t contains the two tangential field components at the ports (e.g., E A  [E y, Ex] , –ˆ –t – t t HA  [H x, H y] ). Combining the fields on opposite ports in supervectors,

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we can write Equation (6.116) in a more compact form:  CD   ˆ zˆAB   H AB     CD   ˆ zˆCD H   CD 

 ˆ   zˆ AB  EAB   AB    ˆ   ˆ AB z E  CD   CD

 ˆ ˆ t ˆ EUV   EUt , EVt   

(6.117)

t ˆ ˆ ˆ  H UV   H Ut , H Vt   

U is identical to A or C, and V to B or D, respectively. The four submatrices in this equation are obtained by open circuiting the ports and placing magnetic walls AB there. If the ports C and D are open circuited, the matrices –zˆ AB and –zˆ AB CD are CD CD obtained. By open circuiting the ports A and B, the matrices –zˆ CD and –zˆ AB can be calculated. Now, we will demonstrate the determination of the two main diagonal submatrices in Equation (6.117). We will assume that the anisotropy is described by diagonal tensors. The general case is described in [17].

6.6.2 MAIN DIAGONAL SUBMATRICES AB Let us start with –zˆ AB . We assume that the ports C and D are open circuited, –ˆ which means HCD  0. Then, the field relation between the ports A and B is described by

ˆ ˆ AB H EAB  zˆAB AB

or

 ˆ   ˆ AB  EA   zAB11    ˆ   zˆ AB  E B   AB21

ˆ  AB   H zˆAB12 A     AB   ˆ  zˆAB22  H   B 

(6.118)

To obtain the four submatrices we again use the technique of open circuiting the –ˆ –ˆ AB –ˆ AB –ˆ AB ports. With H B  0, we obtain the submatrices z AB11 and z AB21. z AB11 is the input impedance matrix in plane or port A (inner side) for the open port B. In our case, the ports A and B are connected via a concatenation of K different waveguide sections. For each of these sections the tangential fields at the ends (or the ports) inside the sections are described by Equations (6.46) and (6.47). We rewrite the relation with open circuit matrix parameters –z for section k as E k   z k  A  1     E Bk   z2k

z2k   H A k  ˆ AB   zABk  z1k  H Bk   

 H k A     H Bk 

(6.119)

The superscript k at the subscript refers to the number of the section. The relation with short circuit matrix parameters –y is analog. Now, assuming impedances at the ports Ak and Bk (inner sides) according to ˆ ˆ EA k, Bk  Zˆ A k, Bk H A k, Bk

© 2006 by Taylor & Francis Group, LLC

(6.120)

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– – the impedance Zˆ Bk has to be transformed to obtain Zˆ Ak using

(

Zˆ A k  zˆ1k  zˆ2k zˆ1k  Zˆ Bk

)

1

zˆ2k

(6.121)

This is done from the end (port Bk) to the beginning (port Ak) of this section. For the transition between the waveguide sections, the formulas in Section 6.3.3 and especially Equation (6.59) hold. The input impedance of section k in plane Ak is – – given in Equation (6.121). From Zˆ Ak, we obtain Zˆ Bk1 with the help of Equation (6.59) by

(

Zˆ Bk1  TEk11, c THk , c Zˆ Ak1TEk1, c

)

1

THk1,c

(6.122)

This impedance matrix is a full matrix. Starting at the end or in section K with – – BK  B and HBK  HB  0 (or Zˆ BK  ), we obtain at AK the impedance Zˆ AK  –zˆ1k. –ˆ With ZBK1 from Equation (6.122) and k  K  1, we can calculate the input impedance of the next section by Equation (6.121). Repeating this process until we – reach the first section, we finally obtain the input impedance Zˆ A1, which is equal AB to zˆAB11. AB For the calculation of the matrix zˆ AB21 , we must now go in the opposite direction. The field transformation in a section k is given by

(

)

ˆ ˆ H Bk  zˆ2k1 zˆ1k  Zˆ A k H A k

(

(6.123)

 ˆ E Bk   zˆ2k  zˆ2k zˆ2k1 zˆ1k  Zˆ A k 

) Hˆ

Ak

ˆ  Zˆ Bk H Bk

(6.124)

–ˆ –ˆ –ˆ The next step is the transformation of H Bk and EBk into the values EAk 1 and –ˆ HAk 1. We obtain (cf., Section 6.3.3): ˆ ˆ EA k1  TˆEk11, c TˆEk , c E Bk

ˆ ˆ H A k1  Zˆ Ak11 EA k1

(6.125)

– AB We start with k  1 and Zˆ A1  –z AB11 . This procedure must be repeated for the other –ˆ sections. Then, from the equation for E former calculated Bk (in which all the –ˆ quantities have been introduced and that is proportional to H A in this case) the AB transmittance matrix –zˆ AB21 is obtained. The analogous procedure holds for the other two submatrices in Equation (6.118): now the port A must be open circuited and the input impedance at port B looking toward A must be calculated. This AB AB input impedance is identical to the submatrix zˆAB22 . For the calculation of zˆAB12 –ˆ we must transform the magnetic field column vector HB from port B through the different waveguide sections to port A taking into account the impedances deter–ˆ –ˆ AB mined before zˆAB12 is the matrix that connects E A with HB. After this procedure AB zˆAB is completely determined.

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Electromagnetic Theory and Applications for Photonic Crystals

The submatrix in –zˆ CD CD in Equation (6.117) is obtained in an analogous way. –ˆ However, in this case the ports A and B must be open circuited with H AB  0. Now we have L different waveguide sections concatenated between ports C and D (not extra marked). AB In the next section we will demonstrate how the off-diagonal submatrix –zˆ CD can be calculated.

6.6.3 OFF-DIAGONAL SUBMATRICES: COUPLING TO OTHER PORTS AB The off diagonal submatrix zˆ CD in Equation (6.117) is defined under the condi–ˆ tion HCD  0 (magnetic walls have to be introduced there) by the equation

ˆ ˆ ECD  zˆ AB CD H AB

 ˆ   ˆ AB  EC   z CD11    ˆ   zˆ AB  ED   CD21

or

 ˆ  zˆ AB CD12   H A      ˆ  zˆ AB CD22   H  B 

(6.126)

–ˆ The two submatrices on the left side are obtained by setting H B  0, and the two on –ˆ the right side by setting HA  0. We will now show the procedure for the two submatrices on the left. The matrices on the right side are obtained analogously. The tangential electric field components on the ports C and D (at the magnetic walls there) are given by (Figure 6.19): E ˆ C,D   ˆ C,D   ˆ C,D   y 2    E y1    E y  ˆ  E  ˆ C,D   ˆ C,D   ˆ C,D  C,D  E x 2   E z1   E z       

(6.127)

Bk Dk Ck

z1 x1 x2

y1 Ak z2

y2

dk

FIGURE 6.19 Subsection of the inner junction to define the coordinates for the coupling from ports Ak and Bk to ports Ck and Dk, enlarged in z1-direction. (Reprinted with permission from R. Pregla, ICTON Conf., vol. 5, 116–121, copyright IEEE 2003.)

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333

Only these electric field components are responsible for the coupling from ports A and B to ports C and D. The magnetic field has only zero tangential field components at the boundaries C and D (because of the magnetic walls). So we have –ˆ C,D –ˆ C,D –ˆ to determine E y and E z caused by HA. These calculations must be done in parts, because the ports C and D must be partitioned in K subsections with subports C k and Dk depending on the sidewall areas of the waveguide sections in the z1-direction. Let us now calculate the tangential electric fields at the subports Ck and Dk. The sub- or superscript k in this subsection always marks the waveguide subsection labeled k. After obtaining the tangential electric fields at all subports we have to arrange each of these field components in a column vector in the order of the subports’ numbers. The field components in a plane at an arbitrary position zk  zk1 between the two subports Ak and Bk of a section k (k  1, 2, … , K) in Figure 6.19 can be calculated from the fields at ports Ak and Bk by ˆ ˆ ˆ F( z k )   dA k FA k   dBk FBk

(6.128)

– where we have zk  0 in the plane of port Ak. Fˆ is the supervector of the trans– – verse electric or magnetic field components. The fields FˆAk and FˆBk both have to be – calculated from the fields at port A (!). Even with HB  0, we have magnetic fields at both subports Ak and Bk in the subsections k, which both have to be cal– culated from HA. Only if Bk is identical to B will we have HBk  0. The diagonal d d matrices Ak and  Bk in Equation (6.128) are given by  dA k 

sinh(kz (d k  z k )) sinh(kz d k )

 dBk 

sinh(kz z k )

(6.129)

sinh(kz d k )

– where d k  k0dk. In the plane of port Ak, we have zk  0. dk is the distance between the ports Ak and Bk. The diagonal matrix  kz of the propagation constants –ˆ for the z direction in section k is obtained as in Equation (6.40). The field E yRk at k k k k ports R  C and R  D can be determined as described in the following. The field Ekyn in the nth column of the cross section at zk is given by Ekyn ( z k )  TE

yn

(

ˆ d E Ak Ak

ˆ   dBk E Bk

)

(6.130)

where TEyn is the nth block of the matrix TEy for the waveguide section k (we will not introduce a sub- or superscript k for this identification). Therefore, we define the k k matrices T CEy and T DE y by

(

T CE  81 9TE k

y

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yN

 TE

yN 1

)

(

T DE  81 9TE  TE k

y

y1

y2

)

(6.131)

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where TEyN (TEy1) and TEyN1 (TEy2) are the last (first) and next to last (second) k block of the matrix TEy, respectively. TREy are rectangular matrices. The relevant field component EyR at ports Rk  Ck or Rk  Dk are now expressed by

(

ˆ ˆ E yR ( z k )  TER  dA k EA k   dBk E Bk y

)

(6.132)

Equation (6.132) must be discretized at the discretization points zki of the k – ports. The field part EAyRk related to Eˆ Ak can be expressed as

ˆ Ak E yR k

 Rk d k   T E  A k ( z1 )  y    ˆ ˆ   E k  Vˆ A k E   yR A k  A    T Rk  d  z k   E y A k  Nkz   

(6.133)

– kk is the last column at the subport C k or Dk in the z1-direction. The field vectors Eˆ Ak zN z – – ˆ k are both determined from H by the procedure described in the previous and E B A subsection. The obtained relations may be expressed by ˆ ˆ EA k  Zˆ  k H A

ˆ ˆ E Bk  Zˆ k H A

(6.134)

The fields on all the subports Ck and Dk (subports k of ports C and D) are now collected in the common column vectors to obtain the total column vectors for –ˆ E y at the ports C and D, respectively.

ˆ E yR

 ˆ B1 ˆ E ˆ A1 ˆ B1   ˆ A1 ˆ  yR1  E yR1   VyR1 ZAA1  VyR1 ZAB1           ˆ     ˆ  H A  Zˆ yAR H   A k k  ˆA E ˆB  k k ˆ ˆ E  VˆyARk ZAA k  VˆyBRk ZABk  yR k   yR k              

(6.135)

This is the upper part in the vector of Equation (6.127). We obtain the tan–ˆ k gential electric field component Eˆ C,D z from the magnetic field H(z ) according to Equation (6.10), or in discretized form ˆ ( z k )  jE1  Dˆ • E z zz  y

ˆ k ˆ ( z k )  jE1  Dˆ • Dˆ   Tˆ H Dˆ x  H zz  y x  H ( z ) (6.136) 

To calculate Ez at the ports Ck and Dk by an extrapolation process, we need the –ˆ values of Ez, that belong to the last two and the first two columns of E z in the k cross section at z , respectively. The n-th column can be calculated by   ˆ Ekzn ( z k )  j(Ekzzn )1  D•y THk Dxn THk  H( z k ) xn yn   © 2006 by Taylor & Francis Group, LLC

(6.137)

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k where Ezzn is the diagonal matrix of the Erzz values for that column. Tˆ Hk is divided into two parts, which are further divided in two subparts. The upper part T kHx belongs to Hkx and the lower part T kH y to Hky with the same number of rows, respectively. T kHxn is the matrix of the nth block of rows related to the nth column. The extrapolation of the columns in the first part of Equation (6.137) to the ports with k the matrices TRxn is constructed analogously to Equation (6.131). The difference •– matrix D y is used at the port as usual. The extrapolation of the second part is completely different because the difference operator D°x– must be applied to the columns. Therefore, D°x–n T kHyn has to be understood in the following way: D°x–n means that this difference operator has to be applied to two neighboring vertical columns of the H ky quantities or the H kyn columns (at ° discretization points). After this application, the columns of the derivatives can be extrapolated analogously to Equation (6.131) because the derivatives have to fulfill Neumann boundary conditions. The columns of the derivatives can be expressed by the central differences of the neighboring columns. The field component Hkyn itself has to fulfill Dirichlet boundary conditions. Using the central differences for this case, we obtain

(

THC  81 10THk k

y

yN

 THk

yN 1

)

(

THD  81 10THk  THk k

y

y1

y2

)

(6.138)

Equation (6.138) can be checked easily. If the behavior of Hky(x) at the boundary k k k k is completely linear, we have, for example, T H  2T H and therefore, T DHy  T H . y2 y1 y1 k k Now, the values at the subports C and D are given by ˆ EkzD  THD H( z k ) z

ˆ EkzC  THC H( z k ) z

k  hx1 THD  y  k  hx1 THC  y 

 k THD  j(Ekzzn )1  D•y THD z x   k THC  j(Ekzzn )1  D•y THC z x 

(6.139)

– In Equation (6.139) hx is the normalized discretization distance in the x-direction. –ˆ The vector of the magnetic field components H in Equation (6.139) has to be –ˆ –ˆ replaced by Equation (6.128). HAk and HBk are obtained with Equations (6.123) to (6.125). The field component EzR at ports Rk  Ck or Rk  Dk is now expressed by

(

ˆ ˆ EkzR ( z k )  THR dA k H A k  dBk H Bk z

)

(6.140)

Equation (6.140) must be discretized at the discretization points zki of the –ˆ k ports. The field part EAzRk related to H Ak can be expressed as

ˆ Ak E zR k

© 2006 by Taylor & Francis Group, LLC

 T R d ( z k )   H z Ak 1    ˆ H ˆ Ak ˆ     A k  ZzR H A k    T R d k ( z k )   H z A N x 

(6.141)

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Electromagnetic Theory and Applications for Photonic Crystals

–ˆ –ˆ – Both field vectors H Ak and HBk are determined from HA by the procedure described in the previous section. The relation may be given by ˆ ˆ H A k  VˆAA k H A

ˆ ˆ H Bk  VˆABk H A

(6.142)

The fields on all the subports Ck and Dk (subports k of ports C and D) are now –ˆ collected in the common column vectors to obtain the total column vectors for E z at ports C and D, respectively.

ˆ E zR

ˆ B1 ˆ  E ˆ A1 ˆ B1   ˆ A1 ˆ  zR1  E zR1   ZzR1VAA1  ZzR1VAB1            ˆ     H A  ZzAR H A (6.143)  k k  ˆA E ˆ B   ˆ Ak ˆ k ˆ E  ZzRk VAA k  Zˆ zBRk VABk  zR k   zR k            

In an analogous manner, Zˆ ByR and Zˆ BzR can be calculated under the condition –ˆ HA  0. In summary the results may be written as follows:

ˆ ECD

 ˆC   E y 2   Tˆ 1 Zˆ A    C yC  ˆ C   ˆ 1 ˆ A  E x 2   TC ZzC    ˆ D   Tˆ 1 Zˆ A E  y 2   D yD     ˆ D   TˆD1 Zˆ zAD  E x 2  

 ˆA TˆC1 Zˆ yBC  H x   ˆ  TˆC1 Zˆ zBC   H Ay  ˆ AB H    zˆCD AB     1 B ˆ ˆ ˆ B TD ZyD   H  x   TˆD1 Zˆ zBD   ˆ B  H y   

(6.144)

We transformed the field column vectors on the left side with the transformation matrices TˆCD and TˆCD for the ports C and D, respectively. The other matrix parameters in Equation (6.117) are obtained in an analogous way. Now to analyze the behavior of a special device, the load impedances at the ports must be calculated. The connecting waveguides are periodic structures. One of the possible procedures for analysis is described in [32].

6.7 NUMERICAL RESULTS 6.7.1 BAND STRUCTURE CALCULATIONS In this section we show some results of band structure calculations. We begin with two-dimensional devices and continue with three-dimensional structures. Figure 6.20 shows a square array of dielectric veins in air ([1], p. 61) as a first example for a bandgap structure. Due to the horizontal and vertical boundaries, this structure is best suited for analysis using the MoL. The computed band structures

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337


r = 1


r = 8.9

FIGURE 6.20 Square array of dielectric veins in air ([1], p. 61).

are shown in Figures 6.21(a) and (b) for the TM and TE modes respectively. As can be seen, we have a gap for the TE modes. The curves agree very well with the results presented in ([1], p. 61). A second two-dimensional device is the array of rods in air shown in Figure 6.22 ([1], p. 56). Due to the round shape of the rods, we applied the algorithm presented in Section 6.5.3 and compared the results with those obtained for a staircase approximation (i.e., approximation with squares). Figure 6.23(a) shows the curves for the TE polarization compared with ([1], p. 56). As can be seen the results agree quite well. However, the agreement between the MoL could be improved by introducing lines of varying length. The situation is different for the TM polarization shown in Figure 6.23(b). Here we present only the results obtained with lines of varying length; because the curves for the staircase approximation are so close to these that they are hardly distinguishable. For this structure we obtain a bandgap for the TM polarization. We will now give the results for a 3D device (with 2D periodicity). Again we have an array of dielectric rods in air but this time with a finite height. This structure is from [33]. The band structures for the even and odd modes (in relation to the vertical direction) are presented in Figure 6.24. As in [33] we have a bandgap for the odd modes. To obtain the bands in the –M region, we applied a search algorithm as described at the end of Section 6.5.1. Since the time for computing the bands is quite long, we did not determine the –M bands for the even modes. Also these curves agree very well with those from the literature.

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0.9 0.8 0.7

a/wavelength

0.6 0.5 0.4 0.3 0.2 0.1 0.0 (a)

Γ

X

M

Γ

Γ

X

M

Γ

0.9 0.8 0.7

a/wavelength

0.6 0.5 0.4 0.3 0.2 0.1 0.0 (b)

FIGURE 6.21 Determined band structure for the square array of dielectric veins in air (Figure 6.20) (period length a): (a) TM-modes; (b) TE-modes.

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2r = 0.4a

a


r = 8.9


r = 1

FIGURE 6.22 Square array of dielectric rods in air (structure proposed in [1], p. 56). (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

As our last example, we will show the results for another two-dimensional photonic crystal of dielectric rods in air [34]. We used this device for further studies that we discuss in the next section. A square cross section of the rods was assumed from the beginning. By removing one column of rods, a waveguide could be designed (Figure 6.25[a]) in the wavelength area between 1.3 m and 1.7 m for the TM polarization. This area was found from the transmission characteristics with the FDTD [34]. The results of band structure calculations are shown in Figure 6.25(b). With a period length of a  0.6 m, we find that the upper and lower frequencies correspond to the wavelengths 1.26 m and 1.7 m, respectively. Therefore, we have in fact a bandgap in the area found by the FDTD [34] allowing us to design a waveguide.

6.7.2 PC WAVEGUIDE DEVICES As mentioned before, further studies were done for the PC waveguide structure shown in Figure 6.25(a).1 The determined effective index is shown in Figure 6.26(a). From this effective index we can compute the group velocity (see, e.g., [34]). The results are presented in Figure 6.26(b). The MoL calculations were done again with the Floquet algorithm. A very good agreement with the FDTD-results that were used in [34] can be observed. 1 The results presented in this section were determined by A. Barcz, who is working in the same institute as the authors.

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0.2

MoL square 0.1 0 Γ

(a)

X

Γ

M

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a/wavelength

0.6 0.5 0.4

Joannopoulus MoL

0.3 0.2 0.1 0

(b)

Γ

X

M

Γ

FIGURE 6.23 Band structure of an array of dielectric rods in air (see Fig. 6.22), period length: a, height of the rods: infinite, (a) TE-polarization; (b) TM-polarization. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

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0.8 Boundary of the lightcone

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a/wavelength

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Γ

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M

Γ

0.8 Boundary of the lightcone

0.7

a/wavelength

0.6 0.5 0.4 0.3 0.2 0.1 0 (b)

Γ

X

M

FIGURE 6.24 Band structure of an array of dielectric rods in air (structure proposed in [33]), period length: a, height of the rods: 2a, (a) odd modes; (b) even modes. (Reprinted with permission from S.F. Helfert, ICTON Conf., vol. 5, 122–125, copyright IEEE 2003.)

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x

ns n0

a/wavelength

1.4 1.0 0.8 0.6 0.4 0.2

z (a)

1.2

0.0 (b)

Γ

X

M

Γ

FIGURE 6.25 (a) PC-waveguide, period length a  0.6 m, length of the rods 0.15 m, ns  3.4, n0  1 [34]; (b) corresponding band structure for the TM polarization.

One of the promising features of PCs is the design of sharp bends with very low losses. Therefore, we examined an S bend as shown in Figure 6.27(a). The transmission characteristic is presented in Figure 6.27(b) [35]. No optimization has been done so far, yet we see that there are areas where nearly the complete field is transmitted. Also, however, stop bands around 1.4 m and 1.6 m are found. An important problem in designing circuits with photonic crystals is the coupling to dielectric waveguides. For this reason we examined the structure shown in Figure 6.28. We have a symmetric dielectric waveguide connected to the PC waveguide that we just analyzed. The refractive index and the width of the dielectric waveguide were varied in order to find a minimum of the losses. Results for a refractive index of 1.45 for the slab waveguide as a function of the dielectric width are shown in Figure 6.28(b). Comparing our results to the results from the literature, we can see that the total power loss computed by the MoL is a little higher than the results obtain by the FDTD [34] in this case. The results for Fresnel loss are similar to these obtained by the FDTD. A three-dimensional graph of the losses as a function of dielectric width of the input waveguide and if refractive index is presented in Figures 6.29(a) and 6.29(b). The total power loss has a local minimum, whereas the Fresnel loss increases with refractive index and dielectric width. A closer look (not presented) shows that the local minimum of the total loss appears for a refractive index of 1.625 and a dielectric width of 0.45 m.

6.7.3 BRAGG GRATINGS The developed algorithms also were used for the analysis of various Bragg gratings. We will show some examples here. Let us begin with the reflectivity of the deep-etched grating presented in Figure 6.7. Figure 6.30 shows the reflectivity as a function of the wavelength with the number of periods as a parameter. No numerical problems were observed in spite of the high number of periods. It is interesting

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Wavelength (µm)

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Group velocity v/c

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FDTD MoL

0.50 0.45 0.40 0.35 0.30 0.25 1.2

1.3

(b)

1.4

1.5

1.6

1.7

1.8

Wavelength (µm)

FIGURE 6.26 (a) Effective index of the PC waveguide structure in Figure 6.25(a); (b) corresponding group velocity. (Reprinted with permission from A. Barcz, S.F. Helfert, and R. Pregla, ICTON Conf., vol. 4, 126–129, copyright IEEE 2002.)

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Transmission (dB)

0 −10 −20 −30 −40

x n0 ns

1.3

z (a)

1.4

1.5

1.6

1.7

Wavelength (µm)

(b)

FIGURE 6.27 (a) S bend made of photonic crystals, (b) transmission characteristic.

d

nd

ns

x

n0 z

(a) 0.5

FDTD

0.4

Power loss

MoL 0.3 Total loss 0.2

Fresnel loss

0.1

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(b)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Waveguide width (µm)

FIGURE 6.28 (a) Coupling between a dielectric waveguide and a crystal waveguide; (b) normalized power loss as function of waveguide width (refractive index nd  1.45).

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0.5

Total power loss

0.45 0.4 0.35 0.3 0.25 0.2 0.15 1.9 1.8 1.7 Refractive index

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1.5 1.4

0

0.2

0.4

0.6

0.8

1

1.2

Dielectric width (µm)

FIGURE 6.29 (a) Normalized total power loss; (b) normalized Fresnel power loss.

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Reflectivity

0.8

0.6

0.4

0.2

0.0 0.6494

0.6496

0.6498

0.6500

0.6502

Wavelength (µm)

FIGURE 6.30 Reflectivity of the Bragg grating in Figure 6.7 vs. wavelength with number of periods as parameter.

B

A nclA

B nclB

A r

ncoA

ncoB

nclA

nclB

LA

LB

z

OFWF1032

FIGURE 6.31 Fiber grating structure.

that the curves are not symmetrical. Many more results and details for this particular grating can be found in [13,15,36 (pp. 101–107)]. A similar Bragg grating was examined in [37]. It acted as a 2D model of PCs and was used to study the out-of-plane scattering. There, too, the results obtained with the MoL were in good agreement with those obtained by other methods. We also examined fiber Bragg gratings [11,38]. For this purpose cylindrical coordinates were introduced. The structure is shown in Figure 6.31. The reflection coefficient of the fundamental mode for this structure is presented in Figure 6.32.

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1.0 M = 8.000 0.9

M = 8.000 aniso nzz = 1.5

0.8 0.7

Reflectivity

0.6 0.5 0.4 0.3 0.2 0.1 0.0 1.5240

1.5242

1.5244

1.5246

1.5248

1.5250

1.5252

1.5254

Wavelength  (µm)

FIGURE 6.32 Reflectivity of the fiber Bragg grating (rcor  2 m).

Also anisotropic material was introduced leading to a slight shift of the reflection curve. Further results obtained by the MoL can be found in the literature, and a few of them shall be briefly mentioned here. For example, various filters made of PCs and for holey fibers have been examined in [35] and [39]. Here we have shown results only for optical devices. Structures from the microwave area also were studied with the Floquet algorithm. Results for a microwave bandgap structure with an anisotropic substrate can be found in ([40] p. 186). Various meanderlines with and without anisotropic materials have been examined in [16] or ([40] p. 187).

6.8 CONCLUSION In this chapter we showed the analysis of bandgap structures using the method of lines. Due to the periodicity of the PCs, Floquet’s theorem was introduced. This allows us to treat periodic sections as quasi-homogeneous ones. Thus, periodic structures like Bragg gratings with very high numbers of periods can be modeled with a moderate numerical effort and without numerical problems. Also devices containing periodic parts and homogeneous sections can be analyzed in a uniform way. A particular application of photonic crystal waveguides is sharp bends.

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In this case we have waves propagating in perpendicular directions. To take this into account, crossed discretization lines were introduced. The MoL was applied to the examination of various structures, some of which were shown in this chapter. We mention here the computation of band structures and the examination of PC waveguides. The results were found to be in good agreement with other methods. This shows that the MoL can be used as a suitable tool for designing PC devices.

ACKNOWLEDGMENT The authors would like to thank Agnieszka Barcz for doing some of the numerical work, especially for the results presented in Section 6.7.2.

REFERENCES [1] J.D. Joannopoulus, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ, 1995. [2] R.M. De La Rue and T.F. Krauss, “Photonic crystals in the optical regime: past, present, and future”, Progr. in Quantum Electron., 23, 51–96, 1999. [3] R. De La Rue, “Modelling issues for waveguide photonic crystals and photonic microstructures: a personal view”, Opt. Quantum Electron., 34, 417–433, 2002, special issue on optical waveguide theory and numerical modeling. [4] R. Pregla and W. Pascher, “The Method of Lines”, in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed., 381–446, J. Wiley, New York, 1989. [5] R. Pregla, “MoL-BPM method of lines based beam propagation method”, in Methods for Modeling and Simulation of Guided-Wave Optoelectronic Devices (PIER 11), W.P. Huang, ed., Progress in Electromagnetic Research, 51–102, EMW Publishing, Cambridge, MA, 1995. [6] U. Rogge and R. Pregla, “Method of lines for the analysis of dielectric waveguides”, J. Lightwave Technol., 11, 2015–2020, 1993. [7] R. Pregla, “The method of lines as generalized transmission line technique for the analysis of multilayered structures”, AEÜ, 50, 293–300, 1996. [8] R. Pregla, “Analysis of planar waveguides with arbitrary anisotropic material”, in U.R.S.I. Intern. Symp. Electromagn. Theo., Victoria, BC, May 2001, 425–427. [9] S.F. Helfert and R. Pregla, “The method of lines: a versatile tool for the analysis of waveguide structures”, Electromagnetics, 22, 615–637, 2002, special issue on optical wave propagation in guiding structures. [10] R. Pregla, “Modeling of waveguide structures with general anisotropy in arbitrary orthogonal coordinate systems”, in Proc. of IGTE Symp., Graz, Austria, 211–216, Sept. 2002. [11] R. Pregla, “Modeling of optical waveguide structures with general anisotropy in arbitrary orthogonal coordinate systems”, IEEE J. of Sel. Topics in Quantum Electron., 8, 1217–1224, 2002. [12] R.E. Collin, Field Theory of Guided Waves, Series of Electromagnetic Waves, IEEE Press, New York, 2nd ed., 1991. [13] S.F. Helfert and R. Pregla, “Efficient analysis of periodic structures”, J. Lightwave Technol., 16, 1694–1702, 1998.

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[14] R. Pregla, “Efficient modeling of periodic structures”, AEÜ, 57, 185–189, 2003. [15] J. Cˇ tyroky´, S. Helfert, and R. Pregla, “Analysis of a deep waveguide Bragg grating”, Opt. Quantum Electron., 30, 343–358, 1998. [16] R. Pregla, “Efficient and accurate modeling of planar anisotropic microwave structures by the method of lines”, IEEE Trans. Microwave Theory Tech., 50, 1469–1479, 2002. [17] R. Pregla, “Analysis of waveguide junctions and sharp bends with general anisotropic material by using orthogonal propagating waves”, in ICTON Conf., vol. 5, 116–121, Warsaw, Poland, 2003. [18] V.K. Tripathi, “On the analysis of symmetrical three-line microstrip circuits”, IEEE Trans. Microwave Theory Tech., MTT–25, 726–729, 1977. [19] S.F. Helfert, “Efficient analysis of non symmetric periodic optical devices”, in OSA Integr. Photo. Resear. Tech. Dig., vol. 4, 372–374, Victoria, BC, Mar. 1998. [20] R. Pregla, “Analysis of gratings with symmetrical and unsymmetrical periods”, in ICTON Conf., vol. 6, Wraclow, Poland, 101–104, 2004. [21] S.F. Helfert, “Determination of Floquet–modes in asymmetric periodic structures”, Opt. Quantum Electron., 2005, submitted for special issue on optical waveguide theory and numerical modeling. [22] Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides”, J. Opt. Soc. Am. A, Opt. Image Sci., 19, 335–338, 2002. [23] S.F. Helfert, “Numerical stable determination of Floquet-modes and the application to the computation of band structures”, Opt. Quantum Electron., 36, 87–107, 2004, special issue on optical waveguide theory and numerical modeling. [24] R. Pregla, “The impedance/admittance transformation: an efficient concept for the analysis of optical waveguide structures”, in OSA Integr. Photo. Resear. Tech. Dig., 40–42, Santa Barbara, CA, 1999. [25] R. Pregla, “Novel FD–BPM for optical waveguide structures with isotropic or anisotropic material”, in European Conf. on Integrated Optics and Technical Exhibit, Torino, Italy, Apr. 1999, 55–58. [26] S.F. Helfert, “The method of lines for the calculation of band structures in photonic crystals”, in ICTON Conf., vol. 5, 122–125, Warsaw, Poland, 2003. [27] S. Helfert, R. Pregla, A. Barcz, and L. Greda, “Numerical analysis of periodic structures in optics and microwaves”, in IGTE Symp., 436–441, Graz, Austria, Sept. 2002. [28] R. Pregla, J. Gerdes, E. Ahlers, and S. Helfert, “MoL-BPM algorithms for waveguide bends and vectorial fields”, in OSA Integr. Photo. Resear. Tech. Dig., vol. 10, 32–33, New Orleans, 1992. [29] W.D. Yang and R. Pregla, “The method of lines for analysis of integrated optical waveguide structures with arbitrary curved interfaces”, J. Lightwave Technol., 14, 879–884, 1996. [30] R. Pregla, “Modeling of planar waveguides with anisotropic layers of variable thickness by the method of lines”, Opt. Quantum Electron., 35, 533–544, 2003. [31] S.F. Helfert, “Modeling of structures with curved boundaries: applied to the determination of band structures in photonic crystals”, in ICTON Conf., vol. 6, Wraclow, Poland, 118–121, 2004. [32] R. Pregla, “Modeling of optical waveguide structures with general anisotropy in arbitrary coordinate systems”, IEEE Journ. sel. top. Qu. El., vol. 8, 1217–1224, 2002.

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[33] S.G. Johnson, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, and L.A. Kolodziejski, “Guided modes in photonic crystal slabs”, Phys. Rev. B, 60, 5751–5758, 1999. [34] R. Stoffer, H.J.W.M. Hoekstra, R.M. de Ridder, E. van Groesen, and F.P.H. van Beckum, “Numerical studies of 2D photonic crystals: waveguides, coupling between waveguides and filters”, Opt. Quantum Electron., 32, 947–961, 2000, special issue on optical waveguide theory and numerical modeling. [35] A. Barcz, S. Helfert, and R. Pregla, “Modeling of 2D photonic crystals by using the method of lines”, in ICTON Conf., vol. 4, 45–48, Warsaw, Poland, 2002. [36] G. Guekos, Photonic Devices for Telecommunications, COST240 Book, Springer, Berlin, 1999. [37] J. Cˇ tyroky´, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. de Ridder, R. Stoffer, G. Klaasse, J. Petrácek, P. Lalanne, J.-P. Hugonin, and R.M. De La Rue, “Bragg waveguide grating as a 1D photonic bandgap structure: COST 268 modelling task”, Opt. Quantum Electron., vol. 34, 445–470, 2002, special issue on optical waveguide theory and numerical modeling. [38] I.A. Goncharenko, S.F. Helfert, and R. Pregla, “General analysis of fibre grating structures”, J. of Optics A: Pure and Appl. Optics, 1, 25–31, 1999. [39] A. Barcz, S.F. Helfert, and R. Pregla, “The method of lines applied to numerical simulation of 2D and 3D bandgap structures”, in ICTON Conf., vol. 5, 126–129, Warsaw, Poland, 2003. [40] R. Pregla and S.F. Helfert, “Modeling of microwave devices with the method of lines”, in Recent Research Developments in Microwave Theory and Techniques, B. Beker and Y. Chen, eds., 145–196, Research Signpost, Kerala, India, 2002.

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of the Finite7 Applications Difference FrequencyDomain Mode Solution Method to Photonic Crystal Structures Chin-Ping Yu and Hung-Chun Chang

CONTENTS 7.1 Introduction ..............................................................................................352 7.2 The FDFD Model ....................................................................................355 7.2.1 Formulae for 2D Waveguide Problems ......................................355 7.2.2 The FDFD Method with PMLs ..................................................358 7.2.3 Numerical Treatment at the Dielectric Interface ........................363 7.3 Modal Analysis of Photonic Crystal Fibers ............................................365 7.3.1 Holey Fibers ..............................................................................366 7.3.2 Two-Core PCFs ..........................................................................372 7.3.3 Honeycomb PCFs ......................................................................373 7.3.4 Hollow PCFs ..............................................................................376 7.4 The FDFD Method for Analysis of Photonic Band Structures ..............379 7.4.1 The FDFD Formulae ..................................................................379 7.4.2 Periodic Boundary Conditions for 2D PCs ................................383 7.5 Calculation of 2D PC Band Diagrams ....................................................385 7.5.1 PCs with Square Lattices ..........................................................385 7.5.2 PCs with Triangular Lattices ......................................................386 7.5.3 Finger Plots of 2D PCs ..............................................................388 7.6 Modal Analysis of Planar PC Waveguides ............................................391 7.7 Conclusion ..............................................................................................395 Acknowledgments ............................................................................................397 References ........................................................................................................397 351

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7.1 INTRODUCTION†

N layers

Since the photonic crystal (PC) concept was proposed [2,3], there has been a growing interest in the development of PC or electromagnetic crystal materials. They are characterized by an amazing ability to control the flow of light or electromagnetic waves within a frequency band called the photonic bandgap (PBG). Except for light guiding, the PBG concept is also adopted in many applications, such as lasers, resonators, and microwave circuit components. The PBG can be generated in periodic dielectric structures for certain geometries. Many research efforts have been devoted to utilizing the PBG material into waveguides in microwave and optical frequencies, and several PC waveguides have been fabricated. These PC waveguides in previous reports were made by introducing a defect in a dielectric array composed of air columns or dielectric rods to form a guided region for propagating light or electromagnetic waves. Instead of being allowed through the effective total internal reflection, the guided modes are restricted in the defects due to the existence of PBGs generated by the infinite array of PC materials outside the defects. There are two major PC waveguides. One is the planar PC waveguide, also named the planar PBG waveguide, which is formed by a dielectric slab placed between two PBG mirrors. Figure 7.1(a) shows the cross section of a planar PBG

2r

a y

2r

y

a

x

z

N layers

z

x

(a)

(b)

FIGURE 7.1 Cross section of (a) a planar photonic crystal waveguide and (b) a holey fiber near the core region, where a is the lattice distance and 2r is the diameter of dielectric rods or air holes. (After Yu and Chang, Opt. Quantum Electron., 36, 145–163, 2004, with kind permission of Springer Science and Business Media.) †

The major part of Section 7.1 has been published in [1]. Numerical examples in [1] will appear in this chapter. However, figures related to these examples were obtained using the Yee mesh-based finite difference scheme that will be described in Section 7.2, while those in [1] were calculated based on the conventional finite difference method described in Section 2 of [1]. The inclusion of material from [1] in this chapter is with kind permission of Springer Science and Business Media.

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waveguide, and the circles represent either dielectric rods or air columns. The guided modes can propagate on such a waveguide by the Bragg reflection resulting from the presence of the PC. Several experimental results have shown that even through sharp bends [4,5] a wide transmission band can be achieved by carefully designing the geometry of the waveguide. For the air columns case, there also exists another important physical guiding mechanism, the effective total internal reflection, which results from the effective index difference between the slab and the PBG mirrors [6]. Similar mechanisms are responsible for light guiding in another PC waveguide, the photonic crystal fiber (PCF). The first PCF reported by Knight [7], also called the holey fiber, was made from undoped fused silica having a triangular array of air holes along its length, with the central hole missing, forming a core area as shown in Figure 7.1(b). It was found by many experimental results and numerical simulations that the PCF can be single-mode guided over a wide frequency range [8]. By appropriate design of the geometry of the PCF, dispersionlessness at a desired frequency can be achieved, which is quite useful in many applications [9,10]. Because the PC waveguides have played an important role in wave transmission, it is necessary to develop a useful tool to understand their propagation properties. A variety of methods have been utilized to calculate the propagating characteristics of PC waveguides. Among these, the most often used are the plane-wave expansion (PWE) method [11,12] and the finite-difference time-domain (FDTD) method [13,14]. The finite element method (FEM) [10,15] and the T-matrix approach [16] also have been utilized recently. In addition to the above-mentioned methods, the waveguide mode solver based on the finite difference method (FDM), which has been widely used to analyze waveguide modes on various optical or dielectric waveguides [17,18], should be a convenient choice [19]. The conventional PWE method, although popularly employed in PC research, suffers from slow convergence of the Fourier transform of the dielectric function [20]. The FEM is also a well-known technique in waveguide mode analysis with excellent flexibility and accuracy, but its formulation is more complicated than the FDM. The conventional FDM is simple in its formulation and its numerical implementation. It requires less memory storage due to the sparsity of the matrix, and it has been well received in conventional waveguide analysis and design. We have applied the conventional FDM to the analysis of various PCFs and planar PC waveguides [1]. The FDM was first employed to solve the scalar waveguide modes under the weakly guiding approximation [21]. For strongly guiding structures, semivectorial equations for optical waveguides with arbitrary index profiles were derived [22,23]. To obtain more accurate mode fields, a full-vectorial finite difference scheme was then proposed [17,18,24]. Each mesh point is surrounded by four subregions whose permittivities are allowed to be different. The sparse matrix is formed by discretizing the Helmholtz equations and matching the boundary conditions (BCs) for all the subregions. Another way is to discretize the wave equations derived from Maxwell’s equations with the finite difference scheme [25–27], which is usually used in the beam propagation method (BPM) [26,27]. Recently, full-vectorial

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finite difference formulations based on the Yee’s mesh, as in the FDTD method [28], and derived directly from Maxwell’s equations have been proposed [29–32]. They are attractive in that the obtained mode fields can be incorporated easily into the FDTD computation. Ando et al. [29] derived a Yee-mesh-based imaginarydistance propagation method with modified FD formulae that take the dielectric interface conditions into account. Zhu and Brown [30] proposed a finite-difference frequency-domain (FDFD) method that directly discretizes Maxwell’s equations using the central difference scheme based on the Yee’s cell. Although most conventional optical waveguides are designed to propagate well-confined guided modes with ideal real-valued propagation constants, calculation of leaky properties of waveguide modes having complex propagation constants has become more important due to recent intensive study on the confinement losses of PCFs [33]. In this chapter, we consider the Yee-mesh-based FD algorithm for formulating the eigenvalue problem for waveguide mode solutions (named the FDFD method) and incorporate into it the perfectly matched layer (PML) absorbing boundary conditions (ABCs) [34] so that leaky waveguide modes can be analyzed. We employ an alternative formulation for the PML by mapping Maxwell’s equations into an anisotropic complex stretched coordinate [35–37]. To obtain high-accuracy results, proper treatment of the fields near the dielectric interface is essential [38–41]. For example, proper matching of interface conditions through the Taylor series expansion of the fields could achieve excellent accuracy [39]. Similar interface matching procedures can be performed in the Yee-mesh-based algorithm [29,42] and will be applied in this chapter. A variety of numerical methods have been utilized to calculate the band structures of two-dimensional (2D) PCs. Among these, again, the most used are the PWE method [43,44] and the FDTD method [45,46]. Also, FD eigenvalue algorithms for analyzing the band structures of 2D PCs have been proposed. Yang [47] proposed an algorithm by discretizing the Helmholtz equation in terms of the longitudinal magnetic field Hz or the longitudinal electric field Ez for the transverseelectric (TE) and transverse-magnetic (TM) modes, using the central difference scheme in a way similar to that of [24] for waveguide analysis. Only square-lattice PCs were discussed, and no curved dielectric interfaces were considered in the unit cell of the PC. Shen et al. [48] derived another algorithm from wave equations in terms of the parallel and transverse components of the magnetic field for the TE and TM modes, respectively, and employed an effective-medium technique for smoothing the field distribution across the dielectric interface. Again, only square PCs were considered. In this chapter, an FDFD eigenvalue algorithm based on the Yee’s cell is derived and is utilized to analyze the band structures of 2D PCs with either square or triangular lattices [49]. Not only the in-plane propagation but the out-of-plane propagation in the PC is considered so that band edge diagrams, or finger plots, which are essential for designing photonic-band-gap (PBG) fibers [50], can be calculated. The sections of this chapter are organized as follows. The derivation of the formulation of the FDFD eigenvalue algorithm for optical waveguides is described in detail in Section 7.2, including the formulation with PML ABCs and the

© 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

355

numerical treatment at the dielectric interface [32]. Then, modal analysis of various PCFs using the FDFD method is presented in Section 7.3. The formulation of the FDFD method for analysis of photonic band structures is given in Section 7.4, and calculated results of 2D PC band diagrams are presented in Section 7.5. Section 7.6 discusses the application of the FDFD method to modal analysis of planar PC waveguides. Finally, the conclusion is drawn in Section 7.7.

7.2 THE FDFD MODEL 7.2.1 FORMULAE

FOR

2D WAVEGUIDE PROBLEMS

Considering 2D waveguide problems with propagation in the z-direction and exp[ j(vt–bz)] field dependence, where v is the angular frequency and b is the propagation constant, Maxwell’s curl equations can be expressed in terms of the six components of the electric and magnetic fields ∂E z

 jvm0 H x 

∂y

 jbE y

 jvm0 H y  jbE x  ∂E y

 jvm0 H z  jve0 eE x 

∂x

∂H z ∂y

∂H y ∂x

∂E z

(7.2)

∂x

∂E x ∂y

(7.3)

 jbH y

(7.4)



jve0 eE y  jbH x  jve0 eEz 

(7.1)



∂H z

(7.5)

∂x

∂H x ∂y

(7.6)

where each field component is a function of x and y, m0 and e0 are the permeability and permittivity of free space, and e  e(x, y) is the relative permittivity of the medium. Applying 2D Yee’s mesh shown in Figure 7.2, (7.1)–(7.6) can be expressed as  jvm0 ( H x )i , j 1  2

∂E z ∂y

i , j 1

 jb( E y )i , j 1

(7.7)

2

2

 jvm0 ( H y )i 1 , j  jb( E x )i 1 , j  2

2

∂E z ∂x

(7.8) i 1 , j 2

© 2006 by Taylor & Francis Group, LLC

356

Electromagnetic Theory and Applications for Photonic Crystals Ey ( i, j+1/2) j+1/2 Hz (i −1/2, j+1/2)

Hz(i +1/2, j+1/2)

Hx (i, j+1/2)


1 Hy (i +1/2, j )

Hy (i −1/2, j ) Ez (i, j )
z

Ex (i −1/2, j )

Ex (i +1/2, j )
2

Ey (i,j −1/2) j−1/2

Hz(i −1/2, j−1/2) i −1/2

Hz(i +1/2, j−1/2) i+1/2

Hx (i, j −1/2)

FIGURE 7.2 2D Yee’s mesh for the FDFD method. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

∂E y

 jvm0 ( H z )i 1 , j 1  2

∂x

2

jve0 (eE x )i 1 , j 

 i 1 , j 1 2 2

∂H z

2

∂y

i 1 , j

∂E x ∂y

(7.9) i 1 , j 1 2

 jb( H y )i 1 , j

2

(7.10)

2

2

jve0 (eE y )i , j 1  jb( H x )i , j 1  2

∂H z ∂x

2

(7.11) i , j 1

2

jve0 (eEz )i , j 

∂H y ∂x

 i, j

∂H x ∂y

(7.12) i, j

where the subscripts (i, j) denote the grid point located at (i, j). Using the central difference scheme, (7.7)–(7.12) become  jvm0 H  jvm0 H

(

i , j 1 2

(

i 1 , j 2

x,

y,

Ez ,(i , j1)  Ez ,(i , j )

)



)

 jbE

© 2006 by Taylor & Francis Group, LLC

y

(

x,

i 1 , j 2

)



 jbE

(

y , i , j 1

2

)

Ez ,(i1, j )  Ez ,(i , j ) x

(7.13)

(7.14)

Applications of the Finite-Difference Frequency-Domain Mode

E  jvm0 H

(

z,

i 1 , j 1 2 2

)



(

2

(

x,

i 1 , j 2

E

) x ,(

(

y , i , j 1

2

(

z , i 1 , j 1

)



)

 jbH

i 1 , j 2

2

2

)

H

(

y,

i , j 1 2

E

) y,(

i , j 1 2

H jve0 e z ,(i , j ) Ez ,(i , j ) 



(

(

(

y , i 1 , j 2

)

(

x,

i , j 1 2

H

)

(

y , i 1 , j 2

)

x

2

E

(

x , i 1 , j 2

)

y

2

2

)

y 

)

x , i 1 , j1

z , i 1 , j 1

H jve0 e

E

)

x H

jve0 e

E

)

y , i1, j 1

357

(

 jb H

z , i 1 , j 1 2

2

H

)

(

y , i 1 , j

(

2

2

(7.16)

)

z , i 1 , j 1

2

)

x H 

(

x , i , j 1

2

)

H

(

x , i , j 1

2

(7.15)

)

y

(7.17)

(7.18)

which can be written in the matrix form as [30]  H  x   jvm0  H y    Hz   e  x  jve0  0    0 

          

 0     jbI   U  y

0 ey 0

jb I 0 Ux

U y   Ex    U x   Ey    0   Ez  

 0       0   Ey    jbI    ez   Ez   Vy  0   Ex 

jb I 0 Vx

(7.19)

Vy   H x    Vx   H y    0   H z  

(7.20)

where I is an identity matrix, E and H (  x, y, or z) are the vectors composed of the field components E and H, respectively, at the grid points, and e (  x, y, or z) is a diagonal matrix representing e at the corresponding grid points of E. Ux, Uy, Vx, and Vy are square matrices determined by the formulae in (7.13)–(7.18) and the BCs at the edges of the computing window. After some mathematical work, one can obtain an eigenvalue equation from (7.19) and (7.20) in terms of the transverse electric fields

A

  E   x      Ey   



 A  xx   A  yx 

Axy   Ex    Ayy  

       Ey   



  E   x 2 b    Ey   

where 2 1 2 1 2 A xx k 0 U x e z Vy Vx U y  (k0 I  U x e z Vx )(e x  k0 Vy U y ) 2 A yy k02 U y ez 1Vx Vy U x  (k02 I  U y ez 1Vy )(ey  k 0 Vx U x ) 2 A xy (I  k02 U x ez 1Vx )Vy U x  U x ez 1Vy (ey  k 0 Vx U x ) 2 A yx (I  k02 U y ez 1Vy )Vx U y  U y ez 1Vx (ex  k 0 Vy U y )

© 2006 by Taylor & Francis Group, LLC

(7.21)

358

Electromagnetic Theory and Applications for Photonic Crystals

with k0 being the wavenumber in free space. By solving (7.21), we can obtain the propagation constants b of the guided modes from the eigenvalues and the field distributions of the guided mode from the eigenvectors. Here, we solve the eigenvalue problem by using the shift inverse power method (SIPM). Similarly, for the transverse magnetic fields, we can have

B

  H   x      Hy   



 B  xx   B  yx 

Bxy   H x    Byy  

       Hy   



  H   x 2 b    Hy   

(7.22)

where 2 2 1 2 1 Bxx k 0 Vx U y U x e z Vy  (e y  k0 Vx U x )(k0 I  U y e z Vy ) 2 2 1 1 2 Byy k 0 Vy U x U y e z Vx  (e x  k0 Vy U y )(k0 I  U x e z Vx ) 2 2 1 1 Bxy  Vx U y (I  k 0 U x e z Vx )  (e y  k0 Vx U x )U y e z Vx 2 2 1 1 Byx  Vy U x (I  k 0 U y e z Vy )  (e x  k0 Vy U y )U x e z Vy

Note that (7.21) is an alternative to (7.22). Both of the two equations would lead to the same answer. The FDFD method has the advantage that, once the transverse electric fields Et or the transverse magnetic fields Ht are determined, the other transverse field components can be easily obtained from (7.21) or (7.22). At the same time, Ez and Hz can also be determined by (7.19) and (7.20) using the four matrices Ux, Uy, Vx, and Vy.

7.2.2 THE FDFD METHOD WITH PMLS To determine the leaky properties or confinement losses of leaky guided modes, the perfectly matched layers (PMLs) are adopted in the FDFD model. Figure 7.3 shows the cross section of an arbitrary waveguide problem with the computing window surrounded by PML regions I, II, and III, each having a thickness of d. Regions I and II have the normal vectors parallel to the x-axis and y-axis, respectively, and Regions III are the four corner regions. In the PML region, using the stretched coordinate transform [35–37], Maxwell’s equations can be written as   ∇′  E  jvm0 H

(7.23)

  ∇′  H  jve0 n 2 E

(7.24)

where the modified differential operator ∇′ is defined as ∇′  xˆ

© 2006 by Taylor & Francis Group, LLC

1 ∂ 1 ∂ ∂  yˆ  zˆ ∂z s x ∂x s y ∂y

(7.25)

Applications of the Finite-Difference Frequency-Domain Mode

III

359

II

III

d

I

I

y x III

II

III

FIGURE 7.3 Cross section of an arbitrary waveguide problem with the computing window surrounded by PML regions.

TABLE 7.1 Values of sx and sy in the PML Regions and Non-PML Region Regions

I

II

III

Non-PML

sx sy

s l

l s

s s

l 1

with the values of sx and sy summarized in Table 7.1 and s is defined as s  1 j

se ve0

n2

 1 j

sm vm0

(7.26)

with n   e being the refractive index of the adjacent computing domain and se and sm being the electric and magnetic conductivities of the PML, respectively. In (7.26), we can see the impedance matching condition of the PML medium is se e0

© 2006 by Taylor & Francis Group, LLC

n2



sm m0

(7.27)

360

Electromagnetic Theory and Applications for Photonic Crystals

which means that the wave impedance of a PML medium exactly equals that of the adjacent medium in the computing window regardless of the angle of propagation. Assuming the electric conductivity of the PML medium has an m-power profile as  r m se (r)  smax    d 

(7.28)

where r is the distance from the beginning of the PML. At the interface of the PML and the computing window, the theoretical reflection for the normal incident wave is [51]  s  R  exp 2 max e0 cn  

   dr   d   

m dr

∫0

(7.29)

and the maximum conductivity smax can then be determined as m  1 e0 cn 1 ln 2 d R

smax 

(7.30)

where c is the speed of light in free space. For the case of m  2, as we will assume in our model, s in (7.26) becomes s  1 j

2 3l  r  1  ln  4 pnd  d  R

(7.31)

Using the modified operator ∇′ defined in (7.25), we have Maxwell’s curl equations for the whole problem  





 i , j 1

 1 ∂E z  jbE y    s y ∂y

 jvm0 ( H x )i , j 1   2

(7.32) 2

 1 ∂Ez   jvm0 ( H y )i 1 , j   jbE x  sx ∂x  1 2  i , j 2

(7.33)

 1 ∂E 1 ∂E x  y  jvm0 ( H z )i 1 , j 1    s y ∂y  1 1 2 2  sx ∂x  i 2 , j 2

(7.34)

 





 i 1 , j

 1 ∂H z  jbH y    s y ∂y

jve0 (e x E x )i 1 , j   2

© 2006 by Taylor & Francis Group, LLC

(7.35) 2

Applications of the Finite-Difference Frequency-Domain Mode

361

 1 ∂H z  jve0 (e y E y )i , j 1   jbH x  sx ∂x  1 2  i , j 2

(7.36)

 1 ∂H 1 ∂H x  y jve0 (e z Ez )i , j    s y ∂y   sx ∂x  i, j

(7.37)

where sx and sy in each PML region and the computing domain are defined in Table 7.1. Equations (7.32)–(7.37) also can be written in the matrix form as  H  x   jvm0  H y    Hz   e  x  jve0  0    0 

          

 0     jbI   B  y

0 ey 0

0 Bx

0   Ex 

 0   ez 

A y   Ex    A x   Ey    0   Ez  

jb I

   Ey     Ez 

       

 0     jbI   D  y

(7.38)

C y   H x    C x   H y    0   H z  

jb I 0 Dx

where ( A x E z )i , j 

( A y E z )i , j 

Ez ,(i1, j )  Ez ,(i , j ) s

s E

2

(

y , i1, j 1

)

2

(

)

)

x

(

(

)

(

2

2

2

)

2

2

s

(

2

)

(

)

y , i 1 , j

2

)

y

(

z , i 1 , j 1 2

2

)

x

H 2

)

x , i+ 1 , j

H

(

2

x

)

x , i , j 1

z , i 1 , j 1

(

y , i , j 1

E

2

z , i 1 , j 1

(

)

2

y , i 1 , j 1

s H

E

2

2

s H

y

x , i 1 , j 1

x , i 1 , j1

(C x H z )i , j 

© 2006 by Taylor & Francis Group, LLC

(

y , i , j 1

s E

(C y H z )i , j 

2

Ez ,(i , j1)  Ez ,(i , j )

( B x E y )i , j 

( B y E x )i , j 

(

x , i 1 , j

)

(

z , i 1 , j 1

y

2

2

)

(7.39)

362

Electromagnetic Theory and Applications for Photonic Crystals

H ( D x H y )i , j =

(

y , i 1 , j 2

)

H

(

y , i 1 , j 2

)

sx ,(i , j ) x H

( D y H x )i , j 

(

x , i , j 1

2

)

H

(

x , i , j 1

2

)

s y ,(i , j ) y

with the subscript (i, j) in the left-hand side denoting the mesh number in the 2D computing domain and those in the right-hand side representing the corresponding grid (point) numbers as shown in Figure 7.2. After some mathematical work, (7.38) and (7.39) result in an eigenvalue equation in terms of the transverse electric fields   E   x    Ey   

P 

 P  xx

 

P  yx 

Pxy   Ex    Pyy  

       Ey   

 b2

  E   x      Ey   

(7.40)

where 1 −1 2 2 2 Pxx k 0 A x e z D y C x B y  (k0 I  A x e z D x )(e x  k0 C y B y ) 2 2 −1 2 1 Pyy k 0 A y e z D x C y B x  (k0 I  A y e z D y )(e y  k0 C x B x ) 2 −1 2 1 Pxy (I  k 0 A x e z D x )C y B x  A x e z D y (e y  k0 C x B x ) 2 −1 2 1 Pyx (I  k 0 A y e z D y )C x B y  A y e z D x (e x  k0 C y B y )

Alternatively, an eigenvalue matrix equation in terms of the transverse magnetic fields can be obtained as   H   x Q    Hy   



 Q  xx   Q  yx 

Qxy   H x 

  Qyy  

       Hy   

 b2

  H   x      Hy   

(7.41)

where 2 1 2 2 1 Qxx k 0 A x D y C x e z B y  (e y  k0 A x D x )(k0 I  C y e z B y ) 2 2 1 Qyy k02 Ay D xC y ez 1Bx  (ex  k 0 A y D y )( k0 I  C x e z B x ) 2 2 1 1 Qxy  A xD y (I  k 0 C x e z Bx )  (e y  k0 A x D x )C y e z B x 2 1 2 1 Qyx  A y D x (I + k 0 C y e z B y )  (e x  k0 A y D y )C x e z B y

One can see that (7.40) and (7.41) are quite similar to (7.21) and (7.22), respectively, except that more submatrices are needed for the computation including PML regions, which is due to the anisotropic property of the PML medium.

© 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

363

7.2.3 NUMERICAL TREATMENT AT THE DIELECTRIC INTERFACE A general situation for a curved dielectric interface in a mesh is shown in Figure 7.2. The conventional treatment for the dielectric interface in the FDM is to use either the graded-index approximation [25,26] or the staircase approximation [17,18,24]. The graded-index approximation is only suitable for situations when the difference of the dielectric constant is small, i.e., weakly guiding cases. As the index difference is large, e.g., n  2.34 between the semiconductor and the air, large errors will occur. In contrast, although the curved dielectric interface can be approximated by staircase approximation, which takes the interface BCs into account, much finer meshes are needed to approach the original structure. Improved formulations for oblique or curved dielectric interfaces were derived by [38,39] using the Taylor series expansion and matching the interface BCs. Higher order convergency was achieved, but more complicated derivations and the coordinate transformation are needed. A simple index averaging (IA) scheme has been shown to be useful and efficient in increasing the numerical accuracy [30,52–54], in which the values of the relative permittivity e at each grid point are determined by an averaging formula. For example, in Figure 7.2, ez(i, j) is defined as ez(i, j)  f  e1  (1  f )e2, where e1 and e2 are the relative permittivities of media 1 and 2, respectively, in the mesh, and f is the filling fraction of medium 1 in the mesh. The IA scheme can provide reasonably good numerical accuracy for most cases. When very high accuracy in the effective index or mode index (i.e., the propagation constant divided by the free-space wavenumber) is required, the following proper BC matching scheme can be employed [31]. It involves Taylor series expansions and uses interpolation and extrapolation to approximate the fields on both sides of the dielectric interface [29,42,55]. By properly matching the BCs at the dielectric interface, a modified FD formulation can be obtained. Figure 7.4 shows

EZ

HZ

Ey ,Hx

Ex ,Hy


1 j+1

EL

rx ∆x

n ER


2

j i

i+1

i +2

FIGURE 7.4 Dielectric interface lying between two sampled points. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

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Electromagnetic Theory and Applications for Photonic Crystals

an example with a dielectric interface lying between two sampled grid points with EL and ER representing the electric fields on the left-hand and right-hand sides of the interface, respectively, and n being the normal unit vector that points outward from medium 1 to medium 2. Using Taylor series expansions, Ey,L and Ey,(i, j12) can be expressed as E y, L  E

E

(

y,

i , j 1 2

)

(

y,

i 1 , j 1 2 2

E

(

y,

)

 rx x

i 1 , j 1 2 2

)



∂E y ∂x

 O[(x )2 ]

(7.42)

i 1 , j 1 2 2

 x ∂E y 2 ∂xx

 O[(x )2 ]

(7.43)

i 1 , j 1 2

2

where Ey,L is the y component of EL and rxx is the distance from the point (i  12, j  12) to the interface along the x direction. The first-order derivative of Ey at point (i  12, j  12) with respect to x becomes ∂E y ∂x

 i 1 , j 1 2

2

  2  E y , L  E y , i , j 1  ( 2 )  (2rx  1)x 

(7.44)

Considering the continuity BCs of the electric field   n  EL  n  E R   n ⋅ (e1EL )  n ⋅ (e2 E R ) at the dielectric interface, we have n y E x , L  nx E y , L  n y E x , R  nx E y , R

(7.45)

e1 (nx E x , L  n y E y , L )  e2 (nx E x , R  n y E y , R )

(7.46)

where nx and ny are the x and y components of the normal unit vector n. Eliminating Ex,R from (7.45) and (7.46) yields E y, L 

(e2  e1)nx n y e2 E  E y , R e1n y2  e2 nx2 e1n y2  e2 nx2 x , L

(7.47)

where Ey,R and Ex,L can be approximated by interpolation and extrapolation as 3   1 E y ,R    rx  E    rx  E  y ,( i1, j 12 )  2  y ,( i2, j 12 )  2

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(7.48)

Applications of the Finite-Difference Frequency-Domain Mode

Ex , L 

1  rx 2

 r   E x , i 1 , j1  E x , i 1 , j   x ( 2 )  2  ( 2 )

365

   E x , i 1 , j1  E x , i 1 , j  ( 2 )   ( 2 )

(7.49)

Substituting (7.48) and (7.49) into (7.47) and (7.44), we obtain the modified FD formula for ∂Ey /∂x, which can be used in (7.3). Similarly, for ∂Hz /∂x at point (i  1, j  12), we have ∂H z ∂x

 i1, j 1

2

  1  H z , i 3 , j 1  H z ,R   (rx  1)x  ( 2 2 )

(7.50)

The continuity BCs for the magnetic field   n  HL  n  HR   n ⋅ ( m0 H L )  n ⋅ ( m0 H R ) result in H z , R  H z, L  (1  rx )H

(

z , i 1 , j 1 2

2

)

 rx ⋅ H

(

z , i 1 , j 1 2

2

)

(7.51)

Following a similar procedure, we can also obtain the derivatives of the electric and magnetic fields with respect to y for the case with a dielectric interface lying in a mesh. We have demonstrated in [31] and [32] that very high accuracy in the effective index calculation for optical waveguides indeed can be achieved by utilizing the proper BC matching scheme. However, the simpler staircase approximation has been employed in obtaining the results to be presented in the next section, which are found to be good enough even for comparison with the results of other methods.

7.3 MODAL ANALYSIS OF PHOTONIC CRYSTAL FIBERS PCFs are novel photonic structures that represent one successful application of the PC concept [7,50,56]. In recent years, many research efforts have been devoted to understanding the propagation characteristics of such fibers based on different theoretical methods and aimed at demonstrating their possible applications. These methods include that based on orthogonal function expansion [57], the PWE method [11,12,58], the localized function method [59], the mutipole method [33,60], the FEM [10,15], the FDTD method [14], and the FDM [1,19]. Unlike planar PC waveguides having the direction of propagation parallel to the plane of the periodic structure, which leads to confinement only in one direction, the plane of the periodic structure in PCFs is perpendicular to the direction of light propagation,

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confining light in the 2D core region. In this section, we demonstrate that the FDFD method can be efficiently utilized to obtain the propagation characteristics, including the effective indices and field distributions, of different kinds of PCFs. Possible confinement losses can also be determined with the help of the PML ABCs.

7.3.1 HOLEY FIBERS We first analyze a PCF that has the radius of the core region being twice the pitch a, i.e., seven air holes are removed for forming the core region. In our numerical calculation, only a quarter of the PCF is considered for the symmetry of this structure and zero BCs are used at the right and top boundaries of the computing window. The size of the computing window is 4a  4a with 120  120 grid points, resulting in the matrix A in (7.21) with 28,800 elements. For other holey fibers in the rest of this section, the size of the computing window used will be smaller due to smaller core areas, and at least 30 grid points are adopted in each lattice distance. In Figure 7.5 we show the computed electric field distributions for the first six guided modes in this holey fiber at the wavelength l  1.0 m with a  2.3 m and the radius of the air hole r  0.2a. The index of the silica material is taken to be nSiO2  1.46. The propagation constants b and effective indices neff of the six modes are listed in Table 7.2. From the guided mode indices, it is observed that the TE01, TM01, and HE21 modes are degenerate, as are the EH11 and HE31 modes.

0

−5

0

−5 −5

0 x (µm) HE11

0 x (µm) TE01

5

−5

0

−5 −5

0 x (µm) HE21

5

−5

0 x (µm) TM01

5

−5

0 x (µm) HE31

5

5 y (µm)

y (µm)

5

0

0

−5 −5

5

5 y (µm)

5 y (µm)

5 y (µm)

y (µm)

5

0

−5 −5

0 x (µm) EH11

5

FIGURE 7.5 Electric field patterns of the first six guided modes in a holey fiber having the core radius twice the lattice distance with a  2.3 m, r  0.2a, and l  1.0 m. (After Yu and Chang, Opt. Quantum Electron., 36, 145–163, 2004, with kind permission of Springer Science and Business Media.)

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Applications of the Finite-Difference Frequency-Domain Mode

367

If the radius of the core area is reduced to a, the contour of the major electric field component of the fundamental x-polarized mode for a  2.3 m and r  0.3 m at the wavelength l  0.6328 m calculated using our FDFD method is shown in Figure 7.6(a). Indeed, the field is confined very well in the core region, for the air holes reduce the refractive index of the cladding region. The result for r  0.7 m is given in Figure 7.6(b). Compared with Figure 7.6(a), we can see that the field is more centrally located for larger air-hole radius, for it reduces the cladding index more, thus increasing the index difference between the core and cladding regions. The major magnetic field distributions along the x axis for l  0.6328, 1.0, and 1.55 m are plotted in Figures 7.7(a), 7.7(b), and 7.7(c), respectively. At shorter wavelengths, the field tends to concentrate in the silica region. A relatively larger field appears in the core region than in the air holes in Figure 7.7(a). At larger wavelengths, more field penetrates into the air holes and the cladding region, as shown in Figures 7.7(b) and 7.7(c). TABLE 7.2 Propagation Constants and Effective Indices of the First Six Guided Modes on the Holey Fiber with Lattice Distance a  2.3 m and Hole Radius r  0.2a. The Core Region Is Formed by Removing Seven Air Holes from the Center and Has the Radius 4.6 m B

neff

HE11 TE01 TM01 HE21 EH11 HE31

9.15490818 9.12734824 9.12703314 9.12709559 9.09333723 9.09338216

1.45704889 1.45266259 1.45261244 1.45262238 1.44724957 1.44725672

4

4

2

2 y (µm)

y (µm)

Mode

0 −2

0 −2

−4

−4 −4

−2

0 x (µm) (a)

2

4

−4

−2

0 x (µm)

2

4

(b)

FIGURE 7.6 (a) The major transverse field (Ex) contours of the fundamental guided mode for the holey fiber with a  2.3 m and r  0.3 m at l  0.6328 m. The circles represent the cross section of the air holes. (b) The same as (a) but for r  0.7 m.

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Electromagnetic Theory and Applications for Photonic Crystals

One peculiar feature of the PCF is that it can maintain as a single-mode guide over a wide wavelength range, called “endlessly single-moded” [8]. Figure 7.8(a) shows the mode index versus the normalized frequency va/2pc of the structure in Figure 7.6(a). Considering a traditional optical fiber with core radius r, core index nco, and cladding index ncd, the normalized frequency V defined as V

2pr l

2  n2 nco cd

(7.52)

must be less than 2.405 for the fiber to remain single moded. As for the PCF, V becomes V

2pa 2  n2 nco 0 l

(7.53)

where n0 is the effective index of the cladding region formed by air-hole arrays. As we have mentioned, at shorter wavelengths the fields concentrate more in the silica region, causing the increase in the effective index of the cladding region and

0.6 0.4 0.2 0

0

1

2

3 4 5 x (µm)

6

7

1.0

1.0

0.8

0.8

Field pattern

Air hole

0.8

Field pattern

Field pattern

1.0

0.6 0.4 0.2

8

0

0

1

2

(a)

3 4 5 x (µm)

6

7

0.6 0.4 0.2 0

8

(b)

0

1

2

3 4 5 x (µm)

6

7

8

(c)

FIGURE 7.7 The major magnetic field distributions along the x-axis at (a) l  0.6328 m, (b) l  1.0 m, and (c) l  1.55 m.

1.46

1.456 1.452

Core index

Core index

Effective cladding index

1.448

Mode index

Mode index

1.460

1.44

Effective cladding index

1.42

1.444 1.440

2 3 4 5 6 7 8 9 10 Normalized frequency (ωa/2πc) (a)

1.40

2 3 4 5 6 7 8 9 10 Normalized frequency (ωa/2πc) (b)

FIGURE 7.8 Modal dispersion curve for a holey fiber with (a) r  0.3 m and (b) r  0.7 m, and a  2.3 m. (After Yu and Chang, Opt. Quantum Electron., 36, 145–163, 2004, with kind permission of Springer Science and Business Media.)

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Applications of the Finite-Difference Frequency-Domain Mode

369

decreasing the index difference between the core and cladding regions in (7.53). This phenomenon helps to keep V smaller than 2.405 and makes the holey fiber single moded over a remarkably large band of frequency. In Figure 7.8(a), there x y are two linearly polarized modes, HE11 and HE11 , with different (but orthogonal) polarization states, which are too close to be distinguished from each other in the figure. Additionally, larger air holes also make the fiber likely multimoded. If we further increase the hole diameter, the gaps between the holes become narrower, isolating the core more strongly from the silica in the cladding. Thus, there will be more guided modes in the holey fiber as shown in Figure 7.8(b) with the same parameters as in Figure 7.6(b). The dispersion properties also can be obtained by calculating the effective index neff of the guided mode over a range of wavelength. The actual index of the pure silica is taken into account by means of four-term Sellmeier formulas [61,62]. The dispersion in ps/nm/km is derived according to the definition D 

l ∂ 2 neff c ∂l 2

(7.54)

which considers both the material and the waveguide dispersions. Figure 7.9 presents the dispersion characteristics of the holey fiber with the air-hole diameters being 0.5, 0.621, 0.75, and 1.0 m, while a is set to be 2.3 m. The figure shows that one can obtain positive, near zero, and negative dispersion just by adjusting the values of r at l  1.2 m. We can also see that the zero-dispersion point can be shifted easily to a desired wavelength by changing the geometrical parameters of the holey fiber. The flatness property of the dispersion curves over some frequency bands also makes the holey fibers useful in the communication applications. 100 2r = 0.1 µm

Dispersion (ps/nm/km)

50 0

2r = 0.75 µm 2r = 0.621 µm

−50

2r = 0.5 µm

−100 −150 −200 −250 −300 0.6

0.8

1.0

1.2

1.4

1.6

Wavelength (µm)

FIGURE 7.9 Dispersion calculated versus wavelength for holey fibers with different air-hole diameters.

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Electromagnetic Theory and Applications for Photonic Crystals

TABLE 7.3 Comparison of the Values of the Dispersion and the Slope of the Dispersion Obtained from the Experimental Results [9], by the FEM Method [10], and by Our FDFD Method as a  2.3 m and r  0.3105 m at L  0.813 m D (ps/nm/km) slope (ps/nm/km/nm)

Experiment

FEM

FDFD

77.7 0.464

78.6 0.450

76.4 0.473

Wx

d d

PML

Wy

PML

y 2r x a

FIGURE 7.10 The computing window with PMLs for the holey fiber with one-ring air holes. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

Table 7.3 lists the dispersion and its slope for the case with r  0.3105 m and l  0.813 m with the experimental values provided by [9], the results obtained by using the FEM [10], and our FDFD method. Both our results and those by the FEM have very good agreement with the experimental values, showing the reliability of our model. We now consider the lossy properties of a holey fiber having one-ring air holes (corresponding to six air holes) in the cladding region. The cross section of this holey fiber is illustrated in Figure 7.10 with the size of the computing window being Wx  Wy, and PMLs with thickness d are placed on the top and the right side of the window. In the following calculation, we adopt the parameters used in [63]: a  2.3 m; Wx  8.0 m; Wy  7.3 m; d  2 m; R  108 in (7.30). For this structure and those discussed in the rest of this section, nSiO2  1.45 is assumed. In [63], Saitoh and Koshiba proposed a finite-element

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Applications of the Finite-Difference Frequency-Domain Mode

371

1.44 1.43

r /a

1.42

= 0 r /a .25 = r /a 0.3 = 0 0. 35 .

Effective index neff

1.45

1.41

FEM [Saitoh and koshiba, 2002] FDFD method

1.40 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Loss (dB/m)

Wavelength (µm) (a) 10

5

10

4

10

3

102 10

1

10

0

.25 =0 a / r .30 =0 r /a .35 = 0. r /a FEM [Saitoh and koshiba, 2002] FDFD method

10

−1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Wavelength (µm) (b)

FIGURE 7.11 (a) Effective indices and (b) losses of the fundamental guided modes in the one-ring holey fiber with a  2.3 m. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

imaginary-distance BPM for studying PCFs. We consider the one-ring case with r/a  0.25, 0.3, and 0.35. The calculated effective indices and losses for l  0.1 to 2.0 m are illustrated in Figures 7.11(a) and 7.11(b), respectively. The effective index in this case decreases with the increase of the wavelength or the hole size. From the magnitude of the loss, we can see that larger air holes can provide better confinement resulting in smaller loss. In Figure 7.11, the solid lines are obtained by employing the FDFD method; they have good agreement with the circular dots representing the results from [63]. We have also analyzed the tworing case (with 18 air holes) and the three-ring case (with 36 air holes), and the same good agreement with [63] is achieved. These two cases have similar neff as the one-ring case, and the losses for the two-ring case are larger than those of the three-ring case and smaller than those of the one-ring case because more air holes

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Electromagnetic Theory and Applications for Photonic Crystals 6

6

4

4

2

2 y (µm)

y (µm)

372

0

0

−2

−2

−4

−4

−6 −6

−4

−2

0

2

4

6

−6 −6

−4

−2

0

x (µm)

x (µm)

(a)

(b)

2

4

6

FIGURE 7.12 Field patterns of (a) the x-polarized even mode and (b) the x-polarized odd mode in the two-core PCF.

can more efficiently restrict the penetration of the field into the cladding regions, yielding smaller losses.

7.3.2 TWO-CORE PCFS Multicore PCFs can be obtained by eliminating two or more air holes in the airhole array and thus forming multicore areas. There are several ways to fabricate two-core PCFs that function as fiber couplers. Here we consider the case with the coupled cores lying in the y-direction, shown as the background air-hole array in Figure 7.12, and the distance between the core centers being 3a. Light can be transmitted into one core and coupled to the other by proper design. Using our FDFD mode solver, the propagation characteristics of the guided modes on twocore PCFs can be easily obtained. Figures 7.12(a) and 7.12(b) show the contours of the x-polarized even and odd modes, respectively, with a  2.3 m, r/a  0.2, and l  1.5 m. The field distributions of these guided modes are similar to those on usual 2  2 fiber couplers. According to the coupled-mode theory [64], the relative coupled power, P, along the propagation direction z is P  sin 2

( ∫ C(z)dz )

(7.55)

where C is the coupling coefficient defined as C

beven  bodd 2

(7.56)

where beven and bodd are the propagation constants of the even and odd modes, respectively. Figure 7.13 shows the experimental results from [65] for the above twocore PCF with a  2.8 m and r  0.17a. The circular dots were the measured

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Applications of the Finite-Difference Frequency-Domain Mode

373

Relative coupled power

1.0

0.8

0.6

0.4

0.2

0 1.520

1.522

1.524 1.526 Wavelength (µm)

1.528

1.530

FIGURE 7.13 The relative coupled power versus the wavelength obtained by our model, plotted in solid line, in comparison to experimental results [65], which are the circular dots. (After Yu and Chang, Opt. Quantum Electron., 36, 145–163, 2004, with kind permission of Springer Science and Business Media.)

relative coupled powers at the output port at different wavelengths after the input field was propagated a distance of 885 mm in this PCF. The solid line is obtained by our model. Our simulation is quite close to the experimental results. Thus, our model can be used to predict the coupled power at the output port of a two-core PCF.

7.3.3 HONEYCOMB PCFS In the holey fibers, although we can observe good confinement in the core region, the guiding mechanism is not solely the PBGs. This can be proved by the endlessly single-moded property of the holey fiber. Even outside the PBGs, we can still find guided modes propagating in the holey fiber owing to the guiding mechanism of the effective total internal reflection caused by the periodically existing air holes in the cladding region. In this and the next subsection, we will discuss some PCFs with the guidance solely by the PBGs in the cladding region. In contrast to the effective total internal reflection guiding mechanism of the holey fibers, light guiding in the core area due to the existence of PBGs in the cladding region has been achieved in PCFs in which the core index was made smaller than the effective index of the cladding region [50] and even in PCFs with a large air core [56]. The inset of Figure 7.14 shows an example of such PCFs, called the honeycomb PCF [50], which has the honeycomb lattice PC in the cladding region and a smaller air-hole defect in the fiber center resulting in a decrease of the effective core index, with r/a  0.275 and the radius of the central air hole ri  0.175a.

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374

Electromagnetic Theory and Applications for Photonic Crystals 1.42

Rad

Effective index (/k0)

PB

ri = 0.125a

1.38

ry

nda

ou Gb

line iation

1.40

1.36 ri = 0.175a

ary

nd

1.34

G

PB

u bo

Core

1.32 1.30 ri = 0.225a

1.28

FDFD method PWE method FDTD method

1.26 5

6

7

8

9

10

11

12

13

14

Normalized propagation constant (a)

FIGURE 7.14 The fundamental guided mode index of the honeycomb PCF with its cross section shown as the inset calculated with different central hole sizes versus ba using different methods. (After Yu and Chang, Opt. Quantum Electron., 36, 145–163, 2004, with kind permission of Springer Science and Business Media.)

The fundamental guided mode effective index vs. the normalized propagation constant ba for this structure has been calculated using the PWE method [11] and the FDTD method [14], shown in Figure 7.14 as the black dots and triangles, respectively. The three solid lines in Figure 7.14 are our results using the FDFD model for three different central air hole sizes, ri  0.125a, 0.175a, and 0.225a. The middle line for ri  0.175a is to be compared with the results of other methods, which appears to be closer to those of the PWE method. The dashed and dotted lines in Figure 7.14 are, respectively, the boundaries of the PBGs and the radiation line obtained from the finger plot to be discussed in Section 7.5. In fiber structures, the radiation line could be regarded as the effective index of the cladding region. Because the effective index of the core region is smaller than that of the cladding region due to the appearance of the central air hole, we cannot find a guided mode that is guided by the effective total internal reflection above the radiation line. Under the radiation line, only the guided mode with the frequency lying in the PBG could be found. As we increase the size of the central hole, ri, the effective index difference between the core and the cladding will become larger, resulting in a larger downward shift of the solid line in Figure 7.14. The field distributions of the fundamental guided modes for ri  0.175a and k0a  14.5 and 10 are presented in Figures 7.15(a) and 7.15(b), respectively, with the white circles denoting the air holes. As in holey fibers, the guided mode field

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Applications of the Finite-Difference Frequency-Domain Mode

375

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 −0.2

−0.2 (a)

(b)

FIGURE 7.15 Field patterns of the fundamental guided modes on a honeycomb PCF at (a) k0a  14.5 and (b) k0a  10.

PML

PML

Loss (dB/m)

106 105

1-ring 2-ring

104

3-ring 4-ring

103 102 1.3

FDFD FEM 1.4

1.5

1.6

a =1.62 µm

Wavelength (µm)

(a)

(b)

1.7

FIGURE 7.16 (a) The computing window with PMLs for a honeycomb PCF with variant numbers of air-hole rings in the cladding with a  1.62 m and r  0.205a. (b) Losses vs. the wavelength of the guided modes with variant numbers of air-hole rings.

is more centrally concentrated at shorter wavelengths and penetrates more into the cladding region at larger wavelengths. To see the lossy properties of the honeycomb PCF, we consider the structures with 1-ring, 2-ring, 3-ring, and 4-ring air holes in the cladding region as shown in Figure 7.16(a) with a  1.62 m and ri  r  0.205a. At the edges of the computing window, the PMLs are adopted as indicated. The losses for the honeycomb PCFs with variant rings of air holes at different wavelengths are shown in Figure 7.16(b). One can see that the increase in the number of air-hole rings helps the confinement of light in the core region, which results in smaller losses than those with fewer air-hole rings. In Figure 7.16(b), the solid lines are our results obtained by adopting the FDFD method; they have a good agreement with the circular dots from the FEM method [66].

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Electromagnetic Theory and Applications for Photonic Crystals PML

PML

FIGURE 7.17 The computing window with PMLs for a hollow PCF with six rings of air holes in the cladding. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

7.3.4 HOLLOW PCFS Although the honeycomb PCF can be regarded as a PBG fiber with guidance by the PBGs, most of its guided field is concentrated in the center silica region as shown in Figure 7.15, resulting neff 1. Another example of PBG fibers is the hollow PCF, also called the vacuum, or air-core, PCF; its cross section is illustrated in Figure 7.17. When the guiding region is filled by air, this kind of PCF has the advantage of low propagation loss [67,68], small nonlinearity [69], and reduced material damage and field breakdown in the core region. Further, the core can also be filled with gases or liquids for some specified functions. Consider the case of Figure 7.17 with the cladding region formed by six rings of air holes with pitch a being 2 m and air filling fraction f in a unit cell being 0.7. The PMLs have d  2 m and R  108 in (7.30). Applying our FDFD method, the effective indices of the fundamental x-polarized guided modes for this hollow PCF are plotted in Figure 7.18(a) with the dashed lines representing the PBG boundaries. The computing window size is 17 m  15.2 m, and for a 170  152 grid division the computing time is 267 seconds for one wavelength using a Pentium IV 3.0 GHz personal computer. There are two modal dispersion curves with flat and steep slopes, respectively, that correspond to the fundamental

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Applications of the Finite-Difference Frequency-Domain Mode

377

1.01 Su

rfa

Effective index

ce

1.00

m

od

e

A 0.99

Core mode

C B D

0.98

Core mode

su

rfa

ce

mo

de

0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 Wavelength (µm) (a)

104

Surface mode

Loss (dB/m)

Core mode 103

102

101 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 Wavelength (µm) (b)

FIGURE 7.18 (a) Effective indices of the surface modes and fundamental core modes of the hollow PCF of Figure 7.17 with air filling fraction f  0.7. (b) Losses of the core modes and the surface modes. (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

core modes and surface modes [67,70–72] of the PCF. Figure 7.19 shows the field distributions of the guided modes along the modal dispersion curves as indicated in Figure 7.18(a). At points B and C, which are located on the flat curve representing the fundamental core modes, one can see that the stronger field (colored white) is located in the air core, with the weaker field (colored black) in the silica around the air core. The other kind of guided modes are the surface modes, labeled A and D, with most of the energy concentrated in the silica region around the air core. Figure 7.18(b) demonstrates the losses of these two kinds of modes within the frequency range of the PBG. Outside the PBG or near the edges of the PBG, the core modes have more field penetrating in the cladding, resulting in larger losses than those of the core modes in the central part of the PBG. As for the surface modes, the losses remain almost the same magnitude and are larger than those of the core modes at most frequencies.

© 2006 by Taylor & Francis Group, LLC

378

Electromagnetic Theory and Applications for Photonic Crystals (B)

(A)

1.0

1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8

(C)

1.0 0.8 0.6 0.4 0.2 0.0 −0.2

0.8 0.6 0.4 0.2 0.0 −0.2

(D)

1.0 0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8

FIGURE 7.19 The field distributions of the surface modes and fundamental core modes at points A, B, C, and D indicated in Figure 7.18(a). (After Yu and Chang, Opt. Express, 12, 6165–6177, 2004.)

Ideally, the hollow PCFs should be lossless due to the much smaller absorption and Rayleigh scattering of the air. However, the recently reported loss of a hollow PCF having a triangular lattice of air holes was 1.7 dB/km [68], which was not as good as in conventional single-mode fibers that have losses of 0.2 dB/km. Except the loss caused by the shape of the air cores or the number of the air-hole rings, some losses result from the coupling between the core modes, the surface modes, and the leaky cladding modes [70–72]. Although most of the core mode field is concentrated in the air core as shown in Figure 7.19, the overlap of field distribution of core modes and surface modes in the silica region around the air core couples the low-loss core modes to the surface modes having larger losses. Compared with the core modes, the surface modes have more overlap field with the leaky cladding modes in the cladding region, and thus easily couple to these modes, yielding the high loss in the hollow PCF. It has been reported that one can reduce the surface modes if the fiber core has fingers of silica protruding into the core region [71], and careful design of the air-core size can help produce a singlemode hollow PCF free of surface modes [72]. Cregan [56] proposed the first hollow PCF with the core region filled with air by removing seven silica glass capillary canes in the fabrication. The sketch of the cross section of this hollow PCF [56] is illustrated in Figure 7.20(a). For the small fusion degree of this fiber, many small air holes can be observed between the larger air holes, making this structure difficult to analyze. The PC structure in

© 2006 by Taylor & Francis Group, LLC

y/a

a

3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 x /a

(a)

(b)

y/a

Applications of the Finite-Difference Frequency-Domain Mode

379

3 2 1 0 −1 −2 −3 −3 −2 −1 0 1 2 3 x /a (c)

FIGURE 7.20 (a) Sketch of the cross section of the hollow fiber [56]. (b) The field distribution of the fundamental even mode and (c) that of the fundamental odd mode at l  0.62 m on the hollow PCF of [56] for f being 0.39.

the cladding had the pitch a  4.9 m, and the air filling fraction f reported by [56] was from 39% to 50%. In the experimental observation, this PCF had three transition bands centering at l  0.48 m, 0.62 m, and 0.81 m, which correspond to k0  13.09, 10.1, and 7.76 m1, respectively. The Hx (major field) contours of the lowest even and odd guided modes at l  0.62 m with a  4.9 m, having the same parameters as in [56], are shown in Figures 7.20(b) and 7.20(c), respectively. The calculated effective indices for these two modes are 0.998803 and 0.997123, respectively. The field is confined very well in the core area even when the index of the cladding is larger than the index of the core that is equal to 1. We have also calculated the guided modes of the same waveguide at l  0.81 m and 0.48 m corresponding to the other two transmission bands. The Hx contours of the lowest even and odd modes look almost like those in Figure 7.20 at these two wavelengths.

7.4 THE FDFD METHOD FOR ANALYSIS OF PHOTONIC BAND STRUCTURES Most applications of PCs are based on the existence of PBGs, which are quite sensitive to the geometry and the material of the devices. To help the design of PC devices, it is very important to develop a simulation tool to calculate the band structures. Many numerical methods have been studied to calculate the band structures of PCs. The most famous are the PWE method [43,44] and the FDTD method [46]. Also, Yang [47] followed the FDM [17] to develop an eigenvalue algorithm for the band structure analysis. In this section, we will use the FDFD method to develop an eigenvalue algorithm for the analysis of band structures of 2D PCs [49].

7.4.1 THE FDFD FORMULAE We first consider 2D PCs with the cross section shown in Figures 7.21(a) and 7.21(b) for the square lattice and the triangular lattice cases, respectively, with the

© 2006 by Taylor & Francis Group, LLC

380

Electromagnetic Theory and Applications for Photonic Crystals y a

x

y

a

x

unit cell r

r

(a)

(b)

unit cell

FIGURE 7.21 The cross-sectional view of a 2D PC and its unit cell with a being the lattice distance and r being the radius of the circles for (a) the square lattice and (b) the triangular lattice. (After Yu and Chang, Opt. Express, 12, 1397–1408, 2004.)

respective unit cell illustrated by dashed lines where a is the lattice distance and r is the radius of the circles, which can be either dielectric rods or air columns. Because the 2D PC is uniform along the z-direction and periodic in the transverse x-y plane, in the following discussion we only consider the in-plane propagation that has zero propagation constant in the z-direction, b  0, so that the wave modes in the PC are either TE or TM to z mode. 7.4.1.1 The TE Mode For the TE mode, we have only Hz, Ex, and Ey components, and Maxwell’s curl equations become  jvm0 mH z  jve0 eE x 

∂E y ∂x



∂E x ∂y

(7.57)

∂H z

(7.58)

∂y

jve0 eE y 

∂H z

(7.59)

∂x

where m is introduced as the relative permeability of the medium. The transverse plane is then discretized using Yee’s mesh for the TE mode as shown in Figure 7.22(a), and (7.57)–(7.59) become  ∂E   y ∂E x   jvm0 ( m z H z )|i 1 , j 1      ∂x ∂y  1 1 2 2 i , j 2

© 2006 by Taylor & Francis Group, LLC

2

(7.60)

Applications of the Finite-Difference Frequency-Domain Mode Ex(i+1/2, j+1) j+1

Ez(i, j+1)

j+1

381 Ez(i +1, j +1)

Hy(i+1/2, j+1)

Ey(i +1, j+1/2)

Ey(i, j+1/2)

Hx(i, j +1/2)

Hz(i+1/2, j+1/2)

Ez(i, j )

Ex(i+1/2, j)

j

Hx(i +1, j +1/2)

Hx(i+1/2, j)

Ez(i+1, j)

j i +1

i

i +1

i

(a)

(b)

FIGURE 7.22 2D Yee’s mesh for the analysis of band structures of 2D PCs. (a) TE and (b) TM modes. (After Yu and Chang, Opt. Express, 12, 1397–1408, 2004.)

jve0 (e x E x )|i 1 , j 

∂H z ∂y

2

(7.61) i 1 , j 2

jve0 (e y E y )|i , j 1  2

∂H z ∂x

(7.62) i , j 1 2

where mz, ex, and ey are the relative permeability or permittivity at the points of Hz, Ex, and Ey. Using the central difference scheme, the above equation can be written as E  jvm0 ( m z H z )|i 1 , j 1  2

(

y , i1, j 1

2

H

(

z , i 1 , j 1 2

2

(

y , i , j 1

2

)

)

H

(

z , i 1 , j 1

2

2

E −

(

)

x , i 1 , j1 2

E

(

x , i 1 , j

y

)

H

(

z , i 1 , j 1

2

2

2

)

H

(

z , i 1 , j 1 2

2

)

(7.63)

(7.64)

y

2

jve0 (e y E y )|i , j 1 

E

x

2

jve0 (e x E x )|i 1 , j 

)

2

)

(7.65)

x

which can be expressed in the matrix form as   m m 0 0   H z   0 U y  0 z       jv  0 e0 ex 0 0   Ex    Vy        0 e0 ey   Ey  Vx 0 0   

© 2006 by Taylor & Francis Group, LLC

  Ux   H z     0   Ex      0   Ey   

(7.66)

382

Electromagnetic Theory and Applications for Photonic Crystals

After some mathematical work, we have {A  k02 I}  H z  0

(7.67)

A mz 1{U y ex 1Vy  U x ey 1Vx }

(7.68)

where

Unlike the eigenvalue equation for waveguide modal analysis, the eigenvalue of (7.67) is the square of k0, which corresponds to the eigen frequency in the 2D PC. 7.4.1.2 The TM Mode For the TM mode, we have only Ez, Hx, and Hy components, and Maxwell’s curl equations become jve0 eEz 

∂H y ∂x

 jvm0 mH x 



∂H x ∂y

(7.69)

∂E z

(7.70)

∂y

 jvm0 mH y 

∂E z

(7.71)

∂x

Following the same derivation for the TE mode and using Yee’s mesh for the TM mode shown in Figure 7.22(b), we obtain  ∂H ∂H x   y jve0 (e z Ez )|i , j      ∂x ∂y  i, j  jvm0 ( m x H x )|i , j 1 

(7.72)

∂E z

2

∂y

(7.73) i , j 1 2

∂E  jvm0 ( m y H y )|i 1 , j  z ∂x 2

(7.74) i 1 , j 2

where mx, my, and ez are the relative permeability or permittivity at the points of Hx, Hy, and Ez. Applying the central difference scheme yields H jve0 (e z Ez )|i , j 

© 2006 by Taylor & Francis Group, LLC

(

y , i 1 , j 2

)

H x

(

y , i 1 , j 2

)

H 

(

x , i , j 1

2

)

H y

(

x , i , j 1

2

)

(7.75)

Applications of the Finite-Difference Frequency-Domain Mode

 jvm0 ( m x H x )|i , j 1 

383

Ez ,(i , j1)  Ez ,(i , j )

 jvm0 ( m y H y )|i 1 , j 

(7.76)

y

2

Ez ,(i1, j )  Ez ,(i , j )

(7.77)

x

2

which can be expressed in the matrix form as

jv

 e e  0 z    0     0 

0 m0 mx 0

     0    m0 my  

0

 E  z   H  x   H  y

          

 0 Vy     Uy 0  U 0  x

  Vx   Ez     0   H x      0   H y   

(7.78)

We can still have an eigenvalue equation in terms of Ez {B  k02 I}  Ez  0

(7.79)

1 B ez 1{Vy mx 1U y  Vx m y U x}

(7.80)

where

7.4.2 PERIODIC BOUNDARY CONDITIONS

FOR

2D PCS

Due to the periodic geometry, the field distribution in a PC should satisfy the Bloch theorem, i.e., the fields can be written in the Bloch form E(r )  e jkr e(r )

(7.81)

H(r )  e jkr h(r )

(7.82)

where e(r) and h(r) are periodic functions in space and k is the wavenumber vector. We can then have the Bloch conditions for a PC E(r  T)  e jkT E(r )

(7.83)

H(r  T)  e jkT H(r )

(7.84)

where T is the lattice vector of the PC. Thus, in the analysis of PCs, we only need to consider the unit cell as the computing domain along with the periodic boundary conditions (PBCs) resulting from the Bloch conditions. For the PC with square lattice as shown in Figure 7.21(a) with the unit cell indicated by a dashed line, the PBCs can be expressed as

© 2006 by Taylor & Francis Group, LLC

( x  a, y)  e jkx a ( x , y)

(7.85)

( x , y  a)  e jk ya ( x , y)

(7.86)

384

Electromagnetic Theory and Applications for Photonic Crystals

where is either the electric or magnetic field in the unit cell and kx and ky are the wavenumbers in the x- and y-directions, respectively. To calculate the band structures of 2D PCs, the values of kx and ky are needed to determine the PBCs. For the periodic property of PCs, only the irreducible Brillouin zone is considered. Along the irreducible Brillouin zone, we have a wavenumber vector k at each point, which returns the values of kx and ky. Substituting kx and ky into (7.85) and (7.86), which can be applied into (7.67) and (7.79), the eigen frequencies of the wave modes in the PCs can be obtained. As for the PC with triangular lattice shown in Figure 7.21(b), we have the PBCs  3a a   PBC1:  x  , y    e j ( kx  2 2   3a a   PBC2:  x  , y    e j ( kx  2 2 

3a / 2k y a / 2)

( x , y)

3a / 2+ k y a / 2)

( x , y)

PBC3: ( x , y  a)  e jk ya ( x , y)

(7.87)

(7.88) (7.89)

Note that (7.87) and (7.88) are sufficient for describing the BCs in an infinitely periodic triangular lattice and (7.89) can be derived from (7.87) and (7.88). PBC1, PBC2, and PBC3 are indicated at the corresponding sides of the unit cell, as shown in Figure 7.23(a). Since the unit cell for the triangular lattice is hexagonal, it is not conveniently fitted by Yee’s rectangular mesh. To overcome this problem, an FDTD scheme in a nonorthogonal coordinate system was proposed in [46]. By using the nonorthogonal mesh, the nonrectangular unit cell can be matched quite well, but a coordinate transformation and more complicated formulae are needed. In our method, we simply move the upper right triangle to the lower left and the lower right triangle to the upper left, as shown in Figure 7.23(b), generating a modified unit cell. The modified unit cell still obeys the same PBCs at its boundaries, as indicated in Figure 7.23(b), e.g., the upper portion of the left side and the lower portion PBC3

PBC2

PB C1

C1

PBC1

PB

C2 PB

PBC2

C2 PB

PBC1

PBC3

PBC3

PBC3

(a)

(b)

FIGURE 7.23 (a) The unit cell of the PC with triangular lattice and its corresponding PBCs. (b) The modified unit cell. (After Yu and Chang, Opt. Express, 12, 1397–1408, 2004.) © 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

385

of the right side, both labeled as PBC1, are related by the PBC of (7.87). Because it occupies a rectangular area, it is more easily fitted by the rectangular mesh.

7.5 CALCULATION OF 2D PC BAND DIAGRAMS 7.5.1 PCS WITH SQUARE LATTICES

Normalized frequency (ωa/2πc)

We first consider the square-arranged 2D PC formed by parallel alumina rods with relative permittivity e  8.9 and radius r  0.2a in the air. The cross-sectional view of the PC is shown in Figure 7.21(a) with the circles representing the dielectric rods. For the periodic geometry of the PC, only the unit cell of the PC as indicated by dashed lines in Figure 7.21(a) needs to be considered with the corresponding PBCs. To obtain the band diagrams of the PC, we have to find out the eigen frequencies of the PC by solving the eigenvalues from the eigenvalue matrix equations with the irreducible Brillouin zone of the square PCs shown in the inset of Figure 7.24(a). Each point along the boundary of the irreducible

FDFD method PWE method

1.0 0.8 0.6 0.4

M 0.2 0.0

 Γ

kx (ky = 0)

X

X ky (kx = /a) M

k x = ky

Γ

k x = ky

Γ

Normalized frequency (ωa/2πc)

(a) FDFD method

1.0

PWE method

0.8 0.6 0.4 0.2 0.0

Γ

kx (ky = 0)

X

ky(kx = /a) M

(b)

FIGURE 7.24 Band diagrams for the 2D PC shown in Figure 7.21(a) with r/a  0.2 and e  8.9. (a) TE mode and (b) TM mode. (After Yu and Chang, Opt. Express, 12, 1397–1408, 2004.) © 2006 by Taylor & Francis Group, LLC

386

Electromagnetic Theory and Applications for Photonic Crystals

Brillouin zone determines the values of kx and ky in the PBCs in (7.85) and (7.86) and thus the characteristic matrices in the eigenvalue equations (7.67) and (7.79). The eigen frequencies of this PC can then be obtained using the SIPM. Figure 7.24(a) shows the band diagram of the TE mode. For this structure, one can see that modes always exist in the transverse direction as the normalized frequency va/2pc varies from 0 to 0.8, and thus no PBG for TE modes is found. In contrast, as shown in Figure 7.24(b), no TM mode can be found for va/2pc falling between 0.323 and 0.444, corresponding to the existence of a PBG. In both Figures 7.24(a) and 7.24(b), the circular dots represent the results obtained using our FDFD algorithm with the IA scheme, and the solid lines represent those obtained using the MIT photonic-bands (MPB) package [73] based on the PWE method with 128  128 resolution. The circular dots match quite well with the solid lines for both polarizations, showing the reliability of our model. For Figures 7.24(a) and 7.24(b), 40 grid points are utilized in each lattice distance, and it takes just a few minutes to obtain these two figures on Pentium IV 2.0 GHz personal computers. Figure 7.25 demonstrates the calculated values and computational time for the TE fifth eigen frequency versus the number of grid points. The wavenumber vector k is fixed at the M point in the irreducible Brillouin zone, i.e., k  (p/a, p/a, 0). The number of grid points refers to the number of total grid points in the computing domain. It is seen that more grid points lead to better convergent numerical results but demand more computational time and computer memory.

7.5.2 PCS WITH TRIANGULAR LATTICES As for the 2D PC with triangular lattice, the band diagrams for both polarizations can be obtained similarly to our FDFD method. Referring to Figure 7.21(b), we

TE fifth eigen frequency

0.98

100

0.97 0.96

10

0.95

Time (s)

Normalized frequency (ωa/2πc)

0.99

0.94 0.93 0.92

1 0

500 1000 1500 2000 2500 3000 3500 4000 4500 Number of total grid points

FIGURE 7.25 The computational values and time for the TE fifth eigen frequency versus the number of grid points as k is at the M point.

© 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

387

consider the cross section of a 2D PC consisting of dielectric cylinders with r  0.2a and e  11.4 in the air. The unit cell of the 2D PC with triangular lattice is indicated by dashed lines in Figure 7.21(b). Since the unit cell of the 2D triangular PC is not rectangular, making it hard to be fitted by our rectangular mesh, the modified unit cell shown in Figure 7.23 is adopted to overcome this problem. Again, along the irreducible Brillouin zone shown as the inset in Figure 7.26(a), one can determine kx and ky in PBCs and the characteristic matrix to calculate the eigen frequencies. Figures 7.26(a) and 7.26(b) show the band diagrams for the TE and TM modes. Similar to the square lattice case, there is no PBG for the TE mode as shown in Figure 7.26(a). For the TM mode, a wide PBG appears as the normalized frequency falling between 0.281 and 0.452. The triangles indicate our results using the FDFD method with the modified unit cell and the IA scheme, and the solid lines indicate the results obtained by the MPB package

Normalized frequency (ωa/2πc)

0.8 0.7 0.6 0.5 0.4

PWE method FDTD method FDFD method

0.3

M K

0.2

Γ

0.1 0.0

Γ

M

K

Γ

(a)

Normalized frequency (ωa/2πc)

0.8 0.7 0.6 0.5 0.4 0.3 0.2

PWE method FDTD method FDFD method

0.1 0.0

Γ

M

K

Γ

(b)

FIGURE 7.26 Band diagrams of (a) the TE mode and (b) the TM mode for the 2D PC formed by triangular-arranged dielectric cylinders with r/a  0.2 and e  11.4 in the air. Our results (triangles) are compared with the results from the FDTD method (circles) and the PWE method (solid lines). (After Yu and Chang, Opt. Express, 12, 1397–1408, 2004.)

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388

Electromagnetic Theory and Applications for Photonic Crystals

Normalized frequency (ωa/2πc)

0.8 0.7 0.6 0.5 PWE method FDFD method

0.4 0.3 0.2 0.1 0.0

Γ

M

K

Γ

FIGURE 7.27 Band diagrams of the TE mode (dashed lines) and the TM mode (solid lines) for the 2D PC formed by triangular-arranged air columns with r/a  0.48 in the background material having e  13. Our results (circles) are compared with the results from the PWE method (lines).

based on the PWE method [73]. The circles represent the results from the FDTD method based on a nonorthogonal coordinate [46], which was proposed to deal with such a 2D PC with a nonrectangular unit cell. There is a visible difference between our results and those from the FDTD method, but fairly good agreement exists between our results and those of the PWE method. In the above 2D PCs with square and triangular lattices, we can only find the PBGs for the TM mode. For most PC applications, a complete PBG, a bandgap for both the polarizations, is needed. Consider the complemented structure shown as the inset in Figure 7.27, which is a 2D PC composed of triangular air-column arrays with the air-hole radius r  0.48a in the background material having e  13. The dashed and solid lines in Figure 7.27 represent the TE and TM modes, respectively, which are obtained by the PWE method [73], and the circular dots represent the results calculated by our FDFD model with the IA scheme along the irreducible Brillouin zone. Again, our results match quite well with the PWE method. We can also see that PBGs exist for both the TE and TM modes. The region where the PBGs of the TE and TM modes overlap, with the normalized frequency ranging from 0.429 to 0.517, is called the perfect PBG of the 2D PC.

7.5.3 FINGER PLOTS

OF

2D PCS

In some applications based on 2D PCs, such as in PCFs, the propagation direction is not parallel to the plane of the 2D PC. This is called out-of-plane propagation. In such cases, the propagation constant b is no more zero. Thus, band diagrams for b  0 need to be considered. When b  0, the existing modes in the 2D PC are no longer simple TE or TM modes but hybrid modes, i.e., the six components of the electromagnetic fields must be taken into account. Instead of using the

© 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

389

eigenvalue equations (7.67) and (7.79) derived for the TE and TM modes, respectively, (7.19) and (7.20) are employed with b set to a specified value. For example, substituting the electric fields in (7.20) into (7.19) yields   H   x     k02  Hy        H   z 

P

  H   x     H   y       H   z 

(7.90)

where  0   P   jbI   U  y

jb I 0 Ux

  0  U y   ex 1 0    U x   0 ey 1 0     0   0 0 ez 1    

 0    jbI   V  y

jbII 0 Vx

Vy   Vx   0  

Alternately, an eigenvalue equation in terms of electric field components can be obtained   E   x     k02  Ey        E   z 

  E   x      Q  Ey        E   z 

(7.91)

where

Q

 1 e  x    0     0 

0 ey 1 0

    0   ez 1  

0

 0    jbI   V  y

jb I 0 Vx

Vy   0   Vx   jbI   0   U y  

jb I 0 Ux

U y   U x   0  

The first case we consider is the 2D PC formed by triangularly arranged air holes in the silica with air filling fraction f being 0.45, which corresponds to r/a  0.352. The band diagram of this PC as ba  8.0 is illustrated in Figure 7.28(a). One can see that there is one PBG appearing between va/2pc  8.054 and 8.189, which is indicated as the gray region. If ba is set to be 10, the band diagram is shown in Figure 7.28(b). Compared with Figure 7.28(a), it can be observed that another PBG at lower frequency occurs. We can calculate the band diagrams for variant values of ba, and the edges of the PBGs vs. ba are plotted in Figure 7.29, which is the so-called “band edge diagram,” or “finger plot.” The regions between solid lines are the PBGs calculated by our FDFD model. For this PC, one can see that more PBGs appear for a larger ba. If this PC is employed as the cladding region of a waveguide, light with frequency lying in the

© 2006 by Taylor & Francis Group, LLC

Electromagnetic Theory and Applications for Photonic Crystals 9.0 8.5 8.0 7.5 7.0 6.5 6.0

a = 8 Γ

M

K

Γ

Normailzed frequency (ωa/2πc)

Normailzed frequency (ωa /2πc)

390

10.0 9.5 9.0 8.5 8.0 7.5

a =10 Γ

M

Γ

K (b)

(a)

FIGURE 7.28 The band diagrams of the 2D triangular air-hole PC with air filling fraction f  0.45 for (a) ba  8 and (b) ba  10. 12 Air filling fraction f = 0.45

Normalized frequency k0a

11

10

9

8 Air line

7

6

6

Radiation line

7

8 9 10 Normalized propagation constant a

11

12

FIGURE 7.29 The finger plot of the 2D triangular air-hole PC with f  0.45.

PBGs will be restricted to penetrate into the PC cladding. Thus, good confinement in the guiding region can be achieved. The radiation line in Figure 7.29 denotes the lowest order mode existing in this PC, and the dashed line denotes the air line representing b  k0, which corresponds to neff  1. In some PC waveguides, the guiding region is composed of air having a refractive index of 1. According to the waveguide theory, the guided modes in the air region must have neff slightly smaller than 1. To support these modes in the PC waveguides, the PC must have PBGs in which b/k0 1; i.e., these PBGs must extend into the lefthand side of the air line. We can see that several PBGs extend into the left side of the air line for this PC, which is suitable to be utilized for air-guiding structures.

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12 11

Air filling fraction f = 0.6

10 9 8 Air line 7

Radiation line

6 6 7 8 9 10 11 12 Normalized propagation constant a (a)

Normalized frequency k0a

Normalized frequency k0a

Applications of the Finite-Difference Frequency-Domain Mode 15 14 13 12 11 10 9 8 7

391

Air filling fraction f = 0.7

Air line Radiation line 7 8 9 10 11 12 13 14 15 Normalized propagation constant a (b)

FIGURE 7.30 The finger plots of the 2D triangular air-hole PC with (a) f  0.6 and (b) f  0.7.

For example, to guide light with l  1.55 m, the spacing of the air holes should be about 2 m, which uses the lowest PBG extending across the air line. Figures 7.30(a) and 7.30(b) show the finger plots with f  0.6 and 0.7, respectively. Larger f makes the PBGs move toward higher frequency, and more and wider PBGs appear to extend to the left side of the air line, which is more suitable for air-guiding structures.

7.6 MODAL ANALYSIS OF PLANAR PC WAVEGUIDES For their amazing property of trapping lightwaves by PBGs, PCs could be employed to create new means of light waveguiding. One of the remarkable applications is the planar PC waveguide with line defects. Instead of the effective total internal reflection in the conventional dielectric waveguides, the guiding mechanism in planar PC waveguides is the Bragg reflection due to the PBG effects, which offer a very high degree of control over the propagation of light. Although in real device applications the waveguide might be in the form of a slab perpendicular to the axes of the air columns or dielectric rods, it is essential to understand the propagation characteristics of the simpler planar waveguides. We first consider a planar PC waveguide that was also studied by Adibi et al. [6]. It was formed by introducing a line defect in a 2D PC as shown in Figure 3.31(a), which is uniform in the z-direction and periodic along the wave propagating direction. The waveguide has width d  0.375a where a is the lattice distance of the 2D PC, which is formed by circular air columns with the radius r being 0.45a in GaAs having e  12.96. The PBG for the TM mode lies in the normalized frequency range 0.220–0.252. One can imagine that the 2D PCs in the cladding regions function as PBG mirrors that prevent the wave from penetrating the cladding regions with its frequency within the PBG and reflect it into the core region. In our numerical calculation, only the “supercell” of the planar PC waveguide is needed for consideration due to the periodic geometry. Figure 3.31(b) shows

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Electromagnetic Theory and Applications for Photonic Crystals

N

PML

2r PBC

PBC

N

d

y z

x a

PML

(a)

(b)

FIGURE 7.31 (a) Cross section of a planar PC waveguide with a square array of air columns having d  0.375a, r  0.45a, and N  3 in the GaAs slab, and (b) the “supercell” of this waveguide with the applied BCs.

a supercell for such a structure that extends one period along the wave propagation direction and N periods in the transverse direction. The PBCs are imposed upon the two boundaries separated by one period, and the PML is considered for the other two. We utilize our FDFD model with the supercell to obtain the guided mode index and field distributions on the planar PC waveguide. By letting N  3, the dispersion diagram of the TM mode on such planar PC waveguide is plotted in Figure 7.32. A thin GaAs layer having thickness 0.5a is placed between the PBG mirrors and the PML on the top and bottom of the waveguide. The circular dots in Figure 7.32 shows the results from [6] using the FDTD method, and the solid lines show those obtained by our model. Both results match quite well with each other. We also can find other guided modes on this waveguide outside the frequency range of the PBG. Although these modes are not guided by the mechanism of the Bragg reflection, they still can be confined in the core region by the effective total internal reflection because the effective index of the cladding region is smaller than that of the core region due to the existence of the air holes. Figure 7.33 shows the Ez field patterns of the guided modes for the normalized frequencies being 0.285, 0.253, 0.233, and 0.2, respectively, with white and black denoting the positive and negative maxima of the field, respectively. For the case va/2pc  0.233 or 0.253, which falls in the PBG or at the edge of the PBG, the field is centrally well guided. If the normalized frequency is far from the PBG, as in the cases va/2pc  0.285 and 0.2, the confinement is not as good for the cases in the PBG, and more field penetrates into the cladding region due to weak effective index guiding, as can be observed in the figure. If the effective index of the core region is made smaller than that of the cladding region, the guidance by the effective total internal reflection will be no

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Applications of the Finite-Difference Frequency-Domain Mode

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0.36

Normalized frequency (a/2c)

0.34 FDFD method FDTD method

0.32 0.30 0.28 0.26 0.24

PBG

0.22 0.20 0.18 0.16 0.0

0.5

1.0 1.5 2.0 Normalized phase kx a

2.5

3.0

FIGURE 7.32 The TM guided mode dispersion curve for the planar PC waveguide of Figure 7.31. a/2c = 0.285

0.253

0.233

0.2

FIGURE 7.33 The guided mode field distributions for the same waveguide of Figure 7.31 at va/2pc  0.285, 0.253, 0.233, and 0.2.

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Electromagnetic Theory and Applications for Photonic Crystals

2r

a

d

FIGURE 7.34 Cross section of a planar PC waveguide having the cladding composed of dielectric rods with radius r and e  11.4.

longer possible. The only way to confine the fields in the core region for such planar PC waveguides is to make use of the PBGs. One example of such PBG waveguides is shown in Figure 7.34. The waveguide is formed by periodic dielectric rods in air and a line defect in the center forming the air-core region with d  a, r  0.375a, and e  11.4, as studied by Yasumoto and Jia [16]. They provided an accurate analysis of such structures based on the lattice sums technique combined with the T-matrix approach for scattering from a cylindrical object. The dispersion curves of the TE and TM modes for this structure are presented in Figures 7.35(a) and 7.35(b), respectively. The solid lines denote those reported in [16], and the square dots denote our results calculated by the FDFD model with PML. Good agreement has been achieved in both Figures 7.35(a) and 7.35(b). Because light cannot be guided by the effective total internal reflection on this waveguide, there does not exist any guided mode outside the PBG. This is why we can only find the even mode over a specified range of frequency for the TE mode as shown in Figure 7.35(a). The same phenomenon appears in the TM case shown in Figure 7.35(b). Both the even and odd modes exist on such waveguides only in a specified frequency band. For this computation, our results are obtained by letting N  10 and adopting 30 grid points in each lattice, where N is the number of dielectric layers in both the upper and lower PBG mirrors. Figure 7.36 shows the relationship between N and the relative error in the normalized frequency for the TM even mode as the normalized phase kxa/2p is 0.35. The relative error is

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Normalized phase (kxa/2)

Applications of the Finite-Difference Frequency-Domain Mode

395

TE mode

0.3

Even mode 0.2

0.1 FDFD method Yasumoto and Jia, 2002

0.0 0.68

0.69

0.70

0.71

0.72

0.73

0.74

Normalized frequency (ωa/2πc) (a) Normalized phase (kx a/2)

0.5 0.4

TE mode Even mode

0.3 0.2 0.1

Odd mode FDFD method Yasumoto and Jia, 2002

0.0 0.72 0.73 0.74 0.75 0.76 0.77 0.79 0.78 Normalized frequency (ωa/2πc) (b)

FIGURE 7.35 Comparison of the modal dispersion with the results from Yasumoto and Jia [16] for the planar PC waveguide formed by dielectric rods with r  0.375a, e  11.4, and d  a. (a) TE mode. (b) TM even and odd modes.

defined as the difference from the calculated mode index with N  20. Since the guiding mechanism for such a waveguide is only PBG reflection, N must be chosen large enough to produce a PBG effect so that a convergent result can be achieved. The relative error decreases to about 6  109 when N  7 in our computation. Yasumoto and Jia [16] took N  32, and we use 10 layers to obtain similar results, as shown in Figures 7.35(a) and 7.35(b).

7.7 CONCLUSION In this chapter, we have demonstrated that the FDFD method is an efficient method for studying photonic crystal fibers and planar photonic crystal waveguides. Vector guided modes of various PCFs, including holey fibers with small

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Electromagnetic Theory and Applications for Photonic Crystals 10−4

Relatived error

10−6

10−8

10−10

10−12

2

3

4

5 6 7 8 Number of periods N

9

10

FIGURE 7.36 The convergence of simulation versus the number of periods N for the structure of Figure 7.34.

core and large core; two-core holey fibers; and PCFs resulting from a pure PBG mechanism, such as the honeycomb PCF and air-core PCF, have been successfully analyzed using the full-vector FDFD mode solver. The effective indices, the mode field distributions, and the dispersion characteristics of these modes have been accurately obtained. We have incorporated the PML ABCs into the FDFD mode solver so that the leaky properties or the confinement losses of the PCF modes can be calculated, and very good agreement with the FEM analysis [63,66] has been demonstrated. We have also described the proper BC matching scheme across the dielectric interface. Including this in the solver can help to achieve very high accuracy in the calculation of the effective index if necessary. We have also shown that the similar FDFD scheme over 2D space can be applied to solve the TE and TM waves in 2D PCs and planar PC waveguides resulting from line defects, as long as suitable periodic boundary conditions are imposed. Our FDFD calculation of the band diagrams of 2D PCs is shown to agree very well with the PWE calculation. For the nonrectangular unit cell in triangular-lattice PCs, a modified unit cell of rectangular shape is introduced for easier numerical treatment. Our FDFD modal analysis of the planar PC waveguide having multilayered periodic arrays of either air columns or circular dielectric rods as its claddings agrees with the analysis using the FDTD method [6] and other analytical approaches [16]. We have also successfully applied the FDFD algorithm to the determination of band diagrams in the situation of out-of-plane propagation in the 2D PC so that the band edge diagrams, or finger plots, useful for designing PBG fibers can be constructed.

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ACKNOWLEDGMENTS This work was supported in part by the National Science Council of the Republic of China under grants NSC90-2215-E-002-028, NSC90-2215-E-002-040, NSC91-2215-E-002-020, NSC91-2215-E-002-031, NSC92-2215-E-002-004, and NSC92-2215-E-002-008 and in part by the Ministry of Economic Affairs of the Republic of China under grant 91-EC-17-B-08-S1-0015.

REFERENCES [1] C.P. Yu and H.C. Chang, Applications of the finite difference mode solution method to photonic crystal structures, Opt. Quantum Electron., 36, 145–163, 2004. [2] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059–2062, 1987. [3] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett., 58, 2486–2489, 1987. [4] A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, High transmission through sharp bends in photonic crystal waveguides, Phys. Rev. Lett., 77, 3787–3790, 1996. [5] T. Baba, N. Fukaya, and J. Yonekura, Observation of light propagation in photonic crystal optical waveguides with bends, Electron. Lett., 35, 654–655, 1999. [6] A. Adibi, Y. Xu, R.K. Lee, A. Yariv, and A. Scherer, Properties of the slab modes in photonic crystal optical waveguides, J. Lightwave Technol., 18, 1554–1564, 2000. [7] J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, All-silica single-mode optical fiber with photonic crystal cladding, Opt. Lett., 21, 1547–1549, 1996. [8] T.A. Birks, J.C. Knight, and P.St.J. Russell, Endlessly single-mode photonic crystal fiber, Opt. Lett., 22, 961–963, 1997. [9] M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C. Knight, and P.St.J. Russell, Experimental measurement of group velocity dispersion in photonic crystal fibre, Electron. Lett., 35, 63–64, 1999. [10] F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method, Opt. Fiber Technol., 6, 181–191, 2000. [11] S.E. Barkou, J. Broeng, and A. Bjarklev, Silica-air photonic crystal fiber design that permits waveguiding by a true photonic bandgap effect, Opt. Lett., 24, 46–48, 1999. [12] A. Ferrando, E. Silvestre, J.J. Miret, P. Andrés, and M.V. Andrés, Full-vector analysis of a realistic photonic crystal fiber, Opt. Lett., 24, 276–278, 1999. [13] H. Benisty, Modal analysis of optical guides with two-dimensional photonic bandgap boundaries, J. Appl. Phys., 79, 7483–7492, 1996. [14] M. Qiu, Analysis of guided modes in photonic crystal fibers using the finitedifference time-domain method, Microwave Opt. Technol. Lett., 30, 327–330, 2001. [15] M. Koshiba and K. Saitoh, Numerical verification of degeneracy in hexagonal photonic crystal fibers, IEEE Photon. Technol. Lett., 13, 1313–1315, 2001. [16] K. Yasumoto and H. Jia, Modal analysis of two-dimensional photonic crystal waveguides using lattice sums technique. 3rd International Conference on Microwave and Millimeter Wave Technology Proceedings, 2002, 1063–1066.

© 2006 by Taylor & Francis Group, LLC

398

Electromagnetic Theory and Applications for Photonic Crystals

[17] P. Lüsse, P. Stuwe, J. Schüle, and H.G. Unger, Analysis of vectorial mode fields in optical waveguides by a new finite difference method, J. Lightwave Technol., 12, 487–494, 1994. [18] G.R. Hadley and R.E. Smith, Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions, J. Lightwave Technol., 13, 465–469, 1995. [19] C.P. Yu and H.C. Chang, Finite difference modal analysis of photonic crystal fibers, Proceedings of 2001 Progress in Electromagnetics Research Symposium (PIERS 2001), Osaka, Japan, 2001, 432. [20] H.S. Sözüer, J.W. Haus, and R. Inguva, Photonic bands: convergence problems with the plane-wave method, Phys. Rev. B, 45, 13,962–13,972, 1992. [21] M.J. Robertson, S. Ritchie, and P. Dayan, Semiconductor waveguides: analysis of optical propagation in single rib structures and directional couplers, Inst. Elec. Eng. Proc. J., 132, 336–342, 1985. [22] M.S. Stern, Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles, Inst. Elec. Eng. Proc. J., 135, 56–63, 1988. [23] C. Vassallo, Improvement of finite difference methods for step-index optical waveguides, Inst. Elec. Eng. Proc. J., 139, 137–142, 1992. [24] K. Bierwirth, N. Schulz, and F. Arndt, Finite-difference analysis of rectangular dielectric waveguide structures, IEEE Trans. Microwave Theory Tech., 34, 1104–1114, 1986. [25] C.L. Xu, W.P. Huang, M.S. Stern, and S.K. Chaudhuri, Full-vectorial mode calculations by finite difference method, Inst. Elec. Eng. Proc. J., 141, 281–286, 1994. [26] W.P. Huang, C.L. Xu, S.T. Chu, and S.K. Chaudhuri, The finite-difference vector beam propagation method: analysis and assessment, J. Lightwave Technol., 10, 295–305, 1992. [27] C.L. Xu, W.P. Huang, and S.K. Chaudhuri, Efficient and accurate vector mode calculations by beam propagation method, J. Lightwave Technol., 11, 1209–1215, 1993. [28] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., AP-14, 302–307, 1966. [29] T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface, J. Lightwave Technol., 20, 1627–1634, 2002. [30] Z. Zhu and T.G. Brown, Full-vectorial finite-difference analysis of microstructured optical fibers, Opt. Express, 10, 853–864, 2002. [31] H.C. Chang and C.P. Yu, Yee-mesh-based finite difference eigenmode analysis algorithms for optical waveguides and photonic crystals, OSA Integrated Photonics Research Digest, San Francisco, 2004, paper IFE4. [32] C.P. Yu and H.C. Chang, Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers, Opt. Express, 12, 6165–6177, 2004. [33] T.P. White, R.C. McPhedran, C.M. de Sterke, L.C. Botten, and M.J. Steel, Confinement losses in microstructured optical fibers, Opt. Lett., 26, 1660–1662, 2001. [34] J.P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 185–200, 1994. [35] W.C. Chew and W.H. Weedon, A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, Microwave Opt. Technol. Lett., 7, 599–604, 1994. © 2006 by Taylor & Francis Group, LLC

Applications of the Finite-Difference Frequency-Domain Mode

399

[36] Ü. Pekel and R. Mittra, An application of the perfectly matched layer (PML) concept to the finite element method frequency domain analysis of scattering problem, IEEE Microwave Guided Wave Lett., 5, 258–260, 1995. [37] C.M. Rappaport, Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space, IEEE Microwave Guided Wave Lett., 5, 90–92, 1995. [38] Y.P. Chiou, Y.C. Chiang, and H.C. Chang, Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices, J. Lightwave Technol., 18, 243–251, 2000. [39] Y.C. Chiang, Y.P. Chiou, and H.C. Chang, Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles, J. Lightwave Technol., 20, 1609–1618, 2002. [40] G.R. Hadley, High-accuracy finite-difference equations for dielectric waveguide analysis I: uniform regions and dielectric interfaces, J. Lightwave Technol., 20, 1210–1218, 2002. [41] G.R. Hadley, High-accuracy finite-difference equations for dielectric waveguide analysis II: dielectric corners, J. Lightwave Technol., 20, 1219–1231, 2002. [42] A. Ditkowski, J.S. Hesthaven, and C.H. Teng, Modeling dielectric interfaces in the FDTD-method: a comparative study, 2000 Progress in Electromagnetics Research Proceedings (PIERS 2000), Cambridge, MA, 2000. [43] R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, Existence of a photonic band gap in two dimensions, Appl. Phys. Lett., 61, 495–497, 1992. [44] J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystal: Modeling the Flow of Light., Princeton University Press, Princeton, NJ, 1995. [45] C.T. Chan, Q.L. Yu, and K.M. Ho, Order-N spectral method for electromagnetic waves, Phys. Rev. B., 51, 16,635–16,642, 1995. [46] M. Qiu and S. He, A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metalic inclusions, J. Appl. Phys., 87, 8268–8275, 2000. [47] H.Y.D. Yang, Finite difference analysis of 2-D photonic crystals, IEEE Trans. Microwave Theory Tech., 44, 2688–2695, 1996. [48] L. Shen, S. He, and S. Xiao, A finite-difference eigenvalue algorithm for calculating the band structure of a photonic crystal, Comp. Phys. Commun., 143, 213–221, 2002. [49] C.P. Yu and H.C. Chang, Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals, Opt. Express, 12, 1397–1408, 2004. [50] J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, Photonic band gap guidance in optical fibers, Science, 282, 1476–1478, 1998. [51] D.S. Katz, E.T. Thiele, and A. Taflove, Validation and extension to three dimensionals of the Berenger PML absorbing boundary condition for FD-TD meshes, IEEE Microwave Guided Wave Lett., 4, 268–270, 1994. [52] Y.F. Li, K. Iizuka, and J.W.Y. Lit, Equivalent-layer method for optical waveguides with a multiple-quantum-well structure, Opt. Lett., 17, 273–275, 1992. [53] S. Ohke, T. Umeda, and Y. Cho, Equivalent-layer method for optical waveguides with a multiple-quantum-well structure: comment, Opt. Lett., 18, 1870, 1993. [54] M. Saini and E.K. Sharma, Equivalent refractive index of MQW waveguides, IEEE J. Quantum Electron., 32, 1383–1390, 1996. [55] K.H. Dridi, J.S. Hesthaven, and A. Ditkowski, Staircase-free finite-difference time-domain formulation for general materials in complex geometries, IEEE Trans Antennas Propagat., 49, 749–756, 2001. © 2006 by Taylor & Francis Group, LLC

400

Electromagnetic Theory and Applications for Photonic Crystals

[56] R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, Single-mode photonic band gap guidance of light in air, Science, 285, 1537–1539, 1999. [57] T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, Holey optical fibers: an efficient modal model, J. Lightwave Technol., 17, 1093–1102, 1999. [58] J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, Photonic crystal fibers: a new class of optical waveguides, Opt. Fiber Technol., 5, 305–330, 1999. [59] D. Mogilevtsev, T.A. Birks, and P.St.J. Russell, Localized function method for modeling defect modes in 2-D photonic crystals, J. Lightwave Technol., 17, 2078–2081, 1999. [60] M.J. Steel, T.P. White, C. Martijn de Sterke, R.C. McPhedran, and L.C. Botten, Symmetry and degeneracy in microstructured optical fibers, Opt. Lett., 26, 488–490, 2001. [61] I.H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Amer., 55, 1205–1209, 1965. [62] B. Brixner, Refractive-index interpolation for fused silica, J. Opt. Soc. Amer., 57, 674–676, 1967. [63] K. Saitoh and M. Koshiba, Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers, IEEE J. Quantum Electron., 38, 927–933, 2002. [64] A. Yariv, Optical Electronics in Modern Communications, 5th ed., Oxford University Press, New York, 1997. [65] B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, and A.H. Greenaway, Experimental study of dual-core photonic crystal fibre, Electron. Lett., 36, 1358–1359, 2000. [66] D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, Leakage properties of photonic crystal fibers, Opt. Express, 10, 1314–1319, 2002. [67] C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Allan, and K.W. Koch, Low-loss hollow-core silica/air photonic bandgap fibre, Nature, 424, 657–659, 2003. [68] B.J. Mangan, L. Farr, A. Langford, P.J. Roberts, D.P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea, and H. Sabert, Low loss (1.7 dB/km) hollow core photonic bandgap fiber, 2004 Optical Fiber Communication (OFC’ 04), Los Angeles, CA, 2004, postdeadline paper PDP24. [69] G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, Hollow core photonic crystal fibers for beam delivery, Opt. Express, 12, 1477–1484, 2004. [70] K. Saitoh, N.A. Mortensen, and M. Koshiba, Air-core photonic band-gap fibers: the impact of surface modes. Opt. Express, 12, 394–400, 2004. [71] J.A. West, C.M. Smith, N.F. Borrelli, D.C. Allan, and K.W. Koch, Surface modes in air-core photonic band-gap fibers, Opt. Express, 12 1485–1496, 2004. [72] H.K. Kim, J. Shin, S. Fan, M.J.F. Digonnet, and G.S. Kino, Designing air-core photonic-bandgap fibers free of surface modes, IEEE J. Quantum Electron., 40, 551–556, 2004. [73] S.G. Johnson and J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis, Opt. Express, 8, 173–190, 2001.

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Time8 Finite-Difference Domain Method Applied to Photonic Crystals Hiroyoshi Ikuno and Yoshihiro Naka

CONTENTS 8.1 Introduction ............................................................................................401 8.2 Method of Solution ..................................................................................402 8.2.1 Formulation of Problem ..............................................................402 8.2.2 Numerical Dispersion and Numerical Errors ............................411 8.3 Photonic Crystal Straight Waveguide ......................................................415 8.3.1 Numerical Results ......................................................................416 8.3.2 Floquet Mode ..............................................................................417 8.3.3 Dispersion Relation ....................................................................420 8.4 Fundamental Optical Circuit Devices Using Photonic Crystals ............423 8.4.1 Directional Coupler ....................................................................423 8.4.2 Sharply Bent Waveguide with Microcavity ................................427 8.5 Wavelength Multi/Demultiplexer ............................................................438 8.5.1 Design Parameters ......................................................................439 8.5.2 Numerical Results ......................................................................440 8.6 Conclusion ..............................................................................................441 References ........................................................................................................443

8.1 INTRODUCTION The photonic crystal [1,2] is a key artificial material for realizing high density integrated optical circuits. By using the strong confinement of the light by the photonic bandgap, it is expected that waveguide devices whose size is the order of the wavelength of light can be realized. In fact, several microscale photonic crystal optical waveguide devices have been proposed [3–6]. It is important for designing high density integrated optical devices to clarify fundamental properties of basic photonic crystal waveguides, such as the straight waveguide, right-angle bend, directional coupler, and so on [5,6]. Furthermore, it is necessary to evaluate 401

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Electromagnetic Theory and Applications for Photonic Crystals

the fundamental properties of the whole system by considering the interaction of each device. In order to model open-region electromagnetic wave interaction on largescale complicated structures such as photonic crystal waveguide devices, a numerical method requires stability and robustness. One of the most promising techniques for the Maxwell solver is the finite-difference time-domain (FDTD) method based on the principles of multidimensional wave digital filters (MD-WDFs) [7,8] because this method can be implemented easily and its algorithm is simple, stable, and suitable for massive parallel computation. Moreover, it has an ideal absorbing boundary condition (ABC) at the edge of the computational window called perfectly matched layers (PMLs) [9]. The finite difference schemes employed on the FDTD method for the dicretization of Maxwell’s equations is the trapezoidal rule. Numerically, the trapezoidal rule has several advantages over the midpoint rule that is used in the conventional FDTD method, such as numerical stability, robustness, and so on [5,7,8,10]. In addition, the FDTD method employed here can treat arbitrary structures under arbitrary incident waves. We show the formulation of an FDTD method based on the principles of MD-WDFs and its application to photonic crystal optical waveguide devices. This FDTD modeling is very effective for estimating the propagation characteristics of the waveguide, such as its propagation constant, field profile, propagation loss, and so on in a straightforward manner. In the first section we show an algorithm of the FDTD method with the PMLs based on the MD-WDFs for discrete-time modeling of Maxwell’s equations. Next, we apply this method to photonic crystal waveguide devices and clarify their fundamental properties, the propagation constants and field profiles of straight waveguides. Then several circuit devices such as directional couplers and sharply bent waveguides are analyzed, and their propagation characteristics are shown. In the final section, we propose and design a wavelength multiplexer/demultiplexer using directional couplers and right-angle bent waveguides.

8.2 METHOD OF SOLUTION 8.2.1 FORMULATION

OF

PROBLEM

Here we formulate the FDTD method based on the multidimensional wave digital filters (MD-WDFs) for the three-dimensional problem where we restrict the computational zone to a finite region by using the PMLs absorbing boundary condition. This method based on the MD-WDFs uses the trapezoidal rule as a numerical integration method. Although the trapezoidal rule is an implicit formula, we can obtain an explicit relation introducing wave quantities [7,8] occurring in relation with the scattering-matrix formalism. It is noted that the discretized fields are arranged at the same grid point in this formulation. Here we show a modified form of Maxwell’s equations in the PMLs and derive their MD-WDFs representation where current-controlled voltage sources [11] play an important role.

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8.2.1.1 Transformation of Maxwell’s Equations According to [8,12] we transform the physical system to the causal system. The coordinate transformation p′  vt  p, where t is time, p is the original spatial coordinates, p′ is the auxiliary coordinates, and v is a constant having the dimension of velocity, is applied because passivity holds only with respect to time, not with respect to spatial coordinates [12]. The transformed Maxwell’s equations for the MD-WDFs in the physical region are as follows:

(e′  4)

(e′  4)

∂( H z  E x ) ∂( H z  E x ) ∂E x 2 2 ∂(vt  y) ∂(vt ) ∂(vt  y)  ∂( H y  E x ) ∂( H y  E x ) 2 2  2 Jx  0 ∂(vt  z ) ∂(vt  z ) ∂E y ∂(vt )

2

∂( H x  E y )

2

(e′  4)

∂E z ∂(vt )

2 2

(e′′  4)

(e′′  4)

z

2

∂(vt  y)

y

∂(vt  x )

 2 J y  0

(8.1b)

 2 Jz  0

(8.1c)

∂( H y  Ez )

2

∂(vt  x ) ∂( H x  Ez ) ∂(vt  y)

∂( Ez  H x ) ∂( Ez  H x ) ∂H x 2 2 ∂(vt  y) ∂(vt ) ∂(vt  y) ∂( E y  H x ) ∂( E y  H x ) 2 2 0 ∂(vt  z ) ∂(vt  z) ∂H y ∂(vt )

2 2

(e′′  4)

∂(vt  x ) ∂( H x  Ez )

∂(vt  z ) ∂( H  E )

2

∂(vt  x ) ∂( H y  Ez )

∂( H x  E y )

2

∂(vt  z ) ∂( H z  E y )

∂H z ∂(vt )

2 2

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∂( E x  H y ) ∂(vt  z ) ∂( Ez  H y ) ∂(vt  x ) ∂( E y  H z ) ∂(vt  x )

∂( E x  H z ) ∂(vt  y)

(8.1a)

2 2

2 2

(8.1d)

∂( E x  H y ) ∂(vt  z ) ∂( E  H ) z

y

∂(vt  x)

0

(8.1e)

0

(8.1f)

∂( E y  H z ) ∂(vt  x )

∂( E x  H z ) ∂(vt  y)

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Electromagnetic Theory and Applications for Photonic Crystals

In (8.1) we introduce an auxiliary positive constant r0 [8] and define Hk, Jk (k  x, y, z), e′, and e″ by H k  r0 Hk , Jk  r0 Jk (k  x , y, z ),

e′  2r0 ve, e′′  2r01vm

(8.2)

where v is a constant having the dimension of a velocity and Jk (k  x, y, z) is electric current density. Next we derive the transformed form of Maxwell’s equations in the PMLs region. Prior to deriving the similar Maxwell’s Equations (8.1) in the PMLs region, we must modify Maxwell’s equations from J.P. Berenger’s original form [13] to obtain an appropriate form for the MD-WDFs representation. It follows from the same transformation in the physical region that we have

(e′  4)

(e′  4)

∂( H z  E x ) ∂( H z  E x ) ∂E x 2 2 ∂(vt ) ∂(vt  y) ∂(vt  y)  ∂( H y  E x ) ∂( H y  E x )  ex  0 2 2 ∂(vt  z ) ∂(vt  z ) ∂E y ∂(vt )

2

∂( H x  E y ) ∂(vt  z )

2

(e′  4)

∂E z ∂(vt )

2 2

(e′′  4)

(e′′  4)

∂(vt  x ) ∂( H x  Ez )

∂(vt  z )

∂( H z  E y )

2

∂(vt  x ) ∂( H y  Ez )

∂( H x  E y )

2

∂( H z  E y )

2

∂(vt  y)

∂(vt  x )

 ey  0

(8.3b)

 ez  0

(8.3c)

∂( H y  Ez )

2

∂(vt  x ) ∂( H x  Ez ) ∂(vt  y)

∂( Ez  H x ) ∂( Ez  H x ) ∂H x 2 2 ∂(vt ) ∂(vt  y) ∂(vt  y) ∂( E y  H x ) ∂( E y  H x )  ex  0 2 2 ∂(vt  z ) ∂(vt  z ) ∂H y ∂(vt )

2 2

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∂( E x  H y ) ∂(vt  z ) ∂( Ez  H y ) ∂(vt  x )

(8.3a)

2 2

(8.3d)

∂( E x  H y ) ∂(vt  z ) ∂( E  H ) z

y

∂(vt  x )

 e y  0

(8.3e)

Finite-Difference Time-Domain Method Applied to Photonic Crystals

(e′′  4)

∂H z ∂(vt )

∂( E y  H z )

2 2

∂(vt  x ) ∂( E x  H z ) ∂(vt  y)

2 2

405

∂( E y  H z ) ∂(vt  x ) ∂( E x  H z ) ∂(vt  y)

 ez  0

(8.3f)

where ex  s ′y E xy  s ′z E xz , e y  s ′z E yz  s ′x E yx

(8.4a)

ez  s ′x Ezx  s ′y Ezy , ex  s ′′y H xy  s ′′z H xz

(8.4b)

e y  s ′′z H yz  s ′′x H yx , ez  s ′′x H zx  s ′′y H zy

(8.4c)

sk′  2r0 sk , sk′′  2r01s∗k

(8.4d)

with the constraints e′ e′ e′ e′′ e′′ e′′

∂E xk  sk′ E xk  2exk  0, k  y, z ∂(vt ) ∂E yk

(8.5a)

 sk′ E yk  2eky  0, k  z , x

(8.5b)

 sk′ Ezk  2ezk  0, k  x , y

(8.5c)

∂H xk  sk′′H xk  2exk  0, k  y, z ∂(vt )

(8.5d)

∂(vt ) ∂Ezk ∂(vt )

∂H yk ∂(vt ) ∂H zk ∂(vt )

 sk′′H yk  2e yk  0, k  z , x

(8.5e)

 sk′′H zk  2ezk  0, k  x , y

(8.5f)

where exy  ezx  e yz 

∂H y

∂H z ∂H x , e yx  ∂z ∂x

(8.6a)

∂E y ∂E ∂H x , exy  z , exz  ∂z ∂y ∂y

(8.6b)

∂E y ∂E z ∂E x ∂E x , e yx  , ezx  , ezy  ∂z ∂x ∂x ∂y

(8.6c)

∂H z ∂y ∂H y ∂x

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, exz  , ezy 

∂z

, e yz 

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Electromagnetic Theory and Applications for Photonic Crystals

where sk and s*k (k  x, y, z) denote electric conductivity and magnetic loss assigned to the PMLs to absorb outgoing waves, respectively, and satisfy the distortionless condition as sk/  s*k /m (k  x, y, z). From (8.3) and (8.5), for example, we have a relation ∂(Ex  E yx  E zx)/∂t  0 from which we can obtain the splitting field relations Ex  E yx  E zx described in J.P. Berenger’s formulation [9], because Ex  E yx  E zx  0 at excitation time t  t0. Note that J.P. Berenger’s formulation is correct in the steady state but not in the transient state. It also k′ should be noted that ek, ek (k  x, y, z) in (8.3) and ek′ k , ek (k  x, y, z, k′  x, y, z, k  k′) in (8.5) are current-controlled voltage sources [11]; the latter sources mean that their electromotive sources are the spatial derivative of the electric or magnetic field. Equations (8.1), (8.3), and (8.5) can be represented by the multidimensional Kirchhoff circuit [8]. The circuit is MD passive if e′  4 and e″  4, i.e., vm 2 r0 min vemin 2

(8.7)

where emin and mmin are minimum values of  and m, respectively. From (8.7) a constant v is required to satisfy the stability condition as follows: v

2

(8.8)

mmin emin

Note that Maxwell’s equations for the two-dimensional problem can be derived by reducing transformed Maxwell’s Equations (8.1), (8.3), and (8.5) [10]. 8.2.1.2 An FDTD Method Based on MD-WDFs We derive an FDTD method based on MD-WDFs. To do so, we make a discretized form of (8.1). Here we rewrite (8.1) as follows: u1  u11  u12  u13  0

(8.9a)

u2  u21  u22  u23  0

(8.9b)

u3  u31  u32  u33  0

(8.9c)

u4  u41  u42  0

(8.9d)

u5  u51  u52  0

(8.9e)

u6  u61  u62  0

(8.9f)

where u1  (e′  4)

∂E x , ∂(vt )

u2  (e′  4)

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∂E y ∂(vt )

,

u3  (e′  4)

∂E z ∂(vt )

(8.10a)

Finite-Difference Time-Domain Method Applied to Photonic Crystals

u4  (e′′  4) u11  2

u21  2

u31  2

u41  2

u51  2

u61  2

∂H x , ∂(vt )

∂( E x  H z ) ∂(vt  y) ∂( E y  H x ) ∂(vt  z ) ∂( Ez  H y ) ∂(vt  x ) ∂( E y  H x ) ∂(vt  z ) ∂( Ez  H y ) ∂(vt  x ) ∂( E x  H z ) ∂(vt  y)

∂H y

u5  (e′′  4) ∂( E x  H z )

2

∂(vt  y) ∂( E y  H x )

2

∂(vt  z ) ∂( Ez  H y )

2

∂(vt  x ) ∂( E y  H x )

2

∂(vt  z ) ∂( Ez  H y )

2

∂(vt  x ) ∂( E x  H z )

2

∂(vt  y)

∂(vt )

,

u6  (e′′  4) ∂( E x  H y )

,

u12  2

,

u22 = 2

,

u32  2

,

u42  2

,

u52  2

,

u62  2

∂(vt  z ) ∂( E y  H z ) ∂(vt  x ) ∂( Ez  H x ) ∂(vt  y) ∂( Ez  H x ) ∂(vt  y) ∂( E x  H y ) ∂(vt  z ) ∂( E y  H z ) ∂(vt  x )

407

∂H z

(8.10b)

∂(vt ) 2

2

2

2

2

2

∂( E x  H y ) ∂(vt  z ) (8.10c) ∂( E y  H z ) ∂(vt  x ) (8.10d) ∂( Ez  H x ) ∂(vt  y) (8.10e) ∂( Ez  H x ) ∂(vt  y) (8.10f) ∂( E x  H y ) ∂(vt  z ) (8.10g) ∂( E y  H z ) ∂(vt  x ) (8.10h)

u13  2 Jx , u23  2 J y , u33  2 Jz

(8.10i)

The Equations (8.9) can be represented by the Kirchhoff circuit [8] of Figure 8.1 in which Ek and Hk (k  x, y, z) appear as currents and where the forcing fields appear as voltage source. We derive a discretized form of (8.10). First, for example, we rewrite u11 and u61 as ′  u11 ′′ , u61  u61 ′  u61 ′′ u11  u11

(8.11a)

where ′ 2 u11

∂(i1  i6 ) ∂(i  i ) ′  u11 ′ , u61 ′′ u11 ′′ ′′  2 1 6 , u61 , u11 ∂(vt  y) ∂(vt  y)

i1  E x , i6  H z

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(8.11b) (8.11c)

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Electromagnetic Theory and Applications for Photonic Crystals

FIGURE 8.1 A multidimensional Kirchhoff circuit representing Maxwell’s equations as written in the form (8.9).

Then we apply the generalized trapezoidal rule to u′k and u″k (k  11, 61) in the coordinate (vt  y) and (vt  y), respectively. The results are as follows: ′ (t )  u11 ′ (t  Ty′ )  R11 ′ [(i1  i6 )(t )  (i1  i6 )(t  Ty′ )] u11

(8.12a)

′′ (t )  u11 ′ (t  Ty′′)  R11 ′′ [(i1  i6 )(t )  (i1  i6 )(t  Ty′′)] u11

(8.12b)

′ (t )  u61 ′ (t  Ty′ )  R61 ′ [(i1  i6 )(t )  (i1  i6 )(t  Ty′ )] u61

(8.12c)

′′ (t )  u61 ′ (t  Ty′′) R61 ′′ [(i1  i6 )(t )  (i1  i6 )(t  Ty′′)] u61

(8.12d)

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Finite-Difference Time-Domain Method Applied to Photonic Crystals

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where R′n and R″n (n  11, 61) are equivalent to port resistance given by R′11  R″ 11  R′61  R″ 61  2/(v t)  2/ y and T′y and T″ y are the shift vectors given by T′y  (0,  y, 0, v t), T″y  (0, y, 0, v t), where k (k  x, y, z, t) denotes the discretization step in the coordinate x, y, z and t. From (8.11) and (8.12) we have b11 (t ) [(a11  a61 )(t  Ty′ )  (a11  a61 )(t  Ty′′)]/2

(8.13a)

b61 (t ) [(a11  a61 )(t  Ty′ )  (a11  a61 )(t  Ty′′)]/2

(8.13b)

where an and bn (n  11, 61) are the forward flowing wave and backward flowing wave according to the port in the equivalent circuit of Maxwell’s equations and are defined by an (t )  un (t )  Rn in (t )

(8.14a)

bn (t )  un (t )  Rn in (t )

(8.14b)

Rn  4 / (v t )  4 / y, n  11, 61

(8.14c)

respectively. The trapezoidal rule can be applied to the other remaining relations in (8.10). This can be derived according to the forward flowing waves and backward flowing waves. Those wave quantities are connected by Kirchhoff’s voltage law corresponding to (8.9) and current law. In a similar way, we can derive MDWDFs expression of Maxwell’s equations in the PMLs region (8.3) and (8.5). Hereafter, we discuss an algorithm of the FDTD method based on MD-WDFs in the PML region. For example, we obtain forward flowing waves and backward flowing waves according to Maxwell’s equations (8.3a) and (8.5a) in the PMLs region. First we rewrite (8.3a) and (8.5a) as follows: u1  u11  u12  ex  0

(8.15a)

k  e k  0 , k  y, z u1k  u11 x

(8.15b)

where u1  (e′  4) u11  2 u12  2

∂E x ∂(vt )

∂( E x  H z ) ∂(vt  y) ∂( E x  H y ) ∂(vt  z )

ex  s ′y E xy  s ′z E xz

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(8.16a) 2

2

∂( E x  H z ) ∂(vt  y) ∂( E x  H y ) ∂(vt  z )

(8.16b)

(8.16c) (8.16d)

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Electromagnetic Theory and Applications for Photonic Crystals

u1k  e′ exy 

∂E xk k  s′ E k , , u11 k x ∂(vt ) ∂H z ∂y

, exz 

k  y, z

∂H y

(8.16e)

(8.16f)

∂z

In the same way we can calculate forward flowing waves and backward flowing waves as in the physical region [8]. The results are as follows: b1 (t ) a1 (t  Tt )

(8.17a)

b11 (t ) [(a11  a61 )(t  Ty′ )  (a11  a61 )(t  Ty′′)]/2

(8.17b)

b12 (t ) [(a12  a52 )(t  Tz′′ )  (a12  a52 )(t  Tz′ )]/2

(8.17c)

bex (t )  (b1  b11  b12 )(t )  2ex (t )

(8.17d)

an (t )  bn (t )  gn (b1  b11  b12  bex )(t )

(8.17e)

gn  2 Rn / ( R1  R11  R12  Rex ), n  1, 11, 12

(8.17f)

R1  R11  R12  4 / (v t )  4 / z ,

(8.17g)

Rex  R1  R11  R12

b1k (t ) a1k (t  Tt )

(8.17h)

k (t )  0 b11

(8.17i)

k (t )  (b k  b k )(t )  2e k (t ) bex 1 11 x

(8.17j)

k  b k )(t ) ank (t )  bnk (t )  gnk (b1k  b11 ex k  R k ), n  1, 11, k  y, z gnk  2 Rnk / ( R1k  R11 ex

R1k  4 / (v t ),

k  s′ , R11 k

k , k  y, z Rex  R1k  R11

(8.17k) (8.17l) (8.17m)

where shift operators Tt and T′k, T″k (k  x, y, z) are given by Tx′  ( x , 0, 0, t ),

Tx′′  ( x , 0, 0, t )

(8.18a)

Ty′  (0,  y, 0, t ),

Ty′′  (0, z , 0, t )

(8.18b)

Tz′  (0, 0,  z , t ),

Tz′′  (0, 0, z , t )

(8.18c)

Tt  (0, 0, 0, t )

(8.18d)

From this, we can calculate forward flowing waves an(t) and backward flowing waves bn(t) on all spatial grid points at time t from forward flowing waves

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Finite-Difference Time-Domain Method Applied to Photonic Crystals

411

an(t  T) at time (t  t) iteratively. In (8.17d), ex(t) is voltage source, which is obtained by  a y (t  Tt )  2exy (t ) a z (t  Tt )  2exz (t )  ′z 1 ex (t )   s ′y 1  s  Rexy Rexz  

(8.19)

where eyx(t) and ezx(t) are current-controlled voltage sources given by eyx(t)  ∂Hz(t)/∂y, ezx(t)  ∂Hy(t)/∂z, respectively. The dependent source ex(t) contains dependent sources eyx(t) and ezx(t) at time t but not those at time t  t. By assuming that eyx(t)  eyx(t  Tt) and ezx(t)  ezx(t  Tt), i.e., Hy(t)  Hy(t  Tt), Hz(t)  Hz(t  Tt) we can calculate ex(t), and other remaining quantities can be calculated iteratively on all grid points. Referring to [8], the MD-WDFs expressions of Maxwell’s equations in the PMLs region are represented by the signal flow diagram as shown in Figure 8.2. Equations (8.17a–8.17c), and (8.17f), which are shown by two parts of wave digital filters in the dashed lines in Figure 8.2, are represented by shift operators, currentcontrolled voltage sources, and series adapters, respectively [8]. In the figure, signal flow diagrams for constraints (8.5) in the PMLs region are shown for only (8.5a). Note that current controlled voltage sources (ex  ez) in the PMLs region are added to the MD-WDFs expressions in physical region [8]. The blocks N′ and N″ are shown in the bottom of figure. Calculating forward flowing waves and backward flowing waves iteratively for a given initial condition, we have the electromagnetic field on all spatial grid points as follows: an (t )  bn (t ) , k  x , y, z , n  1, 2, 3 2 Rn

(8.20a)

a (t )  bn (t ) , k  x , y, z , n  4, 5, 6 H k (t )  n 2r0 Rn

(8.20b)

Ek (t ) 

Note that the electric and magnetic fields here are at the same grid points, while those of the Yee algorithm are at staggered grid points. Thus we can calculate various physical quantities such as Poynting vectors, propagation constants, and so on with ease.

8.2.2 NUMERICAL DISPERSION AND NUMERICAL ERRORS To show the advantages of the FDTD method based on MD-WDFs over the conventional FDTD method, we check the numerical dispersion. The numerical dispersion gives the lower boundary of the numerical errors of the finite difference method [14,15]. The numerical dispersion relation comes from the nontrivial

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Electromagnetic Theory and Applications for Photonic Crystals

FIGURE 8.2 Signal flow diagram of MD-WDFs structures in the PMLs region.

steady state solution of Maxwell’s equations without any sources, i.e., det( Aij )  0

(8.21)

where Aij are coefficients of Maxwell’s equations at the steady state: 6

∑ Aij c j  0, j1

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i 1 6

(8.22)

Finite-Difference Time-Domain Method Applied to Photonic Crystals

413

where cj ( j  1  6) denotes electric and magnetic fields (Ex, Ey, Ez, Hx, Hy, Hz) and A11 

A22 

A33 

  vdt  k dy   vdt  k dy    y y  tan    tan       2 2       vdt  k dz  1   vdt  kz dz   z  tan     tan      2 dz   2  

 vdt  1 (e′  4)   tan  vdt  2  dy

  vdt  k dz   vdt  k dz    z z  tan     tan       2 2       vdt  k dx  1   vdt  k x dx   x   tan    tan     dx   2 2    

(8.23a)

 vdt  1 (e′  4)   tan  vdt  2  dz

  vdt  k dx   vdt  k dx    x x  tan    tan       2 2       vdt  k dy  1   vdt  k y dy   y     tan   tan   dy   2 2    

(8.23b)

 vdt  1 (e′  4)  tan   2  dx vdt

(8.23c)

  vddt  k dy   vdt  k dy    y y  tan    tan       2 2       vdt  k dz  1   vdt  kz dz   z  (8.23d) tan     tan      2 dz   2  

r0 A44 

 vdt  1 (e′′  4)   tan  vdt  2  dy

r0 A55 

 vdt  1 (e′′  4)  tan   2  dz vdt

r0 A66 

 vdt  1 (e′′  4)  tan   2  dx vdt

  vddt  k dz   vdt  k dz    z z  tan    tan       2 2      vdt  k dx  1   vdt  k x dx   x  (8.23e)   tan  tan     dx   2 2   

A15  r0 A51 

  vddt  k dx   vdt  k dx    x x  tan    tan       2 2       vdt  k dy  1   vdt  k y dy   y  (8.23f)    tan   tan   2 dy   2    

1 dz

  vdt  k dz   vdt  k dz    z z  tan     tan       2 2     

© 2006 by Taylor & Francis Group, LLC

(8.23g)

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Electromagnetic Theory and Applications for Photonic Crystals

A24  r0 A42 

1 dz

  vdt  k dz   vdt  k dz    z z  tan     tan       2 2     

(8.23h)

A16  r0 A61 

  vdt  k dy  1   vdt  k y dy   y    tan   tan     dy   2 2   

(8.23i)

A34  r0 A43 

  vdt  k dy  1   vdt  k y dy   y    tan   tan     dy   2 2  

(8.23j)

A26  r0 A62 

1 dx

  vdt  k dx   vdt  k dx    x x  tan    tan      2 2     

(8.23k)

A35  r0 A53 

1 dx

  vdt  k dx   vdt  k dx    x x  tan    tan      2 2     

(8.23l)

for the FDTD method based on MD-WDFs and A11  A22  A33 

 vdt  e  sin  dt  2 

(8.24a)

A44  A55  A66 

 vdt  m  sin  dt  2 

(8.24b)

A15  A51 A24 A42 

 k dz  1 sin  z   2  dz

(8.24c)

A34  A43 A16 A61 

 k dy  1  y  sin    2  dy

(8.24d)

A26  A62 A35 A53 

 k dx  1 sin  z   2  dx

(8.24e)

for the conventional FDTD method [14,15]. The parameter k  (kx, ky, kz) is the wave vector, and dks (k  x, y, z, t) are discretization steps in space and time. The characteristics Equation (8.22) can be solved numerically for wave vector k that corresponds to an eigenvalue. Figure 8.3 shows plots of the relative error about normalized phase velocity v/c0  |1  2p/|k|| (c0 is the speed of light in vacuum) as a function of the grid space cell size for typical propagation directions (1, 0, 0), (1, 1, 0), and (1, 1, 1). For simplicity we use the square unit cells, i.e., d  dx  dy  dz. For each direction the time step relation c0dt  0.4d (i.e., v  c0 /0.4) is maintained. This relation is commonly used in two- and

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Finite-Difference Time-Domain Method Applied to Photonic Crystals

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% error about normalized phase velocity

100 MD-WDFs Yee algorithm 10−1

(1,0,0)

10−2

(1,1,0)

10−3 (1,1,1) 10−4 10−3

10−2 Grid space cell size
/

10−1

FIGURE 8.3 Variation of the numerical errors for the FDTD method based on MD-WDFs and conventional FDTD method with grid cell size at three wave propagation directions.

three-dimensional cases to satisfy the numerical stability criterion with ample safety margin. For simplicity’s sake, the arbitrary parameter r0 in MD-WDFs is selected as r0   m0/e0 to satisfy (8.8). From Figure 8.3 it can be seen that the numerical error increases as grid space cell size increases, and the numerical error of the MD-WDFs is much smaller than those of the conventional method. This error is cumulative (i.e., increases linearly with wave propagation distance) [15]. In order to obtain the solution with the same accuracy, the Yee algorithm requires about half of the MD-WDFs algorithm’s grid cell size as shown in Figure 8.3. The MD-WDFs’ memory requirement is 18 times the total number of grid points, while the Yee algorithm’s memory requirement is 6 times the total number of grid points. Therefore, the MD-WDFs’ memory requirement is approximately 3/8 (18/6  1/8) of the Yee algorithm’s, and the computational time is reduced to at most 3/16 (18/6  1/16) of the Yee algorithm’s. This demonstrates that the FDTD method presented here requires less memory storage and shorter computational time than the FDTD method based on the Yee algorithm.

8.3 PHOTONIC CRYSTAL STRAIGHT WAVEGUIDE Photonic crystal waveguides are promising elements for optical integrated circuits. They are composed of a line defect in photonic crystal that can confine the light if the frequency is inside the photonic bandgap. The line defect in twodimensional(2D) photonic crystals is one of the effective ways to realize the high density optical circuits because the fabrication process of 2D photonic crystals is easier than that of three-dimensional photonic crystals [16–18]. Broadly, 2D

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Electromagnetic Theory and Applications for Photonic Crystals

photonic crystals have two types, pillar type and air-hole type. Although the pillar type photonic crystal waveguide has strong light confinement due to a wide photonic bandgap, it is difficult to confine the light in the vertical direction because the waveguide layer is composed of air. On the other hand, by using the 2D air-hole type photonic crystal in a thin dielectric slab, we can confine the light in the vertical direction by total internal reflection due to the refractive index difference between the dielectric material and the air. However, it is expected that it is difficult to maintain a single mode region in the air-hole type photonic crystal waveguide because many higher order modes can exist. So it is important to clarify the fundamental propagation properties of both types of photonic crystal waveguides for designing optical devices.

8.3.1 NUMERICAL RESULTS We use the FDTD method based on MD-WDFs to photonic crystal optical waveguides for analyzing two types of pure 2D photonic crystal optical waveguides as shown in Figure 8.4. Here, we evaluate two types of photonic crystal waveguides composed of circular dielectric pillars in air and circular air-holes in dielectric material on a squared array with lattice constant a. The waveguide layer is made by removing a few rows of pillars or air-holes. The air-hole type photonic crystal waveguides have many propagation modes because their waveguide layer is made

a a

b

b a

a

ra

ra a

a

W

x y

D

x z

(a)

y

z

(b)

FIGURE 8.4 Two-dimensional photonic crystal waveguide: (a) pillar type, (b) air-hole type.

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Finite-Difference Time-Domain Method Applied to Photonic Crystals

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of dielectric material. In order to operate air-hole type photonic crystal waveguides in the single mode region, we have to narrow the waveguide width. So we analyze the waveguide whose cladding layers are shifted mutually in the propagation direction as shown in Figure 8.4(b). We have to consider the structure when we design waveguide devices using the air-hole type photonic crystal, such as a bent waveguide, because the width of waveguide layer W and lattice constant a are not in agreement. The shift is denoted by D. In both cases, the relative permittivity of the dielectric material is ea  11.56. In order to obtain a wide range of photonic bandgap, we have chosen the radii of pillars and air-holes ra as ra /a  0.175 and 0.475, respectively. The pillar type photonic crystal has a photonic bandgap for E polarized field (Ey, Hx, Hz) which extends from frequency va/2pc  0.320 to 0.462 where c is the speed of light in vacuum. The air-hole type has a photonic bandgap for E polarized field which extends from va/2pc  0.245 to 0.306. Maxwell’s equations for the two-dimensional problem can be derived by assuming ∂/∂y  0, and the MD-WDFs representation can be obtained in the same way as in Section 2. The perfectly matched layers(PMLs) are used as the absorbing boundary condition at the edges of the computational zone for analysis of the photonic crystal waveguide. Taking impedance matching conditions at the interface between the computational region and the PMLs region into account, in the PMLs region we use the same waveguide structure as that of the computational region, but electric conductivity and magnetic loss are assigned to dielectric material and a background medium whose values satisfy the distortionless condition [9]. Here we use the continuous wave for an excitation. In order to suppress transient phenomena by step function as much as possible, we use a raised sine wave form for incident wave as follows:  F ( x ) sin(vt ) sin 2 {vt / (4 N )}, F ( x , z  0, t )   0  F0 ( x ) sin(vt ),

t 2pN /v v t 2pN /v

(8.25)

where F0(x) denotes a transverse profile that is a Gaussian profile, vc is center angular frequency, and N is positive integer. Figures 8.5 and 8.6 show electric field intensities of pillar type and air-hole type photonic crystal waveguides, respectively. The waveguide width and frequency of incident for pillar type and air-hole type are W/a  1.65, va/2pc  0.4 and W/a  0.75, va/2pc  0.26, respectively. The shift between cladding layers is D/a  0.7 in Figure 8.6(b). We can see that electric field intensities oscillate in the propagation direction in both types of waveguide. From Figure 8.6(b), we can also see that electric field intensities oscillate in the propagation direction and the wavefront is tilted along with an array of air-holes.

8.3.2 FLOQUET MODE First we express the guided modes of this waveguide: wavenumbers and amplitudes of the modes. The spatial profile of electric field intensity |Ey| of the fundamental

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Electromagnetic Theory and Applications for Photonic Crystals

FIGURE 8.5 Electric field intensities of a pillar type photonic crystal waveguide. The waveguide width is W/a  1.65, and frequency of incident wave is va/2pc  0.4.

(a)

(b)

FIGURE 8.6 Electric field intensities of an air-hole type photonic crystal waveguide. The waveguide width is W/a  0.75, and incident frequency is va/2pc  0.26. (a) D/a  0.0, (b) D/a  0.7.

mode of these waveguides as a function of wavenumber b in z-direction is shown in Figure 8.7. The spatial wavenumber is calculated by the Prony’s method [19]. In the Prony’s method, Fourier transformed propagation wave c (x, z, v) is approximated by a sum of damped complex exponentials in the following manner: c( x , z , v)  ∑ Ai ( x , v)e jbi (v ) z i

© 2006 by Taylor & Francis Group, LLC

(8.26)

Electric field intensity Ey (x,)(a.u.)

Finite-Difference Time-Domain Method Applied to Photonic Crystals

419

1 0.8

|m0|

0.6 0.4

|m1|

0.2 0 −3

1

|m (−1)| −2

−1

0 x/a

|m (−2)| 1

2

3 −2

−1 −1.5

1.5

2

0.5 a r  be 2 um

0 −0.5 n

ve

Wa

Electric field intensity Ey (x,)(a.u.)

(a) 1

|m0|

0.8 0.6 0.4

|m1|

0.2 0 −3

2

−2

−1

|m (−2)| 0 x/a

1

2

(b) Electric field intensity Ey (x,)(a.u.)

1.5

|m (−1)|

1

−1 3 −1.5

1 0.5

a er 2 b m

0 −0.5 ve

nu

Wa

|m0|

0.8 0.6 0.4

|m1|

0.2 0 −3

2 1.5

|m(−1)| −2

−1 x/a

0

|m (−2)| 1

2

(c)

−1 3 −1.5

1 0.5

a er 2 b m

0 −0.5

ve

nu

Wa

FIGURE 8.7 Electric field intensity |Ey| as a function of wavenumber b: (a) pillar type, (b) air-hole type, (c) air-hole type with shift D. The waveguide widths, shift between cladding layers, and incident frequencies are (a) W/a  1.65, D/a  0, va/2pc  0.4, (b) W/a  0.75, D/a  0, va/2pc  0.26, and (c) W/a  0.75, D/a  0.7, va/2pc  0.26, respectively.

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Electromagnetic Theory and Applications for Photonic Crystals

where Ai(x, v) is the amplitude profile in transverse direction and bi(x, v) is the wave number in propagation direction. The coefficients Ai and bi are complex quantities. Using the Prony’s method, we can determine wavenumber bi(x, v) and amplitude Ai(x, v). It follows from Figure 8.7 that there are two kinds of wavenumber, the propagation constant b and its space harmonics (b  2np/a), where n is a positive integer, regardless of the amount of shift D between cladding layers. Then we can guess from the numerical results that m-th order mode of electric field Eym(x, z) in the photonic crystal waveguide can be expressed in the following form as E ym ( x , z )  cm 0 ( x )e jbm z 

∑

n n0

cmn ( x )e j ( bm 2 np /a ) z

    cm 0 ( x )  ∑ cmn ( x )e j ( 2 np /a ) z  e jbm z , m  0, 1, … , M n   n0  

(8.27)

where cm0(x) and cmn(x) denote complex amplitude of the transverse electric field profile whose wavenumbers are bm and (bm  2np/a), respectively, and M is a finite number depending on frequency and waveguide structure. Note that this expression corresponds to the Floquet mode of waves in periodic structure. We can find from this expression that electromagnetic fields in the photonic crystal waveguides propagate with a propagation constant bm whose intensity oscillates in the period of integer multiples of lattice constant a as described in the brackets of (8.27). The numerical results show that dominant parts of electric fields are cm0 and cm(1), which are amplitudes of the propagation wave and — first order space harmonics, respectively, as shown in Figure 8.7. (Equation (A-2) in reference [6] should be read by the expression (8.27).) Note that cm0(x) and cmn(x) are becoming real numbers for D  0. Figure 8.8 shows phase profile in x-direction of cm0 and cm(1) for several amounts of shift D for air-hole type photonic crystal waveguides. Vertical lines in this figure denote the interface between the waveguide layer and the cladding layer. We can find that the phase of cm(1) is changed in the x-direction according to the amount of shift D, while the phase of cm0 is almost without change as shown in Figure 8.8. The phase change of cm(1) leads to a formation of an electric field intensity profile as shown in Figure 8.6.

8.3.3 DISPERSION RELATION From previous results we can find that the photonic crystal waveguides are characterized by the propagation constant bm. Figure 8.9 shows the propagation constant of pillar type and air-hole type photonic crystal waveguides bma/2p as a function of normalized frequency va/2pc. The waveguide width of the pillar type and air-hole type are W/a  1.65 and W/a  0.75, respectively. In this figure, solid lines and dotted lines denote propagation constants b0 and b0  2p/a,

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Phase (rad.)

/2

0 m0 −/2 m(−1) −

−0.4

−0.2

D/a= 0.6 0.7 0.8 0 x/a

0.2

0.4

FIGURE 8.8 Phase profile in the x-direction of amplitude cm0 and cm1. The waveguide width is W/a  0.75, and incident frequency is va/2pc  0.26.

respectively. We can see that a fundamental mode and higher-order modes exist in the air-hole type photonic crystal waveguide, while the pillar type has only a fundamental mode and has no cutoff region within the photonic bandgap. In the air-hole type photonic crystal waveguide, the light confinement can be accomplished by photonic bandgap and refractive index differences between the waveguide layer and the air-hole. Because of the light confinement as the dielectric waveguide with a periodic boundary, two propagation modes become cutoff near the region bma/2p  0.5, 1.0, … which corresponds to the Bragg condition. Therefore the waveguide behaves as a single mode one in the region where the first order mode becomes cutoff. Figure 8.10 also shows the propagation constant of air-hole type photonic crystal waveguides with a shift between cladding layers. The waveguide width and the shift between cladding layers are W/a  0.75 and D/a  0.7, respectively. From Figure 8.10 we can see that the even mode and odd mode are coupled because the symmetry of the waveguide structure in transverse direction is lost. As a result, the single mode region is slightly narrower than that which has no shift as shown in Figure 8.9(b). The effective index of air-hole type photonic crystal waveguides bm/k where k is wavenumber in free space is close to that of corresponding conventional dielectric waveguides, while the effective index of the pillar type is less than 1.0. Therefore it is expected that highly efficient coupling can be realized if we can realize an impedance matching condition between air-hole type photonic crystal waveguides and conventional dielectric waveguides. In fact, we attain more than 75% coupling efficiency even for a simple structure where an air-hole type

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Electromagnetic Theory and Applications for Photonic Crystals Photonic bandgap

Propagation const. ma/2

1.00

0.75

0.50

m=0 0.25

0.00

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

Normalized frequency a/2c (a) Photonic bandgap 1.00

Propagation const. ma/2

m=0

m=0

0.75

m=1 0.50 m=1

0.25 m=2 Even mode excitation Odd mode excitation 0.00

0.24

0.26

0.28

0.30

0.32

Normalized frequency a/2c (b)

FIGURE 8.9 Dispersion relation of (a) pillar type and (b) air-hole type photonic crystal waveguides. The waveguide widths are (a) W/a  1.65 and (b) W/a  0.75, respectively. © 2006 by Taylor & Francis Group, LLC

Finite-Difference Time-Domain Method Applied to Photonic Crystals

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Photonic bandgap

Propagation const. ma/2

1.00

m=0

m=0

0.75

m=1 0.50 m=1

0.25 m=2 0.00

0.24

0.26

0.28

0.30

0.32

Normalized frequency a/2c

FIGURE 8.10 Dispersion relation of air-hole type photonic crystal waveguide. The waveguide width and shift are W/a  0.75 and D/a  0.75, respectively.

photonic crystal waveguide and a conventional dielectric waveguide are connected directly.

8.4 FUNDAMENTAL OPTICAL CIRCUIT DEVICES USING PHOTONIC CRYSTALS We analyze several waveguide devices constructed with photonic crystals under the single-mode propagation condition for pillar type and air-hole type photonic crystal waveguides.

8.4.1 DIRECTIONAL COUPLER First we investigate the propagation characteristics of a directional coupler as shown in Figure 8.11. The directional coupler consists of two equal waveguides whose waveguide width is WI  WII. The interval between two waveguides is g/a  2ra, which corresponds with one row of crystals, and the shift between cladding layers is denoted by D as shown in Figure 8.11. Figure 8.12 shows the dispersion relation of a pillar type photonic crystal directional coupler. The relative permittivity of the dielectric materials is eb  11.56, the radius of pillars is ra /a  0.175, the waveguide width is WI  WII  1.65, and the interval between two waveguides is g/a  0.35. In this case there is no shift between cladding layers. be and bo denote the propagation constants of the even and odd modes of directional couplers, respectively. bI and © 2006 by Taylor & Francis Group, LLC

424

Electromagnetic Theory and Applications for Photonic Crystals a ra a

Waveguide I

WI

D

g b

WII

Waveguide II

a

x y

z

FIGURE 8.11 Air-hole type photonic crystal directional coupler.

Normalized propagation constant a/2

0.45 0.40 0.35 0.30 o 0.25 0.20 e 0.15 I = II

0.10 0.05 0.00 0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

Normalized frequency a/2c

FIGURE 8.12 Dispersion relation of pillar type photonic crystal directional coupler.

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bII also denote the propagation constant of waveguide I or II, which exist independently. In this case bII is the same as bI because the waveguide structure is the same. We can see from the figure that the difference between the propagation constants of the even mode and odd modes becomes smaller as the frequency becomes higher. The coupling length of a pillar type photonic crystal directional coupler as a function of normalized frequency is shown in Figure 8.13. The coupling length is calculated by L

p |be  bo |

(8.28)

We can see that the coupling length increases with frequency. This is the same characteristic as directional coupler constructed by conventional dielectric waveguides. Figure 8.14 shows the dispersion relation of an air-hole type photonic crystal directional coupler when the waveguide width is WI  WII  0.75 and 0.85. The relative permittivity of the dielectric materials is eb  11.56, the radius of airholes is ra/a  0.475, and the interval between two waveguides is g/a  0.9. We can see from the figure that the even and odd modes in the directional coupler also make a cutoff region at each frequency band because each mode couples onto each higher-order mode. As a result, the dispersion curves of even and odd modes are changed independently. The coupling length of an air-hole type photonic crystal directional coupler as a function of normalized frequency is shown in Figure 8.15.

45 40

Coupling length L/a

35 30 25 20 15 10 5 0 0.34

0.36

0.38

0.40

0.42

0.44

Normalized frequency a/2c

FIGURE 8.13 Coupling length of pillar type photonic crystal directional coupler as a function of normalized frequency.

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Normalized propagation constant a/2

1.00 e

0.95

I = II

o

0.90 0.85

I = II

0.80 0.75 e

0.70

o 0.65 0.60 0.24

0.25

0.26

0.27

0.28

0.29

0.30

0.31

0.30

0.31

Normalized frequency a/2c (a)

Normalized propagation constant a/2

1.00 0.95

I = II

e

o

0.90 0.85 I = II 0.80 0.75

e

0.70 o 0.65 0.60 0.24

0.25

0.26

0.27

0.28

0.29

Normalized frequency a/2c (b)

FIGURE 8.14 Dispersion relation of air-hole type photonic crystal directional coupler. The waveguide width and shift are (a) W/a  0.75, D/a  0.7 and (b) W/a  0.85, D/a  0.8, respectively.

Since the position of the cutoff region of the even and odd modes is different, the frequency spectrum of the coupling length of the air-hole type shows different characteristics than the pillar type, which shows the same characteristics as the conventional dielectric waveguide.

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100 W/a = 0.65 W/a = 0.75

Coupling length L /a

80

60

W/a = 0.85

40

20

0

0.245

0.250

0.255

0.260

0.265

0.270

0.275

0.280

Normalized frequency a/2c

FIGURE 8.15 Coupling length of air-hole type photonic crystal directional coupler as a function of normalized frequency.

8.4.2 SHARPLY BENT WAVEGUIDE WITH MICROCAVITY To realize signal processing integrated waveguides devices, we need sharply bent waveguides without reflection. Here we propose and design a sharply bent waveguide without reflection that has additional dielectric pillars/air-holes in the corner region. These additional dielectric pillars/air-holes form a microcavity in the corner region and act as potential barriers making a resonant tunneling in a quantum wire [20]. We can eliminate reflected waves from the corner region due to resonant tunneling [5,6,21–23]. 8.4.2.1 Pillar Type Photonic Crystal Bent Waveguide First we analyze a pillar type photonic crystal bent waveguide with additional dielectric-pillars in the corner region as shown in Figure 8.16. In order to consider a larger bent angle, we also discuss photonic crystals composed of circular dielectric pillars in a triangular array. The relative permittivity of the pillars and background are ea  11.56 and eb  1.0, respectively. The radius of pillars is ra /a  0.175 for both lattice types. The triangular lattice photonic crystals also have bandgap for E-polarized field (Ey, Hx, Hz) whose frequency range is 0.304 va /2pc 0.495. The triangular lattice type waveguides also have a single mode region. In the corner region, two additional dielectric pillars whose radius and relative permittivity are the same as those of other photonic crystals are placed. Optical power transmission characteristics of a 90° bent waveguide in square lattice and a 120/60° bent waveguide in a triangular lattice together with the

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a b

a

x y

z (a) a a a b

x y

z

(b) a

a a b

x y

z

(c)

FIGURE 8.16 Photonic crystal sharply bent waveguide with additional pillars in the corner region: (a) 90° bend in square lattice, (b) 120° bend, (c) 60° bend in triangular lattice.

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resonant frequency of the unloaded microcavity, which was made by removing one pillar of photonic crystal, are shown in Figure 8.17. The power flow in the waveguide is defined by P ∫ S ⋅ u dS , S

S 

1 T

T

∫0

(E  H) dt

(8.29)

where S denotes the cross section of the waveguide, 〈S〉 means time averaged Poynting vector, and u is the unit vector of propagation direction. By adding additional pillars we can completely eliminate reflected waves from the sharp corner at the resonant frequency for both types of waveguides. We can find that the resonance mode of the unloaded microcavity is excited at the corner region because resonant frequencies of the bent waveguide and the unloaded microcavity are the same. The resonant frequencies of bent waveguide are lower than that of unloaded microcavity because the microcavity can be connected by external waveguides. Figure 8.18 shows the electric field intensity and Poynting vector of the 90° and 120/60° bent waveguides at the resonant frequency va/2pc  0.389 and 407, respectively, where we use the same frequency in the 120° and 60° bent waveguides in a triangular lattice. We can see that there are no reflected waves in the input side and electric fields concentrate at the corner region due to resonant tunneling. We also can see that the optical power flows efficiently through the 1 0.9 0.8

Resonant frequency of unloaded cavity

Transmissivity

0.7 0.6 0.5

Triangular lattice

Square lattice

0.4 0.3 0.2 0.1 0 0.35

0.36

0.37

0.38

0.39

0.4

0.41

0.42

0.43

0.44

Normalized frequency a/2c

FIGURE 8.17 Optical power transmission characteristics of bent waveguide and resonant frequency of unloaded cavity.

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(a)

(b)

(c)

FIGURE 8.18 Electric field intensity and Poynting vector of (a) 90°, (b) 120°, and (c) 60° bent waveguide. The incident frequencies va/2pc are (a) 0.389 in square lattice and 0.407 for (b) and (c) in triangular lattice.

sharply bent waveguide. More precisely, in the transmitted region the mode profile shows the same transverse profile as that of input region. In other words, there is no mode conversion, and the eigenmode profile remains even if the light propagates through the sharply bent waveguide. Next we try to control the transmission frequency range of bent waveguides using the property of an unloaded microcavity. The resonant frequency of an unloaded microcavity can be adjusted by adding another dielectric pillar in that microcavity [24]. We place an additional pillar in the unloaded microcavity as shown at the top of Figure 8.19. The radius and relative permittivity of additional

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a a a2

a2

0.46

Normalized frequency a/2c

0.44 0.42 0.4 0.38

Triangular lattice

0.36

Square lattice

0.34 0.32

1

1.5

2

2.5

3

3.5

4

4.5

5

Relative permittivity of additional pillar a2

FIGURE 8.19 Resonant frequency of cavity as a function of relative permittivity of additional pillar.

the pillar are denoted by ra2 and ea2, respectively. First we change the relative permittivity of the additional pillar. Figure 8.19 shows the resonant frequency of the unloaded microcavities when relative permittivity of the additional pillar is changed. The radius of the additional pillar is ra2/a  0.175. The resonant frequencies are decreased by increasing the relative permittivity of the additional pillar for both a square and a triangular lattice microcavity. By using this property, we can control the transmission frequency range of the bent waveguide. Figure 8.20 shows the optical power transmission characteristics of 90° and 120° bent waveguides when an additional pillar is placed in the center of the cavity. We can see that the resonant frequency decreases as the relative permittivity of additional pillar ea2 becomes larger. Next we vary another parameter, such as the radius of additional pillar [24]. Figures 8.21 and 8.22 show the resonant frequency of an unloaded microcavity as a function of the radius of an additional pillar placed in the center of the

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Electromagnetic Theory and Applications for Photonic Crystals 1

2.25 1.56 a2 = 1.00

3.06

4.00

a

0.9 0.8 a2

Transmissivity

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.34

0.36

0.38

0.4

0.42

0.44

Normalized frequency a/2c (a) 1

4.00

3.06

2.25

1.56 a2 = 1.00

0.9

a

0.8

Transmissivity

0.7

a2

0.6 0.5 0.4 0.3 0.2 0.1 0

0.36

0.38

0.4

0.42

0.44

Normalized frequency a/2c (b)

FIGURE 8.20 Optical power transmission spectra when relative permittivity of additional pillar is changed: (a) square lattice (b) triangular lattice.

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Finite-Difference Time-Domain Method Applied to Photonic Crystals

Dipoles

Monopole

ra2

433

Hexapoles

Quadrupoles 0.46

Normalized frequency a/2c

0.44

Octupoles

2nd order monopole

Dipoles

0.42 0.4 Hexapoles

Monopole

0.38 Quadrupoles 2nd Order dipoles

0.36 0.34 0.32

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Radius of additional pillar ra2 /a

FIGURE 8.21 Resonant frequency of square lattice photonic crystal unloaded microcavity as a function of radius of additional pillar.

cavity for a square lattice and a triangular lattice photonic crystal microcavity, respectively. The relative permittivity of the additional pillar is ea2  11.56. We can find from these figures that resonant frequency is decreased by increasing the radius of the additional pillar and the higher order resonance modes appear when the radius of the additional pillar exceeds that of the pillar of the photonic crystal. The electric field distributions of the resonance modes are shown at the tops of these figures. The resonance modes have symmetry in accordance with the symmetry of the structure. The resonance modes are labeled by the number of nodes of the electric field [24], e.g., the dipole resonance mode has two nodes.

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ra2

Monopole

Dipoles

Quadrupoles

Hexapoles

0.5

Normalized frequency a/2c

2nd Order monopole

Hexapoles

0.45

Monopole

0.4

Dipoles Quadrupoles 0.35

0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Radius of additional pillar ra2/a

FIGURE 8.22 Resonant frequency of triangular lattice photonic crystal unloaded microcavity as a function of radius of additional pillar.

Figure 8.23 shows the optical power transmission characteristics of a 90° bent waveguide when an additional dielectric pillar whose radius is ra2/a  0.475 is placed in the center of the cavity. We can see that optical power is transmitted completely at the frequency that is the same as that of the higher order resonance mode of the unloaded cavity as shown in Figure 8.21. Figure 8.24 shows the electric field intensity profiles at the resonant frequency. The incident frequencies are va/2pc  0.391 and 0.372. We can see that the second order monopole and the quadrupole resonance mode are excited in the microcavity. Figure 8.25 shows the optical power transmission characteristics of 120/60° bent waveguides when an additional dielectric pillar whose radius is ra2 /a  0.575 is placed in the center of the cavity. We can find that a hexapole resonance mode is excited in the microcavity because another resonance mode cannot be excited by the fundamental

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1 0.9 0.8

Transmissivity

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.34

0.36

0.38 0.4 Normalized frequency a/2c

0.42

0.44

FIGURE 8.23 Optical power transmission spectrum of square lattice bent waveguide when additional dielectric pillar whose radius is ra2/a  0.475 is placed in the cavity.

(a)

(b)

FIGURE 8.24 Electric field intensity of square lattice bent waveguide when additional dielectric pillar whose radius is ra2/a  0.475 is placed in the cavity. The incident frequencies are (a) va/2pc  0.391 and (b) va/2pc  0.372.

mode of the input waveguide. Figure 8.26 shows the electric field intensity profiles of 120/60° bent waveguides at the resonant frequency. The incident frequency is va/2pc  0.415 for both bent waveguides. We can see that the hexapole resonance mode is excited in the microcavity.

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Electromagnetic Theory and Applications for Photonic Crystals 1

Transmissivity

0.8

0.6

0.4

0.2

0

0.36

0.38

0.4

0.42

0.44

Normalized frequency a/2c

FIGURE 8.25 Optical power transmission spectrum of triangular lattice bent waveguide when additional dielectric pillar whose radius is ra2/a  0.575 is placed in the cavity.

(a)

(b)

FIGURE 8.26 Electric field intensity of square lattice bent waveguide when additional dielectric pillar whose radius is ra2 /a  0.575 is placed in the cavity. The incident frequency is va/2pc  0.415. (a) 60° bend, (b) 120° bend.

8.4.2.2 Air-Hole Type Photonic Crystal Bent Waveguide The air-hole type photonic crystal sharply bent waveguide is designed as shown in Figure 8.27. In the corner region are placed two additional air-holes whose radius is ra2. Figure 8.28 shows the electric field intensity and Poynting vector of an L-shaped bent waveguide at the resonant frequency va/2pc  0.248. The radius of the air-hole is ra2/a  0.35. The waveguide width and shift are W/a  0.85 and D/a  0.8, respectively. We can see that there are no reflected

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W

ra

ra2

a

x y

z

FIGURE 8.27 Photonic crystal L-shaped bent waveguide with additional air-holes in the corner region.

(a)

(b)

FIGURE 8.28 Electric field intensity and Poynting vector of L-shaped bent waveguide at the resonant frequency va/2pc  0.248. The radius and relative permittivity of additional air-hole are ra2/a  0.35 and ea2  1.0, respectively. (a) Electric field intensity, (b) Poynting vector.

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Electromagnetic Theory and Applications for Photonic Crystals 1.00 ra2 /a = 0.250 ra2 /a = 0.350

Transmissivity

0.75

0.50

ra2 /a = 0.150 0.25

Without air-holes 0.00

0.246

0.248

0.250

0.252

0.254

0.256

0.258

Normalized frequency a/2c

FIGURE 8.29 Optical power transmission characteristics of L-shaped bent waveguide when the radius of additional air-holes ra2 is changed.

waves in the input side and electric fields concentrate at the corner region due to resonant tunneling. The optical power flows efficiently through the right-angle bend, and it makes some vortices at the corner region as shown in Figure 8.28(b). We can see that the mode field profile in the transmitted region shows the same transverse profile as that of input region. Figure 8.29 shows the optical power transmission characteristics when the radius of an additional air-hole is changed. By using additional air-holes we can completely eliminate reflected waves from the right-angle corner at the resonant frequency. We can see from the figure that for an increase of radius ra2 the resonant frequency shifts to a lower side and its quality factor increases. In other words, transmission frequency ranges can be controlled by changing the radii of additional air-holes.

8.5 WAVELENGTH MULTI/DEMULTIPLEXER As a practical signal processing device, we propose and design a wavelength multi/demultiplexer. A compact size wavelength multi/demultiplexer is desired for a wavelength division multiplexed (WDM) communication system. To handle many wavelengths, the size of a wavelength multi/demultiplexer made with conventional dielectric waveguides becomes several centimeters square because bent waveguides in the device have large curvatures to suppress radiation loss. Using the photonic crystal bent waveguide, we can reduce the size of the device.

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L1 1 Port 1

1 2 3

2 Port 2

Port IN

3 Port 3 L2

FIGURE 8.30 The structure of three-wavelength multi/demultiplexer. Black circles denote additional pillars whose radii and relative permittivity are ra2 /a  0.175 and ea2  1.69, respectively. Lengths of coupling region are L1/a  48.0 and L2 /a  24.0, respectively. The parameters of photonic crystal ra /a  0.175 and ea  11.56 are used.

The wavelength multi/demultiplexer designed here is composed of directional couplers and reflectionless right-angle bent waveguides as shown in Figure 8.30. The number of channels of the wavelength multi/demultiplexer determines the number of directional couplers, i.e., M-wavelengths are multi/demultiplexed by using (M  1) directional couplers, which are connected sequentially. As an example, we design a three-wavelength multi/demultiplexer using two directional couplers. The directional coupler is composed of two waveguides which are single mode waveguides whose width is W/a  1.65. The interval between the two waveguides is d/a  0.35, which corresponds to one row of crystals.

8.5.1 DESIGN PARAMETERS For analyzing this device, we need to estimate several parameters of the directional coupler. Figure 8.31 shows the normalized coupling length of the directional coupler L/a as a function of wavelength. In the figure, integral multiples of coupling length nL (n  1, 2, 3, …) are plotted for determining the length of the directional coupler. If the length of the directional coupler is chosen as an evennumbered multiple of the coupling length 2nL (n  1, 2, …), the optical power for the appropriate wavelength is finally transmitted to the straight port. Considering the desired wavelengths that are filtered by the directional couplers, we choose the lengths of the first and second directional couplers as L1/a  48 and L2/a  24, respectively. From Figure 8.31 we can see that the light of wavelength l1 (2.68a) is completely transmitted across the port of the first directional coupler and two different lights of wavelength l2 (2.60a) and l3 (2.74a) that are first transmitted to the straight port of the first directional coupler are finally divided into two output ports through the second directional coupler.

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Electromagnetic Theory and Applications for Photonic Crystals 50

Coupling length of directional coupler L/a

7L 45 40

6L

35 5L 30 5L

25 20

3L 15 2L

10 5 0 2.50

2.55

2.60

Coupling length L 3

1

2 2.65

2.70

2.75

2.80

Wavelength  /a

FIGURE 8.31 Coupling length of the directional coupler as a function of wavelength. The directional coupler is composed of two single mode waveguides whose width is W/a  1.65. The interval between two waveguides is d/a  0.35.

8.5.2 NUMERICAL RESULTS Using these parameters we assemble directional couplers and L-shaped bent waveguides into the three-wavelength multi/demultiplexer as shown in Figure 8.30. In the corner region we place additional pillars denoted by black circles in the figure. In order to obtain wide transmission frequency characteristics, we choose the radii and relative permittivity of additional pillars as ra2 /a  0.175 and ea2  1.69, respectively. The transmission spectra of the three-wavelength multi/demultiplexer are shown in Figure 8.32. Here we evaluate the characteristics of the threewavelength multi/demultiplexer quantitatively by using the extinction ratio and insertion loss defined by  P ′  P  Extinction ratio 10 log  i  , Insertion loss 10 log  i  (8.30)  Pin   Pi  where Pin is input power flow, Pi and P′i indicate output power at port i for a selected port and an unselected port, respectively. In fact, the extinction ratio and insertion loss for ports 1, 2, and 3 are almost greater than 19.6 dB and less than 9  102 dB, respectively, as shown in Table 8.1. We can see from the figure and table that additional pillars in right-angle bends realize a high extinction ratio and low insertion loss. Figure 8.33 shows electric field intensity of the directional coupler for wavelength l/a  2.68, 2.60, and 2.74. We can see that a low insertion

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Finite-Difference Time-Domain Method Applied to Photonic Crystals 2

1

441

3

0

Transmittance (dB)

−5

−10

−15 Port1 Port2 Port3 −20 2.50

2.55

2.60

2.65

2.70

2.75

2.80

Wavelength /a

FIGURE 8.32 Transmission spectra of the three-wavelength multi/demultiplexer.

TABLE 8.1 Extinction Ratio and Insertion Loss of the Three-Wavelength Multi/ Demultiplexer Wavelength L/a

Extinction ratio [dB]

2.68 2.60 2.74

27.5 18.5 19.6

Insertion loss [dB] 2

3.0  10 8.5  102 6.5  102

Output Port 1 Port 2 Port 3

loss and high-extinction-ratio three-wavelength multi/demultiplexer whose overall length is approximately 35lm, where lm is a maximum wavelength of incident wave, can be realized in all incident wavelengths.

8.6 CONCLUSION We have presented the finite-difference time-domain method based on the multidimensional wave digital filters and shown several advantages of the MD-WDFs over the Yee algorithm. We have newly proposed a modified form of the Maxwell’s equations in the perfectly matched layers and its MD-WDFs’ representation by using the current controlled voltage sources. We have evaluated the

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(a)

(b)

(c)

FIGURE 8.33 Electric field intensity of the three-wavelength multi/demultiplexer. The wavelengths of incident wave are (a) l/a  2.66, (b) l/a  2.58, and (c) l/a  2.72.

numerical errors concerning phase velocity in the plane wave propagation problem by examining the numerical dispersion relation and pointed out the capability of the MD-WDFs compared with the Yee algorithm. We have clarified the propagation characteristics of two-dimensional photonic crystal optical waveguide devices. By examining the eigenmode propagation in the photonic crystal waveguides constructed by photonic crystals, we can find that electric and magnetic field intensities in the waveguides oscillate according to the period of the lattice constant of the photonic crystal. We have checked the dispersion relation of the waveguide and confirmed differences in the performance of a single mode propagation in air-hole type photonic crystal waveguide and a pillar type waveguide. Next, we have analyzed typical waveguide devices such as a directional coupler and a sharply bent waveguide. First, we have simulated a directional coupler constructed by photonic crystal. We have also checked the dispersion relation of the directional coupler and confirmed that the even and odd modes in the air-hole type photonic crystal directional coupler cause cutoff at each frequency band because each mode couples onto each higher-order mode. As a result, the frequency spectrum of coupling length shows different characteristics from that of a pillar type photonic crystal directional coupler, which has the same characteristics as

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the conventional dielectric waveguide. Second, we have proposed a reflectionless sharply bent waveguide with a microcavity in the corner region. Numerical results show that reflected waves from the sharp bend can be completely eliminated by adding additional pillars/air-holes due to resonant tunneling, and its transmission bandwidth can be controlled by changing the structure of the microcavity. Finally, we have designed a wavelength multi/demultiplexer that consists of a directional coupler and reflectionless right-angle bent waveguides and demonstrated that the device can work as a low-insertion-loss and high-extinction-ratio device.

REFERENCES [1] E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., 58, 2059–2062, 1987. [2] J.D. Joannopoulos, P.R. Villeneuve, and S. Fan, Photonic crystals: putting a new twist on light, Nature, 386, 143–149, 1997. [3] S.Y. Lin, E. Chow, V. Hietala, P.R. Villeneuve, and J.D. Joannopoulos, Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal, Science, 282, 274–276, 1998. [4] S. Fan, P.R. Villeneuve, J.D. Joannopoulos, and H.A. Haus, Channel drop tunneling through localized states, Phys. Rev. Lett., 80, 960–963, 1998. [5] H. Ikuno, Y. Naka, and A. Yata, Analysis of optical waveguide devices using the FD-TD method based on the principles of multidimensional wave digital filters, Radio Sci., 35, 595–605, 2000. [6] Y. Naka and H. Ikuno. Two-dimensional photonic crystal l-shaped bend waveguide and its application to wavelength multi/demultiplexer, Turkish J. of Electrical Eng. & Comput. Sci., 10, 245–256, 2002. [7] A. Fettweis, Wave digital filters: theory and practice, Proc. IEEE, 74, 270–327, 1986. [8] A. Fettweis, Multidimensional wave digital filters for discrete-time modelling of Maxwell’s equations, Int. J. Numerical Modelling, 5, 183–201, 1992. [9] J.P. Berenger, A perfectly matched layer for the absorption of electromangetic waves, J. Computational Phys., 114, 185–200, 1994. [10] Y. Naka, H. Ikuno, M. Nishimoto, and A. Yata, FD-TD method with PMLS ABC based on the principles of multidimensional wave digital filters for discrete-time modelling of Maxwell’s equations, IEICE Trans. Electron., E81-C, 305–314, 1998. [11] D.E. Johnson, J.L. Hilburn, and J.R. Johnson, Basic Electric Circuit Analysis, Prentice-Hall, New Jersey, 1984. [12] A. Fettweis and G. Nitsche, Transformation approach to numerically integrating PDES by means of WDF principles, Multidimensional Syst. and Signal Process., 2, 127–159, 1991. [13] J.P. Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic waves, J. Computational Phys., 127, 363–379, 1996. [14] D.R. Lynch and K.D. Paulsen, Origin of vector parasitics in numerical Maxwell solution, IEEE Trans. Microwave Theory Tech., 39, 383–394, 1991. [15] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, London, 1995.

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[16] M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide, Appl. Phys. Lett., 76, 952–954, 2000. [17] M. Loncar, D. Nedejikovic, T. Doll, J. Vuckovic, A. Scherer, and T.P. Pearsall, Waveguiding in planar photonic crystals, Appl. Phys. Lett., 77, 1937–1939, 2000. [18] T. Baba, A. Motegi, T. Iwai, N. Fukaya, Y. Watanabe, and A. Sakai, Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate, IEEE J. Quantum Electron., 38, 743–752, 2002. [19] W.L. Ko and R. Mittra, A combination of FD-TD and Prony methods for analyzing microwave integrated circuits, IEEE Trans. Microwave Theory Tech., MTT-39, 2176–2181, 1991. [20] J. Wang and H. Guo, Resonant tunneling through a bend in a quantum wire, Appl. Phys. Lett., 60, 654–656, 1992. [21] Y. Naka and H. Ikuno, Two-dimensional photonic crystal optical waveguides and their application to optical devices, Proc. Int. Union of Radio Science General Assemblies, Maastricht, Netherlands, 2002, D3.O.2. [22] Y. Naka and H. Ikuno, Analysis of two-dimensional photonic crystal sharply bent waveguides, Proc. URSI 2004 Int. Symp. on Electromagnetic Theory, Pisa, Italy, 2004, 748–750. [23] S. Fan, S.G. Johnson, J.D. Joannopoulos, C. Manolatou, and H.A. Haus, Waveguide branches in photonic crystals, J. Opt. Soc. Am. B, 18, 162–165, 2001. [24] P.R. Villeneuve, S. Fan, and J.D. Joannopoulos, Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency, Phys. Rev. B, 54, 7837–7842, 1996.

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