Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media (Springer Series in Optical Sciences)

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Springer Series in

optical sciences founded by H.K.V. Lotsch Editor-in-Chief: W. T. Rhodes, Atlanta Editorial Board: A. Adibi, Atlanta T. Asakura, Sapporo T. W. H¨ansch, Garching T. Kamiya, Tokyo F. Krausz, Garching B. Monemar, Link¨oping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, M¨unchen

For other titles published in this series, go to http://www.springer.com/series/624

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Springer Series in

optical sciences The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics, quantum information, information optics, applied laser technology, industrial applications, and other topics of contemporary interest. With this broad coverage of topics, the series is of use to all research scientists and engineers who need up-to-date reference books. The editors encourage prospective authors to correspond with them in advance of submitting a manuscript. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also www.springer.com/series/624

Editor-in-Chief William T. Rhodes Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Editorial Board Ali Adibi

Bo Monemar

Georgia Institute of Technology School of Electrical and Computer Engineering Atlanta, GA 30332-0250, USA E-mail: [email protected]

Department of Physics and Measurement Technology Materials Science Division Link¨oping University 58183 Link¨oping, Sweden E-mail: [email protected]

Toshimitsu Asakura Hokkai-Gakuen University Faculty of Engineering 1-1, Minami-26, Nishi 11, Chuo-ku Sapporo, Hokkaido 064-0926, Japan E-mail: [email protected]

Theodor W. H¨ansch Max-Planck-Institut f¨ur Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Takeshi Kamiya Ministry of Education, Culture, Sports Science and Technology National Institution for Academic Degrees 3-29-1 Otsuka, Bunkyo-ku Tokyo 112-0012, Japan E-mail: [email protected]

Ferenc Krausz Ludwig-Maximilians-Universit¨at M¨unchen Lehrstuhl f¨ur Experimentelle Physik Am Coulombwall 1 85748 Garching, Germany and Max-Planck-Institut f¨ur Quantenoptik Hans-Kopfermann-Straße 1 85748 Garching, Germany E-mail: [email protected]

Herbert Venghaus Fraunhofer Institut f¨ur Nachrichtentechnik Heinrich-Hertz-Institut Einsteinufer 37 10587 Berlin, Germany E-mail: [email protected]

Horst Weber Technische Universit¨at Berlin Optisches Institut Straße des 17. Juni 135 10623 Berlin, Germany E-mail: [email protected]

Harald Weinfurter Ludwig-Maximilians-Universit¨at M¨unchen Sektion Physik Schellingstraße 4/III 80799 M¨unchen, Germany E-mail: [email protected]

Kurt E. Oughstun

Electromagnetic and Optical Pulse Propagation 2 Temporal Pulse Dynamics in Dispersive, Attenuative Media

With 350Figures

ABC

Professor Dr. Kurt E. Oughstun University of Vermont College of Engineering & Mathematical Sciences School of Engineering Burlington, VT 05405-0156 USA [email protected]

ISSN 0342-4111 e-ISSN 1556-1534 ISBN 978-1-4419-0148-4 e-ISBN 978-1-4419-0149-1 DOI 10.1007/978-1-4419-0149-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009929777 c Springer Science+Business Media, LLC 2009 ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Chosen by beauty to be a handmaiden of the stars, she passes like a silver brush across the lens of a telescope. She brushes the stars, the galaxies and the light-years into the order that we know them. Rommel Drives on Deep into Egypt Poems by Richard Brautigan This volume is dedicated to my best friend Joyce and to our daughters Marcianna & Kristen In memory of my parents Edmund Waldemar Oughstun & Ruth Kinat Oughstun (New Britain, Connecticut) and my grandparents Julies E. Oughstun & Adeline B. Lehmann (Kalwari, Prussia) Heinrich Kinat & Wanda Bucholz (Sladow, Austria-Poland) and my great-grandfather Karl Øvsttun (Øvsttun, Norway)

Preface

This two volume graduate text presents a systematic theoretical treatment of the radiation and propagation of pulsed electromagnetic and optical fields through temporally dispersive and attenuative media. Although such fields are often referred to as transient, they may be short-lived only in the sense of an observation made at some fixed point in space. In particular, because of their unique properties when the initial pulse spectrum is ultrawideband with respect to the material dispersion, specific features of the propagated pulse are found to persist in time long after the main body of the pulse has become exponentially small. The subject matter divides naturally into two volumes. Volume I presents a detailed development of the fundamental theory of pulsed electromagnetic radiation and wave propagation in causal, linear media that are homogeneous and isotropic but otherwise have rather general dispersive and absorptive properties. In Vol. II, the analysis is specialized to the propagation of plane wave electromagnetic and optical pulses in homogeneous, isotropic, locally linear media whose temporal frequency dispersion is described by a specific causal model. Dielectric, conducting, and semiconducting material models are considered with applications to bioelectromagnetics, remote sensing, ground and foliage penetrating radar, and undersea communications. Taken together, these two volumes present sufficient material to cover a two semester graduate sequence in electromagnetic and optical wave theory in physics and electrical engineering as well as in applied mathematics. Either volume by itself could also be used as the text for a single semester graduate level course. Challenging problems are given throughout the text. The development presented in Vol. I provides a mathematically rigorous description of the fundamental time-domain electromagnetics and optics in linear temporally dispersive media. The analysis begins with a general description of macroscopic electromagnetics and the role that causality plays in the constitutive (or material) relations in linear electromagnetics and optics. The angular spectrum of plane waves representation of the pulsed radiation field in homogeneous, isotropic, locally linear, temporally dispersive media is then derived and applied to the description of pulsed electromagnetic and optical beam fields where the effects of temporal dispersion and spatial diffraction are coupled. Volume II begins with a review of pulsed electromagnetic and optical beam field propagation followed by a concise description of modern asymptotic methods of

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approximation appropriate for the description of pulse propagation in dispersive, attenuative media, including uniform and transitional asymptotic techniques. The detailed theory presented here and in Volume I provides the necessary mathematical and physical basis to describe and explain in explicit detail the dynamical pulse evolution as it propagates through a causally dispersive material. This is the subject of a classic theory with origins in the seminal research by Arnold Sommerfield and Leon Brillouin in the early 1900s for a Lorentz model dielectric and described in modern text-books on advanced electrodynamics. This classic theory has been carefully reexamined and extended by George Sherman and myself, beginning in 1974 when I was a graduate student at The Institute of Optics of the University of Rochester and George Sherman was my research advisor. In addition to improving the accuracy of many of the approximations in the classical theory and applying modern asymptotic methods, we have developed a physical model that provides a simplified quantitative algorithm that not only describes the entire dynamical field evolution in the mature dispersion regime but also explains each feature in the propagated field in simple physical terms. This physical model reduces to the group velocity description in the limit as the material loss approaches zero. More recent analysis has extended these results to more general dispersion models, including the RocardPowles extension of the Debye model of orientational polarization in dielectrics and the Drude model of conductivity. Finally, the controversy regarding the question of superluminal pulse propagation is carefully examined in view of recent results establishing the domain of applicability of the group velocity approximation. I would like to acknowledge the financial support I received during my graduate studies by The Institute of Optics, the Corning Glass Works Foundation, the National Science Foundation, and the Center for Naval Research, as well as the encouragement and support from my thesis committee members: Professors Emil Wolf, Carlos R. Stroud, Brian J. Thompson, John H. Thomas, and George C. Sherman. This research continued while I was at the United Technologies Research Center, the University of Wisconsin at Madison, and the University of Vermont. The critical, long-term support of this research by Dr. Arje Nachmann of the United States Air Force Office of Scientific Research and Dr. Richard Albanese of Armstrong Laboratory, Brooks Air Force Base, is gratefully acknowledged. The majority of results presented here have been published in peer-reviewed journals listed in the references. A good portion of this research has been conducted with several of my former graduate students at the University of Wisconsin (Shioupyn Shen) and at the University of Vermont (Judith Laurens, Constantinos Balictsis, Paul Smith, John Marozas, Hong Xiao, and Natalie Cartwright). Their critical insight has been instrumental in several of the theoretical advances presented here. Burlington, VT January 2009

Kurt Edmund Oughstun

Contents

9

Pulsed Electromagnetic and Optical Beam Wavefields in Temporally Dispersive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Real Direction Cosine Form of the Angular Spectrum of Plane Waves Representation . . . . . . . . . . . . . . . . . . . . 9.1.2 Electromagnetic Energy Flow in the Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Homogeneous and Evanescent Plane Wave Contributions to the Angular Spectrum Representation . . . . . 9.1.4 Paraxial Approximation of the Angular Spectrum of Plane Waves Representation . . . . . . . . . . . . . . . . . . . . 9.2 Angular Spectrum Representation of Multipole Wavefields . . . . . . . . . . 9.2.1 Multipole Expansion of the Scalar Optical Wavefield due to a Localized Source Distribution . . . . . . . . . . . 9.2.2 Multipole Expansion of the Electromagnetic Wavefield Generated by a Localized Charge–Current Distribution in a Dispersive Dielectric Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Stationary Phase Asymptotic Approximations of the Angular Spectrum Representation in Free-Space. . . . . . . . . . . . . . . 9.3.1 Approximations Valid Over a Hemisphere . . . . . . . . . . . . . . . . . . . 9.3.2 Approximations Valid on the Plane z D z0 . . . . . . . . . . . . . . . . . . . Q E .r; !/ . . . . . Q H .r; !/ and U 9.3.3 Asymptotic Approximations of U 9.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Separable Pulsed Beam Wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Gaussian Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 The Inverse Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 The Direct Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 6 9 10 17 23 30

37 50 54 58 66 69 70 75 79 79 83 85 88 88 91

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10 Asymptotic Methods of Analysis using Advanced Saddle Point Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 10.1 Olver’s Saddle Point Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.1.1 Peak Value of the Integrand at the Endpoint of Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 10.1.2 Peak Value of the Integrand at an Interior Point of the Path of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.1.3 The Application of Olver’s Saddle Point Method . . . . . . . . . . . . 103 10.2 Uniform Asymptotic Expansion for Two First-Order Saddle Points at Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 10.3 Uniform Asymptotic Expansion for Two First-Order Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 10.3.1 The Uniform Asymptotic Expansion for Two Isolated First-Order Saddle Points . . . . . . . . . . . . . . . . . . 111 10.3.2 The Uniform Asymptotic Expansion for Two Neighboring First-Order Saddle Points . . . . . . . . . . . . . 113 10.3.3 The Transitional Asymptotic Approximation for Two Neighboring First-Order Saddle Points . . . . . . . . . . . . . 125 10.4 Uniform Asymptotic Expansion for a First-Order Saddle Point and a Simple Pole Singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 10.4.1 The Complementary Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.4.2 Asymptotic Behavior for a Single Interacting Saddle Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10.5 Asymptotic Expansions of Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.5.1 Absolute Maximum in the Interior of the Closure of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.5.2 Absolute Maximum on the Boundary of the Closure of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11 The Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.2 The Pulsed Plane Wave Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . 152 11.2.1 The Delta Function Pulse and the Impulse Response of the Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 11.2.2 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . . . . . . . 161 11.2.3 The Double Exponential Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 11.2.4 The Rectangular Pulse Envelope Modulated Signal . . . . . . . . . 163 11.2.5 The Trapezoidal Pulse Envelope Modulated Signal . . . . . . . . . 165 11.2.6 The Hyperbolic Tangent Envelope Modulated Signal . . . . . . . 169

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11.2.7 The Van Bladel Envelope Modulated Pulse . . . . . . . . . . . . . . . . . . 174 11.2.8 The Gaussian Envelope Modulated Pulse . . . . . . . . . . . . . . . . . . . . 177 11.3 Wave Equations in a Simple Dispersive Medium and the Slowly Varying Envelope Approximation . . . . . . . . . . . . . . . . . . . . . 178 11.3.1 The Dispersive Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11.3.2 The Slowly Varying Envelope Approximation . . . . . . . . . . . . . . . 180 11.3.3 Dispersive Wave Equations for the Slowly Varying Wave Amplitude and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.4 The Classical Group Velocity Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 194 11.5 Failure of the Classical Group Velocity Method . . . . . . . . . . . . . . . . . . . . . . . 200 11.5.1 Impulse Response of a Double-Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.5.2 Heaviside Unit Step Function Signal Evolution. . . . . . . . . . . . . . 212 11.5.3 Rectangular Envelope Pulse Evolution . . . . . . . . . . . . . . . . . . . . . . . 214 11.5.4 Van Bladel Envelope Pulse Evolution . . . . . . . . . . . . . . . . . . . . . . . . 215 11.5.5 Concluding Remarks on the Slowly Varying Envelope and Classical Group Velocity Approximations. . . . 222 11.6 Extensions of the Group Velocity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11.7 Localized Pulsed-Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 11.7.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.7.2 Paraxial Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 11.8 The Necessity of an Asymptotic Description . . . . . . . . . . . . . . . . . . . . . . . . . . 244 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 12 Analysis of the Phase Function and Its Saddle Points . . . . . . . . . . . . . . . . . . . . . 251 12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 12.1.1 The Region About the Origin (j!j  !0 ). . . . . . . . . . . . . . . . . . . . 254 12.1.2 The Region About Infinity (j!j  !m ) . . . . . . . . . . . . . . . . . . . . . . 260 12.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 12.2 The Behavior of the Phase in the Complex !-Plane for Causally Dispersive Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 12.2.1 Single-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . 264 12.2.2 Multiple-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . 279 12.2.3 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . 293 12.2.4 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 12.3 The Location of the Saddle Points and the Approximation of the Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 12.3.1 Single-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . 316 12.3.2 Multiple-Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . 353 12.3.3 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . 366 12.3.4 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 12.3.5 Semiconducting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

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12.4

Procedure for the Asymptotic Analysis of the Propagated Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 12.5 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

13 Evolution of the Precursor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.1 The Field Behavior for  < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 13.2 The Sommerfeld Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 13.2.1 The Nonuniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 13.2.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 13.2.3 Field Behavior at the Wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 13.2.4 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . . . . . . . 407 13.2.5 The Delta Function Pulse Sommerfeld Precursor . . . . . . . . . . . . 408 13.2.6 The Heaviside Step Function Pulse Sommerfeld Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics . . . . . . . . . . 416 13.3.1 The Nonuniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 13.3.2 The Uniform Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 13.3.3 The Instantaneous Oscillation Frequency . . . . . . . . . . . . . . . . . . . . 437 13.3.4 The Delta Function Pulse Brillouin Precursor . . . . . . . . . . . . . . . 439 13.3.5 The Heaviside Step Function Pulse Brillouin Precursor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 13.4 The Brillouin Precursor Field in Debye Model Dielectrics . . . . . . . . . . . 445 13.5 The Middle Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 13.6 Impulse Response of Causally Dispersive Materials . . . . . . . . . . . . . . . . . . 454 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 14 Evolution of the Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 14.1 The Nonuniform Asymptotic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 468 14.2 Rocard–Powles–Debye Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 14.3 The Uniform Asymptotic Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 14.4 Single Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . 478 14.4.1 Frequencies below the Absorption Band . . . . . . . . . . . . . . . . . . . . . 479 14.4.2 Frequencies above the Absorption Band . . . . . . . . . . . . . . . . . . . . . 483 14.4.3 Frequencies within the Absorption Band . . . . . . . . . . . . . . . . . . . . . 485 14.4.4 The Heaviside Unit Step Function Signal . . . . . . . . . . . . . . . . . . . . 488 14.5 Multiple Resonance Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . . . . . . 494 14.6 Drude Model Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Contents

xiii

15 Continuous Evolution of the Total Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 15.1 The Total Precursor Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504 15.2 Resonance Peaks of the Precursors and the Signal Contribution . . . . . 507 15.3 The Signal Arrival and the Signal Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 15.3.1 Transition from the Precursor Field to the Signal . . . . . . . . . . . . 509 15.3.2 The Signal Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 15.4 Comparison of the Signal Velocity with the Phase, Group, and Energy Velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 15.5 The Heaviside Step Function Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . 532 15.5.1 Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554 15.5.3 Signal Propagation in a Drude Model Conductor . . . . . . . . . . . . 561 15.5.4 Signal Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 15.5.5 Signal Propagation along a Dispersive Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 15.6 The Rectangular Pulse Envelope Modulated Signal . . . . . . . . . . . . . . . . . . . 572 15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . 574 15.6.2 Rectangular Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . 591 15.6.3 Rectangular Envelope Pulse Propagation in Triply Distilled Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 15.6.4 Rectangular Envelope Pulse Propagation in Saltwater. . . . . . . 603 15.7 Noninstantaneous Rise-Time Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 15.7.1 Hyperbolic Tangent Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . 617 15.7.3 Trapezoidal Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric . . . . . . . . . . . . . . . 619 15.8 Infinitely Smooth Envelope Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric . . . . . . . . . . . . . . . . . 621 15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric . . . . . . . . . . . 638 15.8.3 Brillouin Pulse Propagation in a Rocard– Powles–Debye Model Dielectric; Optimal Pulse Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638

xiv

Contents

15.9

The Pulse Centroid Velocity of the Poynting Vector . . . . . . . . . . . . . . . . . . 643 15.9.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 15.9.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 15.9.3 The Instantaneous Centroid Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 649 15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651 15.10.1 The Singular Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 15.10.2 The Weak Dispersion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 15.11 Comparison with Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 15.12 The Myth of Superluminal Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 669 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 16 Physical Interpretations of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . . . . 679 16.1 Energy Velocity Description of Dispersive Pulse Dynamics . . . . . . . . . . 681 16.1.1 Approximations Having a Precise Physical Interpretation . . 683 16.1.2 Physical Model of Dispersive Pulse Dynamics . . . . . . . . . . . . . . 689 16.2 Extension of the Group Velocity Description . . . . . . . . . . . . . . . . . . . . . . . . . . 701 16.3 Signal Model of Dispersive Pulse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 702 16.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712 17 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 17.1 On the Use and Application of Precursor Waveforms . . . . . . . . . . . . . . . . . 713 17.2 Electromagnetic Energy Dissipation in Causally Dispersive Media . . 716 17.2.1 General Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 17.2.2 Evolved Heat in Lorentz Model Dielectrics . . . . . . . . . . . . . . . . . . 719 17.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720 17.3 Reflection and Transmission Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 17.3.1 Reflection and Transmission at a Dispersive Half-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 728 17.3.2 The Goos–H¨anchen Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 738 17.3.3 Reflection and Transmission at a Dispersive Layer: The Question of Superluminal Tunneling . . . . . . . . . . . . 751 17.4 Optimal Pulse Penetration through Dispersive Bodies . . . . . . . . . . . . . . . . 751 17.4.1 Ground Penetrating Radar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752 17.4.2 Foliage Penetrating Radar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756 17.4.3 Undersea Communications using the Brillouin Precursor . . . 759 17.5 Ultrawideband Pulse Propagation through the Ionosphere . . . . . . . . . . . . 761 17.6 Health and Safety Issues Associated with Ultrashort Pulsed Electromagnetic Radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 17.7 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

Contents

xv

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 F

Asymptotic Expansion of Single Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777 F.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 F.2 Asymptotic Sequences, Series and Expansions . . . . . . . . . . . . . . . . . . . . . . . . 783 F.3 Integration by Parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 F.4 The Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792 F.5 Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794 F.6 Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800 F.7 The Method of Steepest Descents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 810

G Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 H The Radon Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 819 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825

Chapter 9

Pulsed Electromagnetic and Optical Beam Wavefields in Temporally Dispersive Media

The macroscopic electromagnetic field behavior in a homogeneous, isotropic, locally linear (HILL), temporally dispersive medium with no externally supplied charge or current sources is described by the macroscopic Maxwell’s equations1 [see (5.12)–(5.15) of Vol. 1] r  D.r; t / D 0;

   1  @B.r; t /  r  E.r; t / D    c  @t ; r  B.r; t / D 0;      1  @D.r; t /  4     r  H.r; t / D   C  c  Jc .r; t /; c @t taken together with the constitutive relations Z

t

Z

1 t

Z

1 t

D.r; t / D H.r; t / D Jc .r; t / D

O .t  t 0 /E.r; t 0 /dt0 ; O 1 .t  t 0 /B.r; t 0 /dt0 ; O .t  t 0 /E.r; t 0 /dt0 ;

1

as described in (4.85), (4.141), and (4.113) of Vol. 1, respectively, where the conduction current density satisfies the equation of continuity [see (5.8)–(5.11) of Vol. 1] r  Jc .r; t / D 0: 1

As in Vol. 1, the analysis is presented here in such a manner that it can be interpreted in both cgs (Gaussian) units and mksa units. This is accomplished by writing each equation such that it can be interpreted in mksa units provided that the factor * in the double brackets kk is omitted, while the inclusion of this factor yields the appropriate expression in cgs units. If there is no double bracketed quantity, the expression can be interpreted equally in Gaussian or mksa units unless specifically noted otherwise. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 9, 

1

2

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Hence, the electric displacement vector D.r; t /, the electric field intensity vector E.r; t /, the magnetic induction vector B.r; t /, the magnetic field intensity vector H.r; t /, and the conduction current density vector Jc .r; t / are all solenoidal vector fields in the source-free medium. Here O .t / denotes the (real-valued) dielectric permittivity, .t O / denotes the (real-valued) magnetic permeability, and O .t / denotes the (real-valued) electric conductivity response of the simple dispersive medium. By O  t 0 / D 0, and O .t  t 0 / D 0 for t 0 > t , as exhibited in causality, O .t  t 0 / D 0, .t the upper limit of integration in the above three constitutive relations. In any HILL medium, the angular spectrum of plane waves representation expresses any freely propagating electromagnetic wavefield that propagates into the positive half-space z  z0 as a superposition of both homogeneous and inhomogeneous plane waves [1, 2]. The inhomogeneous plane wave components are unimportant in many instances when the material is lossless because they become evanescent in the propagation distance. Because of this evanescence, an electromagnetic beam field in a lossless medium is typically defined, in part, by the requirement that it can be represented by an angular spectrum that contains only homogeneous plane wave components [3], which then typically leads to the paraxial approximation that is so widely used in Fourier optics [4]. As the homogeneous waves do not attenuate in such a lossless medium, this condition ensures that all of the angular spectrum components of the beam field remain in the original proportions throughout the half-space z  z0 . The situation is much more involved in a temporally dispersive, attenuative medium as the plane wave spectral components no longer conveniently separate into homogeneous and evanescent constituents. The detailed solution of this problem is then of central importance to this work as it forms the basis for the theoretical description of ultrashort pulse dynamics in causally dispersive media, particularly since diffraction is itself a dispersive phenomenon. Associated with this rigorous description is the question of the separability of pulsed beam fields given that the intial field is separated into the product of a temporal and a spatial part. Such space and time separability is trivially obtained for pulsed plane wavefields, but not so in the general case of a pulsed beam wavefield. The related topics of multipole expansions, local pulsed beam solutions, electromagnetic “bullets,” and the inverse initial value problem are also introduced and considered with sufficient detail to provide a sound basis for further research. Although these latter topics are not essential for the topics considered in the remainder of the book, and so may be omitted without any loss of pedagogical continuity, they nevertheless do serve to illustrate the sweeping breadth of this research area.

9.1 Angular Spectrum Representation A completely general representation of the propagation of a freely propagating electromagnetic wavefield into the half-space z  z0 > Z of an HILL temporally dispersive medium is now considered. The phrase “freely-propagating” is used

9.1 Angular Spectrum Representation Fig. 9.1 Geometry of the planar electromagnetic boundary value problem

3 E0(rT ,t) = E0(x,y,t) H0(rT ,t) = H0(x,y,t)

v

1z z

Half-Space z z0

Plane z = z0

here2 to indicate that there are no externally supplied charge or current sources for the field present in this half-space, the field source residing somewhere in the region z  Z. It is unnecessary to know what this source is provided that the pair fE0 ; B0 g of electromagnetic field vectors are known functions of time and the transverse position vector rT D 1O x x C 1O y y in the plane z D z0 , as illustrated in Fig. 9.1. The rigorous, formal solution of this planar boundary value problem for the electromagnetic field in the half-space z  z0 forms the basis of investigation for a wide class of pulsed electromagnetic beam field problems in both optics and electrical engineering. Consider an electromagnetic wavefield that is propagating into the half-space z  z0 > Z > 0 and let the electric and magnetic field vectors on the plane z D z0 , denoted by E.rT ; z0 ; t / D E0 .rT ; t /;

(9.1)

B.rT ; z0 ; t / D B0 .rT ; t /;

(9.2)

be known functions of time and the transverse position vector rT 1O x x C 1O y y in the plane z D z0 , as indicated by the 0 subscript, as depicted in Fig. 9.1. It is 2

A freely propagating field is fundamentally different from a free-field whose externally supplied charge and current sources have all been turned off. The spatiotemporal properties of free-fields are considered in Chap. 8 of Vol. 1.

4

9 Pulsed Beam Wavefields in Temporally Dispersive Media

assumed here that the two-dimensional spatial Fourier transform in the transverse coordinates as well as the temporal Fourier–Laplace transform of each field vector on the plane z D z0 exists, where QQ .k ; !/ D U 0 T

Z

Z

1

1

dt dxdy U0 .rT ; t /e i.kT rT !t/ ; (9.3) 1 1 (Z ) Z 1 1 QQ .k ; !/e i.kT rT !t/ ; (9.4) < d! d kx d ky U U0 .rT ; t / D 0 T 4 3 CC 1

where kT 1O x kx C 1O y ky is the transverse wave vector. Here U0 .rT ; t / represents either the initial electric E0 .rT ; t / or magnetic B0 .rT ; t / field vector at the QQ .k ; !/ then represents the corresponding plane z D z0 ; the spectral field vector U 0 T Q Fourier–Laplace transform EQ 0 .kT ; !/ or BQQ 0 .kT ; !/ of the initial electric or magnetic field vector at that plane. If the initial time dependence of either field vector E0 .rT ; t / or B0 .rT ; t / at the plane z D z0 is such that it vanishes identically for all t < t0 for some finite value of t0 , then the transform relations appearing in (9.3) and (9.4) are Laplace transformations and the contour of integration CC is the straight line path ! D ! 0 C ia, with a greater than the abscissa of absolute convergence for the initial time behavior of the field [5] and with ! 0 0 is a positive constant with " < 1=3. The integral in (9.195) can then be written in the form Q J1 .x; y; z0 ; !/ C U Q J 2 .x; y; z0 ; !/ Q J .x; y; z0 ; !/ D U U

(9.223)

with J D H; E, where Q J1 .x; y; z0 ; !/ D U Q J 2 .x; y; z0 ; !/ D U

Z Z

DJ1

DJ 2

QQ 0 .p; q/U.p; q; !/e ik.pxCqyCmz0 / dpdq;

(9.224)

QQ Œ1  0 .p; q/ U.p; q; !/e ik.pxCqyCmz0 / dpdq: (9.225)

9.3 Stationary Phase Asymptotic Approximations

59

The various regions of integration in the p; q-plane appearing in these integral expressions are defined as ˚ DH1 .p; q/ j 1  3"  p 2 C q 2  1 ; ˚ DH 2 .p; q/ j 0  p 2 C q 2  1  "=2 ; ˚ DE1 .p; q/ j 1  p 2 C q 2  1 C 3" ; ˚ DE2 .p; q/ j p 2 C q 2  1 C "=2 ; and are depicted in Fig. 9.18. The asymptotic approximation of each of the integrals appearing in (9.224) and (9.225) is now separately considered.

9.3.2.1

The Region DH2

Q H2 .x; y; z0 ; !/ is found [32] The integrand appearing in the expression (9.225) for U to satisfy all of the conditions required in Theorem 1 of Chako [36]. Because the region DH2 does not contain any critical points of that integrand, and because the QQ amplitude function Œ1  0 .p; q/ U.p; q; !/ and all N of its partial derivatives with respect to p and q vanish on the boundary of DH2 , it then follows that ˚ Q H 2 .x; y; z0 ; !/ D O .kR/N U

(9.226)

as kR ! 1 with fixed k > 0.

9.3.2.2

The Region DE2

Q E2 .x; y; z0 ; !/ is also found The integrand appearing in the expression (9.225) for U [32] to satisfy the conditions in Theorem 1 of Chako [36], but the integration domain DE2 appearing in (9.225) extends to infinity whereas Chako’s proof is given only for a finite domain of integration. Its extension to the case of an infinite domain has been given by Sherman, Stamnes, and Lalor [32] who consider the same integral but with the region of integration now given by 1 C "=2  p 2 C q 2  K, where K > 1 C "=2 is an arbitrary constant that is allowed to go to infinity. This change then introduces two complications into Chako’s proof. For the first, the amplitude QQ function U.p; q; !/ and its partial derivatives with respect to p and q do not, in general, vanish on the boundary p 2 Cq 2 D K for finite K, but they do vanish in the limit as Kp ! 1 because each of these boundary terms contains the exponential factor e z0 K1 with z0 > 0.8 The second complication occurs in the remainder integral 8

Notice that the special case when z0 D 0 can be treated only if additional restrictions are placed QQ on the behavior of the spectral amplitude function U.p; q; !/ and its partial derivatives with respect 2 2 to p and q in the limit as p C q ! 1.

60

9 Pulsed Beam Wavefields in Temporally Dispersive Media

after integration by parts N times, which is now taken over an infinite domain. However, all that is required in the proof is that this integral p is convergent and this is z0 p 2 Cq 2 1 guaranteed by the presence of the exponential factor e in the integrand. Hence, the result of the theorem due to Chako applies in this case, so that ˚ Q E2 .x; y; z0 ; !/ D O .kR/N U

(9.227)

as kR ! 1 with fixed k > 0.

9.3.2.3

The Region DH1

Attention is now turned to obtaining the asymptotic approximation of Q H1 .x; y; z0 ; !/ as kR ! 1 with fixed k > 0. Under the change of variable U p D sin ˛ cos ˇ, q D sin ˛ sin ˇ the integral representation appearing in (9.224) becomes Q H1 .x; y; z0 ; !/ D U

Z

ˇ0 C2 ˇ0

Z

=2

˛10

A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ;

(9.228)

with fixed direction cosines 1 D sin # cos ', 2 D sin # sin ' [see (9.201)] with 0  #p < =2 and 0 p' < 2. Here ˇ0 is an arbitrary real constant, ˛10 arcsin 1  3" D arccos 3", and A0 .˛; ˇ/ 0 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛;

(9.229)

with m D .1  p 2  q 2 /1=2 D cos ˛ and V .p; q; m/ is as defined in (9.204). Since ˇ0 is an arbitrary constant, it is chosen for later convenience to be given by ˇ0 D '  =4. The integral appearing in (9.228) is now in the form of the integral given in (9.202) with phase and amplitude functions f .˛; ˇ/ D sin ˛ cos .ˇ  '/; g.˛; ˇ/ D A0 .˛; ˇ/;

(9.230) (9.231)

respectively. Because both of these functions satisfy all of the requirements necessary for application of the method of stationary phase, this method may now be directly applied. The critical points of the integral in (9.228) occur at the stationary phase points that are defined by the condition [cf. (9.203)] ˇ ˇ @f ˇˇ @f ˇˇ D D 0; @˛ ˇ.˛s ;ˇs / @ˇ ˇ.˛s ;ˇs /

(9.232)

9.3 Stationary Phase Asymptotic Approximations

61

so that both cos ˛ cos .ˇ  '/ D 0 ) ˛ D =2 _ ˇ D ' C =2; ' C 3=2 and sin ˛ sin .ˇ  '/ D 0 ) ˇ D '; ' C . The stationary phase points are then Point a: Point b:

.˛s ; ˇs / D .=2; '/; .˛s ; ˇs / D .=2; ' C /;

both of which occur on the boundary of the integration region DH1 , as illustrated in Fig. 9.19. Additional critical points occur at the two corners .=2; ˇ0 / and .=2; ˇ0 C 2/. Because of the above chioice that ˇ0 D '  =4, neither of the stationary phase points coincide with a boundary corner of the integration domain, as depicted in Fig. 9.19. Consider first the contribution from the two corner critical points which are just an artifact of the change of variables of integration. Because their location along the line ˛ D =2 is completely determined by the choice of the constant ˇ0 , it is expected that, taken together, they do not contribute to the asymptotic behavior of

f/ '' f/

''

b

''

f/

a

'' ''

f/

f/

'

Fig. 9.19 Illustration of the integration region DH1 appearing in (9.228) and the location of the critical points appearing in that integral

62

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Q H1 .x; y; z0 ; !/ as kR ! 1. The verification of this expected result is given by U the following argument [32]. Construct a periodic neutralizer function  00 .ˇ/ with period 2 that is a real, continuous function of ˇ with continuous derivatives of all orders, and is such that  00 .ˇ/ D 1;

when

'  =8  ˇ  ' C 9=8;

and  00 .ˇ/ D 0 in some neighborhood of ˇ0 , as depicted on the left side of Fig. 9.19. Q H1 .x; y; z0 ; !/ may then be exThe integral representation (9.228) of the field U pressed as Q H1 .x; y; z0 ; !/ D U

Z Z

ˇ0 C2 ˇ0 ˇ0 C2

C ˇ0

Z

=2

˛10

Z

=2

˛10

 00 .ˇ/A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ   1   00 .ˇ/ A0 .˛; ˇ/e ikR sin ˛ cos .ˇ'/ d˛dˇ:

Because the integrand vanishes (due to the neutralizer function) at the corner points, the only critical points in the first integral are the stationary phase points labeled a and b in Fig. 9.19. Moreover, because  00 .ˇ/ D 1 in a separate neighborhood of each of the stationary phase points, the asymptotic behavior of this first integral is identical to the stationary phase point contributions to the asymptotic behavior of Q H1 .x; y; z0 ; !/ as kR ! 1. On the other hand, because the quantity .1   00 .ˇ// U vanishes at both of the stationary phase points, the only critical points of importance in the second integral are the corner points. Because the integrand in this second integral is a periodic function of ˇ with period 2, the integrand can be made to vanish at the corner points simply by changing the region of integration so that it extends from '9=8 to 'C7=8. It then from Theorem 1 of the paper [36] ˚ follows by Chako that the second integral is O .kR/N as kR ! 1. Q H1 .x; y; z0 ; !/ may then be expressed as The field U ˚ Q .a/ .x; y; z0 ; !/ C U Q .b/ .x; y; z0 ; !/ C O .kR/N ; (9.233) Q H1 .x; y; z0 ; !/ D U U H1 H1 .a/

.b/

Q .x; y; z0 ; !/ denote the separate contributions to Q .x; y; z0 ; !/ and U where U H1 H1 Q H1 .x; y; z0 ; !/ from the stationary phase points a the asymptotic behavior of U and b, respectively. To obtain the separate asymptotic approximations of both Q .b/ .x; y; z0 ; !/, the asymptotic expansion due to a boundQ .a/ .x; y; z0 ; !/ and U U H1 H1 ary stationary phase point of the hyperbolic type must be used [32]. This has been shown to be of the same form as that for a boundary stationary phase point of the elliptic type, which has been given in Sects. 3.3–3.4 of Bremmerman [38]. Thus Q .j1 / .x; y; z0 ; !/ C U Q .j2 / .x; y; z0 ; !/; Q .j / .x; y; z0 ; !/ D U U H1 H1 H1

(9.234)

9.3 Stationary Phase Asymptotic Approximations

63

with N

Q .j1 / .x; y; z0 ; !/ U H1

.j / 2 ˚ BHn1 .'/ e ˙ikR X D C o .kr/N=4 ; n kR nD0 .kR/

(9.235)

N

.j / 2 1 ˚ BHn2 .'/ e ˙ikR X .j2 / Q UH1 .x; y; z0 ; !/ D C o .kr/N=4 ; 3=2 n .kR/ .kR/ nD0 .j /

(9.236)

.j /

as kR ! 1 with fixed k > 0, where BHn1 .'/ and BHn2 .'/ are both independent of R. The upper sign in the exponential appearing in (9.235)–(9.236) is used when j D a and the lower sign is used when j D b. Explicit expressions for the coeffi.j / .j / cients BH1n .'/ and BH2n .'/ are given in Appendix II of the paper [32] by Sherman, Stamnes, and Lalor.

9.3.2.4

The Region DE 1

Q E1 .x; y; z0 ; !/ as Consider finally obtaining the asymptotic approximation of U kR ! 1 with fixed k > 0. With the change of variable defined by p D cosh  cos ; q D cosh  sin ;

(9.237) (9.238)

m D Ci sinh ;

(9.239)

so that with Jacobian J.p; q=; / D sinh  cosh , the integral appearing in (9.224) becomes Q E1 .x; y; z0 ; !/ D U

Z

ˇ0 C2 ˇ0

where 1 sinh1

Z

1

A00 .; /e ikR cosh ./ cos .'/ dd;

(9.240)

0

p  p  3 D cosh1 1 C 3 and

A00 .; / i 0 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / cosh ;

(9.241)

with p and q as given in (9.237) and (9.238). The integral appearing in (9.240) is now in the form of the integral in (9.202) with amplitude and phase functions g.; / D A00 .; /; f .; / D cosh ./ cos .  '/; respectively.

(9.242) (9.243)

64

9 Pulsed Beam Wavefields in Temporally Dispersive Media

As was found for the integral in (9.228) for the spectral field component Q H1 .x; y; z0 ; !/, all of the critical points lie on the boundary @DE1 of the inteU gration domain. Just as in that case, the critical points on the boundary  D 1 and at the corners .; / D .0; 0/ and .; / D .0; 2/ do not contribute to the asymptotic behavior of the integral in (9.240). The asymptotic behavior of Q E1 .x; y; z0 ; !/ is then completely determined by the the spectral field component U contributions from the stationary phase points (a) at .s ; s / D .0; '/ and (b) at Q E1 .x; y; z0 ; !/ is expressed as .s ; s / D .0; ' C /. As in (9.233), the field U ˚ Q .a/ .x; y; z0 ; !/ C U Q .b/ .x; y; z0 ; !/ C O .kR/N ; (9.244) Q E1 .x; y; z0 ; !/ D U U E1 E1 .a/

.b/

Q .x; y; z0 ; !/ denote the separate contributions to the Q .x; y; z0 ; !/ and U where U E1 E1 Q H1 .x; y; z0 ; !/ from the stationary phase points (a) and (b), asymptotic behavior of U respectively. An analysis similar to that following (9.233) then yields [32] Q .j1 / .x; y; z0 ; !/ C U Q .j2 / .x; y; z0 ; !/; Q .j / .x; y; z0 ; !/ D U U E1 E1 E1

(9.245)

with N

Q .j1 / .x; y; z0 ; !/ U E1

.j / 2 ˚ BEn1 .'/ e ˙ikR X D C o .kr/N=4 ; n kR nD0 .kR/

(9.246)

N

.j / 2 1 ˚ BEn2 .'/ e ˙ikR X .j2 / Q UE1 .x; y; z0 ; !/ D C o .kr/N=4 ; 3=2 n .kR/ .kR/ nD0 .j /

(9.247)

.j /

as kR ! 1 with fixed k > 0, where BEn1 .'/ and BEn2 .'/ are both independent of R. The upper sign in the exponential appearing in (9.246) and (9.247) is used when j D a and the lower sign is used when j D b.

9.3.2.5

.j /

.j /

Relationship between the Coefficients BH 1n .'/, BH 2n .'/, .j / .j / and BEn1 .'/, BEn2 .'/ .j /

.j /

The various relationships between the coefficients BH1n .'/ and BH2n .'/ in Q H1 .x; y; z0 ; !/ and the asymptotic expansion given in (9.235) and (9.236) for U .j1 / .j2 / the coefficients BEn .'/ and BEn .'/ in the asymptotic expansion given in (9.246) Q E1 .x; y; z0 ; !/ are given by Jones and Kline [35], as modified by and (9.247) for U Sherman, Stamnes, and Lalor [32], as9 9

Explicit expressions for all of these coefficients are unnecessary for the final asymptotic expansion and so are not given here. The interested reader should consult Appendices I–III in the 1976 paper [32] by Sherman, Stamnes, and Lalor.

9.3 Stationary Phase Asymptotic Approximations .a /

65

BEn1 .'/ D CBH 1n .'/;

.a /

(9.248)

.a / BEn2 .'/ .b / BEn1 .'/ .b / BEn2 .'/

.a / BH 2n .'/; .b / BH1n .'/; .b / BH2n .'/:

(9.249)

D D D

(9.250) (9.251)

The pair of relations in (9.250) and (9.251) then show that the contribution of Q E1 .x; y; z0 ; !/ is equal in magnitude but opposite in sign with the point b to U Q H1 .x; y; z0 ; !/. Consequently, the point b does the contribution of the point b to U Q H1 .x; y; z0 ; !/ C not contribute to the asymptotic behavior ˚of the field quantity U Q E1 .x; y; z0 ; !/ with order lower than O .kR/N=4 that arises from the sum of U the second terms on the right hand side of (9.233) and (9.244). In a similar manner, (9.249) implies that the contribution of the point a to the series involving inverse half Q E1 .x; y; z0 ; !/ is equal in magnitude but oppopowers of .kR/ in the expansion of U Q H1 .x; y; z0 ; !/. Taken together site in sign to the same contribution of the point a to U with the previous result, this implies that the asymptotic expansion of the field quanQ E1 .x; y; z0 ; !/ does not include terms with half powers of Q H1 .x; y; z0 ; !/ C U tity U ˚ .kR/ of order lower than O .kR/N=4 . Finally, the relation given in (9.248) implies Q E1 .x; y; z0 ; !/ contribute equally Q H1 .x; y; z0 ; !/ and U that the field components U Q to the remaining terms in the asymptotic expansion of the total field U.x; y; z0 ; !/ involving only inverse integral powers of .kR/. It then follows from this result together with (9.194), (9.223), (9.227), and (9.235) that N

.a / 2 ˚ BH 1n .'/ e ikR X Q C o .kr/N=4 ; U.x; y; z0 ; !/ D 2 n kR nD0 .kR/

(9.252)

as kR ! 1 with fixed k > 0 and z D z0 . .a / The coefficients BH 1n .'/ appearing in the asymptotic expansion (9.252) are found [32] to be given by the limiting expression .a /

BH 1n .'/ D

1 lim Bn .#; '/ 2 ! 2

(9.253)

of the coefficients Bn .#; '/ appearing in the asymptotic expansion (9.212). It then follows that the results of this section can be combined with those of Sect. 9.3.1 to yield the asymptotic expansion N

2 ˚ Bn .#; '/ e ikR X Q C o .kr/N=4 ; U.x; y; z0 ; !/ D kR nD0 .kR/n

(9.254)

as kR ! 1 with fixed k > 0 and z0 > 0 that is uniformly valid with respect to the angles # and ' in their respective domains 0  #  =2 and 0  '  2. The zeroth-order coefficient B0 .#; '/ is given in (9.213) and the first-order coefficient is given by B1 .#; '/ D .i=2/L2 B0 .#; '/, where the differential operator L2 is defined in (9.216).

66

9 Pulsed Beam Wavefields in Temporally Dispersive Media

Q H .r; !/ and U Q E .r; !/ 9.3.3 Asymptotic Approximations of U The respective dominant terms in the asymptotic expansion of the homogeneous Q H .r; !/ and in the asymptotic expansion of the evanescent field field component U Q component UE .r; !/ are now derived for each of the three direction cosine regions 0 < 3 < 1, 3 D 0, and 3 D 1. 9.3.3.1

Approximations Valid over the Hemisphere 0 < 3 < 1

To obtain the asymptotic approximation of the homogeneous wave contribution Q H .r; !/ to the total spectral wavefield, the notation and definitions used in (9.221)– U (9.225) are applied with z0 replaced by z. The only critical point of the integral for Q H 2 .r; !/, where the integrand is nonzero, is the interior stationary phase point U Q H 2 .r; !/ is then seen to be iden.ps ; qs / D .1 ; 2 /. The asymptotic expansion of U Q !/ given in (9.212) and (9.217). tical with the asymptotic expansion of U.r; It then follows from the above result together with (9.194) that the asymptotic Q E .r; !/ to the total wavefield is expansion of the evanescent wave contribution U Q H1 .r; !/. The asymptotic expansion of the integral expression identical to that of U Q H1 .x; y; z; !/ is treated in the same manner as was employed for Q H1 .r; !/ D U for U Q H1 .x; y; z0 ; !/ in Sect. 9.3.2. The change of integration variables given by p D U sin ˛ cos ˇ, q D sin ˛ sin ˇ that was used in the proof of Theorem 1 is first made, resulting in the integral [cf. (9.228)] Z

ˇ0 C2

Z

=2

A0 .˛; ˇ/e ikRf .˛;ˇ/ d˛dˇ;

(9.255)

f .˛; ˇ/ D sin # sin ˛ cos .ˇ  '/ C cos # cos ˛:

(9.256)

Q H1 .x; y; z; !/ D U

˛10

ˇ0

with phase function [cf. (9.230)]

The location of the critical points for this phase function are the same as those depicted in Fig. 9.19. As in that case, the points a and b are the only critical points that contribute to the asymptotic behavior of the integral under consideration. However, unlike that case, the saddle points here are not ordinary stationary phase points where both @f =@˛ and @f =@ˇ vanish, but rather they are boundary stationary phase points where the isotimic contours f .˛; ˇ/ D constant are tangent to the boundary of the integration domain; in the present case they are both points on the boundary line ˛ D =2 where @f =@ˇ D 0. The resultant asymptotic expansion is then found to be [32, 34, 35] p i=4 h 0 0 Q H1 .x; y; z; !/ D 2= sin #e V.10 ; 20 ; 0/e ik.x0 1 Cy0 2 / e ikR sin # U 3=2 .kR/ cos # i ˚ 0 0 Ci V. 0 ;  0 ; 0/e ik.x0 1 Cy0 2 / e ikR sin # C O .kR/2 1

2

(9.257)

9.3 Stationary Phase Asymptotic Approximations

67

as kR ! 1 with fixed k > 0 and N  8, where 10 D 1 = sin # D cos '; 20 D 2 = sin # D sin ':

(9.258) (9.259)

Q E .r; !/ are then given by Q H .r; !/ and U The resultant asymptotic expansions for U (for N  8) ikR

Q H .x; y; z; !/ D 2 i e V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / U kR ˚ Q H1 .x; y; z; !/ C O .kR/2 ; CU ˚ Q H1 .x; y; z; !/ C O .kR/2 ; Q E .x; y; z; !/ D U U

(9.260) (9.261)

as kR ! 1 with fixed 1 , 2 , and k > 0. Q E .r; !/ is of These results then show that the evanescent wave contribution U 1 Q higher order in .kR/ than the homogeneous wave contribution UH .r; !/. This then Q H .r; !/ Q E .r; !/ in comparison to U provides a rigorous justification for neglecting U as kR ! 1 with fixed k > 0 when 0 < 3 < 1. In addition, the evanescent Q E .r; !/ is of even higher order in .kR/1 in comparison to the wave contribution U Q H .r; !/ as kR ! 1 in the special case when homogeneous wave contribution U V.p; q; 0/ D 0.

9.3.3.2

Approximations Valid on the Plane z D z0

The asymptotic behavior of the spectral wavefield on the plane z D z0 (or, equivalently, when 3 D 0) is directly obtained from the analysis presented in Sect. 9.3.2, the present analysis focusing on obtaining explicit expressions for the dominant terms in the separate homogeneous and evanescent wave contributions. The asympQ H .x; y; z0 ; !/ is totic expansion of the homogeneous spectral wave contribution U obtained from (9.223), (9.226), and (9.233)–(9.236), taken together with the sej1 j2 ries expressions [32] for the coefficients BH n .'/ and BH n .'/, with the result (for N  8) h Q H .x; y; z0 ; !/ D  i V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / U kR

i ˚ V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / C O .kR/3=2 (9.262)

as kR ! 1 with fixed 1 , 2 , and k > 0. Similarly, the asymptotic expansion of Q E .x; y; z0 ; !/ is obtained from (9.223), the evanescent spectral wave contribution U (9.227), (9.244), (9.248), and (9.250) with the result (for N  8)

68

9 Pulsed Beam Wavefields in Temporally Dispersive Media

h Q E .x; y; z0 ; !/ D  i V.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / U kR

i ˚ CV.1 ; 2 ; 0/e ikR e ik.x0 1 Cy0 2 / C O .kR/3=2 (9.263)

as kR ! 1 with fixed 1 , 2 , and k > 0. These two expressions then show that the homogeneous and evanescent wave contributions are of the same order in .kR/1 as Q E .x; y; z0 ; !/ cankR ! 1. As a consequence, the evanescent wave contribution U not, in general, be neglected in comparison to the homogeneous wave contribution Q H .x; y; z0 ; !/ on the plane z D z0 . U

9.3.3.3

Approximations Valid on the Line x D x0 , y D y0

Along the line x D x0 , y D y0 , one has that 3 D 0 and the homogeneous spectral wave contribution can be written as Q H .x0 ; y0 ; z; !/ D U

Z

Z

2

=2

d˛ A.˛; ˇ/e ikz cos ˛ ;

dˇ 0

(9.264)

0

where A.˛; ˇ/ D V.p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛; (9.265) p 2 2 with p D sin ˛ cos ˇ, q D sin ˛ sin ˇ, and m D 1  p  q D cos ˛ [see the proof of Theorem 1]. Because the phase function in (9.264) does not involve the integration variable ˇ, the asymptotic behavior of the integral can be obtained from the single integral Q H .x0 ; y0 ; z; !/ D U

Z

=2

A.˛/e ikR cos ˛ d˛;

(9.266)

0

where R D z  z0 , with A.˛/ e ikz0 cos ˛

Z

2

A.˛; ˇ/dˇ:

(9.267)

0

Application of the method of stationary phase for single integrals in Appendix F shows that the only contributions to the asymptotic behavior of the integral in (9.266) arise from the endpoints of the integral at ˛ D 0 and ˛ D =2 with the result [see (F.50)] ikR ˚ Q H .x0 ; y0 ; z; !/ D 2 i e V.0; 0; 1/e ikz0 C i A.=2/CO .kR/3=2 (9.268) U kR kR

as kR ! 1 with k > 0 and for N  4.

9.3 Stationary Phase Asymptotic Approximations

69

Because the first term in (9.268) is the dominant term in the asymptotic approxQ 0 ; y0 ; z; !/, (9.194) then shows that the asymptotic behavior of the imation of U.x evanescent wave contribution is given by ˚ Q E .x0 ; y0 ; z; !/ D  i A.=2/ C O .kR/3=2 U kR

(9.269)

as kR ! 1 with k > 0. The homogeneous and evanescent wave contributions are again seen to be of the same order in .kR/1 as kR ! 1 so that, in general, Q H .x0 ; y0 ; z; !/ for large Q E .x0 ; y0 ; z; !/ cannot be neglected in comparison to U U kR ! 1.

9.3.4 Summary The stationary phase asymptotic expansions presented in this section are all valid as kR ! 1 with fixed wavenumber k > 0 and fixed direction cosines 1 D .x  x0 /=R, 2 D .y  y0 /=R with N  12 in a nonabsorptive (and hence, strictly speaking, nondispersive) medium. Less restrictive conditions on N result in special cases. Q !/ is the same for all The asymptotic behavior of the spectral wavefield U.r; 3 D .z  z0 /=R such that 0  3  1 and is given by [cf. (9.254)]   i e ikR 2 Q 1C L V.1 ; 2 ; 3 /e ik.1 x0 C2 y0 C3 z0 / U.r; !/ D 2 i kR 2kR ˚ (9.270) CO .kR/3 as kR ! 1 with fixed k > 0, where L2 is the differential operator defined in (9.216). The asymptotic behavior of the separate homogeneous and evanescent comQ J .r; !/, J D H; E, however, depends on the value of 3 , ponent wavefields U separating into the three cases (a) 0 < 3 < 1, (b) 3 D 0, and (c) 3 D 1, as described in Sect. 9.3.3. The results show that the evanescent wave contriQ E .r; !/ is negligible in comparison to the homogeneous wave contribution U Q H .r; !/ for large kR ! 1 in case (a), but not necessarily in cases bution U (b) and (c). In some important applications of the angular spectrum representation in electromagnetic wave theory, integral representations of the form given in (9.194) and QQ (9.195) are obtained with U.p; q; !/ … TN because of the presence of isolated singularities in the integrand. This occurs, for example, in the analysis of the refelection and refraction of a nonplanar wavefield (e.g., an electromagnetic beam field) at a planar interface separating two different media [37, 40]. A neutralizer function can QQ then be used to isolate each singularity. Because U.p; q; !/ 2 TN , the asymptotic approximation of the resultant integral that does not contain any of the singularities

70

9 Pulsed Beam Wavefields in Temporally Dispersive Media

can then be obtained using the two-dimensional stationary phase results of Sherman, Stamnes, and Lalor [32] presented here. Each of the remaining integrals contains QQ one of the isolated singularities, U.p; q; !/ … TN and so its asymptotic approximation must be obtained using some other technique. In some case, as in the reflection and refraction problem [40], a change of integration variable can result in a transformed integral that is amenable to the stationary phase method presented here. If that is not the case, uniform asymptotic methods [41–43] may then need to be employed.

9.4 Separable Pulsed Beam Wavefields A problem of special interest in both optics and electromagnetics is that in which the spatial and temporal properties of a pulsed electromagnetic (or optical) beam wavefield are separable. The general conditions under which this simplifying assumption is valid are presented in Sect. 7.4.2 of Vol. 1. In this case, the spatial and temporal properties of the initial field vectors at the input plane z D z0 are assumed to be separable in the sense that [44] E0 .rT ; t / D E0 .x; y; t / D EO 0 .x; y/f .t /; B0 .rT ; t / D B0 .x; y; t / D BO 0 .x; y/g.t /;

(9.271) (9.272)

in which case EQQ 0 .p; q; !/ D EQO 0 .p; q; k/fQ.!/; BQQ 0 .p; q; !/ D BQO 0 .p; q; k/g.!/: Q

(9.273) (9.274)

One now has the spatial Fourier transform pair relations QO .p; q; k/ D U 0

Z

1

Z

1

O 0 .x; y/ D U

1 .2/2

1

O 0 .x; y/e ik.pxCqy/ dxdy; U

(9.275)

1

Z

Z

1

1

1

QO .p; q; k/e ik.pxCqy/ k 2 dpdq U 0

(9.276)

1

for both EO 0 .x; y/ and BO 0 .x; y/, and the separate Fourier–Laplace transform pair relations Z 1 Q h.!/ D h.t /e i!t dt; (9.277) 1

1 h.t / D < 

(Z

) i!t Q d! h.!/e CC

(9.278)

9.4 Separable Pulsed Beam Wavefields

71

for both f .t / and g.t /. It is seen from (9.8) that these functions cannot be independently chosen because their spectra must satisfy the relation BQO 0 .p; q; k/g.!/ Q D ..!/c .!//1=2 sO  EQO 0 .p; q; k/fQ.!/;

(9.279)

where sO D 1O x p C 1O y q C 1O z m.!/. The separation of this relation into spatial and temporal frequency components as OQ 0 .p; q; k/; c BQO 0 .p; q; k/ D sO  E 1=2  g.!/ Q D .!/c .!/ fQ.!/;

(9.280) (9.281)

does not provide a unique solution of (9.279) except in special cases. Consider now the propagation of such a separable pulsed wavefield in a (fictitious) dispersive medium that is lossless. Substitution of the generic form of the separibility condition given in (9.273) and (9.274) into the angular spectrum representation given in (9.35) then yields (Z Z 1 i!t Q < d! h.!/e k 2 .!/dpdq U.r; t / D 4 3 CC R< [R>

) Q O 0 .p; q; k/G.p; q; !; z/e ik.!/.pxCqy/ ; U (9.282)

˚ q/j p 2 C q 2 < 1 defines the homogeneous plane wave domain, where R ˚ < D .p; R> D .p; q/j p 2 C q 2 > 1 defines the evanescent plane wave domain, and where [see (7.195)] G.p; q; !; z/ e ik.!/m z D e ik.!/ z.1p

2 q 2 1=2

/

(9.283)

is the propagation factor for a plane wave progressing in the direction specified by the direction cosines .p; q; m/. As described in Sect. 7.4 of Vol. 1, because the evanescent plane wave components in the angular spectrum representation do not provide any time-average energy flow into the positive half-space z > z0 [see (9.37)], an electromagnetic beam field is defined, in part, by the requirement that its angular spectrum does not contain any evanescent plane wave components [3], and consequently can be represenated by an angular spectrum that contains only homogeneous plane wave components. Because the homogeneous plane wave spectral components do not attenuate with propagation distance in a lossless medium, this condition ensures that all of the angular spectrum components of the beam field are maintained in their initial proportion throughout the positive half-space z  z0 . Such wavefields are known as

72

9 Pulsed Beam Wavefields in Temporally Dispersive Media

source-free wavefields [45, 46] and, as such, possess several rather unique properties [see Sect. 7.4 of Vol. 1]. QO .p; q; k/ of the sepAs a consequence, if the initial spatial frequency spectrum U 0 arable pulsed beam wavefield contains only homogeneous plane wave components, then the inner integration domain R< can be extended to all of p; q-space and the propagation factor G.p; q; !; z/ may then be represented by its Maclaurin’s series expansion as [see (7.215–7.216)] G.p; q; !; z/ D

1 X 1 X G .2r;2s/ .0; 0; !; z/ rD0 sD0

.2r/Š.2s/Š

p 2r q 2s ;

(9.284)

where G .m;n/ .p; q; !; /

@mCn G.p; q; !; / : @p m @q n

(9.285)

Substitution of this expansion in (9.282) and interchanging the order of integration and summation then yields ( 1 1 Z XX G .2r;2s/ .0; 0; !; z/ Q 1 < d! U.r; t / D h.!/e i!t 3 4 .2r/Š.2s/Š C C rD0 sD0 ) Z 1Z 1 Q 2r 2s ik.!/.pxCqy/ 2 O U0 .p; q; k/p q e  k .!/dpdq : 1

1

(9.286) Because [see (7.218)] mCn O U0 .x; y/ O .m;n/ .x; y/ @ U 0 @x m @y n Z 1Z 1 1 QO .p; q; k/ .ik.!//mCn p m q n U D 0 2 2 1 1 e ik.!/.pxCqy/ k 2 .!/dpdq;

(9.287) one then obtains ( 1 1 Z XX 1 i!t Q U.r; t / D < d! h.!/e  C C rD0 sD0

) G .2r;2s/ .0; 0; !; z/ O .2r;2s/ U  .x; y/ ; .2r/Š.2s/Š.ik.!//2.rCs/ 0 (9.288)

9.4 Separable Pulsed Beam Wavefields

73

for all z  0. The spatial series expansion appearing in the above expression is due to G. C. Sherman [45, 46] and is correspondingly referred to as Sherman’s expansion. Notice that the factor .ik.!//2.rCs/ appearing in the above series expansion is misleading because the same factor with an opposite-signed exponent is contained in the partial derivative G .2r;2s/ .0; 0; !; z/. It is seen from (9.283) that G .m;n/ .p; q; !; / D .ik.!/ /mCn e ik.!/ .1p where ' .m;n/ .p; q/

2 q 2 1=2

/ ' .m;n/ .p; q/;

1=2 @mCn  1  p2  q2 : m n @p @q

(9.289)

(9.290)

With these substitutions, the above expression for the propagated, separable wavefield becomes U.r; t / D

1 X 1 X . z/2.rCs/ rD0 sD0

' .2r;2s/ .0; 0/ .2r/Š.2s/Š ) (Z 1 .2r;2s/ i.k.!/ z!t/ Q O  < U .x; y/h.!/e d! ; 0  CC (9.291)

for all z  0. The transverse spatial variation of both the source-free electric and magnetic field vectors at any plane z > z0 is thus seen to depend solely upon the transverse spatial variation of the field at the initial plane at z D z0 through all of the even-order spatial derivatives of the corresponding field vector at that plane. At first impression, it would appear that this transverse spatial variation is independent of the wavenumber k.!/, and hence of the angular frequency !, so that the function O .2r;2s/ .x; y/ could be taken out from under the integral in (9.291). This clearly U 0 deserves a more careful examination. O .2r;2s/ .x; y/ is indeed independent of the wavenumber k.!/, then the propaIf U 0 gated source-free pulsed beam field is given by the product of two separate factors [see (7.260) and (7.261) in Vol. 1], one describing the transverse spatial variation and the other describing the longitudinal spatiotemporal variation, each factor dependent upon the propagation distance z D z  z0 . This apparent wavenumber (or frequency) independence is a direct consequence of the two-dimensional ikm z , with Maclaurin p series expansion of the propagation kernel G.p; q; z/ e m D 1  p 2  q 2 , whose explicit dependence upon k.!/ is then transferred to O 0 .x; y/. With this in mind, (9.291) may be the transverse spatial derivatives of U rewritten as (Z ) 1 i.k.!/ z!t/ Q O d! ; (9.292) h.!/e U.r; t / D U.x; y/ <  CC

74

9 Pulsed Beam Wavefields in Temporally Dispersive Media

with [see (7.222) in Vol. 1] O U.x; y/

1 X 1 X . z/2.rCs/ rD0 sD0

.2r/Š.2s/Š

.2r;2s/

O ' .2r;2s/ .0; 0/U 0

.x; y/:

(9.293)

Unfortunately, except in trivial cases (such as for a plane wave), this series representation converges extremely slowly so that an exceedingly large (perhaps even infinite) number of terms is required. The integral representation corresponding to this series expansion of the transverse field distribution is found to be given by (see Problem 9.10) 1 O U.x; y/ D 4 2

Z

1 1

Z

1

QO . ;  /e i .x xCy yC. / z/ d d ; U 0 x y x y

(9.294)

1

1=2  with where   2  x2  y2 QO . ;  / D U 0 x y

Z

1

1

Z

1

O 0 .x; y/e i.x xCy y/ dxdy: U

(9.295)

1

Notice that this integral representation of the source-free transverse field vector O U.x; y/ that is defined by the infinite summation given in (9.293) is independent of the value of the spatial frequency  appearing in the propagation factor in the integrand of (9.294) only in the limit as  ! 1. The reason that this is so is found in the infinite double summation of (9.293). The transverse field variation is exO 0 .x; y/, and pressed there in terms of all of the even-order spatial derivatives of U this, in turn, requires that the transverse spatial variation of this initial field structure be known all the way down to an infinitesimal scale, that is, as 1= ! 0. This, of course, is just the geometrical optics limit. This rather curious result deserves further explanation. First of all, notice that the field must be source-free. This means that the wavefield does not contain any evanescent field components in a lossless medium. This, in turn, means that the wavenumber component [see (9.12)] 1=2  .!/ D k 2 .!/  kT2 QO .k ; k / is nonzero, where k 2 D is real-valued for all points .kx ; ky / at which U 0 x y T 2 2 kx C ky . In general, for each value of !, the value of the wavenumber k.!/ defines the transition circle kT2 D k 2 .!/ in kx ky -space between homogeneous and evanescent plane wave components. This wavenumber value then appears as the upper limit of integration for the homogeneous plane wave contribution to the angular spectrum representation and this, in turn, sets an upper limit on the spatial frequency scale (or equivalently, a lower limit on the spatial scale) for the transverse spatial structure of the wavefield. However, for a source-free wavefield, this upper limit is replaced by infinity as the homogeneous wave propagation factor

9.4 Separable Pulsed Beam Wavefields

75

G.kx ; ky ; !; z/ is replaced by its Maclaurin series expansion [see (9.284)]. The O result is an expression for the transverse field variation U.x; y/ that is indeed independent of the wavenumber. The results presented here then show that, except in special cases (such as for a plane wave), if the initial wavefield is separable in the sense defined in (9.271) and (9.272), then the propagated wavefield will not, in general, remain separable unless it is strictly source-free. In that idealized case, the wavefield remains separable throughout its propagation and its transverse spatial variation is independent of the wavenumber (or frequency). This then demonstrates how a seemingly innocent assumption (in this case, the assumption that the wavefield contains only homogeneous wave components) can lead to extreme results when they are taken to their logical limit. If the initial pulse is strictly quasimonochromatic (in which case [47] !=!c  1, where ! is the bandwidth of the pulse spectrum centered at !c ), then the propagated field is, to some degree of approximation, separable with the spatial frequency  being taken as the wavenumber k.!c / evaluated at this characteristic angular frequency of the pulse. Except in special cases, this approximation does not hold in the ultrawideband case. In that case, the expression (9.292) for the propagated wavefield should be written as (Z ) 1 i.k.!/ z!t/ Q O d! ; U.x; y/h.!/e U.r; t / D <  CC

(9.296)

O with U.x; y/ given by (9.294) with  D k.!/.

9.4.1 Gaussian Beam Propagation An example of central importance in both optics and microwave theory considers the spatial properties of a gaussian beam wavefield. Let the initial transverse field behavior at the plane z D z0 be described as 2 2 2 2 UO 0 .x; y/ D A0 e .x x Cy y / ;

(9.297)

with fixed amplitude A0 and constants x and y that set the initial beam widths in the x- and y-directions, respectively. With use of the integral identity Z

1

2

e ˛ e ˙iˇ d D

1

  1=2 ˛

e ˇ

2 =4˛

;

˛ > 0;

(9.298)

the initial field spectrum [see (9.295)] is found to be given by  .x2 =4x2 Cy2 =4y2 / UQO 0 .x ; y / D A e : x y

(9.299)

76

9 Pulsed Beam Wavefields in Temporally Dispersive Media

The propagated field distribution is then obtained from (9.294) as UO .x; y/ D

A 4x y

Z

1

1

Z

1

e .x =4x Cy =4y / e i .x xCy yC. / z/ dx dy : 2

2

2

2

1

(9.300) With the paraxial approximation    D 

  1=2 1  x2 = 2  y2 = 2 1



x2 C y2

(9.301)

2

of the propagation factor, valid when .x2 C y2 /= 2  1, the propagated field given in (9.300) may be separated into the product of a pair of two-dimensional gaussian beam fields as UO .x; y/ AUO .x/UO .y/; (9.302) with UO .x/

1 2 1=2 x

Z

1

e .x =4x Ci z=2 / e ix x dx 2

2

1

1 x 2 =.1=x2 C2i z= / D  ; 1=2 e 1 C 2i x2 z=

(9.303)

with an exactly analogous expression for UO .y/. Notice that this result depends explicitly on the spatial frequency value ; this is due to the paraxial approximation given in (9.301) used in obtaining (9.303). Furthermore, in the limit as  ! 1, 2 2 UO .x/ ! e x x , which is just the geometrical optics limit. The argument of the exponential factor appearing in (9.303) can be separated into real and imaginary parts as 

x2 2 z= x2 D  x2 C i x2 1=x2 C 2i z= 1 C 4x4 . z/2 = 2 1=x4 C 4. z/2 = 2 D

1  x2 C i x2; w2x . z/ 2Rx . z/

where

 wx . z/ wx0 1 C

2ız w2x0 

(9.304)

2 !1=2 (9.305)

is the beam radius (or “spot size”) at the e 1 amplitude point in the x-direction with beam waist

9.4 Separable Pulsed Beam Wavefields

77

wx0 wx .0/ D and where

 Rx . z/ z 1 C

1 ; x w2x0  2 z

(9.306) 2 ! (9.307)

is the radius of curvature of the phase front10 in the xz-plane in the paraxial approximation. Finally, the square root factor appearing in (9.303) may be written as 1

1

 1=2 1 C 2i x2 z=

e i 2 arctan .2 z=.wx0 // D  1=4 1 C 4. z/2 =.w4x0  2 / r wx0 i x . z/=2 D ; e wx . z/

where

 x . z/ arctan

2

2 z w2x0 

(9.308)

 (9.309)

describes the phase shift with propagation distance z away from the beam waist at z D z0 . With these identifications, the two-dimensional gaussian beam field given in (9.303) becomes r wx0 .x=wx . z//2 .i=2/Œ.=Rx . z//x 2  x . z/

e e UO .x/

; (9.310) wx . z/ with an analogous expression for UO .y/. The gaussian beam wavefront is a plane wave at the beam waist [Rx .0/ D Ry .0/ D 1], and for large j zj it approaches a spherical wavefront with center at the point r0 D .0; 0; z0 / and with radius equal to z, and in between these two limits it is, in general, an astigmatic wavefront. The divergence angle x of the gaussian beam in the xz-plane is obtained from the limiting behavior of the expression tan x D wx . z/=Rx . z/ as j zj ! 1 with the result 2 : (9.311) tan x D wx0 Finally, the collimated beam length or Rayleigh range 2zp R is defined as the distance 2 of its value at the beam over which the beam radius remains less than or equal to p waist, so that w.zR / 2w0 , with solution zR D

1 2 w : 2 0

(9.312)

p   Notice that z D R2  .x 2 C y 2 /  R  x 2 C y 2 =.2R/ in the parabolic approximation of a spherical wavefront with radius R, as appears in the imaginary part of (9.304) due to the paraxial approximation made in (9.301).

10

78

9 Pulsed Beam Wavefields in Temporally Dispersive Media

The gaussian beam approximates a collimated beam over the Rayleigh range j zj  zR on either side of the beam waist, whereas outside of this range it behaves more like a converging spherical (or more generally astigmatic) wave when z < zR and a diverging spherical (or more generally astigmatic) wave when z > zR . All of the paraxial beam parameters for a gaussian beam are explicitly dependendent upon the spatial frequency parameter  which, in effect, sets an upper limit to the spatial frequency scale. With  D k D 2=, the above gaussian beam parameters assume their usual form in the paraxial approximation as [48, 49]  !1=2 z 2 ; wx . z/ D wx0 1 C zR  2 ! zR Rx . z/ D z 1 C ; z   z ; . z/ D arctan x zR 

(9.313)

(9.314) (9.315)

tan x D

 ; wx0

(9.316)

zR D

 2 w :  x0

(9.317)

The physical relation of these parameters to the gaussian beam propagation pattern they describe is illustrated in the contour plot depicted in Fig. 9.20 for the case when w0 = D 20. The symmetric pair of upper and lower dashed curves describe the beam half-width values ˙w. z/ which asymptotically approach the pair of dashed lines at ˙ as z ! ˙1. Notice that if the spot size parameter w0 is decreased from the value used in computing this field pattern, the Rayleigh range will decrease and the angular divergence will increase, resulting in a more divergent field behavior, whereas if it is increased, the Rayleigh range will be lengthened and the divergence angle narrowed, resulting in a more collimated beam behavior.

2w0

Fig. 9.20 Contour plot of the gaussian beam amplitude with propagation distance z about the beam waist when w0 D 20, illustrating the relationship of the beam waist 2w0 , Rayleigh range 2zR , and angular divergence 2 parameters with the main features of the propagation pattern

2q

2zR

9.5 The Inverse Initial Value Problem

79

9.4.2 Asymptotic Behavior Finally, consider the asymptotic behavior as R ! 1 of the integral representation O of the transverse field U.x; y/ defined in (9.294). Under the change of variable x D p, y D q, this representation may be expressed as  2 i z O U.x; y/ D e 4 2

Z

1 1

Z

1

QO .p; q/e i.pxCqyCm z/ dpdq; U 0

(9.318)

1

1=2  . With x0 D y0 D 0 and fixed direction cosines 1 D where m D 1  p 2  q 2 QO .p; q/ [see (9.204)], the x=R, 2 D y=r, 3 D z=R, and with V.p; q; m/ D mU 0 O asymptotic approximation of U.x; y/ is obtained from (9.270) as  z OQ O U0 .x=R; y=R/e iR e i z.z0 =R1/ ; U.x; y/  i 2R2

(9.319)

as R ! 1. With  D k D 2=, the above expression becomes z OQ O U0 .x=R; y=R/e i2R= e i2. z=/.z0 =R1/ ; U.x; y/  i R2

(9.320)

as R= ! 1. As an example, for the gaussian beam wavefield given in (9.297) with initial beam widths (or beam waists) wx0 D 1=x and wy0 D 1=y , the initial field spectrum given in (9.299) becomes QO .x=R; y=R/ D Aw w e . 2 =4R2 /.w2x0 x 2 Cw2y0 y 2 / : U 0 x0 y0 Substitution of this expression in (9.320) then yields wx0 wy0 .=R/2 .w2x0 x 2 Cw2y0 y 2 / i2R= i2. z=/.z0 =R1/ O e U.x; y/  i A 2 e e ; R = z as R= ! 1. An analogous expression is obtained from (9.302) and (9.310) in the limit as z R ! 1 (see Problem 9.11).

9.5 The Inverse Initial Value Problem This chapter concludes with a concise description of the solution to the following time-dependent inverse source problem: Determine the charge sources %.r; t / and currents J.r; t / that are zero everywhere except over the time interval T < t < T

80

9 Pulsed Beam Wavefields in Temporally Dispersive Media

such that for t > T they produce prescribed solutions of the source-free (or homogeneous) Maxwell’s equations    1  @H.r; t /  r  E.r; t / D    c  0 @t ;    1  @E.r; t /  r  H.r; t / D   c  0 @t ; r  E.r; t / D r  H.r; t / D 0;

(9.321) (9.322) (9.323)

in free-space. Although the solution to the inverse problem has been shown to be nonunique [50, 51], Moses and Prosser [52] have shown that by specifying a partial time-dependence in the wave-zone of the radiation field, a unique solution can be found for each such specification. The early analysis of Moses and Prosser [52] begins with the Bateman– Cunningham form of Maxwell’s equations, which is obtained in the following manner. First of all, notice that the pair of curl relations in (9.321) and (9.322) may be expressed as 1 @H.r; t / ; r  E.r; t / D  0 c @t 1 @E.r; t / 0 r  H.r; t / D ; c @t

(9.324) (9.325)

p where 0 0 =0 is the intrinsic impedance of free space. If one then defines the complex vector field .r; t / as .r; t / E.r; t /  i0 H.r; t /;

(9.326)

Maxwell’s equations may be expressed in complex form as r

.r; t / D i

r

.r; t / D 0;

1 @ .r; t / ; c @t

(9.327) (9.328)

which is the Bateman–Cunningham form of Maxwell’s equations when t is replaced by ct . The general solution of the these equations may then be expressed in terms of the eigenfunctions of the curl operator [53]. In a subsequent refinement of their work that is based upon the Radon transform [54], Moses and Prosser [55] have shown that there always exists a vector function of position G.r/ such that for any causal, finite energy solution fE.r; t /; H.r; t /g of Maxwell’s equations, lim .rE.r; t // D G.r  ct; ; /;

(9.329)

lim .rH.r; t // D 0 rO  G.r  ct; ; /;

(9.330)

r!1 r!1

9.5 The Inverse Initial Value Problem

81

with rO  G.r/ D 0;

(9.331)

where rO r=r is the unit vector in the direction of the position vector r D .r; ; / with spherical polar coordinates .r; ; / such that x D r sin  cos , y D r sin  sin , z D r cos  with 0     and 0   < 2. Causality then requires that the radiated field vectors E.r; t / and H.r; t / are obtained from their respective initial temporal field behaviors. The pair of asymptotic limits given in (9.329) and (9.330) then show that, with the exception of the 1=r factor, all causal, finite energy solutions of Maxwell’s equations propagate outward to infinity like one-dimensional electromagnetic waves along rays specified by the polar angles  and . Furthermore, the exact electromagnetic field vectors are given in terms of the wave zone vector field G.r/ as [55] Z 2 Z  @ 1 0 0 sin  d d 0 G.r  uO  ct;  0 ;  0 /; (9.332) E.r; t / D 2 0 @p 0 Z 2 Z  @ 1 G.r  uO  ct;  0 ;  0 /; (9.333) H.r; t / D sin  0 d 0 d 0 uO  2 0 @p 0 where p D r  uO  ct and uO D .sin  0 cos  0 ; sin  0 sin  0 ; cos  0 /. So-called electromagnetic bullets may then be constructed by requiring that the vector function G.r  ct; ; / D G.r  ct; / describing the far-zone behavior be independent of the azimuthal angle  and that it identically vanishes both outside of the cone 0   < c as well as outisde of the radial space–time region a C ct < r < b C ct , where a < b, as depicted in Fig. 9.21. The solution of the inverse source problem then provides the source distribution required to produce this electromagnetic bullet. Unfortunately, the preceeding analysis of the inverse source problem due to Moses and Prosser [52, 55], while of considerable historical importance, is only

z

Fig. 9.21 Space–time domain (shaded region) of an electromagnetic bullet that is defined in the interior region of the cone 0   < c and radial space–time region a C ct < r < b C ct

c

O

82

9 Pulsed Beam Wavefields in Temporally Dispersive Media

applicable to sources with a specific separable space–time dependence. In addition, these solutions have been shown, in general, not to be minimum-energy solutions. These limitations have been addressed by Marengo, Devaney, and Ziolkowski [56] who reconsider the time-dependent inverse source problem with far-field data based upon a limited-view Radon transform, a problem analogous to a limited-view computed tomography reconstruction. Their analysis considers the inverse source problem for the inhomogeneous scalar wave equation 

r2 

1 @2 c 2 @t 2

 U.r; t / D k4kQ.r; t /;

(9.334)

where the radiating source Q.r; t /, which is assumed to be localized within a simply–connected region D 2 0 and the lower sign choice when = ./ < 0, and  Isp .z; s / D q.!sp / 

2 00 zp .!sp ; s /

1=2

e zp.!sp ;s /

r    p  zp.!c ;s /  e zp.!sp ;s / C i erfc i .s / z e C z .s / i e zp.!c ;s / C R1 e zp.!sp ;s / I when =f .s /g D 0; .s / ¤ 0;

(10.87)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

Isp .z; s / D

129

1=2 2 e zp.!sp ;s / zp 00 .!sp ; s /   p 000 .!sp ; s /   q.!sp /    00 !sp .s /  !c 6p .!sp ; s /

 

CR1 e zp.!sp ;s / I

when .s / D 0;

(10.88)

where R1 D O.z3=2 / as z ! 1 uniformly with respect to  for all  2 R, where  is the residue of the simple pole singularity at ! D !c , defined in (10.84), and

1=2 ./ p.!sp ./; /  p.!c ;  / :

(10.89)

1=2 The argument of the quantity  zp 00 .!sp ./; / is defined to be equal to arg.d!/!sp , where d! is a differential element of path length along the path of steepest descent through the saddle point !sp ./, and the argument of . / is defined such that  

   p 00 .!sp . /;  / 1=2 ./ D !c  !sp ./  : !c !!sp ./ 2 lim

(10.90)

p R1 2 Finally, the function erfc. / .2= / e  d  is the complementary error function. The asymptotic behavior of the saddle point integral Isp .z;  / is given by (10.86) with the upper sign choice when the contour P . / lies on one side of the pole (with respect to the original path P 0 ) such that =f . /g > 0, and with the lower sign choice when P . / lies on the other side of the pole (with respect to P 0 ) such that =f ./g < 0. When  D s the pole lies on the contour P .s /, by definition, then =f .s /g D 0 and the asymptotic behavior of the saddle point integral Isp .z; s / is given by (10.87) if .s / ¤ 0, and it is given by (10.88) if .s / D 0, in which case the saddle point coalesces with the pole. Since the order relation R1 D O.z3=2 / for the error term as z ! 1 is satisfied uniformly with respect to  for all  2 R, the apparent discontinuities in the asymptotic behavior of Isp .z;  / exhibited in (10.86)–(10.88) are real. In particular, when the path P . / passes from one side of the pole (again, with respect to the original path P 0 ) to the other, the discontinuous jump in Isp .z; / due to the change in sign of =f . /g in (10.86) is equal to 2 i e zp.!c ;s / . This discontinuity in Isp .z;  / exactly cancels the discontinuity in I.z; / introduced by the contribution of the simple pole singularity when Cauchy’s residue theorem is applied to deform the original contour P 0 to the path of steepest descent P . / through the saddle point, as exhibited in the set of relations given in (10.83). As a result, the asymptotic behavior of I.z;  / is a continuous function of  for all  2 R for fixed, finite values of z. If P . / is an Olver-type path other than the path of steepest descent through the saddle point at ! D !sp . /, then Theorem 5 remains valid provided that P . / is deformable to the path of steepest descent without crossing the pole singularity.

130

10 Asymptotic Methods of Analysis

If the pole is crossed when P . / is deformed to the steepest descent path through the saddle point, then the set of relations given in (10.86)–(10.88) are changed [27] by the addition or subtraction of the term 2 i e zp.!c ;/ . Since the change in the expression for the saddle point integral Isp .z;  / is equal but with opposite sign to the change introduced between I.z;  / and Isp .z;  / when Cauchy’s residue theorem is applied to change the contour of integration from the steepest descent path to the new Olver-type path P . /, the resulting asymptotic expression for I.z;  / remains unchanged. Hence, the uniform asymptotic approximation obtained for I.z;  / is independent of the particular Olver-type path chosen. Nevertheless, in order to apply Theorem 5 to obtain the uniform asymptotic approximation of I.z;  /, it is still necessary to determine the path of steepest descent relative to the position of the pole in order to determine whether or not the residue contribution due to the pole should be added to the right-hand side of the appropriate expression in (10.86)–(10.88). Consider now the determination of the proper argument of the complex quantity ./ defined in (10.89). If the pole encircles the saddle point once as  varies over R, the argument of 2 . / varies over a range of 4 so that the argument of . / varies over a range of 2. Hence, . / is not confined to a single branch of the square root of 2 . / as would be obtained by using a branch cut to restrict the argument of 2 . / to a range of less than 2. To determine the argument of . / that is implied by (10.90), it is useful to apply the following geometrical construction. Let ˛N c denote the angle of slope of the vector from the saddle point !sp. / to the pole !c in the complex !-plane. Equation (10.90) then yields lim

!c !!sp ./

n

 1=2 o C 2 n; arg ./ D ˛N c C arg  p 00 .!sp ./; /

(10.91)

where the limit is taken along the straight line with slope ˛N c and where n is an arbitrary integer. Since arg

n

 p 00 .!sp ./; /

1=2 o

D ˛N sd ;

(10.92)

as required by Theorem 5, where ˛N sd is the angle of slope of a vector tangent to the path of steepest descent at the saddle point, as defined in (10.2) with P taken as the steepest descent path and !1 D !sp ./, then (10.91) becomes lim

!c !!sp ./

 arg . / D ˛N c  ˛N sd C 2 n:

(10.93)

Hence, as the pole approaches the saddle point along a straight line, the argument of ./ approaches 2 n plus the angle that the line makes with the vector tangent to the steepest descent path at the saddle point !sp . /. The integer n can be chosen so that the argument of ı. / lies within the principal range .; 

for all  2 R. For example, in the situation depicted in Fig. 10.9, the limit of arg . / is a small negative angle in part (a) and a small positive angle in parts (b) and (c).

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

131

10.4.1 The Complementary Error Function To obtain a better understanding of the uniform asymptotic nature of the saddle point integral Isp .z; / that is given in (10.86)–(10.88) of Theorem 5, it is necessary to understand the analytic properties of the complementary error function erfc. / 2 for complex . Since e  is an entire function, both the error function erf. /, which p 2 is defined as the integral of the Gaussian distribution function g./ D .2= /e  from 0 to , viz., Z 2 2 erf. / p e  d ; (10.94)  0 and the complementary error function erfc. /, which is defined as the integral of the Gaussian distribution function from to 1, viz., 2 erfc. / p 

Z

1

2

e  d ;

(10.95)

are entire functions of complex , where erfc. / D 1  erf. /. Along the real axis, erf.1/ D 0, erf.0/ D 1=2, and erf.1/ D 1, whereas erfc.1/ D 2, erfc.0/ D 1, and erfc.1/ D 0, as illustrated in Fig. 10.10. The behavior of the complementary error function erfc. / in the complex -plane is more complicated, as illustrated in Fig. 10.11. Part (a) of the figure shows the real part and part (b) the imaginary part. Notice that the real part is even symmetric about the imaginary axis whereas the imaginary part is odd symmetric, that is < ferfc.  /g D < ferfc. /g and = ferfc.  /g D = ferfc. /g.

Error & Complementary Error Functions

2

erfc(ζ) = 1 - erf(ζ) 1.5

1

0.5

0

erf(ζ)

−0.5

−1 −3

−2

−1

0

1

2

3

ζ

Fig. 10.10 The error function erf. / and the complementary error function erfc. / for real

132

10 Asymptotic Methods of Analysis

ℜ{erfc( )}

a 20 10 0

−10 −20

2

1 ℑ{␨}

b

0

−1

−1

−2 −2

0

1

2

ℜ{␨}

ℑ{erfc( )}

20 10 0

−10 −20

2 1 ℑ{␨}

0

−1

−1

−2 −2

0

1

2

ℜ{␨}

Fig. 10.11 Real (a) and imaginary (b) parts of the complementary error function erfc. / for complex

Along the imaginary axis where D i 00 D e ˙i=2 with   0, the integral expression for the complementary error function becomes Z    2 2 erfc e ˙i=2 D 1 i p e t dt; (10.96)  0 which is related to Dawson’s integral FD ./ D e 

2

Z



2

e t dt:

(10.97)

0

  Consequently, erfc e ˙i=2 ! 1 i 1 as  ! 1, as indicated in Fig. 10.12.

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

133

20 15 10 5 0 −5 −10 −15 −20 1 1

0.5 0.5

0 ( )

0

−0.5

−0.5 −1

( )

−1

Fig. 10.12 Cosine C ./ and sine S ./ Fresnel integrals with real argument . Notice that C .1/ D S .1/ D 1=2, C .0/ D S .0/ D 0, and C .1/ D S .1/ D 1=2

Along the diagonal axes where D e ˙i=4 with real-valued , the complementary error function is given by h p  p i p   2= i S 2= ; erfc e ˙i=4 D 1  2e ˙i=4 C

(10.98)

where C./ and S./ are the cosine and sine Fresnel integrals, respectively, defined by C./

Z



  cos .=2/t 2 dt;

(10.99)

  sin .=2/t 2 dt:

(10.100)

0

S./

Z



0

The cosine and sine Fresnel integrals are plotted against each other in Fig. 10.12 with the real argument  plotted along the vertical axis. The projection of this curve onto the horizontal plane then yields the well-known Cornu spiral in  illustrated  Fig. 10.13. Since C.˙1/ D S.˙1/ D ˙1=2, it is seen that erfc e ˙i=4 ! 0   as  ! 1 and that erfc e ˙i=4 ! 2 as  ! 1. The asymptotic expansion of the complementary error function is given by [21] 1

X 1  .n C 1=2/ 2 erfc. / D p e  .1/n  .1=2/ 2nC1  nD0

(10.101)

134

10 Asymptotic Methods of Analysis 0.8 0.6 0.4 0.2

( )

0

−0.2 −0.4 −0.6 −0.8 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

( ) Fig. 10.13 The Cornu spiral, plotting the cosine Fresnel integral C ./ vs. the sine Fresnel integral S ./ at the same value of the real argument  which varies from negative to positive infinity. Notice that C .1/ D S .1/ D 1=2, C .0/ D S .0/ D 0, and C .1/ D S .1/ D 1=2

as j j ! 1 uniformly in j arg. /j < =2. With use of the identity erfc. / C erfc. / D 2;

(10.102)

the asymptotic behavior of erfc. / in the half-plane s ;

(10.109)

(10.110)

n

p p 3 o as j . /j z ! 1 uniformly with respect to  for ./ z

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

137

 p At  D s , arg i .s / z D =2 for real z > 0. In that case, the complementary error function appearing in (10.109) is given by (10.96), so that 1=2 2 e zp.!sp ;s / zp 00 .!sp ; s / r

 p p  zp.!c ;s /  e zp.!sp ;s / C 2 FD .s / z e C z .s /

 I.z; s / D q.!sp / 

i e zp.!c ;s / C R1 e zp.!sp ;s / I

 D s ;

(10.111)

p 2 R 2 as j ./j z ! 1, where FD ./ D e  0 e t dt is Dawson’s integral, defined in (10.97). The numerically determined behavior of Dawson’s integral as a function of real   0 is illustrated in Fig. 10.14. The dashed curve in the figure depicts the behavior of the first two (dominant) terms in the asymptotic expansion of FD ./, given by N 1   1 X  .n C 1=2/ (10.112) C O .2N C1/ FD ./ D 2nC1 2 nD0  .1=2/ as  ! 1. Notice that this two term asymptotic approximation provides an accurate estimate of Dawson’s integral when  > 2.

0.6 FD( r) ~ (1/2){1/r + 1/(2 r3)}

Dawson's Integral FD( r)

0.5

0.4 FD(r) 0.3

0.2

0.1

0 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.14 The functional dependence of Dawson’s integral FD ./ for real  0. The numerically determined behavior is represented by the solid curve and the behavior of the first two (dominant) terms in the asymptotic expansion of FD ./ as  ! 1 is represented by the dashed curve

138

10 Asymptotic Methods of Analysis

 p At  D c , arg i .c / z D =4 for real z > 0. In that case, the complementary error function appearing in (10.110) is given by (10.98), so that 1=2  2 I.z; c / D q.!sp /  00 e zp.!sp ;c / zp .!sp ; c / (  p i p h p C i 2 C 2z=j .c /j  S 2z=j .c /j e zp.!c ;c / ) r  e zp.!sp ;c / C  3i e zp.!c ;c / C R1 e zp.!sp ;c / I  D c ; z .c / (10.113) p as j ./j z ! 1, where C. / and S. / are the cosine and sine Fresnel integrals defined in (10.99) and (10.100). For computational purposes, these two functions may be written as     1 C f . / sin

2  g. / cos

2 ; 2 2 2     1

2  g. / sin

2 ; S. / D  f . / cos 2 2 2 C. / D

(10.114) (10.115)

where the functions f . / and g. / may be accurately computed using the rational approximations [22] 1 C 0:926 C ". /; 2 C 1:792 C 3:104 2 1 C ". /; g. / D 2 C 4:142 C 3:492 2 C 6:670 3

f . / D

(10.116) (10.117)

for all  0, where j". /j  2  103 . The Fresnel integrals graphed in Figs. 10.12 and 10.13 were computed using these rational approximations. The asymptotic expressions given in (10.111) and (10.113) describe the behavior of the integral I.z; / at the two critical values at  D s and  D c , respectively, given as a function of the real variable z in terms of well-known real-valued functions. When the saddle point is far enough away from the pole that (10.105) and (10.107) can be applied in (10.83), then (10.108)–(10.110) become  I.z; / D q.!sp / 

2 zp 00 .!sp ;  /

CR1 e zp.!sp ;/ I

1=2

e zp.!sp ;/

 < s ;

(10.118)

10.4 Uniform Expansion for a Saddle Point and Nearby Singularity

 I.z; s / D q.!sp / 

2 00 zp .!sp ; s /

1=2

139

e zp.!sp ;s /  i e zp.!c ;s /

CR1 e zp.!sp ;s / I  D s ; (10.119) 1=2  2 I.z; / D q.!sp /  00 e zp.!sp ;/  2i e zp.!c ;/ zp .!sp ;  / CR1 e zp.!sp ;/ I

 > s ;

(10.120)

n

p p 3 o , as j . /j z ! 1 uniformly with respect to  for where R1 D O ./ z all  2 R. These results are the same as those obtained with a direct application of Olver’s saddle point method, except that the dependence of the remainder term on the separation between the saddle point and the pole is displayed explicitly through the factor ./ in (10.118)–(10.120). This set of equations provides a continuous asymptotic approximation of the integral I.z;  / for fixed large values of z as  varies continuously over R provided that the quantity zjp.!sp ; s /  p.!c ; s /j is large enough so that the discontinuity at  D s when the steepest descent path through the saddle point crosses the pole is negligible. When this quantity is small enough that this discontinuity is significant, then (10.108)–(10.110) must be employed.

10.4.3 Asymptotic Behavior for Two Isolated Interacting Saddle Points Finally, consider the case in which there are two relevant saddle points !1 ./ and !2 . / of the complex phase function p.!;  / which remain isolated from each other over the entire range R of values of  and interact with the simple pole singularity at ! D !c of the amplitude function q.!/ appearing in the path integral I.z;  / of the form given in (10.1). As in Sect. 10.3.1, the path P can then be deformed into an Olver-type path P . / that is composed of two parts P1 . / and P2 ./ such that P . / D P1 . / C P2 . /, where Pj ./, j D 1; 2, passes through the saddle point !j . / and is an Olver-type path with respect to that saddle point. The integral Isp .z; / taken over the contour P ./ can then be expressed as the sum of the two integrals Isp1 .z; / and Isp2 .z;  / taken over the respective paths P1 ./ and P2 ./. The following Corollary to Theorem 5 due to Cartwright [13] in 2004 then applies. Corollary 2. (Cartwright) In the integral Z

q.!/e zp.!;/ d!

Isp .z;  / D

(10.121)

P ./

considered in Theorem 5 for real z > 0, let all of the conditions stated there hold with the exception that the complex-valued phase function p.!;  / possesses two

140

10 Asymptotic Methods of Analysis

first-order saddle points !1 . / and !2 ./ that are isolated from each other as well as from any other saddle points of p.!;  / for all  2 R. The positions of these two saddle points are assumed to move in a vicinity of the isolated simple pole singularity of the amplitude function q.!/ that is located at ! D !c such that !j . / ¤ !c , j D 1; 2, for all  2 R. In addition, let the contour of integration P . / be composed of the two Olver-type paths P1 . / and P2 . / such that P ./ D P1 . / C P2 . /, where Pj . /, j D 1; 2, is the path of steepest descent through the corresponding saddle point !j ./. Define the functions

1=2 j . / p.!j ;  /  p.!c ;  / ;

j D 1; 2:

(10.122)

It is assumed that only the steepest descent path emanating from ˚ the saddle point !1 . / may cross the simple pole at !c , in which case = 2 ./ ¤ 0 for all  2 R. The appropriate argument of 1 ./ is then determined by (10.90). Under these conditions, the asymptotic behavior of the saddle point integral Isp .z;  / is given by 1=2 2 Isp .z; / D q.!1 /  00 e zp.!1 ;/ zp .!1 ;  / 1=2  2 Cq.!2 /  00 e zp.!2 ;/ zp .!2 ;  / r    p  e zp.!1 ;/ C ˙i erfc i 1 ./ z e zp.!c ;/ C z 1 ./ r    p  zp.!c ;/  e zp.!2 ;/ C ˙i erfc i 2 ./ z e C z 2 ./  zp.!1 ;/  ˚ zp.!2 ;/ CK e I when = j . / ¤ 0; Ce (10.123) ˚ where the ˚ upper sign choice is used when = j ./ > 0 and the lower sign choice when = j . / < 0, for j D 1; 2, and 

1=2 2 Isp .z; / D q.!1 /  00 e zp.!1 ;/ zp .!1 ;  / 1=2  2 Cq.!2 /  00 e zp.!2 ;/ zp .!2 ;  / r    p  zp.!c ;/  e zp.!1 ;/ C i erfc i 1 ./ z e C z 1 ./ r    p  zp.!c ;/  e zp.!2 ;/ C ˙i erfc i 2 ./ z e C z 2 ./   i e zp.!c ;/ C K e zp.!1 ;/ C e zp.!2 ;/ I ˚ ˚ when = 1 ./ D 0; ./ ¤ 0; = 2 . / ¤ 0; 

(10.124)

10.5 Asymptotic Expansions of Multiple Integrals

141

˚ where the upper sign choice is used when = 2 ./ > 0 and the lower sign choice  ˚  when = 2 . / < 0, where K D O z3=2 as z ! 1 uniformly with respect to  for all  2 R.

10.5 Asymptotic Expansions of Multiple Integrals The extension of Laplace’s method (see Appendix F.6) to the two-dimensional case is now considered. This extension has its origin in the analysis due to Hsu [28] in 1948 that was later extended by Fulks and Sather [15] in 1961. The description presented here is based upon the detailed analysis presented by Bleistein and Handelsman [29] in 1975. Consider then the asymptotic behavior as z ! 1 of the function I.z/ defined by the double integral [cf. (F.73)] Z I.z/ D

g./e zh./ d 2 ;

(10.125)

D

where  D .1 ; 2 / with 1 and 2 real, and where the finite, simply connected integration domain D is bounded by a smooth curve  . In particular, the closed contour  is defined by the parametric equation ˚  D .1 ; 2 /j 1 D 1 .s/; 2 D 2 .s/; 0  s  L ;

(10.126)

where s is the arc-length along the curve such that  is traced out in the counterclockwise sense as s increases, and where both of the functions 1 .s/ and 2 .s/ are continuously differentiable on Œ0; L . Finally, it is assumed that the functions g./ D g.1 ; 2 / and h./ D h.1 ; 2 / are both continuous with continuous partial derivatives through at least the second order. The asymptotic behavior of the integral I.z/ as z ! 1 depends upon whether the function h./ possesses an absolute maximum value either in the interior of the closure DN of the integration domain D or on the boundary curve  of DN . Each of these two cases is now considered separately.

10.5.1 Absolute Maximum in the Interior of the Closure of D Assume that the function h./ possesses a single absolute maximum in the interior of the closure DN of the domain D at the point  0 D .10 ; 20 /, so that rh. 0 / D 0;

(10.127)

142

10 Asymptotic Methods of Analysis

with rh./ ¤ 0 at all other points of DN , and "

@2 h./ @12



  2 2 # @2 h./ @ h./  > 0; @1 @2 @22 D 0   2 @ h./ < 0: @22 D 0

(10.128) (10.129)

If several maxima occur in the interior of DN , then the integration domain can be partitioned into a set of subdomains, each containing a single absolute maximum. It is then expected that the dominant contribution to the integral I.z/ arises from the local behavior of h./ about the critical point  D  0 , as described by the Taylor series expansion 1 h./ D h. 0 / C h1 1 . 0 /.1  10 /2 C h1 2 . 0 /.1  10 /.2  20 / 2 1 (10.130) C h2 2 . 0 /.2  20 /2 C    ; 2 where hi j ./ @2 h./=@i @j , i D 1; 2, j D 1; 2. With this in mind, the asymptotic approximation of the integral I.z/ in (10.125) is found to be given by [29] 2g. 0 /e zh. 0 / I.z/  h  2 i1=2 ; z h1 1 . 0 /h2 2 . 0 /  h1 2 . 0 /

(10.131)

as z ! 1.

10.5.2 Absolute Maximum on the Boundary of the Closure of D Consider now the case when rh. 0 / ¤ 0 in DN , so that the absolute maximum of h./ in DN occurs on the boundary  of D . In particular, assume that this maximum occurs at the point   D .1 ; 2 / 2  and that this maximum is unique. For simplicity, let this point occur when s D 0, so that [from (10.126)] 1 D 1 .0/;

2 D 2 .0/:

(10.132)

The first directional derivative of h./ along the unit tangent vector T to the boundary curve  must then vanish at  D   , in which case ˇ rh./  TˇsD0 D h 1 .1 /10 .0/ C h 2 .2 /20 .0/ D 0;

(10.133)

10.5 Asymptotic Expansions of Multiple Integrals

143

where the prime denotes differentiation with respect to the arc length s along the curve  . Equation (10.133) then states that the vector rh./ is orthogonal to the boundary curve  at the point  D   . With the expression   O n.s/ 20 .s/; 10 .s/

(10.134)

for the unit outward normal vector to  , it is seen that ˇ ˇ O rh.  / D ˇrh.  /ˇn.0/;

(10.135)

as rh./ is in the direction of decreasing h./ and the absolute maximum of h./ occurs on the boundary of DN and not in its interior. Define the vector field H./ g./

rh./ ; jrh./j2

(10.136)

so that g./e zh./ D 

  1 e zh./ r  H./ C r  H./e zh./ : z z

(10.137)

Substitution of this result in the integrand of (10.125) followed by application of the divergence theorem (in two dimensions) then results in 1 I.z/ D z

I O H.s/  n.s/e 

zh.s/

1 ds  z

Z

.r  H.// e zh./ d 2 :

(10.138)

D

Because the second integral in this expression is of the same form as the original surface integral over D and because it has a 1=z multiplicative factor, it is typically of lower order than I.z/. As a consequence, the dominant term in the asymptotic expansion of I.z/ as z ! 1 comes from the boundary integral in (10.138). The asymptotic approximation of this term then gives [29] r

I.z/ 

2 g.  /e zh.  / z3  ˇ  ˇh1 1 .  /h22 .  /  2h1 2 .  /h1 .  /h2 .  / ˇ ˇ3 ˇ Ch2 2 .  /h21 .  / .  / ˇrh.  /ˇ ˇ

1=2 ; (10.139)

as z ! 1. Here .  / is the curvature of the contour  at the point  D   , where the upper sign in the above equation is used when  is convex and the lower sign when  is concave at   .

144

10 Asymptotic Methods of Analysis

Comparison of this result with that given in (10.131) shows that when the absolute maximum of the function h./ occurs at an interior point  D  0 of the integration domain [where rh. 0 / D 0] ˚ I.z/e zh. 0 / D O z1 ; whereas

˚ I.z/e zh.  / D O z3=2 ;

as z ! 1;

as z ! 1;

(10.140)

(10.141)

when the absolute maximum of the function h./ occurs at a boundary point  D   of the integration domain and rh.  / ¤ 0. However, when the absolute maximum at  D   occurs at a boundary point of the domain and rh.  / D 0, the asymptotic behavior of the integral I.z/ is found [29] to be given by 1=2 of the expression given in (10.131) for an interior maximum with  0 replaced by   .

10.6 Summary The inherent complexity of the asymptotic method of analysis for a given problem is offset by its ability to accurately describe a complicated physical process with a compact mathematical expression that explicitly displays the dependence on each factor appearing in the physical model used. This is to be contrasted with a purely numerical method of analysis which may be easier to implement and provides quicker results, but is much less capable of providing a deeper understanding of the physical phenomena involved. The philosophical approach to dispersive pulse propagation used in this book is to fully develop the asymptotic description and to then illustrate the predicted results from this description through the use of accurate numerical results. When warranted, the asymptotic theory may be augmented by numerical techniques, resulting in a hybrid asymptotic-numerical procedure with tremendous practical applicability.

References ¨ 1. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 2. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 3. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988.

References

145

6. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 7. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 8. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 9. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 10. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998. 11. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 12. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech, vol. 17, no. 6, pp. 533–559, 1967. 13. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 14. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2007. 15. W. Fulks and J. O. Sather, “Asymptotics II: Laplace’s method for multiple integrals,” Pacific J. of Math., vol. 11, pp. 185–192, 1961. 16. B. Riemann, Gesammelte Mathematische Werke. Leipzig: Teubner, 1876. 17. P. Debye, “N¨aherungsformeln f¨ur die zylinderfunktionen f¨ur grosse werte des arguments und unbeschr¨ankt verander liche werte des index,” Math Ann., vol. 67, pp. 535–558, 1909. ¨ 18. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. 19. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics. New York: McGraw-Hill, 1953. Vol. I. 21. E. T. Copson, Asymptotic Expansions. London: Cambridge University Press, 1965. 22. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 23. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering. Cambridge: Cambridge University Press, 1992. 24. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves. Bristol, England: Adam Hilger, 1986. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 26. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 27. A. Ba˜nos, Dipole Radiation in the Presence of a Conducting Half-Space. Oxford: Pergamon Press, 1966. Sect. 3.3. 28. L. C. Hsu, “On the asymptotic evaluation of a class of multiple integrals involving a parameter,” Duke Math. J., vol. 15, pp. 625–634, 1948. 29. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 30. F. W. J. Olver, Asymptotics and Special Functions. Natick: A K Peters, 1997.

146

10 Asymptotic Methods of Analysis

Problems 10.1. Beginning with the contour integral definition of the Bessel function J . / due to Schl¨afli, given by J . /

1 2 i

Z

1Ci

e sinh   d ;

1i

for argf g < =2, where the integration contour extends from 1  i  to 1 C i  in the domain i    00  i , 0   0  C1, use Olver’s theorem to derive the asymptotic expansion of J . / as j j ! 1 with argf g < =2. See pp. 130–133 of Olver [30]. 10.2. Prove the limiting result given in (10.64). 10.3. Show that C. / D Ai . / along the contour L32 depicted in Fig. 10.6. 10.4. Using the integral representation of the Airy function Ai . / given in (10.73), determine both the Maclaurin series approximation and the asymptotic expansion of the Airy function and its first derivative for real . Estimate the number of terms required in each expansion to provide a matched expansion representation of both the Airy function and its first derivative that is valid for all real , as illustrated in Fig. 10.7. 10.5. Show that the real part of the complementary error function erfc. / is even symmetric about the 00 -axis whereas the imaginary part is odd symmetric, where

D 0 C i 00 ; that is, show that < ferfc.  /g D < ferfc. /g and = ferfc.  /g D = ferfc. /g. 10.6. Use the rational approxiamations of the cosine and sine Fresnal integrals given in (10.114)–(10.117) to generate the graphs of the Cornu spiral given in Figs. 10.12 and 10.13. 10.7. Use the integral definition of the error function given in (10.94) to determine the asymptotic expansion of the complementary error function erfc. / given in (10.101) as j j ! 1 with j arg. /j < =2. 10.8. Determine the asymptotic expansion of Dawson’s integral FD ./, given in (10.97), for real  > 0 as  ! 1.

Chapter 11

The Group Velocity Approximation

Because of its mathematical simplicity and direct physical interpretation, the group velocity approximation has gained widespread use in the physics, engineering, and mathematical science communitites. However, the fundamental assumptions that are used to obtain this description are violated when either the loss component of the material dispersion cannot be neglected or the pulse spectrum becomes ultrawideband, which is taken here to mean that the bandwidth of the pulse spectrum spans at least one critical feature in the material dispersion. This inconsistency then results in intellectual mayhem over such topics as superluminal pulse velocites and superluminal tunneling in the ultrashort pulse dispersion regime. Because of this, it is essential to fully understand this approximate theory so that a better appreciation of the necessity of an asymptotic theory may be gained.

11.1 Historical Introduction Dispersive wave propagation was first considered in terms of a coherent superposition of monochromatic scalar wave disturbances by Sir William R. Hamilton [1] in 1839 where the concept of group velocity was first introduced. In that paper, Hamilton compared the phase and group velocities of light, showing that the phase velocity of a wave is given by the ratio !=k while the velocity of the wave group is given by d!=d k, where ! denotes the angular frequency and k the wavenumber of the disturbance. Subsequent to this definition, Stokes [2] posed the concept of group velocity as a “Smiths Prize examination” question in 1876. Lord Rayleigh then mistakenly attributed the original definition of the group velocity to Stokes, stating that [3] when a group of waves advances into still water, the velocity of the group is less than that of the individual waves of which it is composed; the waves appear to advance through the group, dying away as they approach its anterior limit. This phenomenon was, I believe, first explained by Stokes, who regarded the group as formed by the superposition of two infinite trains of waves, of equal amplitudes and of nearly equal wave-lengths, advancing in the same direction.

Rayleigh then applied these results to explain the difference between the phase and group velocities of light with respect to their observability [4]. K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 11, 

147

148

11 The Group Velocity Approximation

These early considerations are best illustrated by the coherent superposition of two time-harmonic waves with equal amplitudes and nearly equal wave numbers (k and k C ık) and angular frequencies (! and ! C ı!, respectively) traveling in the positive z-direction. The linear superposition of these two monochromatic wave functions then yields the polychromatic waveform [5, 6] U.z; t / D a cos .kz  !t / C a cos ..k C ık/z  .! C ı!/t/   1 N  !t N /; D 2a cos .zık  t ı!/ cos .kz 2

(11.1)

which is an amplitude modulated wave with mean wavenumber kN D k C ık=2 and mean angular frequency !N D ! C ı!=2. The surfaces of constant phase propagate with the phase velocity !N (11.2) vp ; kN while the surfaces of constant amplitude propagate with the group velocity vg

ı! : ık

(11.3)

Notice that these results are exact for the waveform given in (11.1). If the medium is nondispersive, then kN D !=c, N ık D ı!=c, and the phase and group velocities are equal. However, if the medium exhibits temporal dispersion so that k.!/ D .!=c/n.!/ where n.!/ is not a constant, then the phase and group velocities will, in general, be different. In particular, if n.!/ > 0 increases with increasing !  0, then vp  vg > 0 and the phase fronts will advance through the wave group as described by Rayleigh [3]. This elementary phenomenon is illustrated in Fig. 11.1 for the simple wave group described in (11.1). Each wave pattern illustrated in this figure describes a “snapshot” of the wave group at a fixed instant of time. In the upper wave pattern, the coincidence at z D 0 of a particular peak amplitude point in the envelope (marked with a G) with a peak amplitude point in the waveform (marked with a P ) is indicated. As time increases from this initial instant of time (t D 0), these two points become increasingly separated in time, as illustrated in the middle (t D ıt ) and bottom (t D 2ıt ) wave patterns, showing that the phase velocity of the wave is greater than the group velocity of the envelope in this case. The group velocity approximation was precisely formulated by Havelock [7,8] in 1914 based upon Kelvin’s stationary phase method [9]. It is apparent that Havelock was the first to employ the Taylor series expansion of the wavenumber k.!/ about a given wavenumber value k0 that the spectrum of the wave group is clustered about, referring to this approach as the group method. In addition, Havelock stated that [8] “The range of integration is supposed to be small and the amplitude, phase and velocity of the members of the group are assumed to be continuous, slowly varying, functions” of the wavenumber k.!/. This research then established the group velocity method for dispersive wave propagation. Because the method of

11.1 Historical Introduction

149 G,P |

0

z

0

U(z,t)

GP || 0

t = dt

0

z

G P || 0

t=0

0

t = 2dt z

Fig. 11.1 Evolution of a simple wave group in a temporally dispersive medium with normal frequency dispersion (i.e., when d n.!/=d! > 0). In this case, the phase front point P , which is coincident with the wave group amplitude point G at t D 0 in the upper part of the figure, advances through the wave group as t increases

stationary phase [10] requires that the wavenumber be real-valued, this method cannot properly treat causally dispersive, attenuative media. Furthermore, notice that Havelock’s group velocity method is a significant departure from Kelvin’s stationary phase method with regard to the wavenumber value k0 about which the Taylor series expansion is taken. In Kelvin’s method, k0 is the stationary phase point of the wavenumber k.!/ whereas in Havelock’s method k0 describes the wavenumber value about which the wave group spectrum is peaked. This apparently subtle change in the value of k0 results in significant consequences for the accuracy of the resulting group velocity description. Finally, notice that the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic in this formulation, the characteristics then propagating instantaneously [11] instead of at the vacuum speed of light c. The group velocity approximation was later refined and extended during the period from 1950 through 1970, most notably by Eckart [12] who considered the close relationship between the method of stationary phase and Hamilton–Jacobi ray theory in dispersive but nonabsorptive media. Of equal importance are the papers by Whitham [13] and Lighthill [14] on the general mathematical properties of threedimensional wave propagation and the group velocity for ship-wave patterns and magnetohydrodynamic waves. The appropriate boundary value problem is solved in both papers through application of the method of stationary phase to a plane-wave expansion representation. Their approach, however, is useful only for nonabsorbing media, thereby limiting the types of dispersion relations that may be considered. The equivalence between the group velocity and the energy-transport velocity in lossfree media and systems was also established [13–17], thereby providing a physical

150

11 The Group Velocity Approximation

basis for the group velocity in lossless systems with an inconclusive extension to dissipative media [18, 19]. The mathematical basis for the group velocity approximation was completed when the quasimonochromatic or slowly varying envelope approximation was precisely formulated by Born and Wolf [6] in the context of partial coherence theory. More recently published treatments concerned with the propagation of wave packets in dispersive and absorptive media [20–23] have employed Havelock’s technique of expanding the phase function appearing in the integral representation of the field in a Taylor series about some fixed characteristic frequency of the initial pulse. This approach may also be coupled with a recursive technique in order to obtain purported correction terms of arbitrary dispersive and absorptive orders for the resultant envelope function. This analysis again relies upon the quasimonochromatic approximation, and hence, can only be applied to study the evolution of pulses with slowly varying envelope functions in weakly dispersive systems. This approximate approach has since been adopted as the standard in both fiber optics [24] and nonlinear optics in general [25,26] with little regard for either its accuracy or its domain of applicability. In summary, this group velocity description of dispersive pulse propagation is based on both the slowly varying envelope approximation and the Taylor series approximation of the complex wavenumber about some characteristic angular frequency !c of the initial pulse at which the temporal pulse spectrum is peaked, as originally described by Havelock [7, 8]. The slowly varying envelope approximation is a hybrid time and frequency domain representation [26] in which the temporal field behavior is separated into the product of a slowly varying temporal envelope function and an exponential phase term whose angular frequency is centered about !c . The envelope function is assumed to be slowly varying on the time scale tc  1=!c , which is equivalent [27] to the quasimonochromatic assumption that its spectral bandwidth ! is sufficiently narrow that the inequality !=!c  1 is satisfied. Under these approximations, the frequency dependence of the wavenumber may then be approximated by the first few terms of its Taylor series expansion about the characteristic pulse frequency !c with the unfounded assumption [20, 21, 26] that improved accuracy can always be obtained through the inclusion of higher-order terms. This assumption has been proven incorrect [28, 29] in the ultrashort pulse, ultrawideband signal regime, optimal results being obtained using either the quadratic or the cubic dispersion approximation of the wavenumber. Recently published research [28, 29] by Xiao and Oughstun has identified the space–time domain within which the group velocity approximation is valid. Because of the slowly varying envelope approximation together with the neglect of the frequency dispersion of the material attenuation, the group velocity approximation is invalid in the ultrashort pulse regime in a causally dispersive material or system, its accuracy decreasing as the propagation distance z  0 increases. This is in contrast with the modern asymptotic description whose accuracy increases in the sense of Poincar´e [10] as the propagation distance increases. There is then a critical propagation distance zc > 0 such that the group velocity description using either the quadratic or cubic dispersion approximation provides an accurate description of the

11.1 Historical Introduction

151

pulse dynamics when 0  z  zc , the accuracy increasing as z ! 0, while the modern asymptotic theory provides an accurate description when z > zc , the accuracy increasing as z ! 1. This critical distance zc depends upon both the dispersive material and the input pulse characteristics including the pulse shape, temporal width, and characteristic angular frequency !c . For example, zc D 1 for the trivial case of vacuum for all pulse shapes, whereas zc  zd for an ultrashort, ultrawideband pulse in a causally dispersive dielectric with e 1 penetration depth zd at the characteristic oscillation frequency !c of the input pulse. In an attempt to overcome these critical difficulties, Brabec and Krausz [30] have proposed to replace the slowly varying envelope approximation with a slowly evolving wave approach that is supposed to be “applicable to the single-cycle regime of nonlinear optics.” As with the slowly varying envelope approximation, the difficulty with the slowly evolving wave approach is twofold. First, the fundamental hyperbolic character of the underlying wave equation is approximated as parabolic. The characteristics then propagate instantaneously [11]. Second, the subsequent Q imposed Taylor series expansion of the complex wavenumber k.!/ about !c approximates the material dispersion by its local behavior about some characteristic angular frequency of the initial pulse. Because this approximation is incapable of correctly describing the precursor fields, it is then incapable of correctly describing the dynamical evolution of any ultrashort pulse and its accuracy monotonically decreases [28] as z exceeds a single absorption depth zd in the dispersive medium. Recent research has been focused on the contentious topic of superluminal pulse propagation [31–38] in both linear and nonlinear optics. Again, the origin of this controversy may be found in the group velocity approximation which is typically favored by experimentalists. In response, Landauer [32] has argued for more careful analysis of experimental measurements reporting superluminal motions. Diener [33] then showed that “the group velocity cannot be interpreted as a velocity of information transfer” in those situations in which it exceeds the vacuum speed of light c. This analysis is in fact based upon an extension of Sommerfeld’s now classic proof [39, 40] that the signal arrival cannot exceed c in a causally dispersive medium. Kuzmich, Dogariu, Wang, Milonni, and Chiao [36] defined a signal velocity that is operationally based upon the optical signal-to-noise ratio and showed that, in those cases when the group velocity is negative, “quantum fluctuations limit the signal velocity to values less than c.” In addition, they argue that a more general definition of the “signal” velocity of a light pulse must satisfy two fundamental criteria:  “First, it must be directly related to a known and practical way of detecting a

signal.”  “Second, it should refer to the fastest practical way of communicating infor-

mation.” In contrast, Nimtz and Haibel [37] argue regarding superluminal tunneling phenomena that “the principle of causality has not been violated by superluminal signals as a result of the finite signal duration and the corresponding narrow frequency-band

152

11 The Group Velocity Approximation

width.” In addition, Winful [38] argues that “distortionless tunneling of electromagnetic pulses through a barrier is a quasistatic process in which the slowly varying envelope of the incident pulse modulates the amplitude of a standing wave.” In particular, “for pulses longer than the barrier width, the barrier acts as a lumped element with respect to the pulse envelope. The envelopes of the transmitted and reflected fields can adiabatically follow the incident pulse with only a small delay that originates from energy storage.” Unfortunately, each of these arguments neglects the frequency-dependent attenuation of the material comprising the barrier. When material attenuation is properly included, the possibility of evanescent waves is replaced by inhomogeneous waves [41,42], thereby rendering the accuracy of this superluminal tunneling analysis as questionable at best. This fundamental question of superluminal pulse propagation and tunneling provides the impetus for obtaining a deeper and physically correct understanding of the dispersive pulse phenomena that are involved.

11.2 The Pulsed Plane Wave Electromagnetic Field An important class of wave fields that arises in many practical situations is that in which either of the field vectors are transverse to some specified propagation direction. These are the transvers electric (TE) and transverse magnetic (TM) mode fields whose importance arises, for example, in the analysis of reflection and transmission phenomena at a dielectric interface. Common to both of these mode solutions is the plane wave field which also holds a position of fundamental importance in the angular spectrum of plane waves representation of time-domain electromagnetic wave-fields [see Vol. 1]. Because the analysis of plane wave pulse propagation through a dispersive medium yields the fundamental dynamics of pulse dispersion that is unencumbered by diffraction effects, this field type is of central importance to the theoretical development presented here. For a transverse electromagnetic wave field with respect to the z-axis, it is required that both Ez .r; t / and Bz .r; t / vanish for all z  z0 . The appropriate solution may then be obtained from the angular spectrum of plane waves representation given in (9.5). One may, without any loss of generality, choose the plane wave field to be linearly polarized along some convenient direction that is orthogonal to the z-axis. Any other state of polarization may then be obtained through an appropriate linear superposition of such linearly polarized wave fieldes with suitable orientations of the vibration plane [see Sect. 7.2 of Vol. 1]. Let the electric field vector be linearly polarized along the y-axis so that E.r; t / D 1O y Ey .z; t /;

(11.4)

B.r; t / D 1O x Bx .z; t /;

(11.5)

11.2 The Pulsed Plane Wave Electromagnetic Field

153

where 1 Ey .z; t / D < 

(Z CC

kck < Bx .z; t / D  

Q EQ y.0/ .!/e i .k.!/ z!t / d!

(Z CC

) ;

) Q k.!/ Q EQ y.0/ .!/e i .k.!/ z!t / d! ; !

(11.6) (11.7)

with z D z  z0 denoting the propagation distance into the positive half-space z  z0 from the input plane at z D z0 . Let the initial time behavior of the plane wave electric field vector at the plane z D z0 be specified by the dimensionless function f .t / of the time t as (11.8) Ey.0/ .t / D E0 f .t /; .0/ with temporal frequency spectrum EQ y .!/ D E0 fQ.!/, where

fQ.!/ D

Z

1

f .t /e i!t dt

(11.9)

1

is the Fourier–Laplace transform of f .t /. With this substitution, the pair of relations appearing in (11.6) and (11.7) become Z iaC1

1 Q E0 < fQ.!/e i .k.!/ z!t / d! ;  ia Z iaC1

kck Q i .k.!/ z!t / Q n.!/f .!/e d! ; E0 < Bx .z; t / D  c ia

Ey .z; t / D

(11.10) (11.11)

Q with use of the expression k.!/ D .!=c/n.!/. For convenience, this pair of expressions may be rewritten as Z iaC1

1 E0 < fQ.!/e . z=c/.!;/ d! ;  ia Z iaC1

kck Bx .z; t / D  E0 < n.!/fQ.!/e . z=c/.!;/ d! ; c ia

Ey .z; t / D

(11.12) (11.13)

for z D z  z0  0. The complex phase function .!;  / appearing in these equations is defined as [43, 44]   .!; / i ! n.!/   ;

(11.14)

Q where n.!/ D .c=!/k.!/ is the complex index of refraction of the dispersive medium, and where ct (11.15)  z

154

11 The Group Velocity Approximation

is a nondimensional space–time parameter defined for all z > 0. Notice that both the electric and magnetic field vectors given in (11.4) and (11.5) and (11.12)–(11.13) may be obtained from the single vector potential A.z; t / D 1O y A.z; t / kck D 1O y E0 < 

(Z

iaC1 ia

fQ.!/ i .k.!/ z!t Q / d! e !

) (11.16)

as 1 @A.z; t / 1 @A.z; t / D 1O y ; kck @t kck @t @A.z; t / B.z; t / D r  A.z; t / D 1O x ; @z

E.z; t / D 

(11.17) (11.18)

in agreement with the results of Sect. 6.2 of Vol. 1. On the other hand, if the initial time behavior of the plane wave magnetic induction field vector at the plane z D z0 is specified by the dimensionless function g.t / so that (11.19) Bx.0/ .z; t / D B0 g.t /; .0/

with temporal frequency spectrum BQ x .!/ D B0 g.!/, Q where Z

1

g.!/ Q D

g.t /e i!t dt

(11.20)

1

is the Fourier–Laplace transform of g.t /, then, from (9.8) [see also (9.279)] B0 g.!/ Q D kck

Q k.!/ kck E0 fQ.!/ D n.!/E0 fQ.!/; ! c

(11.21)

and the pair of relations given in (11.12) and (11.13) become Z iaC1

g.!/ Q c B0 < e . z=c/.!;/ d! ; kck n.!/ ia Z iaC1

1 . z=c/.!;/ Bx .z; t / D  B0 < g.!/e Q d! ;  ia

Ey .z; t / D

(11.22) (11.23)

for the propagated plane wave field for z D z  z0  0. The corresponding vector potential is then given by A.z; t / D 1O y A.z; t / with A.z; t / D

Z iaC1

g.!/ Q 1 B0 < i e . z=c/.!;/ d! ; Q  k.!/ ia

(11.24)

where the electric and magnetic field vectors are as given in (11.17) and (11.18), respectively.

11.2 The Pulsed Plane Wave Electromagnetic Field

155

Because of the requirement of causality, admissable models for describing the behavior of the complex index of refraction n.!/ in the complex !-plane must satisfy the symmetry relation [see Sect. 4.3 and Problem 5.2 of Vol. 1] n.!/ D n .!  /;

(11.25)

  .!; / D i ! n.!/    

D   .!  ;  /; D i !  n.!  /  

(11.26)

and consequently

so that the complex phase function .!;  / satisfies the same symmetry relation in the complex !-plane. Furthermore, for any real-valued initial pulse function f .t / one has that fQ.!/ D

Z

1

f .t /e i!t dt

1

Z

1



f .t /e i! t dt

D



D fQ .!  /:

(11.27)

1

Since g.t / is real-valued, its spectrum also satisfies this symmetry relation [i.e., g.!/ Q D gQ  .!  /]. It is then seen that the symmetry relation given in (11.25) Q D for the complex index of refraction directly follows from the relation B0 g.!/ .kck=c/ n.!/E0 fQ.!/. Because of these fundamental symmetry relations, the integral expression given in (11.12) may be rewritten as E0 Ey .z; t / D 2

(Z

iaC1

fQ.!/e . z=c/.!;/ d!

ia

Z

iaC1

C

)  fQ .!/e . z=c/ .!;/ d! 

ia

D

E0 2

(Z

iaC1

fQ.!/e . z=c/.!;/ d!

ia

Z

iaC1

C

) fQ.  ! 0 C i ! 00 /e

. z=c/.! 0 Ci! 00 ;/

d.! 0 C i ! 00 / ;

ia

where ! 0 0 whose Fourier–Laplace transform is fQı .!/ D

Z

1

ı.t  t0 /e i!t dt D e i!t0 ; 0

(11.50)

11.2 The Pulsed Plane Wave Electromagnetic Field

161

the propagated scalar wave disturbance is then given by the integral representation 1 < Aı .z; t / D 2

Z e

.z=c/t0 .!;t0 /

d! ;

(11.51)

C

where t0 .!; t0 / i ! .n.!/  t0 /

(11.52)

is the retarded complex phase function with retarded space–time parameter t0

c .t  t0 /: z

(11.53)

The propagated wave field given by (11.51) is called the impulse response of the dispersive medium. Its central importance in dispersive pulse propagation lies in the fact that the propagated wave field due to any other input pulse f .t / is given by the convolution operation A.z; t / D f .t / ˝ Aı .z; t /

(11.54)

for all z  0.

11.2.2 The Heaviside Unit Step Function Signal For a unit step function modulated signal, the initial pulse envelope is given by the Heaviside unit step function 0 I for t < 0 uH .t / I (11.55) 1 I for t > 0 that is, the external current source for the field abruptly begins to radiate harmonically in time at the instant t D 0 and continues indefinitely for all t > 0 with a constant amplitude and frequency. The Fourier–Laplace transform of this initial envelope function is then given by Z 1 i e i!t dt D uQ H .!/ D (11.56) ! 0 for =f!g > 0. The Fourier–Laplace integral representation of the propagated plane wave signal is then given by 1 AH .z; t / D  < 2

Z C

1 e .z=c/.!;/ d! !  !c

(11.57)

for t > 0 and is zero for t < 0, where z  0. This canonical wave field is precisely the signal considered by Sommerfeld [40] and Brillouin [43, 45] in 1914, Baerwald

162

11 The Group Velocity Approximation 2

10

1

10

0

10

−1

c

10

−2

10

−3

10

−4

10

−5

10

−6

10

0

10

1

10

2

10

3

10 (r/s)

4

10

5

10

6

10

Fig. 11.2 Angular frequency dependence of the magnitude of the spectrum for the Heaviside unit step function signal with angular carrier frequency !c D 1  103 r=s

[46] in 1930, and Oughstun and Sherman [47–49] in 1975 in order to give a precise definition of the signal velocity in a dispersive medium. As such, it is one of the most fundamental canonical problems to be considered in this area of research. The angular frequency dependence of the magnitude of the spectrum uQ H .!/ D i=.! !c / for the Heaviside unit step function signal with angular carrier frequency !c D 1  103 r=s is presented in Fig. 11.2. The frequency behavior depicted here illustrates the basic features of an ultrawideband signal, the most important feature being the ! 1 fall-off in spectral amplitude as ! ! 1. Notice that the signal does not have to be ultrashort in order for it to be ultrawideband, as the temporal duration of the Heaviside step function modulated signal is infinite. Notice further that an ultrawideband pulse does not need to be an envelope modulated sinusoidal signal, as the delta function pulse given in (11.49) is certainly both ultrashort and ultrawideband.1 1

In its 2002 report (Tech. Rep. FCC 02-48), the Federal Communications Commission defined an “ultra-wideband” device as any device for which the fractional bandwidth FB 2

fH  fL fH C fL

is greater than 0:25 or otherwise occupies 1:5 GHz or more of spectrum when the center frequency is greater than 6 GHz. Here fH denotes the 10 dB upper limit and fL the 10dB lower limit of the energy bandwidth. The center frequency of the waveform is defined there as the average of the

11.2 The Pulsed Plane Wave Electromagnetic Field

163

11.2.3 The Double Exponential Pulse A pulse shape that is similar in temporal structure to the delta function pulse but that possesses a nonvanishing temporal width is the double exponential pulse   fde .t / a e ˛1 t  e ˛2 t uH .t / (11.58) with ˛j > 0 for j D 1; 2, and where the constant a is chosen such that the peak amplitude of the pulse is unity. The peak amplitude point of the pulse occurs at the instant of time tm > 0 when dfde .t /=dt D 0, so that ˛1 e ˛1 tm  ˛2 e ˛2 tm D 0; with solution tm D

ln ˛1 =˛2 : ˛1  ˛2

(11.59)

Because ude .tm / 1, substitution of (11.59) in (11.58) then gives 1  : a D e ˛1 tm  e ˛2 tm

(11.60)

A measure of the temporal width of the pulse is given by the temporal difference between the e 1 points of the leading and trailing edge exponential functions in (11.58), so that t D j˛1  ˛2 j=.˛1 ˛2 /. Finally, with the result given in (11.56), the spectrum of the double exponential pulse is found to be given by fQde .!/ D a



1 1  ! C i ˛1 ! C i ˛2

 ;

(11.61)

which is clearly ultrawideband.

11.2.4 The Rectangular Pulse Envelope Modulated Signal For a unit amplitude, rectangular pulse envelope modulated signal, the initial pulse envelope is given by the rectangle function uT .t /

n0I 1I

for either t < 0 or t > T I for 0 < t < T

(11.62)

upper and lower 10 dB points, so that fc

1 .fH C fL /: 2

In turn, the energy bandwidth was defined in 1990 by the OSD/DARPA Ultra-Wideband Radar Review Panel (Tech. Rep. Contract No. DAAH01-88-C-0131, ARPA Order 6049) as the frequency range within which some specified fraction of the total signal energy resides.

164

11 The Group Velocity Approximation

AT (0,t)

1

uT (t)

0

−1

-uT (t) 0

T/2 t (s)

T

Fig. 11.3 Temporal field structure of a 10-cycle, unit amplitude, rectangular envelope modulated sinusoidal signal. The dashed curves describe the envelope function ˙uT .t /

that is, the external current source for the wave field abruptly begins to radiate harmonically in time at time t D 0 and continues with a constant amplitude and frequency up to the time T > 0 at which it abruptly ceases to radiate, as illustrated in Fig. 11.3 for a 10-cycle pulse. Notice that this rectangular envelope function can be represented by the difference between two Heaviside step function envelopes displaced in time by the pulse width T . The Fourier–Laplace transform of the rectangular envelope function uT .t / defined in (11.62) is given by Z

T

e i!t dt D

uQ T .!/ D 0

 1  i!T e 1 : i!

(11.63)

The Fourier–Laplace integral representation of the propagated plane wave pulse then becomes (Z 1 1 AT .z; t / D  < e .z=c/.!;/ d! 2 C !  !c ) Z 1 i!c T .z=c/T .!;T / e e d! ; (11.64) C !  !c for t > 0 and is zero for t < 0, where z  0. Here   T .!; T / i ! n.!/  T

(11.65)

11.2 The Pulsed Plane Wave Electromagnetic Field

165

is the generalized complex phase function [cf. (11.52)] with retarded space–time parameter cT c : (11.66) T .t  T / D   z z Notice that the first integral in (11.64) is exactly the same as that given in the integral representation (11.57) for the unit step function modulated signal and that the second integral appearing in (11.64) is of the exact same form except that the phase function is retarded in time by the initial pulse width T . Finally, because the simple pole singularity at ! D !c appearing in each of the integrands in (11.64) can be removed by simply combining these two integrals as ( )   Z sin .!  !c /T =2 .z=c/T =2 .!;T =2 / 1 i!c T =2 AT .z; t / D < i e e :  !  !c C

(11.67)

Because   sin .!  !c /T =2 T lim D ; !!!c !  !c 2 the spectrum uQ T .!/ of the rectangular envelope pulse is actually analytic for all complex values of ! (i.e., it is an entire function of complex !), as it must be because the envelope function uT .t / has compact temporal support (i.e., it identically vanishes ouside of a finite time domain). The ultrawideband character of the spectrum uQ T .!/ of the rectangular envelope pulse is clearly evident in Figs. 11.4 and 11.5. The solid curve in each figure displays the magnitude of the initial pulse spectrum, Fig. 11.4 for a single cycle pulse with carrier frequency !c D 1  103 r=s and Fig. 11.5 for a 10-cylce pulse. The dashed curve in each figure displays the j!  !c j1 frequency dependence of the step function signal for comparison. Notice the manner in which the step function spectrum is approached by the rectangular envelope spectrum as the initial pulse width T is increased. The spectrum of the rectangular envelope pulse is then seen to be ultrawideband for all T > 0.

11.2.5 The Trapezoidal Pulse Envelope Modulated Signal A canonical pulse envelope shape of central importance to both radar and cellular communication systems is the trapezoidal pulse envelope modulated signal of initial duration T > 0 with envelope rise-time Tr > 0 and fall-time Tf > 0. Such a pulse may be described by the time delayed difference between a pair of trapezoidal envelope signals with equal angular carrier frequencies !c and trapezoidal envelope functions given by ( ut rapj .t /

0; .t  Tj 0 /=Tj ; 1;

for t  Tj 0 ; for Tj 0  t  Tj 0 C Tj ; for  Tj 0 C Tj  t;

(11.68)

166

11 The Group Velocity Approximation 2

10

1

10

0

10

~

|uT ( )|

–1

10

c –2

10

–3

10

–4

10

–5

10

–6

10

10

0

10

1

10

2

3

4

10

10

5

10

6

10

(r/s)

Fig. 11.4 Log–log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a single cycle pulse with carrier frequency !c D 1  103 r=s and initial pulse duration T D 2=!c D 6:28 ms. The dashed curve displays the j!  !c j1 angular frequency dependence of the step function signal at the same carrier frequency

10

2

10

1

10

0 c

–1

~

|uT ( )|

10

–2

10

–3

10

–4

10

–5

10

–6

10

0

10

1

10

2

10

3

10 (r/s)

10

4

10

5

10

6

Fig. 11.5 Log–log plot of the angular frequency dependence of the magnitude of the spectrum for the rectangular pulse envelope modulated signal for a 10-cycle pulse with carrier frequency !c D 1  103 r=s and initial pulse duration T D 2=!c D 62:8 ms. The dashed curve displays the j!  !c j1 angular frequency dependence of the step function signal at the same carrier frequency

11.2 The Pulsed Plane Wave Electromagnetic Field

167 uT (t)

1

AT (0,t)

T+(Tr+Tf ) /2

0

–uT (t)

–1 –0.04 –0.02

0

0.02

0.04 t (s)

0.06

0.08

0.1

0.12

Fig. 11.6 Temporal field structure of a 10-cycle (between the half-amplitude points in the envelope function), unit amplitude, trapezoidal envelope modulated sinusoidal signal with !c D 1  103 r=s and equal rise- and fall-times Tr D Tf D 2=fc . The dashed curves describe the envelope function ˙uT .t /

for j D r; f . The total initial pulse duration is then given by T C Tr C Tf and the half-amplitude pulse width is T C .Tr C Tf /=2, as illustrated in Fig. 11.6 with Tr0 D 0s. The temporal angular frequency spectrum of this trapezoidal envelope function is then given by Z uQ t rapj .!/ D

1 1

ut rapj .t /e i!t dt

Z 1 Z Tj 0 CTj 1 i!t .t  Tj 0 /e dt C e i!t dt D Tj Tj 0 Tj 0 CTj   i D sinc !Tj =2 e i!.Tj 0 CTj =2/ ; !

(11.69)

where sinc. /  sin . /= . Notice that in the infinite rise-time (or fall-time) limit as  Tj ! 1, sinc !Tj =2 ! ı.!/ and the initial trapezoidal envelope signal spectrum becomes lim uQ t rapj .!/ D

Tj !1

i ı.!/e i!Tj 0 ; !

168

11 The Group Velocity Approximation

and a monochromatic,   time-harmonic signal is obtained. In the opposite limit as Tj ! 0, sinc !Tj =2 ! 1 and the initial trapezoidal envelope signal spectrum becomes lim uQ t rapj .!/ D

Tj !0

i i!Tj 0 ; e !

which is precisely the ultrawideband spectrum for a step function envelope signal [cf. (11.56)]. Notice that the trapezoidal envelope function is continuous with a discontinuous first derivative at both t D Tj 0 and t D Tj , whereas the Heaviside step function envelope is discontinuous in both its value and its first derivative at t D Tj 0 . The trapezoidal envelope function then retains just the latter feature of the step function envelope, albeit displaced in time by the initial rise-time Tr . In general, the envelope spectrum uQ t rapj .!  !c / described by (11.69) for a trapezoidal envelope signal with fixed angular carrier frequency !c > 0 will be ul> trawideband provided that the inequality 2=Tj  !c is satisfied. In that case, the 1 spectral factor .!  !c / that is characteristic of an ultrawideband signal will remain essentially unchanged over the positive angular frequency domain Œ0; !c , as illustrated in Fig. 11.7. This inequality is equivalent to the inequality
0 is indicative of the rapidity of turn-on of the signal. In contrast with the trapezoidal envelope function, this envelope function is continuous in time with continuous derivatives for all finite, positive values of ˇT . The corresponding rise-time is then seen to be inversely proportional to ˇT , so that Tr  1=ˇT , as seen in Fig. 11.9. In the limit as ˇT ! 1, uht .t / ! uH .t /

170

11 The Group Velocity Approximation

1

b T = 10 b T =1

0.8

uht (t)

b T = 0.1 0.6

0.4

0.2

0 −10

−8

−6

−4

−2

0

2

4

6

8

10

t

Fig. 11.9 Time dependence of the hyperbolic tangent envelope function. For whatever units the time scale t is in, the scale of the inverse rise-time parameter ˇT is given by its inverse so that the product ˇT t remains dimensionless

and the Heaviside unit step function envelope is obtained. In the opposite limit as ˇT ! 0, uht .t / ! 12 which results in a time-harmonic signal with amplitude of 1=2. An example of a hyperbolic tangent envelope modulated signal Aht .0; t / D uht .t / sin .!c t / is illustrated in Fig. 11.10 when ˇT D !c =10. In that case the initial signal rise-time occurs in approximately ten oscillations of the carrier wave. The Fourier transform of the trapezoidal envelope function is given by the integral Z

1

1 e i!t dt 2ˇT t 1 C e 1 Z 1 1 1 x i!=ˇT dx; D 2 ˇT 0 x .x C 1/

uQ ht .!/ D

under the change of variable x D e ˇT t . For convergence, the variable ! is complex-valued and lies in the upper-half of the complex !-plane; that is, ! 00 =f!g > 0. To evaluate this definite integral, consider the following contour integral in the complex z-plane (with z D x C iy) I I.˛/

zi˛ d z; z.z2 C 1/

11.2 The Pulsed Plane Wave Electromagnetic Field

171

1

Aht(0,t)

uht(t)

0

−uht(t) −1 −10

0 t

10

Fig. 11.10 Initial temporal field behavior Aht .0; t / of a hyperbolic tangent envelope modulated signal with inverse rise-time parameter ˇT D !c =10, where !c is the angular carrier frequency of the signal. The dashed curves describe the hyperbolic tangent envelope functions ˙uht .t /

for complex ˛ D ˛ 0 C i ˛ 00 with ˛ 00 > 0, where the integrand has simple poles at z D 0; ˙i and a branch point at z = 0. With the positive real axis chosen as the branch cut, the contour of integration to be used in evaluating this integral is as illustrated in Fig. 11.11. Along the contour C1 , z D x so that, in both the limit as R ! 1 and the limit as  ! 0 (see Fig. 11.11), Z C1

zi˛ dz D z.z2 C 1/

Z

1

0

x i˛ dx; x.x 2 C 1/

whereas along the contour C2 , z D xe i2 so that in the same limits as R ! 1  ! 0, Z C2

zi˛ d z D e 2 ˛ z.z2 C 1/

Z

1 0

x i˛ dx; x.x 2 C 1/

Along the outer circular contour  (see Fig. 11.1), z D Re i , and the following inequality is obtained ˇ Z 2 ˇZ 00 ˇ ˇ zi˛ R˛ 0 ˇ ˇ d z e ˛  d; ˇ z.z2 C 1/ ˇ 2 R C1  0

172

11 The Group Velocity Approximation iy

+i

R C1 C2

Branch Cut

x

–i

Complex z-Plane

Fig. 11.11 Contour of integration used in the evaluation  of the Fourier transform of the hyperbolic tangent envelope function uht .t / D 12 1 C tanh .ˇT t / . The circular contour  has radius R and extends over the angular domain  2 .0; 2/ in the counterclockwise sense, and the circular contour  has radius  and extends over the angular domain  2 .2; 0/ in the clockwise sense. The straight line contours C1 and C2 connect these two circular contours on either side of the branch cut taken along the positive x-axis, C1 extending from  to R in the upper-half plane and C2 extending from R to  in the lower-half plane

where the integral on the right-hand side of this inequality goes to zero as R ! 1 for ˛ 00 < 2. In addition, along the inner circular contour  , z D e i , and the following inequality is obtained ˇ Z 2 ˇZ 00 ˇ ˇ zi˛ ˛ 0 ˇ ˇ e ˛  d; ˇ z.z2 C 1/ d zˇ  2C1  0  where the integral on the right-hand side of this inequality goes to zero as  ! 0 for ˛ 00 > 0. Finally, by application of the residue theorem to the evaluation of the integral I.˛/, taking note that only the simple pole singularities at z D ˙i are enclosed by the contour, there results I

   zi˛ zi˛ Res C zDi zDCi z.z C i /.z  i / z.z C i /.z  i /    3 ˛ ˛ D i  e 2 C e 2 :

zi˛ d z D 2 i z.z2 C 1/



Res



Taken together, these integral evaluations then yield the result Z 1 =2 x i˛ dx D i I 0 < =f˛g < 2: 2 C 1/ .˛=2/ x.x sinh 0

11.2 The Pulsed Plane Wave Electromagnetic Field

173

The temporal frequency spectrum of the hyperbolic tangent envelope function defined in (11.71) is then given by uQ ht .!/ D i

=.2ˇT / I sinh .!=.2ˇT //

0 < =f!g < 2ˇT :

(11.72)

Since sinh .z/ D 0 at z D ˙n i for n an integer, the right-hand side of (11.72) possesses simple pole singularities at ! D !˙n where !˙n ˙2nˇT i I

n D 0; 1; 2; 3; : : : ;

(11.73)

so that the spectrum of the hyperbolic tangent envelope function possesses an infinite number of simple pole singularities evenly spaced along the imaginary ! 00 -axis with spacing 2ˇT . The inequality 0 < =f!g < 2ˇT appearing in (11.72) requires that the contour of integration appearing in the integral representation (11.48) of the propagated pulse wave field lies in the upper half of the complex !-plane between the real axis and the line parallel to the real axis passing through the first (n D 1) simple pole singularity at ! D !1 D 2ˇT i . Notice that in the limit as ˇT ! 1, the spectrum for the hyperbolic tangent envelope function given in (11.72) approaches the limit lim uQ ht .!/ D

ˇT !1

i !

(11.74)

which is precisely the expression for the temporal frequency spectrum of the Heaviside unit step function envelope [cf. (11.56)]. A comparison of the relative angular frequency dependence !=!c of the magnitude of the hyperbolic tangent envelope signal spectrum with that for the Heaviside step function signal for several values of the relative inverse rise-time parameter ˇT =!c is presented in Fig. 11.12. As for the trapezoidal envelope modulated signal, the hyperbolic tangent envelope modulated signal is seen to become ultrawideband when the approximate inequality >

ˇT =!c  1;

(11.75)

is satisfied, where !c is the angular frequency of the carrier wave, and becomes increasingly ultrawideband as ˇT ! 1, as seen in Fig. 11.12. Effectively, this inequality means that the initial envelope rise-time Tr  1=ˇT of the signal Aht .0; t / D uht .t / sin .!c t / occurs in approximately a single period of oscillation of the carrier wave or faster. Notice from (11.73) that the simple pole singularities !˙n D !c ˙ 2nˇT i move away from the real axis toward !c ˙ 1i as ˇT =!c increases above unity and the initial pulse envelope becomes increasingly ultrawideband, leaving just the single simple pole singularity at ! D !c along the positive real axis when ˇT D 1 and the hyperbolic tangent envelope signal has attained its Heaviside step function envelope signal limit.

174

11 The Group Velocity Approximation

102 100 10−2 10−4 10−6 10−8 10−10

0

1

2

3

4

Fig. 11.12 Comparison of the relative angular frequency dependence of the magnitude of the hyperbolic tangent envelope signal spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the inverse rise-time parameter ˇT relative to the angular carrier frequency !c

11.2.7 The Van Bladel Envelope Modulated Pulse An example of an infinitely smooth envelope function with compact temporal support (i.e., one that identically vanishes outside of a finite time domain) is given by the unit amplitude Van Bladel envelope function [50] ( uvb .t /



e

2

1C 4t.t  /

0I



I

when 0 < t <  ; when either t  0 or t  

(11.76)

p with temporal duration  > 0 and full pulse width = 2 at the e 1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a 2-cycle pulse ( D 2Tc ) and in Fig. 11.14 for a 10-cycle pulse ( D 10Tc ), with Tc 1=fc D 2=!c for a cosine carrier wave. This canonical pulse envelope function is of some importance to ultrashort optical pulse dynamics because its properties of infinite smoothness and temporal compactness are common to all experimental pulses. Notice that, although the Van Bladel envelope function equals unity at its midpoint when t D =2, the resultant modulated carrier wave will not unless one of its peak amplitude points coincides with the midpoint of the envelope function, as it does for the examples presented here with a cosine carrier wave.

11.2 The Pulsed Plane Wave Electromagnetic Field

175

1 uvb(t)

Avb(0,t)

_

0

–1 −0.01

–uvb(t) 0

0.01

t - t /2 (s) Fig. 11.13 Temporal field structure of a 2-cycle Van Bladel envelope modulated pulse with angular carrier frequency !c D 1103 r=s and temporal duration  D 2Tc . The dashed curves describe the envelope function ˙uvb .t /

1

Avb(0,t)

uvb(t)

0

–uvb(t) −1 −0.05

0

0.05

t - t /2 (s)

Fig. 11.14 Temporal field structure of a 10-cycle Van Bladel envelope modulated pulse with angular carrier frequency !c D 1  103 r=s and temporal duration  D 10Tc . The dashed curves describe the envelope function ˙uvb .t /

176

11 The Group Velocity Approximation 100 10−1 10−2 10−3

1

−4

10

10−5

/ Tc = 2

−6

10

10−7 /Tc = 10

10−8 10−9 10−10 1 10

102

103

104

105

(r/s)

Fig. 11.15 Comparison of the relative angular frequency dependence of the magnitude of the Van Bladel envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse duration 

Because the Van Bladel envelope function defined in (11.76) possesses compact temporal support, its Fourier transform uQ vb .!/ is an entire function [51] of complex !, where Z    2 1C 4t.t  / i!t e e dt: (11.77) uQ vb .!/ D 0

An accurate numerical evaluation of this Fourier transform integral may be accomplished using the fast Fourier transform (FFT) algorithm, provided that due care is given to both sampling and aliasing [52]. The results are presented in Fig. 11.15 for the spectral magnitude jQuvb .! !c /j with angular carrier frequency !c D 1103 r=s for both the two ( D 2Tc ) and 10-cycle ( D 10Tc ) Van Bladel envelope modulated pulses illustrated in Figs. 11.13 and 11.14, respectively. For comparison, the dashed curve in the figure describes the j!  !c j1 ultrawideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the Van Bladel envelope pulse is not as ultrawideband as its corresponding rectangular envelope modulated pulse is, as seen from a comparison of the two spectral amplitude curves in Fig. 11.15 with their rectangular envelope counterparts in Figs. 11.4 (for a single cycle rectangular envelope pulse) and 11.5 (for a 10-cycle rectangular envelope pulse).

11.2 The Pulsed Plane Wave Electromagnetic Field

177

11.2.8 The Gaussian Envelope Modulated Pulse An example of an infinitely smooth pulse envelope function that does not possess compact temporal support is given by the unit amplitude gaussian envelope function 2 =T 2

ug .t / D e .tt0 /

(11.78)

that is centered at the time t D t0 with initial pulse width t D 2T measured at the e 1 amplitude points, as illustrated in Fig. 11.16 when t0 D 0 and D =2 [which then corresponds to a cosine carrier wave, as described in (11.34)]. Notice that the choice of a cosine carrier wave places a peak at the peak amplitude point, whereas a sine carrier wave places a null at the peak amplitude point. The temporal frequency spectrum of the gaussian envelope function given in (11.78) is given by Z

1

2 =T 2

e .tt0 /

uQ g .!/ D

e i!t dt

1

D

p

1

T e  4 T

2!2

e i!t0 ;

(11.79)

which is another gaussian with angular frequency width ! D 4=T at the e 1 amplitude points. One then has the uncertainty product f t D 2=. For comparison,

1 ug(t)

Ag(0,t)

2T

0

−ug(t) −1 −0.01

0

0.01

t (s) Fig. 11.16 Temporal field structure of a 2-cycle gaussian envelope modulated pulse centered at t0 D 0 with angular carrier frequency !c D 1  103 r=s and temporal duration 2T D 2Tc . The dashed curves describe the envelope function ˙ug .t /

178

11 The Group Velocity Approximation 100

c

T = Tc 10

−5

T = 2Tc

T = 3Tc 10

−10

10

0

10

1

10

2

10

3

10

4

(r/s)

Fig. 11.17 Comparison of the relative angular frequency dependence of the gaussian envelope pulse spectrum (solid curves) with that for the Heaviside step function signal (dashed curve) for several values of the initial temporal pulse width 2T

the dashed curve in the figure describes the j!  !c j1 ultrawideband frequency behavior of the Heaviside step function envelope signal. Because of its infinitely smooth character, the gaussian envelope pulse spectrum is similar to that for the Van Bladel envelope pulse, as seen from a comparison of Figs. 11.15 and 11.17.

11.3 Wave Equations in a Simple Dispersive Medium and the Slowly Varying Envelope Approximation Because of its direct generalizability to pulse propagation in nonlinear media, the wave equations describing electromagnetic pulse propagation in a dispersive medium are of central importance to nonlinear optics in particular [24, 26, 53] and nonlinear wave phenomena in general [54, 55]. This approach parallels the Fourier integral formulation in the linear dispersion regime but, most importantly, it best illustrates the simplifying assumptions that are made in typical treatments of dispersive pulse propagation phenomena.

11.3 Dispersive Wave Equations

179

The analysis is presented here for the case of a simple polarizable dielectric2 for which the electric displacement vector is given by [see (4.50) and (4.91) of Vol. 1] D.r; t / D 0 E.r; t / C k4kP.r; t /;

(11.80)

where P.r; t / is the macroscopic polarization density [see (4.27) of Vol. 1], and where both B.r; t / D 0 H.r; t / and Jc .r; t / D 0 are satisfied. The temporal Fourier transform of the macroscopic polarization density is related to the macroscopic electric field intensity through the electric susceptibility e .!/ of the material as Q !/ Q !/ D 0 e .!/E.r; P.r;

(11.81)

where e .!/ (a macroscopic quantity) is related to the set of molecular polarizabilities ˛j .!/ of the material (microscopic quantities) through the expression P

e .!/ D

Nj ˛j .!/ P 0  .k4k=3/ j Nj ˛j .!/ j

(11.82)

which follows from the Clausius–Mossotti relation [see (4.169) in Vol. 1] X

Nj ˛j .!/ D

j

30 .!/=0  1 ; k4k .!/=0 C 2

(11.83)

which is also referred to as the Lorentz–Lorenz formula. Here Nj is the number density of molecular species with microscopic polarizability ˛j .!/, the latter quantity determined by a dynamical model for the molecular response to an appled timeharmonic electromagnetic wave field.

11.3.1 The Dispersive Wave Equations The source-free, time-domain form of Maxwell’s equations in a simple polarizable dielectric are given by 0 @H.r; t / ; kck @t   4  0 @E.r; t /   @P.r; t / ; C r  H.r; t / D  kck @t c  @t r  E.r; t / D 

2

(11.84) (11.85)

A simple polarizable dielectric is defined here as one for which the quadrupole and all higherorder moments of the molecular charge distribution identically vanish. Equivalently, it is one for which the electric displacement vector is described by (11.80) without approximation

180

11 The Group Velocity Approximation

Rt with r  D.r; t / D r  B.r; t / D 0. Since D.r; t / D 1 .t  t 0 /E.r; t 0 /dt 0 , then Rt r  D.r; t / D 1 .t  t 0 /r  E.r; t 0 /dt 0 D 0, so that r  E.r; t / D 0. The curl of (11.84) then gives, with substitution from (11.85) and the divergenceless character of the electric field intensity vector, r 2 E.r; t / 

   4  @2 P.r; t / 1 @2 E.r; t /   0 D :  c2  c 2 @t 2 @t 2

(11.86)

Similarly, the curl of (11.85) gives, with substitution from (11.84) and the divergenceless character of the magnetic intensity vector, r 2 H.r; t / 

   4  @   1 @2 H.r; t /   r  P.r; t / : D    2 2 c @t c @t

(11.87)

This pair of expressions are the inhomogeneous vector wave equations in a simple dispersive medium. With emphasis typically placed on the electric field component of the electromagnetic wave field, the electric field vector is then taken to be linearly polarized along the 1O x -direction with the direction of propagation in the positive z-direction. In that case E.r; t / D 1O x E.r; t / so that P.r; t / D 1O x P .r; t / and (11.86) becomes    4  @2 P .r; t / 1 @2 E.r; t /   0 r E.r; t /  2 D ;  c2  c @t 2 @t 2 2

(11.88)

which is the inhomogeneous scalar wave equation in a simple dispersive medium. The proper (i.e., without unnecessary simplifying assumptions and approximations) solution of either (11.86) and (11.87) for the electromagnetic wave field or (11.88) for the scalar optical wave field is entirely equivalent to the proper solution of the Fourier–Laplace integral representation for the electromagnetic wave field vectors given, for example, in (11.43) and (11.44), or to the Fourier–Laplace integral representation for the scalar wave field in (11.48). Finally, the central importance of this partial differential equation approach to dispersive wave propagation phenomena is fully realized when the macroscopic polarization density P .r; t / is extended to include nonlinear effects [56–58]. In that case, the polarization is written as P .r; t / D PL .r; t / C PNL .r; t / where PL .r; t / describes the linear response and PNL .r; t / describes the nonlinear response which vanishes as E ! 0.

11.3.2 The Slowly Varying Envelope Approximation The precise definition of the slowly varying envelope (SVE) or quasimonochromatic approximation has its origin in the classical theory of coherence developed by M. Born and E. Wolf [6]. Consider first the complex representation of a real

11.3 Dispersive Wave Equations

181

polychromatic scalar wave w.r; t / that exists for all time t 2 .1; 1/ and is square-integrable, viz., Z 1 w2 .r; t /dt < 1 (11.89) 1

at each point r 2 R . By the Fourier integral theorem [59], the wave field w.r; t / may then be represented by the Fourier integral expression 3

w.r; t / D

1 2

where

Z

1

w.r; Q !/e i!t d!;

(11.90)

w.r; t /e i!t dt:

(11.91)

1

Z

1

w.r; Q !/ D 1

Since w.r; t / is real, it then follows from (11.91) that w Q  .r; !/ D w.r; !/ for real ! and the real part of the wave spectrum is even-symmetric and the imaginary part is odd-symmetric. Because no additional information is then contained in the frequency spectrum w.r; Q !/ for ! < 0, Gabor [60] introduced the complex analytic signal Z 1 1 w.r; Q !/e i!t d! (11.92) v.r; t / 2 0 associated with the real signal w.r; t /. Notice that the complex analytic signal v.r; t / is obtained from its associated wave field w.r; t / at each point r of space simply by suppressing the amplitudes of all of the negative frequency components in the Fourier integral representation of w.r; t / given in (11.90). Because of this, v.r; t / is also referred to as the complex half-range function associated with the real scalar wave field w.r; t /. Let the angular frequency spectrum w.r; Q !/ of the scalar wave field u.r; t / be expressed in the form 1 (11.93) w.r; Q !/ D a.r; !/e i'.r;!/ ; 2 where both a.r; !/ and '.r; !/ are real-valued functions. Substitution of this representation into (11.90) then gives Z 1

Z 1 1 i!t  i!t w.r; t / D w.r; Q !/e d! C wQ .r; !/e d! 2 0 0   Z 1    1 1 i '.r;!/!t i '.r;!/!t Ce d! D a.r; !/ e 2 0 2 Z 1   1 D a.r; !/ cos '.r; !/  !t d!; (11.94) 2 0 whereas substitution of (11.93) in (11.92) gives v.r; t / D

1 4

Z 0

1

a.r; !/e i



'.r;!/!t



d!:

(11.95)

182

11 The Group Velocity Approximation

Comparison of these two expressions then shows that ˚ w.r; t / D 2< v.r; t / :

(11.96)

Application of the Plancherel–Parseval theorem [51] then shows that Z

Z 1 1 w .r; t /dt D Q !/j2 d! jw.r; .2/2 1 1 Z 1 Z 1 1 2 D Q !/j d! D 2 jw.r; jv.r; t /j2 dt 2 2 0 0 Z 1 Z 1 1 1 2 a .r; !/d! D a2 .r; !/d!: D 8 2 0 .4/2 1 1

2

(11.97)

These general results are now applied to the description of the properties of a quasimonochromatic wave field. Let the Fourier spectrum w.r; Q !/ of the real scalar wave field w.r; t / be centered about the angular frequencies C!0 and !0 with effective width !, as illustrated in Fig. 11.18. The wave field is then said to be quasimonochromatic if the effective width ! is small in comparison with the angular frequency !0 ; that is, provided that !  1: !0

(11.98)

~ wrel (r, )

1

0.5

0

–

0

(r/s)

0

Fig. 11.18 Angular frequency dependence of the normalized magnitude of the frequency spectrum of a quasimonochromatic wave centered about the angular frequency !0 with spectral width !

11.3 Dispersive Wave Equations

183

A polychromatic wave field is then defined as a superposition of mutually incoherent quasimonochromatics extending over a finite range of frequencies. As an illustration, consider the case of a strictly monochromatic wave field, as described by the real wave function w.r; t / D A0 .r/ cos .'0 .r/  !0 t / D 0 .r/e i!0 t C 0 .r/ e i!0 t

(11.99)

with 0 .r/ D

1 A0 .r/e i'0 .r/ : 2

(11.100)

The temporal frequency spectrum of this monochromatic scalar wave field is then given by the Fourier transform of (11.99) as 

w.r; Q !/ D 2 0 .r/ı.!  !0 / C 0 .r/ı.! C !0 / :

(11.101)

The associated complex analytic signal to this strictly monochromatic wave field is then given by Z

1

v.r; t / D 0

 0 .r/ı.!  !0 / C 0 .r/ı.! C !0 / e i!t d!

D 0 e i!0 t D

1 A0 .r/e i.'0 .r/!0 t/ : 2

(11.102)

The quasimonochromatic generalization of the scalar wave feld given in (11.99) may be synthesized from it by including temporal dependency in both the amplitude and phase functions as w.r; t / D A.r; t / cos .'.r; t /  !0 t /:

(11.103)

By analogy with (11.102), let the complex analytic signal corresponding to the scalar wave field in (11.103) be of the form v.r; t / D

1 A.r; t /e i.'.r;t/!0 t/ ; 2

(11.104)

v.r; t / D

 1 w.r; t / C i u.r; t / ; 2

(11.105)

which may be written as

where u.r; t / D A.r; t / sin .'.r; t /  !0 t /;

(11.106)

184

11 The Group Velocity Approximation

from (11.103) and (11.104). With (11.103) and (11.106) expressed as w.r; t / D A.r; t / cos ..r; t // and u.r; t / D A.r; t / sin ..r; t //, respectively, the amplitude and phase functions are found to be given by

1=2 A.r; t / D 2jv.r; t /j D w2 .r; t / C u2 .r; t / ; '.r; t / D !0 t C .r; t /; mod.2/;

(11.107) (11.108)

where cos ..r; t // D

w.r; t / ; 2jv.r; t /j

sin ..r; t // D

u.r; t / ; 2jv.r; t /j

(11.109)

at each point r 2 R3 . From (11.92) and (11.104) one has that A.r; t /e i'.r;t/ D

1 

Z

1

w.r; Q !/e i.!!0 /t d!;

(11.110)

0

at each point r 2 R3 . Under the change of variable !N D !  !0 , this transform relation becomes Z 1 1 Q i'.r;t/ N A.r; t /e D d !; N (11.111)

.r; !/e N i !t  !0 Q !/ where .r; N D w.r; Q ! C !0 / for !N  !0 describes the Fourier spectrum of a field which is essentially contained in the low frequency domain .!0 ; !0 /. Because the original wave field was assumed to be quasimonochromatic, the spectral width Q !/ ! of .r; N satisfies the inequality !  !0 and this then implies that the quantity A.r; t /e i'.r;t/ essentially contains only low frequency components. The quantity A.r; t /e i'.r;t/ , which is referred to as the complex envelope of the wave field, is then essentially constant over a time interval t satisfying the inequality t !  1, so that with !  !0 , the complex envelope is seen to be essentially constant over the time interval T0 D 2=!0 . Hence, the complex envelope of a quasimonochromatic wave field changes by only a negligible amount over a few oscillations of the carrier wave, so that the two inequalities ˇ ˇ ˇ ˇ 2 ˇ ˇ ˇ @ A.r; t / ˇ ˇ  !0 ˇ @A.r; t / ˇ ; ˇ ˇ @t ˇ ˇ @t 2 ˇ ˇ 2 ˇ ˇ ˇ ˇ @ '.r; t / ˇ ˇ ˇ ˇ ˇ  !0 ˇ @'.r; t / ˇ ; ˇ @t 2 ˇ ˇ @t ˇ are both satisfied.

(11.112) (11.113)

11.3 Dispersive Wave Equations

185

11.3.3 Dispersive Wave Equations for the Slowly Varying Wave Amplitude and Phase Let the quasimonochromatic scalar wave field E.r; t / have the complex phasor representation (see Sect. 5.1.2 of Vol. 1) E!c .r; t / A.r; t /e i.'.r;t/!c t/ ;

(11.114)

˚ where E.r; t / D < E!c .r; t / with A.r; t / and '.r; t / both being real-valued functions of position r and time t . The phasor representation of the induced polarization density [see (11.81)] is then given by P!c .r; t / 0 e .!c /E!c .r; t / D 0 e .!c /A.r; t /e i.'.r;t/!c t/ ;

(11.115)

˚ where P .r; t / D < P!c .r; t / . With these two substitutions, the scalar wave equation given in (11.88) becomes r 2 E!c .r; t / 

1 @2 E!c .r; t / .1 C k4k .! // D 0; e c c2 @t 2

which may be written as r 2 E!c .r; t / 

n2 .!c / @2 E!c .r; t / D 0; c2 @t 2

(11.116)

where n.!/ D .1 C k4ke .!//1=2 is the complex index of refraction of the dispersive medium [see (4.92) of Vol. 1]. This phasor form of the scalar wave equation is then seen to be characterized by the complex phase velocity vp .!/

c : n.!/

(11.117)

Finally, notice that (11.116) has been obtained without any approximation. This then makes it an ideal starting point for any approximate description of dispersive scalar wave propagation phenomena. Consider now the approximation of (11.116) in the SVE approximation as specified by the two inequalities in (11.112) and (11.113). Because @E!c D @t



 @A @' C iA  i !c A e i.'!c t/ ; @t @t

186

11 The Group Velocity Approximation

then @2 E!c D @t 2

"

 2 @2 ' @2 A @A @' @' C iA C 2i  A 2 2 @t @t @t @t @t

# @A @' 2i !c C 2!c A  !c2 A e i.'!c t/ @t @t "  ! #  2  @' @A @' @' 2  !c C 2!c A 

2i C !c A e i.'!c t/ : @t @t @t @t

The SVE approximation of the scalar wave equation given in (11.116) is then found to be given by   r 2 C kQ 2 .!c / A.r; t /e i'.r;t/ "   @A.r; t / n2 .!c / @'.r; t /  2i  !c c2 @t @t #   @'.r; t / i'.r;t/ @'.r; t /  2!c A.r; t /

0; C e @t @t (11.118) Q where k.!/ D .!=c/n.!/ is the complex wavenumber in the dispersive medium with complex index of refraction n.!/. This is then the general form of the scalar wave equation in the SVE approximation. Because   r 2 A.r; t /e i'.r;t/ D r  r A.r; t /e i'.r;t/   D r  .rA/e i' C iA.r'/e i'   D r 2 A C 2i.rA/  .r'/ C iAr 2 '  A.r'/2 e i' ; the SVE wave equation given in (11.118) becomes 

 r 2 C kQ 2 .!c / A.r; t / C 2i.rA.r; t //  .r'.r; t // CiA.r; t /r 2 '.r; t /  A.r; t /.r'.r; t //2 "   @A.r; t / n2 .!c / @'.r; t /  !  2i c c2 @t @t #   @'.r; t / @'.r; t /  2!c A.r; t / C

0; @t @t (11.119)

11.3 Dispersive Wave Equations

187

after the common factor e i'.r;t/ has been cancelled. As no additional approximations have been made, this form of the SVE wave equation is equivalent to that given in (11.118). Because A.r; t / and '.r; t / are both real-valued functions, then together with the facts that n2 .!c / D .nr .!c / C i ni .!c //2 D n2r .!c /  n2i .!c / C 2i nr .!c /ni .!c / with nr .!/ 0 for finite ! 0 D 0, (12.7) will yield a function "r .!/ that is consistent with all physical requirements, i.e., one which is in principle possible. This makes it possible to use (12.7) even when the function "i .!/ is approximate. On the other hand, (12.8) does not yield a physically possible function "i .!/ for an arbitrary choice of "r .!/, since the condition "i .! 0 / > 0 for finite ! 0 > 0 is not necessarily fulfilled.

Hence, in any serious attempt at obtaining the approximate behavior of the causally interrelated real and imaginary parts of the dielectric permittivity in some specified region of the complex !-plane, special care must be given to the mathematical form of the dispersion relation pair given in (12.7) and (12.8) in order that physically meaningful results are obtained.

254

12 Analysis of the Phase Function and Its Saddle Points

For a nonconducting medium as considered here, the material absorption identically vanishes at zero frequency [most simply because i .! 0 / is an odd function of real ! 0 ] so that, from (12.8), one immediately obtains the sum rule Z 1 "r .! 0 /  1 0 P d! D 0: (12.9) !0 1 The material absorption also identically vanishes at infinite real frequency, as can be seen from the limiting behavior of the dispersion relations given in (12.7) and (12.8) lim "r .! 0 / D 1;

(12.10)

lim "i .! 0 / D 0:

(12.11)

! 0 !˙1

! 0 !˙1

Hence, with little or no loss in generality, it is safe to assume that the angular frequency dependence of "i .! 0 / along the positive ! 0 -axis is such that the loss is significant only within a finite angular frequency domain Œ!0 ; !m with 0  !0 < !m  1:

(12.12)

For all nonnegative values of ! 0 outside of this frequency domain, the material absorption is then negligible by comparison. Attention is now focused on the two special regions of the complex !-plane wherein the dielectric permittivity is reasonably well behaved, these being the circular region about the origin where j!j  !0 and the annular region about infinity where j!j  !m . The analysis presented here follows that given by the author [10] in 1994.

12.1.1 The Region About the Origin (j!j  !0 ) As "i .!/ identically vanishes at the origin ! D 0, one may expand the denominator in the integrand of (12.7) for small j!j in a Maclaurin series as Z 1 "i . / 1 .1  != /1 d "r .!/ D 1 C P 

1 Z 1 1 X 1 "i . / Š 1C !j P d : (12.13) j C1  1 j D0 The validity of this expansion relies upon the property that when j j  j!j and the Maclaurin series expansion of the factor .1  != /1 in the integrand breaks down, "i . / is very close to zero and serves to neutralize the effect. Due to the odd symmetry of "i . / [i.e., "i . / D "i .  /], one then obtains the expansion "r .!/ Š 1 C

1 X j D0

ˇ2j ! 2j

(12.14)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

with coefficients ˇ2j

1 P 

Z

1 1

"i . / d ;

j C1

255

(12.15)

which is valid for j!j  !0 . Since "r . / does not vanish when D 0, the same expansion technique cannot be used to obtain a low frequency expansion of the dispersion relation given in (12.8). However, because of the even symmetry of "r . / [i.e., "r . / D "r .  /], that equation may be rewritten as "i .!/ D !

2 P 

Z

1 0

"r . /  1 d ;

2  !2

(12.16)

which explicitly displays the property that "i .0/ D 0. In addition, for an attenuative medium one must have that "i .! 0 /  0 for all ! 0  0. Based upon these two results, one may then take the approximation "i .!/ 2ı1 !

(12.17)

for j!j  !0 , where ı1 is a nonnegative real number. Hence, for sufficiently small values of complex ! such that j!j  !0 , the complex relative dielectric permittivity ".!/ .!/=0 may be approximated as ".!/ "s C 2i ı1 ! C ˇ2 ! 2 ; where "s 1 C ˇ0 D 1 C

1 P 

Z

1 1

"i . / d

(12.18)

(12.19)

is the static relative dielectric permittivity of the dispersive material. Notice that, because of the odd symmetry of "i . /, the static relative dielectric permittivity satisfies the inequality "s  1 where the equality holds [i.e., where "s D 1] only in the case of the ideal vacuum [i.e., when "i .!/ D 0 for all !]. The complex index of refraction in the small frequency region about the origin is then given by  1=2 ı1 1 n.!/ D ".!/

0 C i ! C 0 20

  ı12 ˇ2 C 2 ! 2 ; 0

(12.20)

where D n.0/ 0 "1=2 s

(12.21)

is the static index of refraction of the dispersive dielectric material. The saddle points of the complex phase function .!;  / D i !.n.!/ / defined in (12.3) are determined by the condition  0 .!; / D 0, where the prime denotes differentiation with respect to !. The saddle points are then given by the roots of the saddle point equation (12.22) n.!/ C !n0 .!/ D ;

256

12 Analysis of the Phase Function and Its Saddle Points

which explicitly depend upon the space–time parameter  D ct =z. With the approximation given in (12.20) for the complex index of refraction in the region about the origin, the saddle point equation becomes !2 C i

4ı1 20 ! .  0 / 0; 3˛1 3˛1

(12.23)

where ˛1 ˇ2 Cı12 =02 . The roots of (12.23) then yield the approximate near saddle point locations 2ı1 ˙ ./ ˙ ./  i ; (12.24) !SP n 3˛1 with

 1=2 ı12 1 0 6 .  0 /  4 2 . / : 3 ˛1 ˛1

(12.25)

This approximate result is precisely in the form of the first approximation for the location of the near saddle points in a single-resonance Lorentz model dielectric [2, 3, 6, 7] as well as in multiple-resonance Lorentz model dielectrics [11, 12]. The saddle point dynamics (i.e., the evolution of the saddle point locations with  ) are thus seen to depend upon the sign of the quantity ˛1 ˇ2 C ı12 =02 . For a Lorentz model dielectric, ˇ2 is typically positive so that ˛1 > 0; such a medium is classified here as a Lorentz-type dielectric. On the other hand, for a Debye model dielectric (as well as for the Rocard–Powles extension of the Debye model), ˇ2 is typically negative so that ˛1 < 0; such a medium is classified here as a Debye-type dielectric. Of course, it may just happen that ˛1 D 0; in this highly unusual event, a so-called transition-type dielectric is obtained. The dynamical evolution of the near saddle points are now separately treated for these three cases.

12.1.1.1

Case 1: The Lorentz-Type Dielectric (˛1 > 0)

For Lorentz-type dielectrics, the saddle point solution separates into two subluminal (i.e.,  > 1) space–time domains separated by the critical space–time value 1 0 C

2ı12 ; 3˛1 0

(12.26)

where 1 > 0 for nonvanishing ı1 . Application [1–3, 6, 7] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A.0; t / of the plane wave pulse at the input plane z D 0 is zero for all time t < 0, then the propagated wavefield in a Lorentz-type dielectric is zero for all superluminal space–time points  < 1 with z > 0. The saddle point dynamics in a Lorentz-model dielectric then need to be considered for such finite duration pulsed signals over just the subluminal space–time domain  > 1.

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

257

For initial values of the space–time parameter  2 .1; 1 , the near saddle point locations are given by ˙ ./ !SP n

  2ı1 :

i ˙ j ./j  3˛1

(12.27)

The two near, first-order saddle points SP˙ n are then seen to initially lie along the imaginary angular frequency axis, symmetrically situated about the point ˙ .1 / D .2ı1 =3˛1 /i , approaching each other along the imaginary axis as !SP n  increases from unity to 1 . Detailed analysis (see Sect. 12.3.1) shows that only the upper near saddle point SPC n is relevant to the subsequent asymptotic analysis of the integral representations given in (12.1) and (12.2) over this -domain [6, 7]. The approximate behavior of the complex phase at the upper near saddle point is then obtained from (12.3) with the substitution of the approximate expressions from (12.20) and (12.27) as ( 2ı 1 C ; /   j . /j   0 .!SP n 3˛1      ) ı12 2ı1 0 2ı1 ı1   j ./j  ˇ2 C 2  j . /j 2 3˛1 ˛1 3˛1 0 

(12.28) for 1 <  < 1 . At the critical space–time point  D 1 , .1 / D 0 and the two near, first-order saddle points SP˙ n have coalesced into a single second-order saddle point SPn at !SPn .1 / 

2ı1 i: 3˛1

(12.29)

The approximate value of the complex phase at this critical point is given by .!SPn ; 1 / 

4ı13 9˛13 0

  ı2 ˇ2 C 12 ; 0

(12.30)

which directly follows from (12.28) with substitution of (12.26). Notice that .!SPn ; 1 / vanishes as ı1 ! 0. Finally, for  > 1 the two near, first-order saddle points SP˙ n have moved off of the imaginary axis and are now symmetrically situated about the imaginary axis in the lower-half of the complex !-plane. The near saddle point locations are now given by 2ı1 ˙ ./ ˙ ./  i; (12.31) !SP n 3˛1

258

12 Analysis of the Phase Function and Its Saddle Points

where . / is real-valued over this space–time domain. The approximate behavior of the complex phase at these two near saddle points is then found to be given by (   ı12 3 2ı 1 1 ˙ 2 ˛1  ˇ2  2 .!SPn ; /  ./   0 C 3˛1 0 2 0   ) 2ı12 ı12 C 2 ˇ2  3˛1 C 2 9˛1 0 ) (    2 2 8ı ı12 8ı 1 2 ˇ2 C 2 ./  12 C 1 ˙i ./ 0   C 20 9˛1 0 9˛1 (12.32) for  > 1 . Hence, the complex phase function .!;  / at the relevant near saddle points is nonoscillatory for 1 <   1 while it has an oscillatory component for all  > 1 . Notice that the accuracy of these approximate solutions for the near saddle ˇ ˙ pointˇ . /ˇ dynamics rapidly diminishes as the quantity j  0 j increases, because ˇ!SP n will then no longer be small in comparison to !0 . An accurate description of the near saddle point dynamics for a Lorentz-type dielectric that is valid for all  > 1 can only be constructed once the behavior of the complex index of refraction n.!/ is explicitly known in the region of the complex !-plane about the first absorption peak at !0 , as has been done [6, 7] for a single-resonance Lorentz model dielectric. These results remain valid in the special case when ı1 D 0. In that case, the !-dependence of "i .!/ varies as ! 3 or higher about the origin. The approximate saddle point equation given in (12.23) is then still correct to order O.! 2 / and, for values of ! about the origin such that the inequality j!j  !0 is well satisfied, the approximate saddle point locations are now given by ˙ .!/ !SP n



20

˙ .  0 / 3ˇ2

1=2 :

(12.33)

The same dynamical behavior then results, but with the two near, first-order saddle points SP˙ n , which are now symmetrically situated about the origin, coalescing into a single and second-order saddle point SPn at the origin when  D 0 . Clearly, (12.28), (12.30), and (12.32) for the complex phase behavior at the near saddle points remain valid in this case with ı1 D 0. This is the only special situation that can arise for a Lorentz-type dielectric since neither ˇ0 nor ˇ2 can vanish for a causally dispersive dielectric, the trivial case of a vacuum being excluded. 12.1.1.2

Case 2: The Debye-Type Dielectric (˛1 < 0)

For a Debye-type dielectric, ˛1 < 0 so that 1 D 0 

2ı12 3j˛1 j0

(12.34)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

259

and 1 < 0 . Application [12, 14] of the method of proof of Jordan’s lemma [13, Sect. 6.222] shows that if the initial time behavior A.0; t / of the plane wave pulse at the input plane z D 0 is zero for all time t < 0, then the propagated wavefield in a Debye-type dielectric is zero for all space–time points  < 1 with z > 0; a detailed derivation is given in Sect. 13.1. This due to the fact that the absorption does not go to zero as ! ! 1 for a Debye-type dielectric. As a consequence, the saddle point dynamics in a Debye-model dielectric need to be considered for such finite duration pulsed signals over just the space–time domain  > 1 > 1. From (12.24)–(12.25) the two near, first-order saddle point locations are seen to be given by   2ı1 ˙ (12.35) !SPn . / i ˙j ./j C 3j˛1 j for  > 1 . These two saddle points are symmetrically situated about the fixed point ! D i 2ı1 =3j˛1 j and move away from each other along the imaginary axis as  increases above 1 . Because the upper saddle point moves away from the origin as  increases (and consequently increasingly violates the condition j!j  !), only the lower near saddle point SP n is relevant to the asymptotic analysis over this small angular frequency domain about the origin. The saddle point SP n then moves down the imaginary axis and crosses the origin at  D 0 . The complex phase behavior at this first-order near saddle point is then given by (  2ı1    0 .!SPn ; /  j . /j  3j˛1 j   )    ı12 2ı1 2ı1 1 2ı1  ˇ2 C 2 j . /j  j . /j   20 3j˛1 j 3j˛1 j 0 (12.36) for  > 1 . Hence, the complex phase behavior at the near saddle point SP n of a Debye-model dielectric is nonoscillatory for all  > 1 .

12.1.1.3

Case 3: The Transition-Type Dielectric (˛1 D 0)

In the highly unusual event that ˛1 D 0, in which case ˇ2 D ı12 =02 , a so-called transition-type dielectric is described. The approximation given in (12.20) for the complex index of refraction then becomes n.!/ 0 C i

ı1 ! C O.! 3 /; 0

(12.37)

which then results in a single first-order near saddle point that moves down the imaginary axis linearly with increasing  as !SPn ./ i

0 .  0 /: 2ı1

(12.38)

260

12 Analysis of the Phase Function and Its Saddle Points

A more accurate description of the near saddle point dynamics in this transitional case requires that the expansion of the complex index of refraction given in (12.37) be extended to include the ! 3 term.

12.1.2 The Region About Infinity (j!j  !m ) Since "i . / vanishes as ! ˙1, the denominator in the integrand of (12.7) may be expanded for large j!j in a Laurent series so that 1 "r .!/ D 1  P 

Š 1

Z

1

"i . / .1  =!/1 d !

1

1 X

1

1 j C1 !  j D0

Z

1

"i . / j d :

(12.39)

1

The validity of this expansion relies on the fact that when j j  j!j and the expansion of the quantity .1  =!/1 that appears in the integrand of the above principal value integral breaks down, the quantity "i .!/ is vanishingly small and serves to neutralize this behavior. Because of the odd symmetry of "i . /, the relation given in (12.39) can be rewritten as "r .!/ Š 1 

1 X a2j ; ! 2j j D1

(12.40)

which is valid for j!j  !m , with coefficients a2j

Z

1 

1

"i . / 2j 1 d :

(12.41)

1

Notice that the first coefficient, a2 , is nonvanishing for any lossy dielectric, namely, 1 a2 

Z

1

"i . / d > 0:

(12.42)

1

Because the quantity ."r . /  1/ also vanishes as ! ˙1, the same expansion procedure can be applied in the integrand of (12.8), which can be rewritten as "i .!/ D !

2 P 

Z 0

1

"r . /  1 d ;

2  !2

(12.43)

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics

261

to yield 2 "i .!/ D P  Š

1 X j D0

Z

1 0

1 "r . /  1  1  2 =! 2 d !

bj ; 2j ! C1

(12.44)

which is valid for j!j  !m , with coefficients 2 bj 

Z

1

."r . /  1/ 2j d :

(12.45)

0

There are then two distinct classes of dielectrics that are distinguished by the value of the zeroth-order coefficient Z 2 1 ."r . /  1/ d : b0 (12.46)  0 For the first class, b0 ¤ 0, which is characteristic of a Debye-type dielectric. For the second class, b0 D 0, which is characteristic of a Lorentz-type dielectric These two cases are now treated separately.

12.1.2.1

Case 1: The Debye-Type Dielectric (b0 ¤ 0)

For a Debye-type dielectric, b0 ¤ 0, and the complex-valued relative dielectric permittivity in the annular region j!j  !m about infinity is given approximately by ".!/ 1 C i

a2 b0  2; ! !

(12.47)

and the associated complex index of refraction n.!/ D .".!//1=2 is then given approximately by a2  b02 =4 b0  : (12.48) n.!/ 1 C i 2! 2! 2 With this substitution in (12.22), the saddle point equation becomes  1

a2  b02 =4

0: 2! 2

(12.49)

The location of these distant saddle point solutions in the complex !-plane is then seen to depend critically upon the sign of the quantity .a2  b02 =4/.

262

12 Analysis of the Phase Function and Its Saddle Points

For a simple Debye-model dielectric with frequency-dependent relative permittivity [see (4.179) of Vol. 1] ".!/ D "1 C ."s  "1 /=.1  i ! / with relaxation time  , static relative permittivity "s , and high-frequency limit "1 D 1, the coefficients appearing in (12.47) are found to be given by b0 D ."s  1/= and a2 D .1  "s /= 2 , in which case it is found that .a2  b02 =4/ < 0 and is equal to zero only when "s D 1 (i.e., only in the trivial of a vacuum). The approximate distant saddle point locations are then given by ˙ ./ ˙i ; (12.50) !SP d .  1/1=2 q where .b02 =4  a2 /=2. These distant saddle point solutions are symmetrically situated about the origin along the imaginary axis and move in toward the origin as  increases from unity, evolving into the near saddle points for a Debye-type dielectric [see (12.35)]. They are then of no further interest in the asymptotic theory.

12.1.2.2

Case 2: The Lorentz-Type Dielectric (b0 D 0)

A Lorentz-type dielectric is further distinguished by the fact that the expansion coefficient b0 appearing in the approximate description of the angular frequency dependence of the relative dielectric permittivity in the region about infinity is identically zero, so that the sum rule [15] Z

1

  "r . /  1 d D 0

(12.51)

0

is satisfied. The complex-valued relative dielectric permittivity in the region j!j  !m is then given approximately by [16] b2 a2 Ci 3 !2 ! a2 ;

1 !.! C ib2 =a2 /

".!/ 1 

(12.52)

and the associated complex index of refraction is then given by n.!/ 1 

a2 : 2!.! C ib2 =a2 /

(12.53)

With this substitution in (12.22), the saddle point equation becomes 1 C

a2

0: 2!.! C ib2 =a2 /2

(12.54)

The roots of this equation then yield the approximate expression for the distant saddle point locations as

12.1 General Saddle Point Dynamics for Causally Dispersive Dielectrics ˙ !SP . / ˙ d



b2 a2  22 2.  1/ 4a2

1=2 i

b2 : 2a2

263

(12.55)

This is precisely the form of the first approximate expressions for the distant saddle point locations in both single-resonance [2, 3, 6, 7] and multiple-resonance [11,12] Lorentz model dielectrics. As can be seen from (12.55), these distant saddle points are symmetrically situated about the imaginary axis and lie along the line ! D ! 0  i b2 =2a2 in the lower-half of the complex !-plane. As  increases from unity, they move in from infinity along this line. Notice that the accuracy of this approximation for the distant saddle point locations diminishes as  increases away ˙ . /j will then no longer be large in comparison with !m . from unity because j!SP d In that case, higher-order approximations that are valid for all   1 need to be obtained. However, such an accurate description of the distant saddle point dynamics for a Lorentz-type dielectric that is valid for all  > 1 can only be constructed once the behavior of the complex index of refraction n.!/ is explicitly known in the region of the complex !-plane about the upper edge of the uppermost absorption band at !m , as has been done [6, 7] for a single-resonance Lorentz model dielectric. If this is not possible, then accurate numerical methods need to be used. With substitution of the approximate expression given in (12.55) into (12.3), together with the approximation of the complex index of refraction given in (12.53), the approximate complex phase behavior at the distant saddle point locations in a Lorentz-type dielectric is found to be given by ˙ ; /  .!SP d

   1=2 b2 b2 1 C 23 .  1/ : .  1/ i 2a2 .  1/ a2 a2

(12.56)

˚ ˙ ˙ ; / < .!SP ;  / vanishes at the speed of light point Notice that  .!SP d d  D 1 and then decreases linearly with  as  increases above unity.

12.1.3 Summary This general, but very approximate, description of the saddle point dynamics of the complex phase function .!;  / in a causally dispersive dielectric has shown that, as far as linear dispersive pulse propagation is concerned, only Lorentz-type and Debye-type dielectrics need be considered. Because of its highly unusual likelihood, the analysis of the transition-type dielectric is left as an exercise (see Prob. 12.1). The necessary detail of the saddle point dynamics in both Lorentz-model and Debye-model dielectrics, as well as in Drude model metals, then forms the subject matter of the remainder of this chapter.

264

12 Analysis of the Phase Function and Its Saddle Points

12.2 The Behavior of the Phase in the Complex !-Plane for Causally Dispersive Materials   The behavior of the complex phase function .!;  / i ! n.!/   in the complex !-plane at any fixed real-value of the space–time parameter   1 is dictated by the analytic form of the complex index of refraction n.!/. Because causality is an essential, physical feature of dispersive pulse dynamics, only causal, physical models of the complex index of refraction are considered here. The analysis begins with a detailed review and extension of Brillouin’s classical analysis [2, 3] of this problem for a single-resonance Lorentz-model dielectric, followed by that for a multiple-resonance Lorentz-model dielectric, a Rocard Powles Debye model dielectric, and finally for a Drude model metal.

12.2.1 Single-Resonance Lorentz Model Dielectrics The complex index of refraction for a single-resonance Lorentz model dielectric is given by (see Sect. 4.4.4 of Vol. 1)  n.!/ D 1 

b2 ! 2  !02 C 2iı!

1=2 ;

(12.57)

p where b D .k4k=0 /N qe2 =m is the plasma frequency of the dispersive medium with number density N of harmonically bound electrons of charge magnitude qe and mass m with angular resonance frequency !0 , and where ı is the associated phenomenological damping constant of the bound electron system. The values of these material parameters as chosen by Brillouin in his analysis [2, 3] for a medium possessing a single near-ultraviolet resonance frequency are !0 D 4:0  1016 r/s; p b D 20  1016 r/s; 16

ı D 0:28  10

(12.58)

r/s:

Although this choice of the medium parameters corresponds to an extremely dispersive, absorptive dielectric medium, it is used in many of the numerical examples presented here to facilitate a comparison with Brillouin’s results. To facilitate comparison with experimental results in the optical region of the spectrum which primarily are conducted in highly transparent glasses, the weak dispersion limit as N ! 1 is considered throughout the analysis as deemed necessary. The singular dispersion limit, obtained as ı ! 0, is also considered. Finally, this analysis is equally applicable to dispersive dielectrics that possess resonance frequencies from the infrared through the ultraviolet regions of the electromagnetic spectrum.

12.2 The Behavior of the Phase in the Complex !-Plane

265

Attention is now turned to the description of the analytic structure of the complex phase function .!;  / in the complex !-plane. This analysis is simplified by the general symmetry relation [see (11.26)] .!; / D   .!  ;  /;

(12.59)

so that the real part  .!;  / D 0, is zero at  D 0, and is positive for  < 0. Near the lower branch point !C , however,  .!;  / is negative on the upper side of the branch cut and positive on the lower side for all  , as depicted in part (b) of Fig. 12.3. From the behavior of  .!;  / in the region of the complex !-plane about the lower branch point !C , it is seen that the zero isotimic1 contour  .!;  / D 0 must pass through the branch point !C from above [since  .!;  / ! C1 as ! ! !C along 1

From the Greek isotimos, of equal worth.

274

12 Analysis of the Phase Function and Its Saddle Points

the line ! D ! 0  i ı from below] and then continues on from the lower side of 0 for  > 0, as described by the dashed curve the branch cut between !C and !C in Fig.12.3b. For  < 0, the zero isotimic contour  .!;  / D 0 continues on from 0 , and for  D 0, this contour the upper side of the branch cut between !C and !C 0 . continues on from the upper branch point !C 12.2.1.5

Numerical Results

This simple sketch of the behavior of  .!;  /  .!SPd ; 1:501/

12.2 The Behavior of the Phase in the Complex !-Plane

277

'' (x1016r/s)

(+1.0x1016) (−1.0x1016) SPn+

'

(+1.0x1016)

(0)

' (x1016r/s)

SPd+

(−1.0x1016)

Fig. 12.8 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 1:65 after the near saddle point pair has moved off of the imaginary axis into the lower-half of the complex !-plane. The shaded area indicates the region of the complex !-plane below the dominant near saddle point pair where ˙  .!; 1:65/ <  .!SP ; 1:65/, and the darker shaded area indicates the region below the distant n ˙ ˙ ˙ ; 1:65/, where  .!SP ; 1:65/ <  .!SP ; 1:65/ saddle point pair where  .!; 1:65/ <  .!SP n d d

parameters, and then coalescing into a single second-order saddle point at  D 1 , as approximately described by (12.27) and (12.29) for a general Lorentz-type dielectric, where the value of this critical space–time point for Brillouin’s choice of the medium parameters is just slightly larger than the space–time value  D 1:501 illustrated in Fig. 12.7. As  increases above 1 , the two near first-order saddle points are seen to separate from each other, symmetrically situated about the imaginary axis, approaching the inner branch points !˙ , respectively, as  ! 1. The two distant saddle points SPd˙ , on the other hand, are located in the lowerhalf of the complex !-plane for all   1 and are located at ˙1  i ı at the luminal space–time point  D 1, as approximately described by (12.55) for a general Lorentz-type dielectric. As  increases from unity, these two saddle points symmetrically move in from infinity and approach the respective outer branch points 0 as  increases to infinity, as evident in Figs. 12.8 and 12.9. !˙ Initially, the distant saddle points SPd˙ have less exponential decay associated with them than does the upper near saddle point SPC n in Figs. 12.4 and 12.5, that is C ˙ ; / >  .!SP ; /  .!SP d n

when 1   < SB :

(12.96)

278

12 Analysis of the Phase Function and Its Saddle Points

''

(x1016r/s)

(2.0 x1016)

(−3.4x1016)

(0)

'

(x1016r/s)

'

SPn+

SPd+

(2.0x1016)

(−3.4x1016)

Fig. 12.9 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / in the right-half of the complex !-plane at the fixed space–time point  D 5:0. The shaded area indicates the region of the complex !-plane below the dominant near saddle point pair where ˙ .!; 5/ < .!SP ; 5/, and the darker shaded area indicates the region below the distant saddle n ˙ ˙ ˙ ; 5/, where  .!SP ; 5/ < .!SP ; 5/. Notice the approach point pair where .!; 5/ <  .!SP n d d C of the near saddle point SPn to the lower branch point !C and the approach of the distant saddle 0 point SPC d to the upper branch point !C

Because the original contour of integration C appearing in the integral representation of the propagated plane wave pulse given in either (12.1) or (12.2) is not deformable into an Olver-type path through the lower near saddle point SP n over the initial space–time domain 1   < 1 , that saddle point is irrelevant for the present analysis for  below and bounded away from 1 . At the critical space–time point  D SB Š 1:33425, illustrated in Fig. 12.6, the upper near saddle point SPC n has precisely the same exponential decay associated with it as do the two distant saddle points, that is C ˙ ; SB / D  .!SP ; SB /  .!SP d n

when  D SB :

(12.97)

 Consequently, at the space–time point  D SB , those three saddle points (SPC d , SPd , C and SPn ) are of equal importance (or dominance) in the asymptotic description of the propagated wavefield. The remaining figures show that for values of  2 .SB ; 1 / the upper near saddle point SPC n is dominant over the two distant saddle points SP˙ d , so that C ˙ ; / >  .!SP ; /  .!SP d n

when SB   < 1 :

(12.98)

12.2 The Behavior of the Phase in the Complex !-Plane

279

At  D 1 , the two near first-order saddle points SP˙ n have coalesced into a single second-order saddle point SPn which is dominant over the distant saddle point pair, so that ˙ ; 1 /  .!SPn ; 1 / >  .!SP d

when  D 1 :

(12.99)

Finally, for all  > 1 , the near saddle points SP˙ n are dominant over the distant saddle points SPd˙ , so that ˙ ˙ ; / >  .!SP ; /  .!SP n d

when  > 1 ;

(12.100)

as evident in Figs. 12.8–12.9. Notice the change in scale of the real and imaginary coordinate axes in Fig. 12.9, demonstrating how the topography of  .!;  / 0 as  becomes increasingly concentrated about the branch cuts !0 ! and !C !C increases above the critical space–time value 1 with the near saddle points SP˙ n approaching the lower branch points !˙ , respectively, and the distant saddle points 0 . SPd˙ approaching the upper branch points !˙ These detailed numerical results demonstrate the necessity of obtaining approximate analytic expressions for both the near and distant saddle point locations that accurately describe their dynamical evolution in the complex !-plane for all   1 as well as the complex phase behavior at them. In addition, accurate analytic expressions are needed for each of the critical space–time points encountered here. These include the critical value SB at which the upper near and distant saddle points are of equal importance (see Fig. 12.6) and the critical value 1 at which the two near firstorder saddle points coalesce into a single second-order saddle point (see Fig. 12.7). Moreover, approximate (if an exact solution is unattainable) analytic expressions for each of these quantities need to be obtained that are accurate over the entire space–time domain of interest for both the strong, intermediate, and weak dispersion limits, the former being of central interest to bioelectromagnetics and the latter being of central interest to optics.

12.2.2 Multiple-Resonance Lorentz Model Dielectrics For a double-resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by (see Sect. 4.4.4 of Vol. 1)  n.!/ D 1 

b02 b22  2 2 ! 2  !0 C 2i ı0 ! ! 2  !2 C 2i ı2 !

1=2 ;

(12.101)

where !j denotes the undamped angular p resonance frequency, ıj is the phenomenological damping constant, and bj D .k4k=0 /Nj qe2 =m is the plasma frequency with number density Nj of Lorentz oscillators denoted by the indices j D 0; 2.

280

12 Analysis of the Phase Function and Its Saddle Points

The branch points of this double-resonance complex index of refraction function, and consequently of the complex phase function .!;  /, may be determined by rewriting the expression given in (12.101) as     31=2 .3/  !  !.1/ !  !C !  !.3/   n.!/ D 4   5 : .0/  .2/  !  !C !  !.0/ !  !C !  !.2/ 2

.1/

!  !C

.0/

(12.102)

.2/

The branch point singularities !˙ and !˙ appearing in this expression are given by the zeros of the denominator in (12.102), so that q .0/ !˙ D ˙ !02  ı02  i ı0 ; q .2/ !˙ D ˙ !22  ı22  i ı2 : .1/

(12.103) (12.104)

.3/

Unfortunately, the branch point zeros !˙ and !˙ appearing in (12.102) are much more difficult to determine, even in approximate form. These four zeros are given by the roots of the quartic equation ! 4 C 2i.ı0 C ı2 /! 3  .!12 C !32 C 4ı0 ı2 /! 2 2i.ı2 !12 C ı0 !32 /! C !02 !22 C b02 !22 C b22 !02 D 0; (12.105) where q !1 C !02 C b02 ; q !3 C !22 C b22 :

(12.106) (12.107)

Approximate analytic solutions to this quartic equation may be obtained in the following manner. Because the zeros must appear in symmetric pairs about the imaginary axis, let .1/

!˙ D ˙1  i ı0 ; .3/

!˙ D ˙3  i ı2 ; .0/

.1/

where it has been assumed that the branch points !˙ and !˙ lie along the line .2/ .3/ ! 00 D ı0 and that the branch points !˙ and !˙ lie along the line ! 00 D ı2 . In that case, the quartic equation given in (12.105) must then be factorizable as .!  .1  i ı0 //.!  .1  i ı0 //.!  .3  i ı2 //.!  .3  i ı2 // D 0;

12.2 The Behavior of the Phase in the Complex !-Plane

281

which then results in ! 4 C 2i.ı0 C ı2 /! 3  .12 C ı02 C 32 C ı22 C 4ı0 ı2 /! 2   2i ı0 .32 C ı22 / C ı2 .12 C ı02 / ! C .12 C ı02 /.32 C ı22 / D 0: Comparison of the terms in this quartic equation with the corresponding terms in (12.105) then yields the set of relations 12 C 32 D !12  ı02 C !33  ı22 ; ı0 .32 C ı22 / C ı2 .12 C ı02 / D ı0 !32 C ı2 !12 ; .12 C ı02 /.32 C ı22 / D !02 !22 C b02 !22 C b22 !02 ; which are overdetermined. This overdetermination of the solution implies that the .1/ .3/ zeros !˙ and !˙ do not, in general, lie along the lines ! 00 D ı0 and ! 00 D ı2 , respectively, as was assumed in constructing this solution. Nevertheless, if ı0 ı2 , then they approximately lie along these lines and the first pair of the above set of equations are nearly identical, resulting in the reduced system of equations 12 C 32 D !12  ı02 C !33  ı22 ; .12 C ı02 /.32 C ı22 / D !02 !22 C b02 !22 C b22 !02 : The solutions to this pair of equations then yields the approximate branch point locations2 s ! 4 C 2.!12  ı02 /!32  b02 b22 C ı03 .1/ !˙ ˙ !12  ı02 C 3  i ı0 ; (12.108) !12 C !32  2ı02 s ! 4 C 2.!32  ı22 /!32  b02 b22 C ı23 .3/ !˙ ˙ !32  ı22 C 1  i ı2 : (12.109) !12 C !32  2ı22 The branch cuts chosen here are the straight line segments !.3/ !.2/ and !.1/ !.0/ .0/ .1/ .2/ .3/ in the left-half plane, and !C !C and !C !C in the right-half plane, as depicted in Fig.12.10. It is typically assumed that 0  !0 < !1  !2 < !3 so that the outer and inner branch cuts do not overlap each other. Finally, the complex index of refraction n.!/ and the complex phase function .!; / are analytic throughout the .0/ .1/ .2/ .3/ complex !-plane with the exception of the branch points !˙ , !˙ , !˙ , and !˙ . The limiting behavior of the complex index of refraction n.!/ and the real part  .!;  / of the complex phase function .!; / for a double-resonance Lorentz model dielectric is quite similar to that of a single-resonance Lorentz model dielectric, especially when the branch cuts are sufficiently separated from each other, and 2

Notice that these approximate expressions for the branch point zero locations are different from those given in an earlier paper [11].

282

12 Analysis of the Phase Function and Its Saddle Points

''

' (1)

(0)

branch

branch

cut

cut

(3) cut

(1)

(0)

(2) cut

(2)

(3)

.j /

.3/ .2/ Fig. 12.10 Location of the branch points !˙ , j D 0; 1; 2; 3 and the branch cuts ! ! and .0/ .1/ !C !C

.2/ .3/ !C !C

.1/ .0/ ! ! in the left-half and and in the right-half of the complex !-plane for a double-resonance Lorentz model dielectric with undamped resonance frequency !0 , damping q constant ı0 , and plasma frequency b0 with !1 D !02 C b02 for the lower resonance line, and with undamped resonance frequency !2 , damping constant ı2 , and plasma frequency b2 with !3 D q

!22 C b22 for the upper resonance line

even more so if one of the resonance lines is much stronger than the other [e.g., when either b2  b0 or b0  b2 ]. In particular, the limiting behavior as j!j ! 1 described in (12.84) and (12.85) remains valid in the multiple-resonance case, as .1/ does the behavior in the vicinity of the branch points given in (12.91) for !˙ and .3/ .0/ .2/ !˙ and in (12.93) for !˙ and !˙ , provided again that the outer and inner branch cuts are sufficiently separated from each other. The real angular frequency dependence of the real and imaginary parts of the complex index of refraction for a double-resonance Lorentz model dielectric is depicted in Fig. 12.11 for a highly absorptive material with visible and near-ultraviolet resonance lines with parameters !0 D 1:0  1016 r=s; p b0 D 0:6  1016 r=s; ı0 D 0:1  1016 r=s;

!2 D 7:0  1016 r=s; p b2 D 12:0  1016 r=s; ı2 D 0:28  1016 r=s:

Notice that the real index of refraction nr .! 0 / varies rapidly with ! 0 within each region of anomalous dispersion and that these two regions essentially coincide with each region Œ!0 ; !1 and Œ!2 ; !3 where the imaginary part ni .! 0 / of the complex index of refraction peaks to a local maximum and the absorption is strongest. The cross along the graph of the function nr .! 0 / in the upper graph of Fig. 12.11 indicates the (numerically determined) inflection point in the real refractive index in the pass-band between the two resonance frequencies where the second derivative @2 nr .! 0 /=@! 02 changes sign from negative to positive values as ! 0 increases, and the cross along the graph of the function ni .! 0 / in the lower graph of Fig. 12.11

12.2 The Behavior of the Phase in the Complex !-Plane

283

nr ( )

2

1

0 1015

1016

(r/s)

1017

1018

1017

1018

102

ni ( )

100 10–2 10–4 10–6 15 10

1016

(r/s)

Fig. 12.11 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for apdouble-resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, b0 D p 0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:28  1016 r=s for the upper resonance line. The dotted line indicates the nondispersive limit of the vacuum [n.!/ D 1]

indicates the point at which the imaginary part of the index of refraction is a minimum in that pass-band. Notice that these two points occur, in general, at different frequency values. The inclusion of this single additional resonance feature in the Lorentz model results in the appearance of four additional saddle points of the complex phase function .!; / in the complex !-plane. In addition to the distant saddle point pair SP˙ d and the near saddle point pair SP˙ n , there are now four additional first-order saddle ˙ points SP˙ m1 and SPm2 , symmetrically situated about the imaginary axis, that evolve with   1 in the intermediate frequency domain between the lower and upper resonance frequencies, i.e., such that
 , as depicted in Figs. 12.17–12.19. From the near saddle points SP˙ 1 n perspective of the lower resonance line at !0 , the dynamical evolution of the first˙ over the space–time domain  > N1 imitates order middle saddle point pair SPm1 that of the distant saddle point pair in a single-resonance dielectric as they approach .1/ the outer branch points !˙ , respectively, as  ! 1. Similarly, from the perspective of the upper resonance line at !2 , the dynamical evolution of the first-order middle ˙ for  > N1 imitates that of the near saddle point pair for saddle point pair SPm2  > 1 in a single-resonance dielectric as they approach the inner branch points .2/ !˙ , respectively, as  ! 1. Attention is now turned to the determination of the space–time sequence of saddle point dominance over the subluminal space–time domain   1. Because the original contour of integration C appearing in the integral representation of the

288

12 Analysis of the Phase Function and Its Saddle Points 3

2

(0)

(−1.0x1016)

+

+

SPn

(0)

0

+

'' (x1016r/s)

1

−

+ (0)

SPn

(1)

(0)

−1

+ SPm1 +

+ SPm2

(0)

(2)

(+1.0x1016)

−2

−3

0

1

2

3

4

5

6

7

' (x1016r/s) Fig. 12.16 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed ˙ space–time point  D N1  1:25 when the two pairs of first-order middle saddle points SPm1 and ˙ SPm2 come into closest proximity with each other in the left- and right-half planes and dominate ˙ both the upper near saddle point SPC n and the distant saddle points SPd

propagated plane wave pulse given in either (12.1) or (12.2) is not deformable into an Olver-type path through the lower near saddle point SP n over the initial space– time domain 1   < 1 , that saddle point is irrelevant for the present analysis for  below and bounded away from 1 . Similarly, because the integration path C is not ˙ deformable into an Olver-type path through the lower middle saddle point pair SPm2 over the space–time domain 1   < N1 , those two saddle points are irrelevant for values of  below and bounded away from N1 . However, for  > N1 the contour C can be deformed into an Olver-type path through this lower middle saddle point pair ˙ , but they are then dominated by (i.e., possess greater exponential attenuation SPm2 ˙ over this entire space–time domain. than) the upper middle saddle point pair SPm1 Finally notice that the simple fact that a saddle point does not become the dominant saddle point does not necessarily mean that it does not influence the asymptotic field behavior (e.g., see Sect. 10.3.2). ˙ do not necessarily become the dominant The upper middle saddle points SPm1 saddle points in all cases. The necessary condition [11] for whether or not they do become the dominant saddle points over some subluminal space–time domain is

12.2 The Behavior of the Phase in the Complex !-Plane

289

3

2

+ +

''

SP n

+

0

SP n−

(−1.0x1016)

(0)

+

16

(x10 r/s)

1

(0)

(1)

+SP m1

+ + SP m2

(0)

−1

(2)

(0)

(+1.0x1016)

−2

−3

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.17 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:3 when the upper near saddle point SPC n is dominant over both the middle and distant saddle points

obtained from a consideration of the energy transport velocity vE .!/ D

c E .!/

(12.113)

for a monochromatic electromagnetic plane wave in a double-resonance Lorentz model dielectric, where [see (5.212) of Vol. 1] 1 E .!/ D nr .!/ C nr .!/

"

# b02 ! 2 b22 ! 2 C :  2 2 ! 2  !02 C 4ı02 ! 2 ! 2  !22 C 4ı22 ! 2 (12.114)

Let p 2 .1; 0 / denote the space–time value at which  .!SP ˙ ;  / at the upper m1

˙ middle saddle point pair SPm1 has a local maximum. If the upper middle saddle ˙ point SPm1 has less exponential decay associated with it at the space–time point  D p than does the upper near saddle point SPC n , that is, if

 .!SP ˙ ; p / >  .!SPnC ; p /; m1

(12.115)

290

12 Analysis of the Phase Function and Its Saddle Points 3

2

(−1.0x1016) ++

0

''

(x1016r/s)

1

(0)

SP n+ SP n−

+

(0)

(1)

+SP m1

+ + SP m2

(0)

(2)

(0)

−1

(1.0x1016)

−2

−3

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.18 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 0 ' 1:3583, just prior to the coalescence of the two first-order near saddle points SP˙ n into a single second-order saddle point SPn at  D 1 . At this space–time value 0 the upper near saddle point SPC n is dominant over both the middle and distant saddle points

˙ then the upper middle saddle point pair SPm1 will be the dominant saddle points in a small  -neighborhood about that space–time point. As shown in Sect. 15.3.2, this condition is equivalent to the condition that the maximum value of the energy transport velocity in that intermediate frequency interval between the upper and lower absorption bands be greater than the value of the energy velocity at zero frequency. Let !p denote the real angular frequency value at which this maximum value occurs, in which case

p D E .!p /:

(12.116)

Because vE .0/ D c=n.0/ D c=0 , then the condition vE .!p / > vE .0/ for the ˙ over the upper near saddle dominance of the upper middle saddle point pair SPm1 over a small space–time interval about  becomes point SPC p n p < 0 :

(12.117)

12.2 The Behavior of the Phase in the Complex !-Plane

291

2

1

16

(x10 r/s)

(−1.0x1016)

(0)

+

SPn

0

''

+

(0)

(0)

(1)

SP + + m1

+

+ SPm2

(2)

( 0) −1

−2

(1.0x1016)

0

1

2

3

'

4 (x1016r/s)

5

6

7

Fig. 12.19 Isotimic contours of the real part  .!;  / of the complex phase function .!;  / for a double-resonance Lorentz model dielectric in the right-half of the complex !-plane at the fixed space–time point  D 1:5 when the near saddle points SP˙ n are dominant over both the middle and distant saddle points

This is then the necessary condition for the dominance of the upper middle saddle point pair. If this condition is not satisfied, the middle saddle points never become the dominant saddle points for all   1.

Case 1: p < 0

12.2.2.1

Initially, the distant saddle points SPd˙ have less exponential decay associated with them than does either the upper near saddle point SPC n or the upper middle saddle ˙ , as seen in Figs. 12.12 and 12.13, and this remains the case until  points SPm1 reaches the space–time point SM , that is  .!SP ˙ ; / >  .!SPnC ; / >  .!SP ˙ ;  /

when 1   < BM ;

 .!SP ˙ ; / >  .!SPnC ; / D  .!SP ˙ ;  /

when  D BM ;

 .!SP ˙ ; / >  .!SP ˙ ; / >  .!SPnC ;  /

when BM <  < SM ;

m1

d

m1

d

d

m1

(12.118)

292

12 Analysis of the Phase Function and Its Saddle Points

where SM is defined by the condition ˙ ˙ ; SM / D  .!SP ; SM /:  .!SP m1 d

(12.119)

Notice that the upper near saddle point SPC n is initially dominant over the middle ˙ , this dominance switching at the space–time point  D BM . saddle point pair SPm1 ˙ is then dominant over both the upper near SPC The middle saddle point pair SPm1 n ˙ and distant SPd saddle points over the space–time domain SM <  < MB , as seen in Figs. 12.14–12.16, where  .!SP ˙ ; / >  .!SP ˙ ; / >  .!SPnC ;  /

when SM   < SB ;

 .!SP ˙ ; / >  .!SPnC ; / D  .!SP ˙ ;  /

when  D SB ;

 .!SP ˙ ; / >  .!SPnC ; / >  .!SP ˙ ;  /

when SB <  < MB ;

m1

d

m1

d

m1

d

(12.120)

where MB is defined by the condition  .!SP ˙ ; MB / D  .!SPnC ; MB /: m1

(12.121)

The upper near saddle point SPC n is then the dominant saddle point over the space– time domain MB <   1 , as seen in Figs. 12.17 and 12.18, where  .!SPnC ; / >  .!SP ˙ ; / >  .!SP ˙ ;  / m1

d

when MB <   1 ;

(12.122)

the two near first-order saddle points coalescing into a single second-order saddle point at  D 1 . The near saddle point pair SP˙ n is then dominant over both the ˙ and distant SPd˙ saddle point pairs over the final space–time domain middle SPm1  > 1 , as seen in Fig. 12.19, where  .!SP ˙ ; / >  .!SP ˙ ;  / >  .!SP ˙ ;  / n

m1

d

when  > 1 :

(12.123)

Notice that the critical space–time points encountered in this saddle point evolution are ordered such that 1 < SM < SB < N1 < MB < 0 < 1

(12.124)

when both ıj > 0 and Nj > 0 are bounded away from zero.

12.2.2.2

Case 2: p > 0

When the inequality given in (12.117) is not satisfied, then the upper middle saddle ˙ never become the dominant saddle points over the entire space–time points SPm1 domain   1. In that case, the saddle point dominance sequence is the same as

12.2 The Behavior of the Phase in the Complex !-Plane

293

that for a single-resonance Lorentz model dielectric. However, this does not mean that these middle saddle points can never influence the dynamical field evolution. Indeed, it can happen that  .!SP ˙ ;  / at the upper middle saddle point pair is m1 just below the value  .!SP ˙ ;  / at the near saddle point pair and interferes with n that contribution over the space–time domain  > 1 , as occurs for triply distilled water [17]. Because of this, the middle saddle points play an important role in the dynamical field evolution and their detailed behavior must be determined even when they are never the dominant saddle points.

12.2.3 Rocard–Powles–Debye Model Dielectrics The complex index of refraction for a single relaxation time Rocard–Powles–Debye model dielectric is given by (see Sect. 4.4.3 of Vol. 1)  n.!/ D 1 C

a0 .1  i !0 /.1  i !f 0 /

1=2 ;

(12.125)

where 1  1 denotes the large frequency limit of the relative dielectric permittivity due to the Rocard–Powles–Debye model alone (typically when ! exceeds  1  1012 r/s). Here 0 denotes the effective relaxation time [see (4.177) of Vol. 1] with the associated friction time f 0 [see (4.184) of Vol. 1] introduced in the Rocard– Powles extension [18] of the Debye model [19], where a0 s  1 with s .0/ denoting the relative static dielectric permittivity of the material. Values of these model coefficients for triply distilled water at 25ı C are given by 1 D 2:1; a0 D 74:1; 0 D 8:44  1012 s; 14

f 0 D 4:62  10

(12.126)

s;

as determined by an rms fit to the numerical data presented in Figs. 4.2 and 4.3 of Vol. 1. Although a double relaxation time model provides a near-optimal fit to the numerical data (see Sect. 4.4.5 of Vol. 1), the complications introduced by the inclusion of a second (and comparatively weaker) relaxation mode does not justify its inclusion in the analysis presented here. Nevertheless, this secondary feature is included in numerical simulations when necessary. Attention is now turned to the description of the analytic structure of both the complex index of refraction n.!/ and the complex phase function .!;  / D i !.n.!/   / in the complex !-plane for the Rocard–Powles–Debye model [20]. Note first that the general symmetry relations given in (12.59)–(12.61) hold here, so that only the right-half of the complex !-plane needs to be considered here. The branch points of n.!/, and consequently of .!;  /, can be directly determined

294

12 Analysis of the Phase Function and Its Saddle Points

by rewriting the expression given in (12.125) for the frequency-dependence of the complex index of refraction as  n.!/ D  D

1 0 f 0 ! 2 C i 1 .0 C f 0 /!  s 0 f 0 ! 2 C i.0 C f 0 /!  1 .!  !z1 /.!  !z2 / .!  !p1 /.!  !p2 /

1=2

1=2 :

(12.127)

The branch point singularities !pj , j D 1; 2 are given by the two zeros of the denominator of the above expression as !p1 

i ; 0

!p2 

i f 0

;

(12.128)

which are both situated along the negative imaginary axis, and the branch point zeros !zj , j D 1; 2 are given by the two zeros of the numerator of the above expression as !zj

0 C f 0 20 f 0

(

) 1=2 0 f 0 s 1 i ; ˙ 4 1 .0 C f 0 /2 

(12.129)

where the upper sign choice is used for j D 1 and the lower sign choice for j D 2. Notice that these two branch point zeros are symmetrically situated about the point !z i

0 C f 0 ; 20 f 0

(12.130)

which also happens to be the midpoint of the two branch point singularities !p1 and !p2 . There are then three possibilities for the location of the branch point zeros, dependent upon the sign of the quantity appearing in the square root of the above expression, as follows:  If 40 f 0 =.0 C f 0 /2 < 1 =s , then

!zj

0 C f 0 D i 20 f 0

(

s 1

0 f 0 s 14 1 .0 C f 0 /2

) ;

(12.131)

and the two branch point zeros are located along the imaginary axis, symmetrically situated about the point !z , as depicted in Fig. 12.20a.  If 40 f 0 =.0 C f 0 /2 D 1 =s , then !zj D !z , j D 1; 2, and there is just a single branch point zero, located along the negative imaginary axis, as depicted in Fig. 12.20b.

12.2 The Behavior of the Phase in the Complex !-Plane

295

''

a

'

= −i/

cut

p1

z1

= −i (

f

)/(2

f

)

cut

z z2

= −i/

p2

b

f

'' ' = −i/

cut

p1

=

z

= −i/

f

= −i(

f

)/(2

)

f

cut

z1,2

p2

c

''

cut

'

branch branch

z2

p1

= −i/

cut z1

z p2

= −i/

f

Fig. 12.20 Location of the branch point singularities !pj and branch point zeros !zj , j D 1; 2, in the complex !-plane for a single relaxation time Rocard–Powles–Debye model dielectric with relaxation time 0 and associated friction time f 0 . The branch cuts are chosen as the line segments (a) !p1 !z1 and !z2 !p2 when 40 f 0 =.0 C f 0 /2 < 1 =s , (b) !p1 !z and !z !p2 when 40 f 0 =.0 C f 0 /2 D 1 =s , and (c) !p1 !p2 and !z1 !z2 when 40 f 0 =.0 C f 0 /2 > 1 =s . Notice that the classical Debye model is obtained in the limit as f 0 ! 0, in which case part (a) of the figure applies. In that limiting case there are just two branch points located along the negative imaginary axis at !p D i=0 and !z D i.s =1 /=0

 If 40 f 0 =.0 C f 0 /2 > 1 =s , then

!zj

0 C f 0 D 20 f 0

) 0 f 0 s 1i ; ˙ 4 1 .0 C f 0 /2

( s

(12.132)

296

12 Analysis of the Phase Function and Its Saddle Points

and the two branch point zeros are located in the lower-half of the complex !-plane, symmetrically situated about the imaginary axis along the line ! 00 D .0 C f 0 /=.20 f 0 /, as depicted in Fig. 12.20c. Notice that in the limit as f 0 ! 0, the Rocard–Powles–Debye model reduces to the classical Debye model. In that limiting case one obtains the pair of branch points !p 

i ; 0

!z i

s =1 ; 0

(12.133)

where !z is not to be confused with the symmetry point defined in (12.130) for just the Rocard–Powles extension of the Debye model. Because s =1 > 1, the branch cut extends along the line segment !p !z down the negative imaginary axis. Because the inequality 40 f 0 =.0 C f 0 /2 < 1 =s is satisfied for the set of Rocard–Powles–Debye model parameters given in (12.126), the branch points for that case are as depicted in part (a) of Fig. 12.20. Because this case also represents the limiting behavior of the classical Debye model, it is the focus of the remaining analysis for the Rocard–Powles–Debye model presented here. If the conditions require it, either of the other two cases may be treated in a similar manner.

12.2.3.1

Behavior Along the Real !0 -Axis

The complex index of refraction given in (12.125) for a single relaxation time Rocard–Powles–Debye model dielectric along the real angular frequency axis may be expressed in phasor form as

n.! 0 / D

h

1=4   i2 2 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 / Ca02 .0 Cf 0 / ! 02 p 2 q 2 e i =2 ; 1C0 ! 02 1Cf 0 ! 02

where

"

D arctan

a . C /! 0  0 0 f 0 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 /

(12.134)

# (12.135)

is the phase angle of n2 .! 0 /. The real and imaginary parts of the complex index of refraction along the real ! 0 -axis are then given by nr .! 0 / D 0

ni .! / D

ˇ ˇ ˇn.! 0 /ˇ cos . 1 .! 0 //; 2 ˇ ˇ ˇn.! 0 /ˇ sin . 1 .! 0 //; 2

(12.136) (12.137)

where ˇ ˇ ˇn.! 0 /ˇ D

h

1=4   i2 2 1 .1C02 ! 02 / 1Cf2 0 ! 02 Ca0 .10 f 0 ! 02 / Ca02 .0 Cf 0 / ! 02 p 2 q 2 : 1C0 ! 02 1Cf 0 ! 02

(12.138)

12.2 The Behavior of the Phase in the Complex !-Plane

297

9

Real & Imaginary Parts of the Complex Index of Refraction

8 7 nr ( ')

6 5 4

ni ( ')

3 2 1 0 108

109

1010

x 1011

1012

1013

x

f 1014

' (r/s)

Fig. 12.21 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for a single relaxation time Rocard–Powles–Debye model of triply-distilled H2O with medium parameters 1 D 2:1, a0 D 74:1, 0 D 8:44  1012 s, f 0 D 4:62  1014 s

As seen in Fig. 12.21, the real index of refraction is nearly constant over both the low frequency domain j! 0 j  1=0 and the high frequency domain j! 0 j  1=f 0 . In the intermediate frequency domain 0 < ! 0 < f 0 , the real index of refraction rapidly p decreases from its near static value n.0/ D s to its high-frequency limit n1 D p 1 . The imaginary part of the complex index of refraction peaks to its maximum value near the lower end of this intermediate frequency domain 0 < ! 0 < f 0 , whereas it is comparatively small in both the low frequency domain j! 0 j  1=0 and the high frequency domain j! 0 j  1=f 0 , approaching zero as ! 0 ! 0 and as ! 0 ! 1.

12.2.3.2

Limiting Behavior as j!j ! 1

Consider now the behavior of  .!;  / 1 ,  .!;  / is equal to C1 in the upper-half of the complex !plane, zero at the real ! 0 -axis [i.e.,  .! 0 ;  / D 0 at ! 0 D ˙1], and is equal to 1 in the lower-half of the complex !-plane.

12.2.3.3

Behavior Along the Imaginary Axis

The behavior of the complex index of refraction for a single relaxation time Rocard– Powles–Debye model dielectric along the imaginary axis is obtained from (12.125) with the substitution ! D i ! 00 , with the result  n.! 00 / D 1 C

a0 00 .1 C 0 ! /.1 C f 0 ! 00 /

1=2 :

(12.141)

It is then seen that n.! 00 / is real-valued everywhere along the imaginary axis excluding the branch cuts !p1 !z1 and !z2 !p2 [see part (a) of Fig. 12.20)]; that is, when either ! 00 > 1=0 , ! 00 < 1=f 0 , or j!z2 j < ! 00 < j!z1 j. The index of refraction n.! 00 / is also positive-valued over each of these intervals. In particular, as ! 00 increases over the upper interval ! 00 > 1=0 , n.! 00 / monotonically decreases from its positive infinite branch point singularity !p1 to its zero frep value at the p quency value n.0/ D 1 C a0 D s , approaching its infinite frequency limit p n.i 1/ D 1 as ! 00 ! 1, as illustrated in Fig. 12.22. Similarly, as ! 00 decreases over the lower interval ! 00 < 1=f 0 , nr .! 00 / monotonically decreases from its positive infinite value at the branch point singularity !p2 , approaching its infinite p frequency limit n.i 1/ D 1 as ! 00 ! 1. Finally, in the intermediate interval j!z2 j < ! 00 < j!z1 j between the two branch cuts, nr .! 00 / increases from zero to a local maximum and then decreases back to zero, as illustrated in Fig. 12.22. Notice that ni .! 00 / D 0 in those regions where nr .! 00 / is nonvanishing and that nr .! 00 / D 0 in those regions where ni .! 00 / is nonvanishing. Since .! 00 ; / D ! 00 .n.! 00 /  / along the imaginary axis, then  .! 00 ;  / D ! 00 .nr .! 00 /   /:

(12.142)

12.2 The Behavior of the Phase in the Complex !-Plane

299

Real & Imaginary Parts of the Complex Index of Refraction

5

4

3 ni ( '' ) 2

ni ( '' )

nr( '')

1

0 −5

nr( '' )

nr ( '' ) f z2

z1

5

0 13

'' (x10 r/s)

Fig. 12.22 Frequency dependence of the real (solid curve) and imaginary (dashed curve) parts of the complex index of refraction n.! 00 / D nr .! 00 / C i ni .! 00 / along the imaginary angular frequency axis for a single relaxation time Rocard–Powles–Debye model of triply distilled H2 O with medium parameters 1 D 2:1, a0 D 74:1, 0 D 8:441012 s, f 0 D 4:621014 s. Notice that nr .! 00 / D 0 and ni .! 00 / > 0 when either 1=f 0 < ! 00 < j!z2 j or j!z1 j < ! 00 < 1=0 , and that ni .! 00 / D 0 and nr .! 00 / > 0 when either ! 00 < 1=f 0 , j!z2 j < ! 00 < j!z1 j, or 1=0 < ! 00

As nr .! 00 / D 0 along the two branch cuts !p1 !z1 and !z2 !p2 , then  .! 00 ;  / D ! 00  . Notice that ! 00 < 0 along the two branch cuts so that  .! 00 ;  / < 0 for all   1 when either 1=f 0 < ! 00 < j!z2 j or j!z1 j < ! 00 < 1=0 . 12.2.3.4

Behavior in the Vicinity of the Branch Points

Consider finally the limiting behavior of both n.!/ and  .!;  / in the immediate vicinity of each of the four branch points depicted in part (a) of Fig. 12.20. In the region about the upper branch point pole !p1 , the complex angular frequency may be written as (12.143) ! D !p1 C r1 e i'1 ; where the ordered-pair .r1 ; '1 / denotes the polar coordinates about the point !p1 D i=0 . Substitution of this expression into (12.127) results in the limiting behavior   .!p1  !z1 /.!p1  !z2 / 1=2 n.r1 ; '1 / D .!p1  !p2 /r1 e i'1 s s =1  1 1 i.=4'1 =2/ e (12.144) Š 0  f 0 r 1=2 1

300

12 Analysis of the Phase Function and Its Saddle Points

as r1 ! 0 about the upper branch point !p1 . Similarly, in the region about the lower branch point pole !p2 , the complex angular frequency may be written as ! D !p2 C r2 e i'2 ;

(12.145)

where the ordered-pair .r2 ; '2 / denotes the polar coordinates about the point !p2 D i=f 0 . Substitution of this expression into (12.127) results in the limiting behavior  .!p2  !z1 /.!p2  !z2 / 1=2 n.r2 ; '2 / D .!p2  !p1 /r2 e i'2 s s =1  1 1 i.=4C'2 =2/ Š e 0  f 0 r 1=2 

(12.146)

2

as r2 ! 0 about the upper branch point !p2 . In the region about the upper branch point zero !z1 , the complex angular frequency may be expressed as ! D !z1 C R1 e i 1 ;

(12.147)

where the ordered-pair .R1 ; 1 / denotes the polar coordinates about the point !z1 given in (12.131) with the upper sign choice. Substitution of this expression into (12.127) results in the limiting behavior 1=2 .!z1  !z2 /R1 e i 1 .!z1  !p1 /.!z1  !p2 / v s u u 0 C f 0 0 f 0 s 1=2 t Š 14 R e i. s =1  1 1 .0 C f 0 /2 1 

n.R1 ;

1/ D

1 =2C=4/

(12.148)

as R1 ! 0 about the upper branch point !z1 . Similarly, in the region about the lower branch point zero !z2 , the complex angular frequency may be expressed as ! D !z2 C R2 e i 2 ;

(12.149)

where the ordered-pair .R2 ; 2 / denotes the polar coordinates about the point !z2 given in (12.131) with the lower sign choice. Substitution of this expression into (12.127) results in the limiting behavior 1=2 .!z2  !z1 /R2 e i 2 .!z2  !p1 /.!z2  !p2 / v s u u 0 C f 0 0 f 0 s 1=2 t Š 14 R e i. s =1  1 1 .0 C f 0 /2 2 

n.R2 ;

2/ D

as R2 ! 0 about the upper branch point !z2 .

2 =2=4/

(12.150)

12.2 The Behavior of the Phase in the Complex !-Plane

301

Fig. 12.23 Limiting behavior of the complex index of refraction n.!/ about the branch point poles !pj and branch point zeros !zj , j D 1; 2, for a single relaxation time Rocard–Powles–Debye model dielectric

''

'

p1

(1– i)

(1+i) −i

+i

−i0 +i0 (1+i)0

(1– i)0 z1

−0

+0 (1+ i)0

z2

+i0 −i0

+i (1+ i)

−i

(1− i)0

(1− i) p2

The limiting behavior of the complex index of refraction for a single relaxation time Rocard–Powles–Debye model dielectric about each of the branch point poles !pj and branch point zeros !zj , j D 1; 2, as described by (12.144), (12.146), (12.148), and (12.150), is illustrated in Fig. 12.23. Because the near saddle point for a Debye-type dielectric moves down the imaginary axis as the space–time parameter  increases [see (12.35) and the discussion following], crossing the origin p at the space–time point  D 0 , where [see (12.21)] 0 s , it primarily interacts with the upper branch point pole !p1 , which also happens to be the only branch point in the finite !-plane in the Debye model limit. From these results, the limiting behavior of the real part  .!;  / D 0, where A.z; t / is given by the Fourier– Laplace integral representation in either (12.1) or (12.2), the saddle points of the complex phase function .!; / must be located in the complex !-plane and the behavior of .!;  / at these points determined. The condition that .!;  / be stationary at a saddle point is  0 .!;  / D i .n.!/  / C i!n0 .!/ 0; where the prime denotes differentiation with respect to !. The saddle point equation is then given by n.!/ C !n0 .!/ D : (12.177) Approximate analytic expressions for A.z; t / then require approximate analytic solutions of (12.177) for the dynamical evolution of the saddle point locations with  as well as for the complex phase behavior at each of these saddle points. In those exceptional cases when reasonably accurate, approximate analytic expressions are unavailable, numerical results alone will have to suffice.

12.3.1 Single-Resonance Lorentz Model Dielectrics An exact analytic expression for the location of the saddle points of .!;  / for a single-resonance Lorentz model dielectric is considered first. With the complex index of refraction given by (12.57), namely,  n.!/ D 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

12.3 The Location of the Saddle Points and the Approximation of the Phase

317

with the first derivative b 2 .! C i ı/

0

n .!/ D  ! 2  !02 C 2i ı!

 2 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

the saddle point equation (12.177) becomes  1

b2 ! 2 !02 C2iı!

1=2

C



b 2 !.!Ciı/ 2

.! 2 !02 C2iı! /

1

b2 ! 2 !02 C2iı!

1=2

D : (12.178)

To eliminate the square root factors appearing in this expression, it may be rewritten as ! 2  !12 C 2i ı! b 2 !.! C i ı/ C  2 D  2 ! 2  !0 C 2i ı! ! 2  !02 C 2i ı!



! 2  !12 C 2i ı! ! 2  !02 C 2i ı!

1=2 ;

where !12 D !02 C b 2 . Squaring both sides of this equation then yields  ! 2  !12 C 2i ı! C

2 b 2 !.! C i ı/ ! 2  !02 C 2i ı!    D  2 ! 2  !12 C 2i ı! ! 2  !02 C 2i ı! : (12.179)

Since both of the expressions given in (12.178)–(12.179) are complicated functions of the complex variable !, it is difficult (if not indeed impossible) to determine exact analytic expressions for the saddle point locations as a function of  for all   1. However, from the computer-generated contour plots illustrating the topography of  .!;  / in the complex !-plane given in Figs. 12.4–12.9, it is found that there are in general a pair of saddle points that evolve with   1 in the region j!j  !0 about the origin and a pair of saddle points that evolve with   1 in the region j!j  !1 removed from the origin. These two regions may then be considered separately in develop approximate analytic expressions for the respective saddle point locations. These approximate solutions may then be used as initial values in a numerical solution of the exact saddle point equation to test both their accuracy and numerically determine more accurately the roots of that equation. Before proceeding with this approximate analysis, notice that two exact roots of the saddle point equation given in (12.174) are readily obtained in the limit as  ! 1. In that limit, either ! 2  !02 C 2i ı! D 0, yielding the roots q ! ! !˙ D ˙ !02  ı 2  i ı;

as

 ! 1;

(12.180)

318

12 Analysis of the Phase Function and Its Saddle Points

or n.!/ D 0, yielding the roots !!

0 !˙

q D ˙ !12  ı 2  i ı;

as

 ! 1:

(12.181)

Thus, in the limit as  ! 1, the saddle points move into the branch points !C and 0 in the right-half plane and ! and !0 in the left-half plane. !C Moreover, an exact polynomial equation describing the location of the saddle points can be obtained as follows: (12.178) is rewritten to eliminate the square roots as  

! 2  !12 C 2i ı! ! 2  !02 C 2i ı!

1=2

 2 2 !  !02 C 2i ı!

    D ! 4 C 4i ı! 3  2 !02 C 2ı 2 ! 2  i ı 4!02 C 3b 2 ! C !12 !02 :

Upon squaring both sides of this equation, there results   3  2 ! 2  !12 C 2i ı! ! 2  !02 C 2i ı!   2

  D ! 4 C 4i ı! 3  2 !02 C 2ı 2 ! 2  i ı 4!02 C 3b 2 ! C !12 !02 : After a bit of algebraic manipulation, one finally obtains the following exact polynomial equation for the saddle point locations in a single-resonance Lorentz model dielectric [4, 7]:     

   2   1 ! 8 C 8i ı  2  1 ! 7  4 !02 C 6ı 2  2  1 C b 2  2 ! 6    2i ı 12!02 C 3b 2 C 16ı 2  2  1 ! 5 

  C 6!04 C .48ı 2 C 2b 2 /!02 C 12b 2 ı 2 C 16ı 4  2  1   Cb 2 !02  2  12ı 2 ! 4 

  C4i ı 6!04 C 4b 2 !02 C 8ı 2 !02 C 4ı 2 b 2  2  1    C !02 C 2ı 2 2!02  2  b 2 ! 3 

   !02 4!04 C 3b 2 !02 C 24ı 2 !02 C 12ı 2 b 2  2  1   b 2 !04 C 20ı 2 !02 C 9ı 2 b 2 ! 2      2i ı!02 4!02 C 3b 2 !02  2  1  b 2 !      C!04 !02 C b 2 !02  2  1  b 2 D 0: (12.182) Because this eighth-order polynomial is extremely formidable as well as difficult to approximate, the approximate solution of the saddle point equation as given in either (12.178) or (12.179) is now developed for both the distant and near saddle points that are a characteristic of a single-resonance Lorentz model dielectric.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.1.1

319

The Region Removed from the Origin (j!j  !1 )

The First Approximation To permit comparison with the classical asymptotic theory due to Brillouin [2, 3], a critical review of this first approximation is now considered. For sufficiently large values of j!j the quantity !02 can be neglected in comparison to the quantity ! 2 in the expression (12.57) for the complex index of refraction of a single-resonance Lorentz model dielectric, so that n2 .!/ 1 

b2 ; !.! C 2iı/

(12.183)

provided that j!j2  !02 . Notice that this approximation simplifies the highfrequency response of a Lorentz model dielectric by the exact behavior of a Drude model conductor [cf. (12.153)]. Because the magnitude of the second term in the above expression for n2 .!/ is small in comparison to unity for j!j2  b 2 , the complex index of refraction may then be further approximated as n.!/ 1 

b2 : 2!.! C 2iı/

(12.184)

Notice that the accuracy of these approximations only improves in the weak dispersion limit as b ! 0 (more fundamentally, in the limit as N ! 0, where N is the number density of Lorentz oscillators). Differentiation of the approximation given in (12.184) with respect to ! yields n0 .!/ b 2

! C iı .! 2

C 2iı!/2

:

(12.185)

Substitution of these approximate expressions into the saddle point equation given in (12.177) then results in the approximate saddle point equation 1

b 2 .! C iı/ b2 C

 2!.! C 2iı/ ! .! C 2iı/2

(12.186)

in the region removed from the origin, with solutions b !SP ˙ ./ ˙ p  2iı; d 2.  1/

(12.187)

for   1. This result, first obtained by Brillouin [2, 3], is referred to as the first approximation of the distant saddle point locations [cf. (12.55)]. The distant saddle points are then seen to be symmetrically located about the imaginary axis, lying

320

12 Analysis of the Phase Function and Its Saddle Points

along the line ! D 2iı. At the luminal space–time point  D 1 these two saddle points are at !SP ˙ .1/ D ˙1  2iı, and as  increases away from unity they d move in toward the imaginary axis that is 0 increases as the number density N decreases. A more accurate approximation that is valid over the entire space–time domain  2 Œ1; 1/ is then seen to be desirable.

The Second Approximation To obtain a more accurate description of the distant saddle point locations, particularly for large values of  , the exact saddle point equation given in (12.179) is first rewritten as 1  2 .! 2 !02 C2iı! /

 ! 2  !12 C 2iı! C

b 2 !.!Ciı/ ! 2 !02 C2iı!

2

D ! 2  !12 C 2iı!: (12.188)

This particular form of the saddle point equation explicitly displays the desired limiting behavior as  ! 1, because in that limit, the right-hand side of this equation must approach zero, so that q lim !SP ˙ ./ D ˙ !12  ı 2  iı:

!1

(12.189)

d

With this limiting behavior in mind, the rational function appearing in the squared term of (12.188) may be approximated as    2  ı b 2 !.! C iı/ 2 1 C iı=! 2

b C O ! 1  i ; (12.190)

b 1 C 2iı=! ! ! 2  !02 C 2iı! provided that j!j  !0 and j!j  ı . As a first approximation [for the purpose of comparison with Brillouin’s first approximation given in (12.187)], let the above expression be approximated by the first term on the right-hand side of (12.190). In that case, the saddle point equation (12.188) becomes 2  1   ! 2  !02 C 2iı! ! 2  !12 C 2iı!;  2 ! 2  !02 C 2iı!

12.3 The Location of the Saddle Points and the Approximation of the Phase

with solution

321

r

b2 2  iı; (12.191) d 2  1 for   1, which is to be compared with the expression given in (12.187). Although the distant saddle points SPd˙ now lie along the line ! D iı in this first-order approximation, residing at ˙1  iı at  D 1, they do approach the respective outer 0 as  ! 1, in agreement with the exact result [see (12.181)]. branch points !˙ Consider now obtaining the second-order approximation of the distant saddle point locations, in which case (12.190) is approximated as !SP ˙ . / ˙ !02  ı 2 C

ıb 2 b 2 !.! C iı/ 2 :

b  i ! ! 2  !02 C 2iı! With this substitution, the saddle point equation given in (12.179) becomes  2 2  ı2b4 ıb 2  2 !  !02 C 2iı!  2i !  !02 C 2iı!  2 !   !  D  2 ! 2  !02 C 2iı! ! 2  !12 C 2iı! : The term ı 2 b 4 =! 2 may be neglected in comparison to the other two terms on the left-hand side of this equation with the result   ıb 2 b2 2 ! C 2i 2

0: ! 3 C 2iı! 2  !02 C 2  1  1

(12.192)

The zeros of this cubic equation can be obtained by first determining the form of its reduced equation as follows: Define the coefficients a1 2iı;   b2 2 ; b1  !02 C 2  1 ıb 2 c1 2i 2 :  1 Then, under the change of variable ! 

2 a1 D   iı; 3 3

(12.193)

the cubic equation given in (12.192) is reduced to the form  3 C a2  C b2 0;

(12.194)

322

12 Analysis of the Phase Function and Its Saddle Points

where 1 b2 2 4 a2 b1  a12 D !02  2 C ı2; 3  1 3   2 C 3 2 3 1 2 8 a1  a1 b1 C c1 D iı !02  ı 2 C b 2 2 : b2 27 3 3 9  1 To construct the solutions to this reduced cubic equation, let 0

b2 A˙ @ ˙ 2

s

11=3 b22 a23 A ; C 4 27

where s "  2  2 2  2  2 2 2  2  a23 b22 i 4 2 2 3!0  2ı !0  C 2ı 9!0  8ı C D p !0 !0  ı C b 4 27 2  1 3 3 #1=2  2  4  2  2 2 b6 6 4 3!0  ı  C 9ı 2 C 3 Cb C : . 2  1/2 . 2  1/3 Notice that the algebraic expression appearing under the square root operation in the above expression is positive for all   1. If one then defines the real valued quantities   2 ı 8 2 2 2 C 3 !0  ı C b 2 ; (12.195) ˇ1 3 9  1 "  2  2 2  2  2 2 2  2  1 4 2 2 3!0  2ı !0  C 2ı 9!0  8ı ˇ2 p !0 !0  ı C b 2  1 3 3 #1=2  2  4  2  2 2 6 6 3!  2  ı C 9ı C 3  b 0 Cb 4 C ; . 2  1/2 . 2  1/3 (12.196) one finds that A˙ D i .ˇ1 ˇ2 /1=3 ;

(12.197)

where the principal branch 0  arg .A˙ / < 2 has been chosen. The three solutions of the reduced cubic equation given p in (12.194) are then given by  D AC C A and ˙ D .AC CA /=2˙i.AC A / 3=2. Because there are only two distant saddle points which are symmetrically situated about the imaginary axis, it is seen that the last two solutions are of the proper form. Thus, the two sought-after solutions are given by

12.3 The Location of the Saddle Points and the Approximation of the Phase

323

p ˙ D ˙

 3

.ˇ1 C ˇ2 /1=3  .ˇ1  ˇ2 /1=3 2  i

 .ˇ1 C ˇ2 /1=3 C .ˇ1  ˇ2 /1=3 : 2

(12.198)

Substitution of this solution into (12.193) then results in the approximate expression for the distant saddle points locations p  3

!SP ˙ . / ˙ .ˇ1 C ˇ2 /1=3  .ˇ1  ˇ2 /1=3 d 2    2 1 1=3 1=3 .ˇ1 C ˇ2 / C .ˇ1  ˇ2 / : (12.199) i ıC 3 2 This complicated expression is rather formidable to work with, however, and a more simplified expression that possesses greater accuracy than either of the two first-order approximations given in (12.187) and (12.191) is desired. Because ˇ2  ˇ1 for all   1, the following two approximations can then be made:   ˇ1 1=3 ˇ1 1=3 1C

ˇ2 C 2=3 ; ˇ2 3ˇ2  1=3 ˇ1 1=3 ˇ1 1=3 D ˇ2 1

ˇ2 C 2=3 : ˇ2 3ˇ 1=3

.ˇ1 C ˇ2 /1=3 D ˇ2 .ˇ1  ˇ2 /1=3

2

With substitution of these two approximations, (12.199) for the distant saddle point locations becomes ! p 1=3 ˇ1 2 ı C 2=3 : !SP ˙ . / ˙ 3ˇ2  i (12.200) d 3 3ˇ 2

1=3

Finally, the quantity ˇ2 p

may be approximated as

#1=2  2  4   ! ! b2 2 1 2 0 C !02  ı 2 C !02  ı 2

C 04 !02  ı 2 2  1 3 3 b2 3b r b2 2 ;

!02  ı 2 C 2  1 "

1=3 3ˇ2

to a fair degree of approximation [cf. (12.191)]. In addition, with this result one finds that # " ˇ1 ı 3b 2 

: 1C  2 2=3 3 !0  ı 2 . 2  1/ C b 2  2 3ˇ2

324

12 Analysis of the Phase Function and Its Saddle Points

The distant saddle point locations may then be expressed as   !SP ˙ . / D ˙./  iı 1 C . /

(12.201)

d

with the second approximate expressions3 r

b2 2 ; 2  1 b2  ./  2 : !0  ı 2 . 2  1/ C b 2  2 ./

!02  ı 2 C

(12.202) (12.203)

  Notice that ./ D b 2 = . 2  1/ 2 ./ to this order of approximation. The expressions given in (12.201)–(12.203) then comprise the second approximation of the distant saddle point locations. For values of  close to unity, the above expressions simplify to ./ ! p

b 2.  1/

;

. / ! 1; so that the second approximation reduces to the first approximation [cf. (12.187)] in this limit. In particular, in the limit as  approaches unity from above lim !SP ˙ ./ D ˙1  2iı:

!1C

(12.204)

d

On the other hand, for sufficiently large values of  the second approximate expressions given in (12.202) and (12.203) become q ./ ! !12  ı 2 ; . / ! 0; so that in the limit as  approaches infinity q 0 lim !SP ˙ ./ D ˙ !12  ı 2  iı D !˙ ; !1

(12.205)

d

and the distant saddle points SPd˙ respectively approach the outer branch points 0 . The second approximation to the distant saddle point locations then captures !˙ the exact limiting behavior in the two opposite extremes at either  D 1 or  D 1. A sketch of the respective paths followed by these two distant saddle points in the complex !-plane is presented in Fig. 12.39. 3

Notice that the second approximate expression for . / that is used here is slightly modified from that given in earlier publications [4, 6, 7].

12.3 The Location of the Saddle Points and the Approximation of the Phase

325

''

' ' cut −

' cut +

SPd

SPd

Fig. 12.39 Depiction of the behavior of the distant saddle points SPd˙ in the complex !-plane for a single-resonance Lorentz model dielectric. The dotted curves indicate the respective directed paths that these saddle points follow as  increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP ˙ ;  / through d that saddle point, the shaded region indicating the local region about each saddle point where the inequality  .!;  / <  .!SP ˙ ;  / is satisfied, the vectors indicating the local direction of ascent d along the lines of steepest descent and ascent through each saddle point

An analytic approximation of the complex phase behavior of the phase function .!;  / that is valid in the region of the complex !-plane traversed by the distant saddle points as  varies from unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in (12.184) is sufficiently accurate so that the complex phase behavior in the region j!j  !1 of the complex !-plane that is removed from the origin may be approximated as b2 : (12.206) .!;  / i !.1  /  i 2.! C 2iı/ To obtain the approximate behavior of  .!;  / 1 , the exact saddle point equation given in (12.179) is again employed. This saddle point equation may be rewritten in the form    2 ! 2  !02 C 2iı! D ! 2  !12 C 2iı! C 2b 2

!.! C iı/ ! 2  !02 C 2iı!

! 2 .! C iı/2 Cb 4   2 : ! 2  !12 C 2iı! ! 2  !02 C 2iı! (12.216) For j!j small in comparison to !0 , the two expansions !.! C iı/ 1 ! 2 C iı! D  2 ! 2  !0 C 2iı! !02 1  i 2ı!2  ! 22 !0 !0       ı2 ı2 1 ı

 2 iı! C 1  2 2 ! 2 C i 2 3  4 2 ! 3 ; !0 !0 !0 !0 and   ! 2 ! 2 C 2iı!  ı 2 ! 2 .! C iı/2     2

2iı!02 2!12 C !02 !  !12 !04 ! 2  !12 C 2iı! ! 2  !02 C 2iı!    !2 2ı 3 

2 4 ı 2  2iı! C i 2 2 2!12 C !02 ! ; !1 !0 !1 !0

12.3 The Location of the Saddle Points and the Approximation of the Phase

329

are useful. With substitution of these approximations, the exact saddle point equation given in (12.216) assumes the approximate polynomial form      b2 ıb 2 ı2 ı2b2   2 ! 2  !02 C 2iı! 2i 4 3 C 2  4 2  4 2 2!12 C !02 ! 3 !0 !1 !0 !1 !0   2  2 ı b ı2b2 !2 C 1 2 24 2  2 2 !0 !0 !1 !0   b2 C2iı 1  2 !  !02  b 2 : !0 Because the coefficient of the cubic term in ! is small in comparison to the other terms appearing in this polynomial expression, it may be neglected. The approximate saddle point equation for the near saddle point locations then becomes 2

2

! C 2iı

 2  02 C 2 !b 2 0

!

2

 2  02 C 3˛ !b 2 0

  !02  2  02 2

 2  02 C 3˛ !b 2

0;

(12.217)

0

where 0 is as defined in (12.212) and where the parameter ˛ has been redefined in this second-order approximation as [cf. (12.213)] ˛ 1

 ı2  2 4!1 C b 2 : 2 2 3!0 !1

(12.218)

Notice that for values of  very close to 0 , the coefficients appearing in the second approximation of the saddle point equation given in (12.217) reduce to those appearing in the first approximation given in (12.211). This then shows that the first approximation of the near saddle point locations is valid only in the immediate space–time region about the value 0 . The near saddle point locations may then be expressed as 2 !SP˙ ./ D ˙ ./  iı ./ n 3

(12.219)

with the second approximate expressions 2 6 . / 4

!02 2



 2    02 02

C

2 3˛ !b 2 0

0  ı2 @

2

 2  02 C 2 !b 2 0

2



02

C

2 3˛ !b 2 0

12 31=2 A 7 5 ;

(12.220)

2

b 2 2 3   0 C 2 !02 :

. /

2  2  02 C 3˛ b 22

(12.221)

!0

The expressions given in (12.219)–(12.221) then comprise the second approximation of the near saddle point locations.

330

12 Analysis of the Phase Function and Its Saddle Points

For values of  close to 0 , the above expressions simplify to s 0 ! 4 1 ı2 . / ! 6 20  4 2 ; 3 ˛b ˛ 1

. / ! ; ˛ so that the second approximation reduces to the first approximation [cf. (12.214)]. On the other hand, in the limit as  approaches infinity q lim !SP˙ ./ D ˙ !02  ı 2  iı D !˙

!1

n

(12.222)

and the near saddle points SP˙ n , respectively, approach the inner branch points !˙ , in agreement with the numerical results presented in Sect. 12.2.1. To analyze the behavior of the near saddle points as described by this second approximation, it is again necessary to first determine the algebraic sign of the argument of the square root in (12.220). This amounts to determining a more accurate value of the critical space–time point  D 1 at which the two near first-order saddle points coalesce into a single second-order saddle point, where 0

  !02 12  02 2

12  02 C 3˛ !b 2 0

 ı2 @

2

12  02 C 2 !b 2 0

2

12  02 C 3˛ !b 2

12 A 0;

0

in this second-order approximation, which simplifies to    2  2  ı2  ı2b4 !0  ı 2 12  02 C b 2 3˛  4 2 12  02  4 4 0: !0 !0 Because 1 is greater then 0 for positive-definite values of the phenomenological damping constant ı [see (12.26)], the appropriate solution of this binomial equation gives v 3 2v u   u u 2 2 ! 2  ı2 2 u ı 3˛!  4ı u 2 0  4t1 C 16  1 t0 C b 2 2  0 2 2  15; 2!0 !0  ı 2 3˛!02  4ı 2

(12.223)

where the positive values of both square roots appearing in this expression are to be taken. To compare this second approximation to the value of the critical space–time point 1 with that given in the first approximation by (12.215), the square of the above expression may be approximated as

12.3 The Location of the Saddle Points and the Approximation of the Phase

331

#   2 2 ı 2 !02  ı 2 2 2 2 3˛!0  4ı   1 C 8 1 0 C b 2  1 2!02 !02  ı 2 3˛!02  4ı 2 "

D 02 C

4ı 2 b 2  ; !02 3˛!02  4ı 2

(12.224)

so that 1 0 C

 2

!0

2ı 2 b 2 ; 3˛!02  4ı 2

(12.225)

which reduces to the first approximate expression given in (12.215) through neglect of the term 4ı 2 in comparison to 3˛!02 in the denominator. Because of its simplicity, the approximate expression given in (12.224) for 12 is used in subsequent calculations concerning the behavior of the near saddle points at that critical space–time value. An analytic approximation of the complex phase behavior .!;  / that is valid in the region of the complex !-plane traversed by the near saddle points as  varies from unity to infinity is now considered. For this analysis, the complex index of refraction that is given by the first approximate expression in (12.210) is sufficiently accurate. With (12.213), this approximate expression may be written as n.!/ 0 C

b2 !.˛! C 2iı/; 20 !04

(12.226)

so that the complex phase behavior in the region j!j  !0 of the complex !-plane may be approximated as .!;  / i ! .0  / C

b2 ! 2 .i˛!  2ı/: 20 !04

(12.227)

the dynamical behavior of the near saddle points and the local complex phase behavior about them, as described by this second approximation, is now considered for the three separate cases 1   < 1 ,  D 1 , and  > 1 . Case 1 (1   < 1 ) Over this initial space–time domain the near saddle point locations are given by  !SP˙ ./ D i ˙ n

2 o ./  ı ./ 3

 (12.228)

332

12 Analysis of the Phase Function and Its Saddle Points

with the second approximate expressions 31=2  2  2   0 7 6 2@ 0 A  ; (12.229) o . / 4ı 2 5 b2 2 2 2   0 C 3˛ ! 2   02 C 3˛ !b 2 2

0

2

 2  02 C 2 !b 2

12

!02

0

2

. /

02

0

2 2 !b 2 0

3   C ; 2  2  02 C 3˛ b 22

(12.230)

!0

that are appropriate over this domain. As depicted in Fig. 12.40, the near saddle points SP˙ n are located along the imaginary axis, symmetrically situated about the point ! 00 D  23 ı ./, where . / varies slowly over this space–time domain. To obtain the approximate local behavior of  .!;  / 1 . The dotted curves indicate the respective directed paths that these first-order saddle points follow as  increases to infinity. The dashed lines through each saddle point indicate the local behavior of the isotimic contour  .!;  / D  .!SP˙ ;  / through that saddle point, the shaded region indicating n the local region about each saddle point where the inequality  .!;  / <  .!SP˙ ;  / is satisfied, n the vectors indicating the local direction of ascent along the lines of steepest descent and ascent through each saddle point

336

12 Analysis of the Phase Function and Its Saddle Points

To obtain the approximate behavior of the real phase function  .!;  / in the local vicinity of these near saddle points, ! is again expressed in polar coordinates .r; '/ about the specific saddle point. Because  .!;  / is symmetric about the imaginary axis, only the behavior about the near saddle point SPC n in the right-half of the complex !-plane needs to be considered. Hence, let ./ C re i' ! D !SPC n D

2 ./  i ı ./ C re i' : 3

With substitution of this expression into (12.227), the approximate complex phase behavior about the near saddle point SPC n for  > 1 is found to be given by   2 .r; '; /

ı ./ C i ./ C irei' .0   / 3 (   2   ˛ 8 3 2 b

. / 1  C ı

. / C 2ı 2 ./ ˛ . /  1 4 9 3 20 !0     4 2 3 ı . / . / 2  ˛ . / C ˛ ./ Ci 3        4 2 2 C 4ı . / ˛ . /  1 C i 3˛ ./ C ı . / 2  ˛ . / re i' 3 )

   2 i2' 3 i3' C 2ı ˛ . /  1 C 3i ˛ ./ r e C i ˛r e : (12.240) The real part of this equation then yields the result   2  .r; ';  / r sin '  ı ./ .0  / 3 (     ˛ 8 3 2 b2 ı . / 1  . / C 2ı 2 ./ ˛ . /  1 C 4 9 3 20 !0       4 C4ı . / ˛ . /  1 r cos '  3˛ 2 ./ C ı 2 . / 2  ˛ . / r sin ' 3 )   2 2 3 C2ı ˛ . /  1 r cos 2'  3˛ ./r sin 2'  ˛r sin 3' ; (12.241) from which it is seen that  .r; ';  / attains its maximum variation about the near saddle point SPC n when ' D =4; 3=4; 5=4; 7=4. The lines of steepest descent through this saddle point are at ' D =4; 7=4 and the lines of steepest ascent are at ' D 3=4; 7=4. Because of the even symmetry of  .!;  / about the imaginary axis, the lines of steepest ascent and descent through the near saddle point SP n are reversed, as illustrated in Fig. 12.42.

12.3 The Location of the Saddle Points and the Approximation of the Phase

12.3.1.3

337

Determination of the Dominant Saddle Points

The asymptotic description of dispersive pulse dynamics in a given medium relies upon the determination of the saddle point (or points) that give the least exponential decay as the propagation distance z ! 1. Such a saddle point SP at which  .!SP ; / b  ı, and that !SB b C !02 =b when b > !0  ı. Hence, when considered as a function of the number density N p of Lorentz oscillators comprising the material, !SB is seen to be bounded below by 2!0 in the weak dispersion limit and bounded above by the plasma frequency b.

12.3.1.4

Comparison with Numerical Results

A numerical determination of the exact saddle point locations and the exact behavior of the real and imaginary parts of the complex phase function .!;  / D  .!;  / C i $ .!; / at these saddle points is now presented. These results are then compared to both the first and second approximations for the distant and near saddle points developed in the preceding sections. This comparison is done over a reasonable representation of the entire range of values of the space–time parameter   1 of importance in order that the range of values of  over which a given approximation closely describes the exact, numerically determined behavior may be ascertained. Because of its historical importance, Brillouin’s choice of thepsingle-resonance Lorentz medium parameters (namely, !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s) are used in the first part of this comparison. Because this choice corresponds to a highly absorptive medium, the second part of this numerical comparison investigates the behavior in the weak dispersion limit as b ! 0 (i.e., as N ! 0). With the complex index of refraction given by (12.57), the exact locations of the saddle points of .!;  / are given by the roots of (12.178), which may be written more simply (and more suitably for the purposes of numerical analysis) as b 2 !.! C iı/   0: F .!;  / D n.!/ C  2 ! 2  !02 C 2iı! n.!/

(12.261)

12.3 The Location of the Saddle Points and the Approximation of the Phase

343

The numerical solution of this equation can then be accomplished using Newton’s method at each fixed value of  with either the second approximate solution at that same  -value as an initial guess or the exact, numerical solution at a neighboring  value as the initial guess. Suppose then that ! is an approximate solution of (12.261) at some fixed  -value and let ! be a small correction that is to be determined such that F .! C !;  / D 0. If this equation is expanded in a Taylor series about the approximate solution ! and then truncated after the first-order term in !, there results F .!;  / ; (12.262) ! D  @F .!; /=@! from which the next approximation is obtained and the process repeated, if necessary. That is, let !1 denote either the second approximate solution of the desired saddle point location at a given fixed  -value or the exact, numerically determined solution at the previous -value, and let !1 , as determined from (12.262), denote the associated correction factor. The second trial solution of (12.261) is then given by !2 D !1 C !1 , and after j such iterations, the .j C 1/th solution is given by !j C1 D !j 

F .!j ;  / : @F .!; /=@!j!D!j

(12.263)

This numerical iteration procedure is terminated at a value !k at which jF .!k ;  /j < ", where " > 0 sets the accuracy of the numerical procedure. For the numerical results presented here, " D 11011 . Finally, the exact expression for @F .!; /=@! appearing in (12.263), given by @F .!;  / b2 D  2 @! ! 2  !02 C 2iı! n.!/ 2

0

13 b2 4n.!/ C ! 2 ! 2 C2iı! . /n.!/ A5 0   43! C 2iı  !.! C iı/2 @  2 ; !  !02 C 2iı! n.!/ (12.264)

is employed in the calculation of the exact saddle point equations. Once the numerically determined exact saddle point location at a given value of  is obtained, the exact values of both  .!;  / and $ .!; / at that space–time point are calculated using the exact expressions given in (12.68) and (12.69), respectively. Notice that Newton’s method fails when j@F .!;  /=@!j becomes exceedingly small, as may occur near the critical space–time point  D 1 as the two near firstorder saddle points approach each other and coalesce into a single second-order saddle point at  D 1 and then separate and move apart. In that case, the method of bisection may be used to numerically determine the saddle point location along the imaginary axis. The numerically determined saddle point locations as a function of  are illustrated in Figs. 12.44–12.46 along with the first and second approximate results, as

344

12 Analysis of the Phase Function and Its Saddle Points 20

30

5.0 4.0 3.0 2.0 2.0

3.0

s

1.009

1.010 1.009

1.010

1.020

1.025

1.030

1.050

1.040

1.10

1.15

1.50 1.40 1.30 1.25 1.20

at io n First Approximation

1.015

im

1.015

ox

on

1.020

pr

ati

1.02 5

nd

1.1 0

co

Ap

Loc

1.03 0

Se

0 1.2

0

0 1.5 40 1. 30 E 1. .25 x a ct 1

1.0 50

1.50 1.40 0 1.3

−0.5

−0.6

25

branch cut

1.04

−0.4

ω ‘ (x1016r/s)

15

1.1 0

ω“ (x1016r/s)

−0.3

10

1. 20 1. 15

−0.2

5

Fig. 12.44 Comparison of the exact (numerically determined) distant saddle point locations C !SP . / in the right-half of the complex !-plane as a function of  > 1 with that given by the d first and second approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

4 3 + SPn

'' (x1016r/s)

2 1 0 −1

1.1

1.2

1.3

1.4

1.5

−2 −3

SPn−

−4 ˙ Fig. 12.45 Comparison of the exact (solid curves) near saddle point locations !SP . / along the n imaginary axis as a function of  2 Œ1; 1 with that given by the first (short dashed curves) and second (long dashed curves) approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

12.3 The Location of the Saddle Points and the Approximation of the Phase

345

ω' (x1016r/s) 4.0

3.0

2.5

2.0

1.90

1.80

1.70

80 1.

1. 90

2. 0

1. 90

1.

2. 0

p p ro at

2.5

m io

3.0

xi

ca

ω” (x1016r/s)

A

2.

Lo

on

4.0

n

ti s

−0.28

6

nd

5

t

−0.26

5

co

ac

−0.24

4

First Approximation

Ex

−0.22

3

Se

1.7 0 80

1.60

2

1.7 0

1.51 1.52 1.5 1.5 3 1.5 4 5 1.6 0

−0.20

1

1.51

−0.18

0

3.0

5.0 4.0 10

5.0 20 50 10 0 2

ω+

b r a n c h c u t

ω‘+

C !SP . / n

Fig. 12.46 Comparison of the exact (solid curve) near saddle point locations in the righthalf of the complex !-plane as a function of  1 with that given by the first (short dashed line) and second (long dashed curve) approximations for a single-resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters

indicated in each figure. Consider first the  -evolution of the distant saddle point locations !SP C in the right-half of the complex !-plane that is depicted in Fig. 12.44. d The  -values corresponding to the saddle point locations marked by the crosses along the respective path are indicated along each curve in the figure. The second approximate locations of this distant saddle point are seen to be in close agreement with the exact locations over the entire range of values of  considered. Furthermore, it is seen that these second approximate locations move at approximately the same rate with respect to  as do the exact locations. As regards, this rate of motion of the distant saddle points with increasing , both the exact and the second approximations show that for  very close to unity, a small increase in the value of  produces a large change in the distant saddle point location, and that as  increases away from unity, the change in location of the saddle point with  steadily diminishes, so that 0 , as the distant saddle points SPd˙ approach ever closer to the outer branch points !˙ respectively, a large increase in the value of  produces only a small change in the saddle point location. Finally, the second approximate distant saddle point locations are seen to provide a marked improvement in accuracy over the first approximate lo˙ . / approaches near to cations, which are seen to fail rapidly as the real part of !SP n 0 the real part of the branch point !˙ , which, for Brillouin’s choice of the medium parameters, corresponds to values of  greater than approximately 1:05. Consequently, the second approximation of the distant saddle point behavior accurately describes

346

12 Analysis of the Phase Function and Its Saddle Points

the exact behavior over the entire -domain of interest [i.e.,  2 Œ1; 1/], whereas the first approximation is valid only for values of  close to unity. Consider next the  -evolution of the near saddle point locations in the right-half of the complex !-plane. For  2 Œ1; 1 the near saddle points SP˙ n are situated along the imaginary axis, approximately symmetric about the value ! 00 D 2ı=.3˛/, as illustrated in Fig. 12.45. Notice that the second approximation provides only a slightly more accurate description of the near saddle point locations over this initial -domain than does the first approximation. The exact and approximate locations are seen to be in excellent agreement for values of  in the range 1:3    1 . The critical space–time value  D 1 when the two near first-order saddle points SP˙ n coalesce into a single second-order saddle point SPn is found numerically to lie in the range 1:50275 < 1 < 1:50300; where (12.225) yields the first approximate value 1 1:50414 and (12.223) yields the second approximate value 1 1:50275, in very close agreement with the numerically determined lower bound given above. At  D 1:502752 the numerically determined saddle point locations are found to be separated by a very small distance along the imaginary axis. For  > 1 the near saddle points move off symmetrically from the imaginary axis and into the lower-half of the complex !-plane, as illustrated in Fig. 12.46 for the near saddle point SPC n in the right-half plane. Notice that the first approximation of the near saddle point behavior rapidly fails as  increases away from 1 because . /j quickly becomes comparable to !0 . The second approximate locations, j!SPC n on the other hand, closely follow along with the exact near saddle point locations as  increases away from 1 , SP˙ n approaching the inner branch points !˙ , respectively, as  ! 1. Notice that the path traced out by the second approximation lies closely adjacent to the path traced out by the exact near saddle point locations for all  > 1 , but that the positions predicted by the second approximation lie slightly ahead of the actual positions. Furthermore, the second approximate locations plotted in Fig. 12.46 are seen to move with increasing  at approximately the same rate as do the exact locations over the entire range of  -values depicted. As in the case for the distant saddle points, this rate of motion is rapid at first, but as the near saddle point locations !SP˙ ./ approach closer to the inner branch points n !˙ , respectively, their rate of motion with  rapidly decreases. Consequently, taken together with the previous results over the initial space–time domain  2 Œ1; 1 , the second approximation of the near saddle point behavior accurately describes the exact behavior over the entire  -domain of interest (namely,  > 1), whereas the first approximation is valid only for values of  within the limited space–time interval 1:3    1 . Finally, consider the behavior of the complex phase function .!;  / at these distant and near saddle points as a function of , as illustrated in Figs. 12.47–12.49. Figures 12.47 and 12.48 describe the real part  .!;  / of the complex phase behavior at the saddle points and Fig. 12.50 describes the imaginary part $ .!; / of

12.3 The Location of the Saddle Points and the Approximation of the Phase

347

1.2 1.0

0.8 0.6 SPn−

0.4

SP

(x1016 /s)

0.2 0

SB

1.2

1.5

+ SPn−

−0.2 −0.4 −0.6

2.0

+ SPd− SPn+

−0.8 −1.0

Fig. 12.47 Comparison of the exact behavior (solid curves) of the real part  .!;  / of the complex phase function .!;  / at the near and distant saddle points of a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 1    2:2 with that given by the first (dotted curves) and second (dashed curves) approximations

the complex phase behavior at the saddle points. The real and imaginary parts of the complex phase behavior described by the first approximation is indicated by the dotted curves, that by the second approximation by the dashed curves, and the exact, numerically determined behavior by the solid curves in each figure. Consider first the  -dependence of the real phase function  .!;  / at the saddle points, as depicted in Figs. 12.47 and 12.48. At the two first-order distant saddle points SPd˙ ,  .!SP ˙ ; / is seen to identically vanish at  D 1, and then to decrease d monotonically as  increases away from unity, where lim!1  .!SP ˙ ;  / D 1. d The first approximation to  .!SP ˙ ;  / is seen to rapidly diverge away from the d exact behavior as  increases above 0 , whereas the second approximation closely follows the exact behavior over the entire range of values of  considered. At the upper near saddle point SPC n , both the first and second approximations to ; / are very close to each other and are in fair agreement with the exact  .!SPC n

348

12 Analysis of the Phase Function and Its Saddle Points

0

2

4

6

8

10

12

14

−1

−2 + SPd−

+ SPn−

SP

(x1016/s)

−3

−4

−5

−6

−7

−8

Fig. 12.48 Comparison of the exact behavior (solid curves) of the real part  .!;  / of the complex phase function .!;  / at the near and distant saddle points of a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 2    14 with that given by the first (dotted curves) and second (dashed curves) approximations

behavior for 1   < SB the agreement becoming excellent over the space–time interval SB    1 . The numerically determined space–time value SB at which ; SB / is found to be given by  .!SP ˙ ; SB / D  .!SPC n d

SB Š 1:334; where the first two terms of (12.257) result in the rough estimate SB 1:495, all three terms in (12.257) yielding the first-order approximate value SB 1:255, and (12.256) yielding the second approximate value SB 1:295, which is in error by only 2:9%. The related angular frequency value !SB defined in (12.258) is found to be given by !SB Š 8:70  1016 r/s; where the simplified expression given in (12.260) yields the estimate !SB

7:22  1016 r/s and (12.259) gives !SB 8:41  1016 r/s, which is in error by only

12.3 The Location of the Saddle Points and the Approximation of the Phase

349

3:3%. For all  > SB , the near saddle point SPC n , and then both near saddle points for    >  are dominant over the distant saddle points, the second apSP˙ 1 SB n ; / for  2 Œ ; 

and  .! ;  / for   1 providing proximation of  .!SPC SB 1 SP˙ n n an accurate description of the exact, numerically determined behavior. For values of  in a small neighborhood about 0 , the first approximation is also seen to accurately describe the exact behavior at the near saddle points SP˙ n , as was expected, but as  increases further and further away from 0 , the accuracy of the first approximation is seen to steadily diminish. Finally, notice from Fig. 12.48 that for values of  sufficiently greater than 1 ,  .!SP ;  / at both the near and distant saddle points decreases steadily in a nearly linear relationship to , with  .!SP˙ ;  / >  .!SP ˙ ;  /. n d Consider next the -dependence of the imaginary part $ .!; / of the complex phase behavior at the saddle points in the right-half of the complex !-plane, illustrated in Fig. 12.49. Because $ .! 0 C i ! 00 ;  / D $ .! 0 C i ! 00 ;  /, the negative

1

2

3

4

5

0

−2

−4

SP

(x1016/s)

−6

−8

−10

+ SPd

SPn+

−12

−14

−16

Fig. 12.49 Comparison of the exact behavior (solid curves) of the imaginary part $ .!;  / of the complex phase function .!;  / at the near and distant saddle points in the right-half of the complex !-plane for a single-resonance Lorentz model dielectric (with Brillouin’s choice of the medium parameters) as a function of  for 1    5 with that given by the first (dotted curves) and second (dashed curves) approximations

350

12 Analysis of the Phase Function and Its Saddle Points

of this behavior is exhibited in the left-half plane. As is evident from the figure, the first approximate behavior at the distant saddle point SPdC rapidly diverges away from the exact behavior as  increases away from unity, and the first approximate behavior at the near saddle point SPC n rapidly diverges away from the exact behavior as  increases away from 1 , where $ .!SP˙ ;  / D 0 for  2 Œ1; 1 . Notice n that the first approximate curves for the near and distant saddle points cross each other at  4:3, a behavior that is not exhibited by the exact solutions at these saddle points. The second approximate behavior for both the distant and near saddle points, however, is seen to provide a significant improvement over the respective first approximate behavior over the entire  -domain illustrated. Consider finally, the saddle point behavior in the weak dispersion limit as the number density N of Lorentz oscillators decreases, resulting in a decrease in the p plasma frequency b D .k4=0 /N qe2 =m of the dispersive medium which approaches vacuum in the vanishing dispersion limit as N ! 0. The change in the frequency dependence of the complex index of refraction along the positive real angular frequency axis as the number density N is decreased is illustrated in Fig. 12.50, where the initial case is for Brillouin’s choice of the medium parameters

3

nr ( )

2.5 2

N

1.5 N/10 1

N/100

0.5 0

10

ni ( )

10

0

5

' ( x1016r/s)

10

15

10

15

1

0

N –1

10

N/10

–2

10

N/100

–3

10 0

5

' ( x1016r/s)

Fig. 12.50 Frequency dependence of the real (upper graph) and imaginary (lower graph) parts of the complex index of refraction n.! 0 / D nr .! 0 / C i ni .! 0 / along the positive real angular frequency axis for a single-resonance Lorentz p model dielectric with medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100.

12.3 The Location of the Saddle Points and the Approximation of the Phase

351

[N D .0 =k4k/.m=qe2 /b 2 6:275  1029 ] and the other two cases correspond to the number densities N=10 6:275  1028 and N=100 6:275  1027 . Notice that the zero frequency value n.0/ D 1:5 in the initial case is reduced to n.0/ 1:0607 in the N=10 case and is further reduced to n.0/ 1:0062 in the N=100 case. Although the real frequency dispersion nr .!/ may be considered weak when the number density has been reduced by 100, the material absorption ˛.!/ D .!=c/ni .!/ is still significant about the medium resonance frequency !0 . As regard to the behavior of n.!/ in the complex !-plane, notice that [see (12.64) and (12.65)] 0 lim !˙ D !˙ I (12.265) N !0

0 that is, the outer branch points !˙ move in toward the inner branch points !˙ as the number density decreases to zero, cancelling each other out at N D 0 when the vacuum is obtained. The  -dependence of the real and imaginary parts of the distant saddle point locations, as described by the second approximate expressions given in (12.219)– (12.221) for   1 is illustrated in Fig. 12.51 for the N , N=10, and N=100 cases.

18

' (x1016r/s)

10

N

17

10

N/10 N/100 16

10

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

6

'' (x1016r/s)

5.5 5

N

4.5 4

N/10

3.5

N/100

3 2.5

1

1.1

1.2

Fig. 12.51 space–time  -dependence of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPdC evolution for a single-resonance Lorentz model dielectric with p medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100

352

12 Analysis of the Phase Function and Its Saddle Points

Notice that, as the number density decreases, the large  limiting behavior is attained at earlier values of  . This is reflected in the numerical value of the critical space– time parameter SB which decreases from the initial (second approximate) value SB 1:2949 to the value SB 1:0347 at N=10 and then to the value SB

1:00298 at N=100. Accompanying this change, the angular frequency value !SB decreases from its initial (second approximate) value !SB 8:41  1016 r/s to the  1016 r/s value !SB 7:18  1016 r/s at N=10 and then to the value !SB 7:05q

at N=100, while at the same time the angular frequency value !1 !02 C b 2 that sets the scale for the outer branch points decreases from its initial value !1 D 6  1016 r/s to the value !1 4:24  1016 r/s at N=10 and then to the value !1

4:025  1016 r/s at N=100, where !1 ! !0 as N ! 0. Finally, the  -dependence of the real and imaginary parts of the near saddle point locations, as described by the second approximate expressions given in (12.239) and (12.220)–(12.221) for   1 is illustrated in Fig. 12.52 for the N , N=10, and N=100 cases. Notice that, just as for the distant saddle point behavior, the large  limiting behavior is attained at earlier values of  as the number density decreases. This is q reflected in the numerical values of the critical space–time values 0 1 C b 2 =!02 and 1 , whose second approximation is given in (12.223). Initially, these critical

' (x1016r/s)

4

N/100 N/10

3

N 2 1 0

1

2

3

4

5

6

7

8

9

10

4

5

6

7

8

9

10

0

'' (x1016r/s)

−0.5 −1 −1.5

N

−2

N/10 N/100

−2.5 −3

1

2

3

Fig. 12.52 space–time  -dependence of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPC dielectric n evolution for  1 for a single-resonance Lorentz model p with medium parameters !0 D 4  1016 r/s, ı D 0:218  1016 r/s, and b D 20  1016 r/s with number density N (solid curves). The dashed curves indicate the behavior when the number density is reduced by 10 and the dotted curves when it is reduced by 100. Notice that the value of 1 decreases to unity as the number density is decreased

12.3 The Location of the Saddle Points and the Approximation of the Phase

353

 -values are given by 0 D 1:500 and 1 1:50275 for Brillouin’s choice of the medium parameters. Their values are reduced to 0 1:06066 and 1 1:06105 when the number density is decreased to N=10, and are further reduced to 0

1:00623 and 1 1:00627 when the number density is decreased to N=100. Hence, in the weak dispersion limit as N ! 0, the near and distant saddle point dynamics become increasingly compressed about the luminal space–time point  D 1. Because of this, it is seen best to describe dispersive pulse dynamics in terms of the critical space–time parameters that characterize a specific dispersive medium model, such as SB , 0 , and 1 for a single-resonance Lorentz model dielectric. For a given propagation distance z into the dispersive medium, these space–time parameters then set the appropriate time-scale in which the pulse dynamic may be best described.

12.3.2 Multiple-Resonance Lorentz Model Dielectrics For a double-resonance Lorentz model dielectric with two isolated resonance frequencies, the complex index of refraction is given by [cf. (12.101)]  n.!/ D 1 

b22 b02  2 2 ! 2  !0 C 2iı0 ! ! 2  !2 C 2iı2 ! .1/

1=2 ;

(12.266)

.2/

where it is assumed here that j!˙ j  j!˙ j; see (12.103), (12.104) and (12.108), (12.109) for the branch point locations. Approximate expressions for the distant saddle point SPd˙ dynamics, the near saddle point SP˙ n dynamics, and the middle ˙ , j D 1; 2, dynamics may then be obtained by approximating saddle point SPmj the complex index of refraction in a manner that captures the essential frequency behavior in the appropriate region of the complex !-plane.

12.3.2.1

The Region Above the Upper Resonance Line (j!j  !3 )

q For sufficiently large values of j!j  !3  !1 , where !3 !22 C b22 and !1 q !02 C b02 , the complex index of refraction given in (12.266) may be approximated as [cf. (12.184)] n.!/ 1 

b22 b02  ; 2!.! C 2iı0 / 2!.! C 2iı2 /

(12.267)

with derivative n0 .!/

b02 .! C iı0 / b22 .! C iı2 / C : ! 2 .! C 2iı0 /2 ! 2 .! C 2iı2 /2

(12.268)

354

12 Analysis of the Phase Function and Its Saddle Points

The first-order approximation to the saddle point equation (12.177) is then given by 2.1   /.! C 2iı0 /2 .! C 2iı2 /2 C b02 .! C 2iı2 /2 C b22 .! C 2iı0 /2 0; which may be approximated as 2 2 N 2  b0 C b2 0; .! C 2iı/ 2.  1/

(12.269)

provided that ı0 ı2 , where ıN .ı0 C ı2 /=2. The solution of this equation then gives the first approximate distant saddle point locations [11] s !SP ˙ ./ ˙ d

b02 C b22 N  2iı; 2.  1/

(12.270)

for   1. Based upon the analogy of this result with (12.187) and its corresponding second approximate expression given in (12.201)–(12.203), the second approximation of the distant saddle point locations for a double-resonance Lorentz model dielectric is found to be given by   !SP ˙ . / D ˙./  2iıN 1 C . / ;

(12.271)

d

with s ./

./

.b 2 C b 2 / 2 ; !22  ıN2 C 0 2 2  1

(12.272)

b02 C b22 b02 C b22 D    ; (12.273)  . 2  1/  2 ./ !22  ıN2 . 2  1/ C b02 C b22  2

for all   1. For values of  close to but not less than unity, these two second approximate expressions for ./ and . / may be further approximated as [noting that  2  1 D . C 1/.  1/ 2.  1/] s ./

b02 C b22 ; 2.  1/

. / 1; and this second approximation reduces to the first approximation given in (12.270), with limiting value N lim !SP ˙ ./ D ˙1  2iı: (12.274) !1C

d

12.3 The Location of the Saddle Points and the Approximation of the Phase

355

On the other hand, for sufficiently large values of  ! 1, q ./ ! !32  ıN2 C b02 ; . / ! 0; so that

q .3/ lim !SP ˙ . / D ˙ !32  ıN2 C b02  iıN !˙ ;

!1

(12.275)

d

.3/

where !˙ denotes the outer branch point locations in the left- and right-half planes given in (12.109). Comparison of this second approximate expression for the distant saddle point SPdC locations in the right-half of the complex !-plane with the exact, numerically determined locations [as determined from repeated application of Newton’s method, as described in (12.262) and (12.263)] is provided in Fig. 12.53. Notice that both the real and imaginary parts of the distant saddle point locations are described by this second approximation, as given in (12.271)–(12.273), with sufficient accuracy over the entire space–time domain of interest. In particular, the limiting values described in (12.274) and (12.275) are both realized by the exact solution.

6

' (x1017r/s)

5 4 3 2

Exact Solution

1 0

Second Approximation 1

1.1

1.2

1.3

1.4

1.5

1.4

1.5

'' (x1015r/s)

−2.5 −3

Exact Solution

−3.5

Second Approximation

−4 −4.5 −5 −5.5

1

1.1

1.2

1.3

Fig. 12.53 Comparison of the exact and second approximate  -dependences of the real (upper graph) and imaginary (lower graph) parts of the distant saddle point SPdC evolution for a doublep resonance Lorentz model dielectric with medium parametersp!0 D 1:0  1016 r/s, b0 D 0:6  16 16 16 16 10 r/s, ı0 D 0:1010 r/s, and !2 D 7:010 r/s, b2 D 12:010 r/s, ı2 D 0:281016 r/s

356

12 Analysis of the Phase Function and Its Saddle Points

12.3.2.2

The Region Below the Lower Resonance Line (j!j  !0 )

For sufficiently small values of j!j  !0  !2 , the complex index of refraction given in (12.266) for a double-resonance Lorentz model dielectric may be approximated as    2  b0 ı2 b22 b22 1 i ı0 b02 C 4 !C C 4 !2; (12.276) n.!/ 0 C 0 !04 20 !04 !2 !2 with derivative n0 .!/

i 0



ı0 b02 ı2 b 2 C 42 4 !0 !2

 !C

1 0



b02 b2 C 24 4 !0 !2

 !;

(12.277)

q where 0 n.0/ D 1 C b02 =!02 C b22 =!22 . The first-order approximation of the saddle point equation (12.177) is then found to be given by !04 !24 4 ı0 b02 !24 C ı2 b22 !04 2  !2 C i C .0  / 0; 0 3 b02 !24 C b22 !04 3 b02 !24 C b22 !04

(12.278)

with solution4 "

20 ! 4 ! 4 4  2 4 0 22 4  .  0 /  !SP˙ . / ˙ n 9 3 b0 !2 C b2 !0   2 4 2 ı0 b0 !2 C ı2 b22 !04  :  i 3 b02 !24 C b22 !04



ı0 b02 !24 C ı2 b22 !04 b02 !24 C b22 !04

2 #1=2

(12.279)

This then constitutes the first approximation of the near saddle point locations for a double-resonance Lorentz model dielectric. The critical space–time point  D 1 at which the argument of the square root expression appearing in (12.279) vanishes is given by 2  2 ı0 b02 !24 C ı2 b22 !04  : 1 0 C 30 !04 !24 b02 !24 C b22 !04

(12.280)

axis, symmetrically The two near saddle points SP˙ n then      lie along the imaginary situated about the point ! D 2i ı0 b02 !24 C ı2 b22 !04 = 3 b02 !24 C b22 !04 for  2 Œ1; 1 /, approaching each other as  increases. These two first-order saddle points then coalesce into a single second-order saddle point SPn at the critical space–time point  D 1 , where 4

Notice that this result is somewhat different from (and more accurate than) that given in [11].

12.3 The Location of the Saddle Points and the Approximation of the Phase

!SPn .1 / i

357

 4



2 ı0 b02 !24 C ı2 b22 !0  :  3 b02 !24 C b22 !04

(12.281)

Finally, as  increases above 1 , the two near saddle points SP˙ n separate as they move off of the imaginary axis and into the lower-half of the complex !-plane, ap     proaching ˙1  2i ı0 b02 !24 C ı2 b22 !04 = 3 b02 !24 C b22 !04 as  ! 1. However, when  > 1 becomes sufficiently large such that the inequality j!SP˙ j  !0 is no n longer satisfied, then this first-order approximation is no longer valid. Based upon the analogy of this result with (12.214) and its corresponding second approximate expression given in (12.219)–(12.221), the second approximation of the near saddle point locations for a double-resonance Lorentz model dielectric is found to be given by 2 !SP˙ ./ D ˙ ./  iı0 . / (12.282) n 3 for   1, where 2

 2    02 6  2 . / 4 b  2  02 C 3 !02 C

31=2

!02

0

b22 !02 !24

4 7   ı02 2 . /5 9

; (12.283)

with   b02 ı2 b22 !04 2 2 1 C    C 2 0 3 !02 ı0 b02 !24   :

. /

2 b 2  2   2 C 3 0 1 C b22 !04 2 2 4 0 ! b ! 0

(12.284)

0 2

By construction, this second approximation of the near saddle point locations reduces to the first approximation given in (12.279) for values of  close to 0 . On the other hand, for sufficiently large values of  ! 1, ./ !

. / ! so that

q !02  ı02 ; 3 ; 2

q .0/ lim !SP˙ ./ D ˙ !02  ı02  iı0 D !˙ ;

!1 .0/

n

(12.285)

where !˙ denotes the inner branch point locations in the left- and right-half planes [see (12.103)]. Notice also that this second approximation of the near saddle point

358

12 Analysis of the Phase Function and Its Saddle Points 8

6

'' (x1016r/s)

4

2

Exact Solution

0

Second Approximation

−2

Exact Solution

−4 −6 −8

1

1.1

1.2

1.3

Fig. 12.54 Comparison of the exact (solid curves) and second approximate (dashed curves)  -dependences of the imaginary part of the near saddle point SP˙ n evolution over the space–time domain  2 Œ1; 1 for a double-resonance Lorentz model dielectric with medium parameters p !0 Dp1:0  1016 r/s, b0 D 0:6  1016 r/s, ı0 D 0:10  1016 r/s, and !2 D 7:0  1016 r/s, b2 D 12:0  1016 r/s, ı2 D 0:28  1016 r/s

locations for a double-resonance Lorentz model dielectric reduces to that given in (12.219)–(12.221) for a single-resonance Lorentz model dielectric when b2 D 0. Comparison of this second approximate expression for the near saddle point SP˙ n locations with the exact, numerically determined locations [as determined from repeated application of Newton’s method, described in (12.262)–(12.263)] is presented in Fig. 12.54 for  2 Œ1; 1 when the near saddle points SP˙ n are situated along the imaginary axis, and in Fig. 12.55 for the near saddle point SPC n in the righthalf of the complex !-plane when   1 . Notice that both the real and imaginary parts of the near saddle point locations are described with sufficient accuracy by this second approximation over the entire space–time domain of interest. In particular, the limiting values described in (12.285) is indeed realized by the exact solution.

12.3.2.3

The Region Between the Upper and Lower Resonance Lines (!0 < j!j < !3 )

To analyze the saddle point behavior in the angular frequency region !0 < j!j < !3 between the upper and lower resonance lines in the right-half of the complex !-plane (the behavior in the left-half plane being given by symmetry), consider

12.3 The Location of the Saddle Points and the Approximation of the Phase

359

10

' (x1015 r/s)

Second Approximation 8

Exact Solution 6 4 2 0

1.5

2

2.5

3

3.5

4

4.5

5

'' (x1014 r/s)

−6 −7 −8

Exact Solution −9

Second Approximation −10

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 12.55 Comparison of the exact and second approximate  -dependences of the real (upper graph) and imaginary (lower graph) parts of the near saddle point SPC n evolution over the space– time domain  1 for a double-resonance Lorentz model dielectric with medium parameters p !0 Dp1:0  1016 r/s, b0 D 0:6  1016 r/s, ı0 D 0:10  1016 r/s, and !2 D 7:0  1016 r/s, b2 D 12:0  1016 r/s, ı2 D 0:28  1016 r/s

the change of variable ! D !N s C !; N

(12.286)

where !N s

1 .!0 C !2 / 2

(12.287)

is the mean angular resonance frequency of the double-resonance Lorentz model dielectric, where !N s2  !02 > 0 and !N s2  !22 < 0. With this change of variable, the two resonance terms appearing in the complex index of refraction become ! 2  !02 C 2iı0 ! D !N 2 C 2.!N s C iı0 /!N C !˛2 ; ! 2  !22 C 2iı2 ! D !N 2 C 2.!N s C iı2 /!N C !ˇ2 ; where the two complex-valued quantities !˛2 !N s2  !02 C 2iı0 !N s ;

(12.288)

!ˇ2

(12.289)



!N s2



!22

C 2iı2 !N s :

360

12 Analysis of the Phase Function and Its Saddle Points

have been introduced for notational convenience. With this substitution, the square of the complex index of refraction given in (12.266) for a double-resonance Lorentz model dielectric becomes n2 .!/ N D 1

b02 =!˛2  1 C 2.!N s C iı0 /=!˛2 !N C !N 2 =!˛2 

b22 =!ˇ2   1 C 2.!N s C iı2 /=!ˇ2 !N C !N 2 =!ˇ2 ! # " b02 .!N s C iı0 /2 1 !N s C iı0

1 2 12 !N C 2 4  1 !N 2 !˛ !˛2 !˛ !˛2 ! # " b22 .!N s C iı2 /2 1 !N s C iı2  2 12 !N C 2 4  1 !N 2 !ˇ !ˇ2 !ˇ !ˇ2 ( " # N s C iı0 N s C iı2 2 2 2! 2! D nN s 1 C 2 b0 C b2 !N nN s !˛4 !ˇ4 ! !# ) " b22 .!N s C iı0 /2 .!N s C iı2 /2 1 b02 1 C 4 4  1 !N 2 ; 4  2 nN s !˛4 !˛2 !ˇ !ˇ2 

N < j!ˇ j, where provided that j!j N < j!˛ j and j!j b2 b2 nN s n.!N s / D 1  02  22 !˛ !ˇ

!1=2 :

(12.290)

Notice that nN s is, in general, complex-valued. The complex index of refraction in the intermediate frequency domain between the two resonance lines may then be approximated as " # N s C iı0 N s C iı2 1 2! 2! C b2 b !N n.!/ N nN s C nN s 0 !˛4 !ˇ4 ! ! " b22 .!N s C iı0 /2 .!N s C iı2 /2 1 b02 1 C 4 4 1  4 2nN s !˛4 !˛2 !ˇ !ˇ2 !2 # 1 N s C iı0 N s C iı2 2! 2! C C b2 b0 !N 2 ; nN s !˛4 !ˇ4 (12.291)  3 which is correct to O !N . For notational convenience, this approximate expression may be expressed as ˝1 ˝2 2 n.!/ N nN s C !N  !N ; (12.292) nN s 2nN s

12.3 The Location of the Saddle Points and the Approximation of the Phase

361

with derivative [noting that dn.!/=d! D dn.!/=d N !] N n0 .!/ N

˝1 ˝2 !N  !; N nN s nN s

(12.293)

where !N s C iı0 !N s C iı2 C b22 ; !˛4 !ˇ4 ! b02 b22 .!N s C iı0 /2  1 C ˝2 4 4 !˛ !˛2 !ˇ4

˝1 b02

(12.294) ! ˝12 .!N s C iı2 /2  1 C : 4 nN s !ˇ2 (12.295)

N 0 .!/ N D  for the middle The transformed saddle point equation n.!/ N C .!N s C !/n 5 saddle points in the right-half plane then becomes !N 2 

 2 .2˝1  ˝2 !N s / 2nN s    N0 0; !N C 3˝2 3˝2

where

˝1 N0 nN s C !N s nN s

(12.296)

(12.297)

is a complex-valued space–time value. Notice that, like the critical space–time point 0 D n.0/ for the near saddle points, the complex space–time point is, in part, given by the value of the complex index of refraction at the mean angular frequency !N s where !N D 0. The solution of this transformed saddle point equation, together with the change of variable given in (12.286), then gives the first approximate middle saddle point locations in the right-half of the complex !-plane as 2 2˝1 C .1/j !SP C . / !N s C mj 3 3˝2

"

 .2˝1  ˝2 !N s /2 2nN s    N0  2 3˝2 9˝2

#1=2 ; (12.298)

for j D 1; 2. The middle saddle point locations in the left-half plane are then given  . / D !  by !SPmj C . /. The critical (but complex-valued) space–time value SPmj

 D N1 at which the argument of the square root expression appearing in (12.298) vanishes is given by .2˝1  ˝2 !N s /2 : (12.299) N1 N0 C 6˝2 nN s 5

This result is an extension of that given in [11].

362

12 Analysis of the Phase Function and Its Saddle Points

C C Because N1 is complex-valued in general, the middle saddle points SPm1 and SPm2 do not coalesce into a single second-order saddle point, but rather come into close proximity with each other at the space–time point  D ;  / is satisfied, where SP> denotes the dominant saddle point (or points) and the darker shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP< ;  / is satisfied, where SP< denotes the nondominant saddle point (or points) over the indicated space–time interval

380

12 Analysis of the Phase Function and Its Saddle Points

being connected along the branch cuts. Although this is a perfectly valid deformed contour of integration, it is unnecessarily complicated and places unnecessary importance to the steepest descent path in the resultant asymptotic description. Because of this, it is not used in this analysis, Olver’s method being used instead. In accordance with the method of analysis described in Sect. 10.3.1, the integral I.z; / is expressed as the sum of integrals with the same integrand over the various subpaths, so that for a single-resonance Lorentz model dielectric I.z; / D Id .z; / C InC .z;  / C IdC .z;  /I I.z; / D

Id .z; /

C

In .z;  /

C

InC .z;  /

C

for 1    1 ; IdC .z;  /I

(12.336)

for  > 1 ; (12.337)

where Id˙ .z; / and In˙ .z; / denote the contour integrals taken over the Olver-type paths Pd˙ and Pn˙ , respectively. The same set of relations hold for a doubleresonance Lorentz model dielectric when the inequality p > 0 is satisfied (see Fig. 12.59). However, when the opposite inequality p < 0 is satisfied, then the upper middle saddle points do become the dominant saddle points (see Figs. 12.57 and 12.58) and the above set of relations is modified to read I.z; / D Id .z; / C Im .z;  / C InC .z;  / C ImC .z;  /IdC .z;  /

(12.338)

for 1    1 , and I.z; / D Id .z; / C Im .z; / C In .z;  / C InC .z;  / C ImC .z;  /IdC .z;  / (12.339) for  > 1 , where Im˙ .z; / denote the contour integrals taken over the Olver-type paths Pm˙ , respectively. To obtain an asymptotic approximation of the integral representation of the propagated wavefield A.z; t / in a Lorentz model dielectric, it now only remains to obtain asymptotic approximations of the various contour integrals appearing on the right-hand sides of either (12.336) and (12.337) for a single resonance medium or (12.338) and (12.339) for a double resonance medium. If the distant saddle points SPd˙ do not pass too near to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2), then the results of Sect. 10.2 can be applied to obtain an asymptotic approximation of the quantity Id .z; / C IdC .z; / in the form Id .z; / C IdC .z;  / D As .z; t / C Rd .z;  /;

(12.340)

where As .z; t / is obtained from (10.24) and an estimate of the remainder Rd .z;  / as z ! 1 is given by (10.25). The expression given in (12.340) is uniformly valid for all   1 so long as both of the distant saddle points remain isolated from any poles of either fQ.!/ or uQ .!  !c /. For values of  bounded away from unity from above, (12.340) reduces to the result obtained by application of Olver’s theorem directly to both Id .z; / and IdC .z;  / and adding the results.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

381

˙ If the near saddle points SPC n for 1 <   1 and SPn for  > 1 do not pass too close to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .! !c / for (12.2), then the results of Sect. 10.3.2 can be applied to obtain asymptotic approximations of InC .z;  / for 1 <   1 and In .z;  / C InC .z;  / for  > 1 in the form

In .z; /

C

InC .z; / D Ab .z; t / C Rn .z;  /I

for 1    1 ; (12.341)

InC .z; /

for  > 1 ;

D Ab .z; t / C Rn .z;  /I

(12.342)

where the expression for Ab .z; t / and an estimate of the remainder term Rn .z;  / as z ! 1 are obtained from (10.51). Taken together, (12.341) and (12.342) yield an asymptotic approximation of the quantity I.z;  /  Id .z;  /  IdC .z;  / for a single-resonance Lorentz model dielectric that is valid uniformly for all   1 as Q long as the near saddle points SP˙ n remain isolated from any poles of either f .!/ or uQ .!  !c /. For values of  bounded away from 1 from below, the expression in (12.341) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximation of InC .z;  /. Similarly, for values of  bounded away from 1 from above, the expression in (12.342) reduces to the result that would be obtained by applying Olver’s method directly to obtain the asymptotic approximations of In .z; / and InC .z;  / and summing the results. C , j D 1; 2, of a double-resonance Lorentz model If the middle saddle points SPmj dielectric do not pass too close to any poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2), then the uniform asymptotic method of Sect. 10.3.1 can be directly applied to both Im .z;  / and ImC .z;  / and the results summed to obtain an asymptotic approximation of Im .z;  / C ImC .z;  / in the form (12.343) Im .z; / C ImC .z;  / D Am .z; t / C Rm .z;  /; where  Im .z;  / D Im1 .z;  /;

(12.344)

C ImC .z;  / D Im1 .z;  /;

(12.345)

  Im .z;  / D Im1 .z;  / C Im2 .z;  /;

(12.346)

C C ImC .z;  / D Im1 .z;  / C Im2 .z;  /;

(12.347)

for 1   < p , and

˙ for   p . Notice that each component contour integral Imj .z;  / may be obtained from a direct application of Olver’s method. An estimate of the remainder term Rm .z; / as z ! 1 may be obtained from (10.12). Consider now the situation when either one of the distant saddle points SPd˙ approaches (as  varies) a pole of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) that is located in a region of the complex

382

12 Analysis of the Phase Function and Its Saddle Points

!-plane bounded away from the limiting values at plus or minus infinity in the lower-half of the complex !-plane approached by SPd˙ as  ! 1C [see (12.204) and (12.274)]. The methods of analysis described in Sects. 10.2 and 10.4 can then be applied to obtain an asymptotic approximation of the quantity Id .z;  / C IdC .z;  / in the form Id .z; / C IdC .z; / D As .z; t / C Cd˙ .z; t / C Rd .z;  /;

(12.348)

where As .z; t / is obtained from (10.24), just as for (12.340). Because the pole is bounded away from the infinite limiting values of !SP ˙ . / as  ! 1C , the sadd dle point and pole do not interact for values of  near unity. Hence, the asymptotic approximation of Id .z; /CIdC .z;  / is determined by applying the uniform asymptotic methods of Sect. 10.2, Cd˙ .z; t / is asymptotically negligible, and the expression given in (12.348) reduces to that in (12.340) for values of  near unity. For values of  bounded away from unity from above, the results of Sect. 10.4 are applicable and the right-hand side of (12.348) is obtained from (10.86) of Theorem 5 with As .z; t / being given by the first term and Cd˙ .z; t / being given by the second term. The resulting expression for As .z; t / is then the same as before. Hence, As .z; t / in (12.348) is given by the same expression as is As .z; t / in (12.340) for all   1. Finally, notice that the quantity Cd˙ .z; t / appearing in (12.348) is asymptotically negligible if either of the distant saddle points SPd˙ does not approach a pole of either fQ.!/ or uQ .!  !c /, in which case (12.348) reduces to (12.340). In a similar manner, consider the situation when either of the near saddle points Q SP˙ n approaches (as  varies) a pole of either the spectral function f .!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) with location bounded away from the critical point where the two near first-order saddle points coalesce to form a single second-order saddle point SPn when  D 1 . The methods of analysis presented in Sects. 10.3–10.4 can then be applied to obtain a asymptotic approximations of InC .z; / for 1 <  < 1 and In .z;  / C InC .z;  / for  > 1 in the form InC .z; / D Ab .z; t / C CnC .z; t / C Rn .z;  /;

(12.349)

for 1   < 1 , and In .z; / C InC .z; / D Ab .z; t / C Cn˙ .z; t / C Rn .z;  /;

(12.350)

for  > 1 . In both cases, the expression for Ab .z; t / is the same as that in (12.341) and (12.342). The quantity CnC .z; t / or Cn˙ .z; t / is asymptotically negligible if the corresponding saddle point does not approach a pole; in that case, the expressions in (12.349) and (12.350) reduce to the corresponding expressions in (12.341) and (12.342). ˙ , j D 1; 2, Consider next the situation when any of the middle saddle points SPmj Q approaches (as  varies) a pole of either the spectral function f .!/ for (12.1) or the spectral function uQ .!!c / for (12.2). The method of analysis presented in Sect. 10.4

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

383

can then be applied to obtain an asymptotic approximation of Im .z;  / C ImC .z;  / in the form Im .z; / C ImC .z; / D Am .z; t / C Cm˙ .z; t / C Rm .z;  /;

(12.351)

where Im˙ .z; / is as given in (12.344)–(12.347). The expression for Am .z; t / is the same as that in (12.343). The quantity Cm˙ .z; t / is asymptotically negligible if both C C and SPm2 or in the saddle points in either the pair of middle saddle points SPm1   pair SPm1 and SPm2 do not approach a pole; in that case, the expression in (12.351) reduces to that in (12.343). Combination of (12.333), (12.336)–(12.343), and (12.348)–(12.351) results in the general expression A.z; t / D As .z; t / C Am .z; t / C Ab .z; t / C Ac .z; t / C R.z;  /

(12.352)

for the asymptotic approximation of the integral representation of the propagated wavefield A.z; t / as z ! 1 in a Lorentz model dielectric. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the limiting locations taken by the distant saddle points SPd˙ as  ! 1C and by SPC n as  ! 1 from below. For a single resonance Lorentz model dielectric, the field term Am .z; t / is set equal to zero in (12.352). The contribution Ac .z; t / appearing in (12.352) is obtained by adding all of the terms that involve the poles, namely Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C CnC .z; t /;

(12.353)

for 1    1 , and Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C Cn .z; t / C CnC .z; t /;

(12.354)

for  > 1 . For a single-resonance Lorentz model dielectric, the terms Cm˙ .z; t / are all set equal to zero in these two expressions. An estimate of the remainder term R.z; / as z ! 1 is obtained by taking the largest estimate of the remainder terms appearing in (12.348)–(12.351). An important feature of the general expression given in (12.352) is that the asymptotic behavior of the propagated wavefield A.z; t / in a Lorentz model medium is expressed as the sum of three to four terms which are essentially uncoupled so that they can be treated independently of one another. Each term is determined both by the dynamical behavior of specific saddle points that are a characteristic of the dispersive medium as well as by the analytic behavior of the input pulse spectrum, as described in the paragraphs to follow.

384

12 Analysis of the Phase Function and Its Saddle Points

The dynamic behavior of As .z; t / is determined by the dynamical evolution of the distant saddle points SPd˙ and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial space–time domain feither  2 Œ1; SB / for a single resonance medium or  2 Œ1; SM / for a double-resonance medium that satisfies p < 0 g, the propagated wavefield component As .z; t / describes the dynamical space–time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle points SP˙ n and the value of the input pulse spectrum at these saddle points. Because the near saddle points are dominant immediately following the distant saddle point dominance in a single resonance medium, the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The dynamic behavior of Am .z; t / in a double-resonance Lorentz model dielec˙ and the tric is determined by the dynamic evolution of the middle saddle points SPm1 value of the input pulse spectrum at these saddle points. Because the middle saddle points are dominant (provided that the inequality p < 0 is satisfied) in the space– time domain  2 .SM ; MB / between the first and second precursor dominance, the propagated wavefield component Am .z; t / describes the dynamical space–time behavior of the middle precursor field. This middle precursor field is asymptotically negligible during most of the first and second precursor evolution. If the opposite inequality p > 0 is satisfied, then the middle precursor is asymptotically negligible during the entire field evolution. The dynamic behavior of Ac .z; t / is determined by the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!!c / for (12.2) and the dynamics of the saddle points that interact with them. The wavefield component Ac .z; t / is nonzero only if fQ.!/ or uQ .!  !c / has poles. If the envelope function u.t /, defined in (11.34), of the initial plane wavefield A.0; t / at the plane z D 0 is bounded for all time t , then its spectrum uQ .!  !c / can have poles only if u.t / does not tend to zero too fast as t ! 1.8 Hence, the implication of nonzero Ac .z; t / is that the wavefield A.z; t / oscillates with angular frequency !c for positive time t on the plane z D 0 and will tend to do the same at larger values of z for sufficiently large time t . As a result, the propagated wavefield component Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution. For most values of , only one of the terms As .z; t /, Ab .z; t /, Am .z; t /, and Ac .z; t / appearing in (12.352) is important at a time. There are short space–time intervals in , however, during which two or more of these terms are significant for fixed values of z. These space–time intervals mark the transition periods when the wavefield is changing its character from one form to another and the presence 8

If u.t / is bounded and tends to zero rapidly enough such that the Fourier transform of u.t / converges uniformly for all real !, then uQ .!  !c / is an entire function of complex !.

12.4 Procedure for the Asymptotic Analysis of the Propagated Field

385

of both terms in the expression leads to a continuous transition in the space–time behavior of the propagated wavefield. As a result, (12.352) displays the entire evolution of the field through its various forms in a continuous manner. Analogous results are obtained for Debye model dielectrics and Drude model conductors, as well as for composite models describing semiconducting materials. In particular, (12.355) A.z; t / D Ab .z; t / C Ac .z; t / C R.z;  / for the asymptotic approximation of the integral representation of the propagated wavefield A.z; t / as z ! 1 in a Rocard–Powles–Debye model dielectric. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the origin. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle point SPn (see Figs. 12.25–12.30) and the value of the input pulse spectrum at this saddle point. Because this near saddle point evolution is analogous to the upper near saddle point evolution along the imaginary axis for a Lorentz model dielectric, the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the Brillouin precursor field. This precursor field is asymptotically negligible during most of the signal evolution Ac .z; t / which is determined by the poles of either spectral function fQ.!/ or uQ .!  !c / and the dynamics of the saddle point SPn that interacts with them. As before, if present, Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution. Notice that this same general description also applies to the simple model of a semiconducting medium given by the Debye model with static conductivity (see Figs. 12.63–12.65). The asymptotic approximation of the integral representation of the propagated wavefield A.z; t / in a Drude model conductor is given by A.z; t / D As .z; t / C Ab .z; t / C Ac .z; t / C R.z;  /;

(12.356)

as z ! 1. Notice that this general expression is the same as that for a singleresonance Lorentz model dielectric [cf. (12.352)]. This approximation is uniformly valid for all subluminal space–time points   1 provided that all of the poles of either the spectral function fQ.!/ for (12.1) or the spectral function uQ .!  !c / for (12.2) are bounded away from the origin. The dynamic behavior of As .z; t / is determined by the dynamical evolution of the distant saddle points SPd˙ (see Figs. 12.36–12.38) and the value of the input pulse spectrum at these saddle points. Because the distant saddle points are dominant over the initial space–time domain  2 Œ1; SB /, the propagated wavefield component As .z; t / describes the dynamical space–time behavior of the first or Sommerfeld precursor field. This first precursor field is asymptotically negligible during most of the remaining field evolution. The dynamic behavior of Ab .z; t / is determined by the dynamical evolution of the near saddle point SPn and the value of the input pulse spectrum at it. Because this near

386

12 Analysis of the Phase Function and Its Saddle Points

saddle point is dominant immediately following the distant saddle point dominance (see Fig. 12.62), the propagated wavefield component Ab .z; t / describes the dynamical space–time behavior of the second or Brillouin precursor field. This second precursor field is asymptotically negligible during most of the first precursor and remaining field evolution. The signal contribution Ac .z; t / is determined by the poles of either spectral function fQ.!/ or uQ .!  !c / and the dynamics of the saddle points that interacts with them. If present, Ac .z; t / describes the dynamic behavior of the signal contribution oscillating with angular frequency !c . This contribution to the total field evolution is negligible during most of the precursor field evolution.

12.5 Synopsis The results presented in this rather lengthy chapter are critical to the full understanding of dispersive pulse dynamics. As the propagation distance increases, the observed pulse dynamics are increasingly determined by the dynamics of the saddle points that are a characteristic of the dispersive medium as well as by the analytical behavior of the initial pulse spectrum at them. Each feature in the observed pulse distortion can then be traced back to a specific saddle point behavior, much in the same way as each feature in a scattering process can be traced back to some specific feature in the scattering object. Because of this, specific pulse types can be designed to either strongly interact with a particular set of saddle points (for selective heating, imaging, and remote sensing applications) or to weakly interact with an obscuring medium (for communication and imaging through barriers). On the other hand, specific materials can be designed that weakly interact with specific radar pulses for low-observable (stealth) applications.

References ¨ 1. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914. ¨ 2. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 3. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 4. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 5. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 6. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 7. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: Springer, 1994. 8. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1.

Problems

387

9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media. Oxford: Pergamon, 1960. Ch. IX. 10. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum, 1994. 11. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 12. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum, 1999. 13. E. T. Whittaker and G. N. Watson, Modern Analysis. London: Cambridge University Press, fourth ed., 1963. p. 133. 14. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 15. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, “Superconvergence and sum rules for the optical constants,” Phys. Rev. B, vol. 6, pp. 4502–4509, 1972. 16. H. Xiao and K. E. Oughstun, “Hybrid numerical-asymptotic code for dispersive pulse propagation calculations,” J. Opt. Soc. Am. A, vol. 15, no. 5, pp. 1256–1267, 1998. 17. K. E. Oughstun, J. E. K. Laurens, and C. M. Balictsis, “Asymptotic description of electromagnetic pulse propagation in a linear dispersive medium,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 223–240, New York: Plenum, 1992. 18. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic, 1980. 19. P. Debye, Polar Molecules. New York: Dover, 1929. 20. J. E. K. Laurens, Plane Wave Pulse Propagation in a Linear, Causally Dispersive Polar Medium. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/05-02 (May 1, 1993). 21. M. A. Messier, “A standard ionosphere for the study of electromagnetic pulse propagation,” Tech. Rep. Note 117, Air Force Weapons Laboratory, Albuquerque, NM, 1971. 22. P. Drude, Lehrbuch der Optik. Leipzig: Teubner, 1900. Chap. V. 23. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007. 24. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 25. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Section 6.1.

Problems 12.1. If ı1 D 0, then the imaginary part of the relative dielectric permittivity varies as "i .!/ ı3 ! 3 about the origin, where ı3 is a nonnegative real number. With this term included in (12.18) with ı1 set equal to zero, determine the near saddle point dynamics. 12.2. Derive the approximate expressions given in (12.80) and (12.83) for !mi n in a single-resonance Lorentz model dielectric. 12.3. Derive the approximate expressions given in (12.108) and (12.109) for the .1/ .3/ branch point locations !˙ and !˙ in a double-resonance Lorentz model dielectric.

388

12 Analysis of the Phase Function and Its Saddle Points

12.4. Derive (12.144), (12.146), (12.148), and (12.150). 12.5. Derive (12.168), (12.171), and (12.174) describing the limiting behavior of the complex index of refraction about the branch points !p˙ and !z˙ for a Drude model conductor. Use these expressions to reproduce the limiting behavior depicted in part (a) of Fig. 12.35. 12.6. Derive (12.169), (12.172), and (12.175) describing the limiting behavior of the real part  .!;  / 0. By applying the method Sommerfeld [1,2] used to examine the wavefront evolution of a step-function modulated signal in a causally dispersive medium (the Lorentz medium in particular), it is shown here [3,4] that for wavefields with an initial pulse function f .t / that identically vanishes for all times t < 0, the propagated wavefield is identically zero for all superluminal space–time points  < 1, in complete agreement with the relativistic principle of causality [5]. The remainder of the chapter is devoted to the determination of the evolutionary properties of the precursor fields that, because of their intimate connection to the evolutionary properties of the saddle points, are a charcteristic of the dispersive medium. The analysis follows the now classic approach pioneered by Brillouin [6,7] in his treatment of the Heaviside step-function modulated signal with fixed angular carrier frequency !c > 0 propagating in a single resonance Lorentz medium. That analysis was based upon the then recently developed method of steepest descent (see Sect. F.7 of Appendix F) due to Debye [8]. The analysis presented here is based upon the advanced saddle point methods described in Chap.10. When combined with the more accurate approximations of the saddle point locations and the complex phase behavior at them developed in Chap. 12 for both Lorentz- and Debye-type dielectrics as well as for Drude model conductors and semiconducting materials, accurate asymptotic approximations of the associated precursor fields result that are uniformly valid over the entire space–time domain of interest. If necessary, greatly improved accuracy can always be obtained by using numerically determined saddle point locations in the asymptotic expressions. As a result of this detailed analysis, each feature appearing in the propagated wavefield sequence illustrated in Fig. 13.1 may be traced back to the dynamical behavior of a particular saddle point (or points) together with their interaction with any pole singularity in the initial pulse envelope spectrum. The numerically determined propagated wavefield sequence presented in this figure is due to an initial Heaviside unit step function modulated signal with below resonance carrier frequency !c D !0 =2 at 0, 1, 2, and 3 absorption depths K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 13, 

389

390

13 Evolution of the Precursor Fields

0

z=0

0

A(z,t)

z = zd

0

z = 2z d

0

z = 3z d 0

2

4

6

8

t (fs) Fig. 13.1 Numerically determined propagated wavefield evolution due to an initial Heaviside unit step function modulated signal f .t / D uH .t / sin .!c t / with below resonance carrier frequency !c D !0 =2 at 0, 1, 2, and 3 absorption depths in a single resonance Lorentz model dielectric. Notice that the vertical (wave amplitude) scale of the initial wavefield structure (z D 0) is in units of 1. For the remaining wavefield plots, the vertical scale is in units of 0:5

[zd ˛ 1 .!c /] in a single resonance Lorentz model dielectric. Notice that the steady-state wave structure oscillating at the input angular carrier frequency !c at each propagation distance z has amplitude given by the attenuation factor e z=zd . The complicated field structure preceding this steady-state behavior is then due to the saddle points and is referred to as the first and second precursor fields. Of particular interest here is the observation that the peak amplitude of the second precursor field attenuates with increasing propagation distance z at a significantly smaller rate than does the remainder of the propagated wavefield. This unique feature may then be exploited in both imaging and communications systems. In addition, its impact on health and safety issues concerning exposure to ultrawideband electromagnetic radiation may have far-reaching implications, particularly in regard to digital cellular telephony. In the asymptotic analysis that follows in both this chapter and later chapters, the expression f .z; t /  g.z;  / is used to mean that g.z;  / is an approximation of the dominant term in the asymptotic expansion of f .z; t / as z ! 1 with fixed  ct =z. The reason that g.z;  / is not equal to the dominant term exactly is that approximations of the saddle point locations and the complex phase behavior at them have been used in determining g.z;  /. When the exact (numerically

13.1 The Field Behavior for  < 1

391

determined1 ) saddle point locations and phase behavior are employed so that g.z;  / is equal to the dominant term exactly, then the asymptotic relation is written ˚ f .z; t / D g.z; / C O z3=2 as z ! 1.

13.1 The Field Behavior for  < 1 If the initial pulse function f .t / of the plane wavefield on the plane z D 0 is zero for all time t < 0, then the propagated wavefield A.z; t / can be zero for space–time values  ct =z < 1 only if the wavefront propagates with a velocity greater than the speed of light c in vacuum, in direct violation of the special theory of relativity (or the principle of relativistic causality) [5]. In his 1914 paper, Sommerfeld [2] proved that for a Heaviside step-function envelope signal f .t / D uH .t / sin .!c t / where [see (11.55)] uH .t / D 0 for t < 0 and uH .t / D 1 for t > 0, the propagated wavefield in a Lorentz model dielectric is identically zero for all superluminal space–time points  < 1. The extension of this proof to a broad class of pulse functions f .t / that vanish for all time t < 0 on the plane z D 0 in a linear, causally dispersive medium is now presented [3, 4]. The proof begins with the exact integral representation of the propagated plane wavefield given in (12.1), viz., A.z; t / D

1 2

Z

fQ.!/e .z=c/.!;/ d!;

(13.1)

C

where A.0; t / D f .t /. Here C denotes the Bromwich contour ! D ! 0 C i a where ! 0 0. In addition, if its derivative df .t /=dt is bounded for all t , then integration of the integral in (13.2) by parts shows that the magnitude of uQ .!/ tends to zero uniformly with respect to the angle arg .˝/ for 0    as j˝j ! 1, where ˝ D !  i a with a > 0. 1

Because exact analytic solutions for the saddle point locations are rarely, if ever, available, precise numerical solutions have to suffice.

392

13 Evolution of the Precursor Fields

Because the spectral function fQ.!/ satisfies the above conditions, it is now possible to express A.z; t / by the integral representation given in (13.1) with the change that the integration is now taken over the closed contour that encircles the region ! 00 > a > 0 of the complex !-plane. All that is required is to show that I.z; ; j˝j/ ! 0 uniformly with respect to both z and  for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and " such that 0 < " < 1, where Z

fQ.!/e .z=c/.!;/ d!;

I.z; ; j˝j/

(13.3)

C˝

  with C˝ denoting the semicircular contour !  i a D j˝je i with 0  for fixed j˝j. The proof makes use of the proof of Jordan’s lemma [9]. It directly follows from (13.3) that the inequality Z

ˇ ˇ ˇfQ.!/ˇe .z=c/.!;/ d j!j

jI.z; ; j˝j/j 

(13.4)

C˝

is satisfied, where  .!;  / ˝0 . Hence, for j˝j > ˝0 and 0   ,  .!;  /  ! 00 .nr .!/  / :

(13.6)

Furthermore, for any lossy medium there exists a positive constant ˝1 such that nr .!/  1 for j˝j > ˝1 ; notice that for Debye-model dielectrics, this inequality p may be refined to nr .!/  1  1 [see the discussion following (12.34)]. Henceforth, ˝0 is chosen to be larger than ˝1 . It then follows from (13.6) that  .!;  /  ! 00 .1  /

(13.7)

 . Consequently, for   1  " with 0 < " < 1, for j˝j > ˝0 with 0  combination of the inequalities in (13.4) and (13.7) yields Z jI.z; ; j˝j/j 

ˇ ˇ ˇfQ.!/ˇe .z=c/"! 00 d j!j

(13.8)

C˝

for j˝j > ˝0 . From here on, the proof follows exactly the proof of Jordan’s lemma as given by Whittaker and Watson [10]. Hence, I.z; ; j˝j/ ! 0 uniformly with respect to both z and  for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and " such that 0 < " < 1. Consequently, that semicircular contour integral can be

13.2 The Sommerfeld Precursor Field

393

added to the integral in (13.1) in order to express A.z; t / as an integral over a closed contour that encircles the region ! 00 > a of the complex !-plane, viz., 1 A.z; t / D 2

I

fQ.!/e .z=c/.!;/ d!;

(13.9)

C CC˝

for z  Z > 0 and   1  ". Because the integrand in this integral is a regular analytic function of complex ! for ! 00 > a > 0, it then follows from Cauchy’s residue theorem [11,12] that this integral is identically zero for z  Z and   1  " as j˝j ! 1 for arbitrary Z > 0 and arbitrarily small " > 0. This then proves the following generalized form [3, 4] of the theorem due to Sommerfeld [2]: Theorem 6. Sommerfeld’s Relativistic Causality Theorem. If the initial time behavior A.0; t / D f .t / of the plane wavefield at the plane at z D 0 is zero for all time t < 0 and if the model of the linear material dispersion is causal, then the propa1=2 gated wavefield identically vanishes for all t < "1 z=c with z > 0, where "1  1 denotes the high-frequency limit of the relative complex dielectric permittivity of the material. This fundamental theorem can then be applied to any portion of a pulse in order to prove that neither energy nor information can move forward in the pulse body at a superluminal rate, as has essentially been done by Landauer [13] and Diener [14]. In spite of this, the debate concerning superluminal pulse velocities persists (see Sect. 15.12).

13.2 The Sommerfeld Precursor Field The symmetric contributions of the two distant saddle points to the asymptotic behavior of the propagated wavefield A.z; t / for sufficiently large values of the propagation distance z > 0 yield the dynamical space–time evolution of the first or Sommerfeld precursor field. This contribution to the asymptotic behavior of the total wavefield A.z; t / is denoted by As .z; t / and is dominant over the second or Brillouin precursor field in single resonance Lorentz model dielectrics (as well as in double resonance Lorentz model dielectrics that satisfy the inequality p > 0 ) and Drude model conductors over the initial space–time domain  2 Œ1; SB /, whereas it is dominant over both the middle and Brillouin precursors over the space–time domain  2 Œ1; SM / in double resonance Lorentz model dielectrics when the inequality p < 0 is satisfied. Because the pair of first-order distant saddle points SPd˙ remain isolated from each other and do not change their order throughout their evolution, a straightforward application of Olver’s method (see Sect. 10.1) is applied in Sect. 13.2.1 to obtain the asymptotic behavior of the Sommerfeld precursor evolution for  > 1. However, as the space–time parameter is allowed to approach unity from above, these two distant saddle points approach infinity, condition 2 of Olver’s theorem [15] (Theorem 2) is no longer satisfied and Olver’s

394

13 Evolution of the Precursor Fields

method breaks down. To obtain a valid description of the initial behavior of the first precursor field (the wavefront) for values of  in a neighborhood of unity, the uniform asymptotic expansion due to Handelsman and Bleistein [16] given in Theorem 3 is then applied in Sect. 13.2.2. This uniform expansion is valid for all   1 and reduces to the nonuniform result obtained using Olver’s method for all  > 1 bounded away from unity.

13.2.1 The Nonuniform Approximation The asymptotic behavior of the first or Sommerfeld precursor field As .z; t / as z ! 1 for a given initial pulse envelope function u.t / is obtained from the asymptotic expansion of the integral representation of the propagated wavefield [see (11.48)]

Z 1 < i e i uQ .!  !c /e .z=c/.!;/ d! (13.10) A.z; t / D 2 C about the two distant saddle points SPd˙ . The more general integral representation given in (11.45) for the propagated wavefield due to the intial pulse A.0; t / D f .t / at the plane˚z D 0 may be directly obtained from (13.10) with the identification that fQ.!/ D < i e i uQ .!  !c / , the integral representation given in (13.10) resulting when the initial pulse function is given by f .t / D u.t / sin .!c t C /. Because most of the pulse types considered here are expressed in this envelope-modulated carrier wave form, this form of the integral representation is explicitly considered here, the other (more general) case then being obtained through the above identification. The distant saddle point locations may be expressed as !SP ˙ . / D ˙./  i ı .1 C . //

(13.11)

d

for both Lorentz model dielectrics and Drude model conductors. The second approximations for the functions ./ and . / are respectively given by (12.202) and (12.203) for a single resonance medium and by (12.272) and (12.273) for a double resonance medium with ı D 2ıN D ı0 C ı2 . For a Drude model conductor, ./ is given by (12.310) and ./ is given by (12.311) with ı D =2. For each of the examples considered in this text, both the spectral function uQ .!  !c / and the complex phase function .!;  / appearing in the integrand of (13.10) are analytic about these two distant saddle points for all   1. For a single resonance Lorentz model dielectric, as well as for a double resonance Lorentz model dielectric that satisfies the condition p > 0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P ./ D Pd . / C PnC ./ C PdC . / for  2 .1; 1 and then to the path P ./ D Pd . / C Pn . / C PnC . / C PdC . / for  > 1 , where Pd . / is an Olver-type path with respect to the distant saddle

13.2 The Sommerfeld Precursor Field

395

point SPd and PdC . / is an Olver-type path with respect to the distant saddle point SPdC (see Fig. 12.66). For a double resonance Lorentz model dielectric satisfying the condition p < 0 , the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P . / D Pd . / C PnC ./ C PdC ./ for  2 .1; SM , to the path C  . / C PnC ./ C Pm1 ./ C PdC ./ for  2 ŒSM ; p /, to the P . / D Pd . / C Pm1 C C   . / C Pm2 ./ C PnC ./ C Pm1 ./ C Pm2 ./ C PdC . / path P . / D Pd . / C Pm1   . / C Pm2 . / C for  2 .p ; 1 , and then to the path P ./ D Pd ./ C Pm1 C C C   C Pn . / C Pn . / C Pm1 . / C Pm2 ./ C Pd ./ for  > 1 , where Pd . / is an Olver-type path with respect to the distant saddle point SPd and PdC . / is an Olver-type path with respect to the distant saddle point SPdC . For a Drude model conductor, the conditions of Olver’s theorem are satisfied at the two distant saddle points SPd˙ when the original contour of integration C is deformed to the path P . / D Pd . / C Pn . / C PdC ./ for all  > 1, where Pd ./ is an Olver-type path with respect to the distant saddle point SPd and PdC ./ is an Olver-type path with respect to the distant saddle point SPdC . In each case, (10.18) applies for each of the two distant saddle points with (10.3) and (10.4) taken as Taylor series expansions about these saddle points. Because SPd˙ are first-order saddle points,  D 2, and because uQ .!  !c / is regular at these two saddle points,  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the two Olver-type paths Pd˙ ./ yields the first or Sommerfeld precursor field As .z; t /, which is given by [17–19]  As .z; t / D

c z

(

1=2 < ie

i

a0 .!SP C /e

.z=c/.!

C ;/ SPd

d

Ca0 .!SPd /e

  1 C O z1

.z=c/.!SP  ;/ d

  1 C O z1

)

(13.12) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. To evaluate the pair of coefficients a0 .!SP ˙ / D a0 .!SP ˙ . // appearing in d d (13.12), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  /, as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about the distant saddle points SPd˙ must first be determined. The latter quantity is given by   q0 .!SP ˙ .// D uQ !SP ˙ ./  !c ; d

(13.13)

d

the specific form of which depends upon the particular initial pulse envelope function u.t /. With  D 2 and  D 1 and the observation that the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by

396

13 Evolution of the Precursor Fields

p0 .!SP ˙ ; / D  00 .!SP ˙ ; /=2Š, the coefficients a0 .!SP ˙ ;  / appearing in the d d d asymptotic expansion (13.12) are found to be given by [see (10.9)]   uQ !SP ˙ ./  !c d a0 .!SP ˙ ;  / D h i1=2 : d 2 00 .!SP ˙ ;  /

(13.14)

d

With this substitution, the nonuniform asymptotic expansion (13.12) of the Sommerfeld precursor becomes ("

As .z; t /  < i e

i

#1=2  .z=c/.!SP C ;/  c=z d uQ !SP C . /  !c e  00 d 2 .!SP C ;  / d " ) #1=2  .z=c/.!SP  ;/  c=z d C  uQ !SPd ./  !c e 2 00 .!SPd ;  / (13.15)

as z ! 1 uniformly for all   1C" with arbitrarily small " > 0. Although numerically determined distant saddle point positions (as a function of  ) may always be used in the exact expressions for the complex pahse function and its second derivative appearing in this equation in order to obtain the precise asymptotic behavior of the Sommerfeld precursor for a given input pulse, approximate analytic expressions of these quantities are useful in their own right. This approximate analysis of the complex phase behavior at the distant saddle points is now treated separately for the single and double resonance cases.

13.2.1.1

The Single Resonance Case

From (12.184) for the approximate behavior of the complex index of refraction in a single resonance Lorentz model dielectric, the approximate behavior of the complex phase function .!;  / i !.n.!/  / in the region j!j > !1 of the complex !-plane above the absorption band is given by .!;  / i !.1  /  i

b2 : 2.! C 2i ı/

The same approximate expression applies to a Drude model conductor [see (12.308)] with b D !p , and ı D =2. Differentiation of this approximate expression twice with respect to ! then yields

13.2 The Sommerfeld Precursor Field

397

 0 .!/ i.1  / C i  00 .!/ i

b2 ; 2.! C 2i ı/2

b2 : .! C 2i ı/3

Thus, at ! D !SP ˙ . / with !SP ˙ ./ given by (13.11), one obtains d

d

"   .!SP ˙ ; / ı 1 C . / .  1/ C

 2   # b =2 1  . /  2  2 . / C ı 2 1  . / " # b 2 =2

i./   1 C  2 ; (13.16)  2 ./ C ı 2 1  . /

d

and

b2  00 .!SP ˙ ; / i

 3 : d ˙./ C i ı 1  . /

(13.17)

The second coefficient in the Taylor series expansion (10.3) of the complex phase function .!;  / is then given by p0 .!SP ˙ ; / d

 00 .!SP ˙ ;  / d

2Š

b 2 =2

i

 3 : ˙./ C i ı 1  . /

(13.18)

  The proper value of the quantity ˛N 0˙ arg  p0 .!SP ˙ ;  / must now be d determined according to the convention defined in Olver’s method [see (10.7)]. C For simplicity, the Olver-type path Pd ./ through the distant saddle point SPdC in the right-half of the complex !-plane is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.2. From (12.208),

'

+

SPd

_

+

Pd ( )

Fig. 13.2 Illustration of the steepest descent choice of the Olver-type path PdC . / through the distant saddle point SPdC . The shaded area in the figure indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / <  .!SP C ;  / is satisfied d

398

13 Evolution of the Precursor Fields

the angle of slope of the contour at the saddle point is then given by ˛N C D =4. Therefore, because  arg .z/ D 0, the proper value of ˛N 0C , as determined by the inequality in (10.7), is ˛N0C ' =2. Through a similar argument, the proper value of ˛N 0 arg p0 .!SPd ; / is ˛N 0 ' =2. With these approximate results, the coefficients a0 .!SP ˙ ;  / given in (13.14) for d the asymptotic expansion (13.12) are found to be given by

 3 1=2 1

˙ ./ C i ı 1  . / d 2i b 2    i  3=2  1=2 1  3  4  . / ˙ ıi 1  . /  ./ ;

p uQ !SP ˙ . /  !c e d 2 2b (13.19)

  a0 .!SP ˙ ; / uQ !SP ˙ . /  !c d



where the facts that 0 < 1  . / < 1 and ./  ı for all  2 .1; 1 have been employed in the expansion of the square root expression. With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a single resonance Lorentz model dielectric as [3, 4, 17] s

   z  .b 2 =2/.1  . // c./ exp ı 1 C . / .  1/ C 2 2z c  . / C ı 2 .1  . //2 (      3  < i e i uQ !SP C ./  !c ./ C ıi 1  . / d 2  

 z b 2 =2   exp i ./   1 C 2 C c  ./ C ı 2 .1  . //2 4       3 CQu !SPd . /  !c ./  ıi 1  . / 2 

)   b 2 =2  z ./   1 C 2 C  exp i ; c  ./ C ı 2 .1  . //2 4

1 As .z; t /  b

(13.20) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. This expression also describes the asymptotic behavior of the Sommerfeld precursor in a Drude model conductor with b D !p and ı D =2.

13.2.1.2

The Double Resonance Case

From (12.267) for the approximate behavior of the complex index of refraction of a double resonance Lorentz model dielectric, the approximate behavior of the

13.2 The Sommerfeld Precursor Field

399

complex phase function .!; / in the region j!j > !3 of the complex !-plane above the upper absorption band is given by .!;  / i !.1  /  i

b22 b02 i : 2.! C 2i ı0 / 2.! C 2i ı2 /

Differentiation of this expression twice with respect to ! then yields b02 b22 C i ; 2.! C 2i ı0 /2 2.! C 2i ı2 /2 b02 b22  00 .!/ i  i : .! C 2i ı0 /3 .! C 2i ı2 /3  0 .!/ i.1   / C i

Thus, at ! D !SP ˙ . / with !SP ˙ ./ given by (13.11) with ı D ı0 Cı2 , one obtains d

d

"   .!SP ˙ ; / .ı0 C ı2 / 1 C . / .  1/ 

1 2

 2  b0 C b22 . /

#

 2 ./ C .ı0 C ı2 /2 2 . / #   1 b02 C b22 2

i. /   1 C ; (13.21)  2 ./ C .ı0 C ı2 /2 2 ./

d

"

and  00 .!SP ˙ ; / i d

b02 C b22 Œ˙./  i.ı0 C ı2 /. / 3

:

(13.22)

The second coefficient in the Taylor series expansion (10.3) of the complex phase function .!;  / is then given by p0 .!SP ˙ ; /

 00 .!SP ˙ ;  / d

2Š

d

i

1 2

 2  b0 C b22

Œ˙./  i.ı0 C ı2 /. / 3

:

(13.23)

  As in the single resonance case, the proper value of ˛N 0˙ arg p0 .!SP ˙ ;  / , as d

determined by the inequality in (10.7), is ˛N 0˙ ' ˙=2. With these approximate results, the coefficients a0 .!SP ˙ ;  / given in (13.14) for d the asymptotic expansion (13.12) are found to be given by  

uQ !SP ˙ . /  !c 1

3 1=2 d ˙ ./  i.ı0 C ı2 /. / a0 .!SP ˙ ; /

q d 2i b02 C b22     uQ !SP ˙ . /  !c 3 i 4 3=2 1=2 d  ./ i.ı0 C ı2 /. / ./ :

q e 2 2.b 2 C b 2 / 0

2

(13.24)

400

13 Evolution of the Precursor Fields

With these substitutions in the asymptotic expansion (13.12), one obtains the nonuniform asymptotic approximation of the Sommerfeld precursor field in a double resonance Lorentz model dielectric as s c./ As .z; t /  2.b02 C b22 /z ( " #) 1 2 2  .b C b /./ z  0 2  exp .ı0 C ı2 / 1 C . / .  1/  2 2 c  ./ C .ı0 C ı2 /2 2 . / (     3  < i e i uQ !SP C ./  !c ./  i.ı0 C ı2 /. / d 2 ( " !# ) 1 2 2 .b C b / z  0 2 2  exp i ./   1 C 2 C c  ./ C .ı0 C ı2 /2 2 . / 4     3 CQu !SPd . /  !c ./ C i.ı0 C ı2 /. / 2 ( " !# )) 1 2 .b C b22 / z  2 0 ./   1 C 2  exp i C ; c  ./ C .ı0 C ı2 /2 2 . / 4 (13.25) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0.

13.2.2 The Uniform Approximation As the space–time parameter  ct=z is allowed to approach unity from above, the asymptotic approximations given in (13.20) and (13.25) lose their validity and each must be replaced by the uniform asymptotic representation presented in Theorem 3 of Sect. 10.2. In their 1969 paper [16], Handelsman and Bleistein performed the required analysis for the step-function envelope signal using the first approximation of the distant saddle point locations, obtaining a result derived by Sommerfeld [2] in 1914. As Handelsman and Bleistein pointed out in this paper, their result is not truly uniform because the first approximation of the distant saddle point locations is useful only for very small, positive values of the quantity   1 so that their asymptotic approximation of the Sommerfeld precursor As .z; t / is valid only for space–time values  in the vicinity of  D 1. As shown by Oughstun and Sherman [4, 20] in 1989, when the second approximation of the distant saddle point locations is used, a uniform asymptotic approximation of the first precursor field As .z; t / is obtained that is valid uniformly for all   1. The results of this modern asymptotic theory are now presented. From Theorem 3 (due to Handelsman and Bleistein) and the results of Sect. 12.4, the uniform asymptotic expansion of the contour integral appearing in the integral

13.2 The Sommerfeld Precursor Field

401

representation (13.10) taken over the two Olver-type paths PdC ./ and Pd . / through the distant saddle points SPd and SPdC , respectively, results in a uniform asymptotic expansion of the first or Sommerfeld precursor field that is given by [17, 20] ( z

As .z; t / D < e i c ˇ./ e i h

 0 J

z c

h i   2˛./e i 2 

˛. / C 2˛./e

i 2

1 JC1

z c

i ˛. /

) C R1 .z;  /; (13.26)

as z ! 1 for all   1. The remainder term R1 .z;  / is bounded by the inequality given in (10.25) of Theorem 3 for all   1 with the constant K independent of  , (13.26) then providing the dominant term in the asymptotic expansion of the first precursor field that is uniformly valid for all   1. The real parameter  which sets the order of the Bessel functions J . / and JC1 . / appearing in the uniform expansion (13.26) is defined by (10.21). Finally, the coefficients appearing in (13.26) are defined as [from (10.26)–(10.29)]  i .!SP C ;  /  .!SPd ;  / ; d 2  i ˇ./ .!SP C ;  / C .!SPd ;  / ; d 2 ˛. /

(13.27) (13.28)

and "

#1=2 ˛ 3 ./ i  00 .!SP C ;  / d " #1=2 uQ .!SPd  !c / ˛ 3 . / C

; .1C/  00 i  .!SP C ;  /  2˛./ d " ( #1=2 3 u Q .! C  !c / 1 ˛ ./ SPd 1 . /

.1C/ 00 2˛./ i  .!SP C ;  / 2˛./ d " #1=2 ) uQ .!SPd  !c / ˛ 3 ./ 

; .1C/  00 i  .!SP C ;  /  2˛./

uQ .!SP C  !c / d 0 . /

.1C/ 2˛./

(13.29)

(13.30)

d

h i1=2 where the branch of the square root expression ˙˛ 3 . /=i  00 .!SP ˙ ;  / appeard ing in (13.29) and (13.30) is determined by the limiting relation given in (10.32) of Theorem 3. Explicit expressions for these coefficients in the single and double resonance cases are now given.

402

13 Evolution of the Precursor Fields

13.2.2.1

The Single Resonance Case

From the second approximate expressions given in (13.16) and (13.17) for the complex phase behavior at the distant saddle points in a single resonance Lorentz model dielectric (as well as for a Drude model conductor), one obtains the approximate expressions [20]   b 2 =2 ; ˛. / ./   1 C  2 ./ C ı 2 .1  . //2 "  2   #   b =2 1  . / ˇ./ i ı 1 C . / .  1/ C  2 ;  2 ./ C ı 2 1  . /

(13.31) (13.32)

and   . 12  / b 2 =2  1=2 . / 0 . / .1C/ ./   1 C 2 b  2 ./ C ı 2 .1  . //2 (   3  uQ .!SP C  !c / ./ C i ı.1  . // d 2 )  3 C.1/.1C/ uQ .!SPd  !c / ./  i ı.1  . // ; 2   . 12 C / b 2 =2  1=2 . / 1 . / .2C/ ./   1 C 2 b  2 ./ C ı 2 .1  . //2 (   3  uQ .!SP C  !c / ./ C i ı.1  . // d 2 )  3 .1C/ .1/ uQ .!SPd  !c / ./  i ı.1  . // : 2

(13.33)

(13.34)

h i1=2 The branch of the square root expression ˙˛ 3 ./=i  00 .!SP ˙ ;  / appearing in d (13.29) and (13.30) has been determined by the limiting relation given in (10.32) in the following manner. In accordance with Theorem 3, the argument of the quantity ˙i  00 .!SP ˙ ; / b 2 = Œ./ ˙ i ı.1  . // 3 as determined by the inequality d

given in (10.7) with  arg .i z/ D =2 and ˛N C D =4 is approximately zero [because ./  ı.1  . // for all   1]. Then, according to (10.32), arg .˛. // D 23 arg .a1 / in the limit as  approaches unity from above, where [from the Laurent series expansion given in (10.20) applied to (12.184)], a1 D b 2 =2, which is real and positive. Hence, arg .a1 / D 0 so that lim!1C farg .˛. //g D 0. By continuity, this result is approximately valid for all   1. This branch

13.2 The Sommerfeld Precursor Field

403

requirement, together with the inequality ./  ı.1  . // for all   1, has been used in obtaining the approximations given in (13.33) and (13.34) for the coefficients 0 . / and 1 . /, respectively. Substitution of these second approximate expressions for the coefficients ˛. /, ˇ./, 0 . /, and 1 . / into the uniform asymptotic expansion of the Sommerfeld precursor given in (13.26) then yields [4, 17, 20]  1=2 b 2 =2 ./  1C As .z; t /  2b  2 ./ C ı 2 .1  . //2 ( "  #)  2   b =2 1  . / z   exp ı 1 C . / .  1/ C  2 c  2 . / C ı 2 1  . / ((   3 i . 2 C /  1 bounded away from unity.

13.2.2.2

The Double Resonance Case

From the second approximate expressions given in (13.21) and (13.22) for the complex phase behavior at the distant saddle points in a double resonance Lorentz model dielectric, one obtains the approximate expressions

404

13 Evolution of the Precursor Fields

" ˛. / ./   1 C "

1 2 .b 2 0

 2 ./

#

C b22 / ; C .ı0 C ı2 /2 2 ./

  ˇ./ i.ı0 C ı2 / 1 C . / .  1/ 

(13.36)

# C b22 /./ ; (13.37)  2 ./ C .ı0 C ı2 /2 2 . / 1 2 .b 2 0

and " !#. 12  / 1 2 .b C b22 /  1=2 . / 2 0 ./   1 C 2 0 . /

q  ./ C .ı0 C ı2 /2 2 . / 2.1C/ b02 C b22 (   3  uQ .!SP C  !c / ./  i.ı0 C ı2 /./ d 2  ) 3 C.1/.1C/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /./ ; (13.38) 2 !#. 12 C / " 1 2 .b C b22 /  1=2 . / 2 0 1 . /

./   1 C 2 q  ./ C .ı02 C ı22 /2 ./ 2.2C/ b02 C b22 (   3  uQ .!SP C  !c / ./  i.ı0 C ı2 /./ d 2  ) 3 .1C/ .1/ uQ .!SPd  !c / ./ C i.ı0 C ı2 /. / ; (13.39) 2 where all branch choices are determined in the same manner as in the single resonance case. With these substitutions, the uniform asymptotic expansion of the Sommerfeld precursor given in (13.26) yields " #1=2 1 2 .b C b22 / ./ 2 0  1C 2 As .z; t /  q  ./ C .ı0 C ı2 /2 2 ./ 2 b02 C b22 ( " #) 1 2  .b C b22 /./ z  2 0  exp ı 1 C . / .  1/ C 2 c  ./ C .ı0 C ı2 /2 2 ./ ((   3   1 bounded away from unity.

13.2.3 Field Behavior at the Wavefront By Sommerfeld’s relativistic causality theorem (Theorem 6 of Sect. 13.1), the propagated plane wavefield due to any initial pulse f .t / at the plane z D 0 that identically vanishes for all t < 0 will identically vanish for all superluminal space– time points  ct =z < 1 for all z > 0. If the initial pulse f .t / is then abruptly turned on at time t D 0, the propagated wavefront arrival then occurs at the luminal space–time point  D 1. To investigate the propagated wavefield behavior at this point, attention is now given to the limiting behavior of the uniform asymptotic approximation [as given in either (13.35) for the single resonance case or in (13.40) for the double resonance case] of the Sommerfeld precursor field As .z; t / as  approaches unity from above. In this limit, the functions ./ and . / attain the limiting forms [see (12.202) and (12.203) for the single resonance case and (12.272) and (12.273) for the double resonance case] b lim ./ D p ; C !1 2.  1/ lim . / D 1;

!1C

q where b D b02 C b22 for the double resonance case. In this limit, the argument of the Bessel functions J . / and JC1 . / appearing in the uniform asymptotic expansion of the Sommerfeld precursor becomes  lim

!1C

  b 2 =2 z z p ./   1 C D b 2.  1/: 2 c c  2 ./ C ı 2 .1  . //

406

13 Evolution of the Precursor Fields

Consequently, for subluminal space–time values  very close to unity, the argument of the Bessel functions appearing in the uniform asymptotic expansion of the Sommerfeld precursor is sufficiently small that the small argument limiting form of these Bessel functions may be employed, where (for integer values of the order ) [21] J . / 

 1 

2 ;  . C 1/

as ! 0 with  fixed and nonnegative. For negative values of the order , the relation J . / D .1/ J . / may be employed in conjunction with the above asymptotic expression to obtain  1 

2 ; J . /  .1/  . C 1/ 

as ! 0 with  fixed and nonnegative. For integer   0, substitution of the above results into either (13.25) for the single resonance case or (13.40) for the double resonance case yields the limiting behavior lim As .z; t / 

!1C

z b e 2ı c .1/ p 4  1 ( h i  < e i . 2 C / uQ .!SP C  !c / C .1/1C/ uQ .!SPd  !c / d

 p  bz 1 2.  1/   . C 1/ 2c h i C uQ .!SP C  !c /  .1/1C/ uQ .!SPd  !c / d  p C1 ) i 2 bz e 2.  1/ (13.41)   . C 1/ 2c as z ! 1. Because the initial envelope function u.t / is real-valued, its spectrum satisfies the symmetry property uQ .!/ D uQ  .!  /, and because 0. The same result is obtained from the nonuniform asymptotic expression in (13.20) with substitution from (13.60). The dynamical evolution with  D ct =z of the first, or Sommerfeld, precursor field AH s .z; t / for the Heaviside unit step function envelope modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s is illustrated in Figs. 13.4 and 13.5 at a fixed observation distance z D zd of one absorption depth, where zd ˛ 1 .!c /, the field evolution illustrated in Fig. 13.5 being a close-up view of the wave form immediately following the wavefront. This temporal field evolution was computed using the uniform asymptotic approximation given in (13.61) for a single resonance Lorentz model dielectric with numerically

414

13 Evolution of the Precursor Fields 0.003

0.002

AHs(z,t)

0.001

0

−0.001

−0.002

−0.003

1

1.1

1.2

Fig. 13.4 Temporal evoution of the Sommerfeld precursor field AH s .z; t / at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s 0.003

0.002

AHs(z,t)

0.001

0

−0.001

−0.002

−0.003

1

1.001

1.002

Fig. 13.5 Close-up of the leading edge of the Sommerfeld precursor field AH s .z; t / depicted in Fig. 13.4. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

13.2 The Sommerfeld Precursor Field

415

determined distant saddle point locations using Brillouin’s choice of the medium parameters [see (12.58)]. The accuracy of these results is remarkably good when compared with purely numerical results and only improve as the propagation distance z increases. The rapid amplitude build-up of the first precursor field envelope from its zero value on the wavefront at  D 1 to a maximum value is clearly evident in both figures. For larger values of  , the amplitude damps out exponentially with increasing  . The instantaneous angular frequency of oscillation is also seen to rapidly decrease as  increases away from unity, this decrease becoming less rapid as  continues to increase, as seen in Fig. 13.4. Finally, as the propagation distance z increases away from the initial plane at z D 0, the peak amplitude of the Sommerfeld precursor diminishes and shifts to earlier space–time points, approaching the wavefront at  D 1 as z ! 1. Finally, notice that the relatively small peak amplitude of the Sommerfeld precursor field depicted in Figs. 13.4 and 13.5, as well as in Fig. 13.1, is due to the fact that the input carrier frequency !c of the signal is below the medium resonance frequency !0 . As !c is increased above !0 , the spectral amplitude uQ .!  !c / increases there and so the peak amplitude of the Sommerfeld precursor also increases relative to the remainder of the propagated wavefield. For example, at one absorption depth in the same medium with above resonance carrier frequency !c D 7  1016 r=s, the peak amplitude of the Sommerfeld precursor is found to be ŒAH s .z; t / peak 0:24. As pointed out earlier, the nonuniform asymptotic expression given in (13.62) is not a valid asymptotic approximation of the first precursor field in the limit as  ! 1C for fixed values of the propagation distance z > 0. To establish connection with the now classical result obtained by Brillouin [6, 7] for the first precursor field, however, the behavior of this expression in that limit is now examined. For space– time values  approximately equal to but greater than unity, (13.62) simplifies to s

3=4 2ı z .1/ bc  2.  1/ e c 2z (" b .2.  1//1=2  !c   2 b .2.  1//1=2  !c C 4ı 2

AH s .z; t /  

#  zp  b .2.  1//1=2 C !c 2.  1/ C  cos b  2 c 4 b .2.  1//1=2 C !c C 4ı 2 " 1 C2ı  2 1=2 b .2.  1//  !c C 4ı 2 # )  zp 1   2.  1/ C sin b 2 c 4 b .2.  1//1=2 C !c C 4ı 2 (13.63)

as z ! 1 with subluminal  1. This same result would be obtained using the first approximate expressions of the distant saddle point locations [see (12.187)]

416

13 Evolution of the Precursor Fields

in the nonuniform asymptotic approximation given in (13.20). This expression can be further simplified by noting that for space–time values  very close to unity, any finite angular carrier frequency !c > 0 will be negligible in comparison to the p quantity b= 2.  1/. Hence, in the limit as  ! 1C , the nonuniform asymptotic approximation given in (13.63) simplifies to s

 1=4  zp 2bc !c 2.  1/  2ı cz .1/ cos b 2.  1/ C AH s .z; t /  e z b 2 C 8ı 2 .  1/ c 4 (13.64) as z ! 1 and  ! 1C . This expression is precisely Brillouin’s result for the first forerunner [6]; see also page 73 of Ref. [7]. Hence, Brillouin’s asymptotic approximation of the first precursor field is an approximation, valid for  near 1, of an expression [(13.62)] that is not valid for  near 1.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics The contributions of the near saddle points to the asymptotic behavior of the propagated wavefield A.z; t / for sufficiently large values of the propagation distance z > 0 yield the dynamical space–time evolution of the second or Brillouin precursor field. This contribution to the asymptotic behavior of the total wavefield A.z; t / is denoted by Ab .z; t / and is dominant over the first or Sommerfeld precursor field in single resonance Lorentz model dielectrics, as well as in double resonance Lorentz model dielectrics that satisfy the inequality p > 0 ), for all  > SB , whereas it is dominant over both the Sommerfeld and middle precursor fields for all  > MB in double resonance Lorentz model dielectrics when the inequality p < 0 is satisfied. For Debye model dielectrics, it is the only precursor field and, unlike that for Lorentz model dielectrics, it is due to a single saddle point that moves down the imaginary axis. These two cases must then be treated separately, the more complicated Lorentz model case being treated here first because of its central importance to the classical theory due to Brillouin [6, 7]. From the results presented in Sect. 12.3, the two first-order near saddle points SPn˙ , which are initially isolated from each other at the luminal space–time point  D 1, SPnC situated along the positive imaginary axis and SPn situated along the negative imaginary axis, approach each other along the imaginary axis as  increases to the critical value 1 and coalesce into a single second-order saddle point SPn along the negative imaginary axis when  D 1 , after which they separate into two first-order saddle points and symmetrically move away from each other in the lower half of the complex !-plane as  increases above 1 , approaching in the inner branch points !˙ as  ! 1. A straightforward application of Olver’s theorem is first presented in Sect. 13.3.1 to determine the asymptotic behavior of the Brillouin precursor field in each of the separate space–time domains 1 <  < 1 ,  D 1 , and  > 1 . Because these results are nonuniform in a neighborhood of the critical space–time point  D 1 when the two near first-order saddle points coalesce into a single second-order saddle point and the saddle point order abruptly changes, one

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

417

must then resort to the uniform asymptotic expansion due to Chester, Friedman, and Ursell [25] described by Theorem 4 in Sect. 10.3.2 in order to obtain an asymptotic approximation of the Brillouin precursor that is continuous for all  > 1. This is done in Sect. 13.3.2.

13.3.1 The Nonuniform Approximation The asymptotic behavior of the second or Brillouin precursor field Ab .z; t / for a particular initial pulse envelope function u.t / is derived from the asymptotic expansion of the integral representation of the propagated wavefield given in (13.10) taken about the near saddle points. The near saddle point locations may be expressed as [see (12.245), and (12.246), and (12.282)] (

 i ˙ 0 ./  23 ı ./ ; !SP ˙ . / D n ˙ ./  i 23 ı ./;

1    1 ;   1

(13.65)

where [see (12.220) and (12.221)] 2 . / 4

  !02  2  02 2

 2  02 C 3˛ !b 2

31=2 4 2 2 5  ı . / ; 9

(13.66)

0

2

b 2 2 3   0 C 2 !02

. /

; 2  2  02 C 3˛ b 22

(13.67)

!0

for a single resonance Lorentz model dielectric with ˛ given by (12.218), and where [see (12.283) and (12.284)] 31=2  2  2   0 4 7 6   ı02 2 . /5 ;  2 . / 4 2 2 b b ! 9 2 0 2 0 2   0 C 3 ! 2 C ! 4 0 2   2 b ı b22 !04 2 2 2 0 1 C    C 2 2 2 4 0 3 !0 ı0 b0 !2   ;

. /

2  2   2 C 3 b02 1 C b22 !04 0 !2 b2 ! 4 2

!02

0

(13.68)

(13.69)

0 2

for a double resonance Lorentz model dielectric with ı D ı0 in (13.65). In either case, 02 . / D  2 . /. For each of the examples considered in this text, both the spectral function uQ .!  !c / and the complex phase function .!;  / appearing in the integrand of (13.10) are analytic about the two near saddle points for all   1. Because the near saddle point behavior separates into the three separate space–time domains 1 <  < 1 ,  D 1 , and  > 1 , each case is now separately examined.

418

13.3.1.1

13 Evolution of the Precursor Fields

Case 1: 1 <  < 1

For space–time values in the subluminal space–time domain  2 .1; 1 /, the conditions of Olver’s theorem are satisfied at the upper near saddle point SPnC when the original contour of integration C is deformed to the path P ./ D Pd ./CPnC . /C PdC . /, where PnC . / is an Olver-type path with respect to the upper near saddle point SPnC , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the upper near saddle point SPnC when the original contour of integration is deformed to the path P . / D Pd ./ C PnC ./ C PdC . / for  2 .1; SM , to the C  path P . / D Pd . / C Pm1 . / C PnC ./ C Pm1 ./ C PdC ./ for  2 ŒSM ; p /, C    . / C PnC . / C Pm1 ./ C and then to the path P . / D Pd ./ C Pm1 ./ C Pm2 C C Pm2 . / C Pd . / for  2 .p ; 1 /. In either case, (10.18) applies for the upper near saddle point SPnC with (10.3) and (10.4) taken as Taylor series expansions about this first-order saddle point, in which case  D 2. Furthermore, because uQ .!  !c / is analytic at this saddle point, then  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the Olvertype path PnC . / yields the asymptotic expansion of the initial space–time behavior of the second or Brillouin precursor field Ab .z; t / given by [17–19]  Ab .z; t / D

c z

1=2

< i e i

n  o .z=c/.! C ;/

SPn a0 .!SPnC /e 1 C O z1

(13.70)

as z ! 1 uniformly for all 1 <   1  " with arbitrarily small " > 0. To evaluate the coefficient a0 .!SPnC / appearing in (13.70), the first two coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  / as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about the upper near saddle point SPnC must first be determined. The latter quantity is given by q0 .!SPnC .// D uQ .!SPnC ./  !c /;

(13.71)

the specific form of which depends upon the particular pulse envelope function u.t /. Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 .!SPnC ;  / D  00 .!SPnC ;  /=2Š, the coefficient a0 .!SPnC ; / is found to be given by [see (10.9)] uQ .!SPnC ./  !c / a0 .!SPnC ;  / D h i1=2 : 2 00 .!SPnC ;  /

(13.72)

With this substitution, the nonuniform asymptotic expansion (13.70) of the Brillouin precursor becomes

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics Fig. 13.6 Illustration of the steepest descent choice of the Olver-type path PnC . / through the upper near saddle point SPnC for 1 <  < 0 . This first-order saddle point crosses the origin at  D 0 n.0/. The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / < .!SPnC ;  / is satisfied

Ab .z; t /  < i e i

419 ''

Pn SPn '

8" 9 #1=2 < = c=z .z=c/.! C ;/ SPn uQ .!SPnC ./  !c /e  00 : 2 .!SP C ;  / ; n

(13.73) small " > 0. as z ! 1 uniformly for all 1 <   1  " with arbitrarily  The proper value of ˛N 0 arg   00 .!SPnC ;  / must now be determined according to the convention defined in Olver’s method [see (10.7)]. For simplicity, the Olver-type path PnC . / through the upper near first-order saddle point SPnC is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.6. From (12.232), the angle of slope of the contour at the saddle point is given by ˛N D 0. Because  arg .z/ D 0, the proper value of ˛N 0 , as determined by the inequality in (10.7), is ˛N 0 D 0.

The Single Resonance Lorentz Model Dielectric From (12.227), the approximate behavior q of the complex phase function .!;  / in the below resonance region j!j < !02  ı 2 of the complex !-plane about the origin is given by .!;  / i !.0  / C

b2 ! 2 .i ˛!  2ı/: 20 !04

Differentiation of this expression twice with respect to ! then yields  0 .!;  / i.0  / C  00 .!;  /

b2 .3i ˛! 2  4ı!/; 20 !04

b2 .3i ˛!  2ı/; 0 !04

420

13 Evolution of the Precursor Fields

so that at ! D !SPnC . / D i .!SPnC ; /



 2 0 ./  3 ı ./ , there results

 1 2ı . /  3 0 ./ .0  / 3  2  b2  2ı ./  3 0 ./ 2ı 3  ˛ . / C 3˛ C 4 540 !0

0 ./

 ;

(13.74)  b2   00 .!SPnC ; /  2ı 1  ˛ . / C 3˛ 4 0 !0

 0 ./ :

(13.75)

Substitution of these approximate expressions into (13.73) then gives the asymptotic approximation of the second or Brillouin precursor over the initial subluminal space–time domain  2 .1; 1 / as [3, 4, 17] #1=2 " n o !02 0 c=.z/   Ab .z; t /  < i e i uQ .!SPnC  !c / b 4ı 1  ˛ . / C 6˛ 0 ./ z .2ı ./  3 0 .//  exp 3c i h 2 0 ./

(13.76)  0   C b .2ı ./3 0 .//Œ2ı.3˛ .//C3˛ 18 ! 4 0 0

as z ! 1 uniformly for 1 <   1  " with arbitrarily small " > 0. The dynamical evolution of the wavefield described by (13.76) depends on the algebraic sign of the exponential argument appearing in that expression for z > 0. Because the quantity .2ı ./  3 0 .// is negative for 1 <  < 0 , vanishes at  D 0 , and is positive for 0 <  < 1 , and because the quantity Œ2ı.3  ˛ . // C 3˛ 0 . / is positive for 1 <  < 1 , and because the inequality j0  j 

b 2 j2ı . /  3

0 ./j Œ2ı.3

 ˛ .// C 3˛ 180 !04

0 ./

is satisfied for all  2 .1; 1 /, with the equality holding only at  D 0 when both sides of this equation vanish, the argument of this exponential function is negative for 1 <  < 0 , vanishes identically at  D 0 , and is again negative for   0 <  < 1 . Consequently, the second precursor field Ab .z; t / first grows with increasing  as the exponential argument decreases with increasing  2 .1; 0 /, becoming exponentially dominant over the first precursor field when  > SB , where 1 < SB < 0 . At  D 0 the exponential argument identically vanishes (because the approximate expressions for both the upper near saddle point and the complex phase behavior at this saddle point become exact when it crosses the origin) and the wavefield given in (13.76) varies only as z1=2 for ı > 0,3 making the wavefield behavior at this 3

The wavefield behavior in the special case when ı D 0 is examined in Case 2. In that case, 1 D 0 and the two near saddle points SPn˙ coalesce into a single second-order saddle point SPn at the origin.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

421

space–time point strikingly different from the wavefield at any other space–time point in the dynamical field evolution. Finally, for increasing values of  2 .0 ; 1 /, the exponential argument decreases and the field is again exponentially attenuated with propagation distance z > 0. Although the functional form of the second precursor gives it the appearance of being nonoscillatory over this initial space–time domain, it does have an effective oscillation frequency with half period given by the temporal width at the e 1 amplitude point and the peak amplitude point at  D 0 .

The Double Resonance Lorentz Model Dielectric From (12.276), the approximate behavior q of the complex phase function .!;  / in the below resonance region j!j < !02  ı02 of the complex !-plane about the origin is given by .!;  / i!.0   / 

1 0



ı0 b02 ı2 b 2 C 42 4 !0 !2



!2 C

i 20



b02 b2 C 24 4 !0 !2



!3:

Differentiation of this expression twice with respect to ! then yields   2  ı0 b02 b0 ı2 b22 b22 3i ! C !2; C C 20 !04 !04 !24 !24     ı2 b22 b22 3i b02 2 ı0 b02 00  .!;  /  C 4 C C 4 !; 0 !04 0 !04 !2 !2  0 .!;  / i.0   / 

so that at ! D !SPnC . / D i .!SPnC ; /



2 0



0 ./

  23 ı0 . / , there results

 1 2ı0 . /  3 0 ./ 3(    ı0 b02 ı2 b22 1  2ı0 . /  3 0 . /  0   C C 4 30 !04 !2  2  )  b22  1 b0  C 4 2ı0 . /  3 0 ./ ; 6 !04 !2 (13.77)

2  .!SPnC ; /  0 00



ı0 b02 ı2 b22 C !04 !24



1  0



b02 b22 C !04 !24





3

0 ./

  2ı0 . / : (13.78)

Substitution of these approximate expressions into (13.73) then gives the asymptotic approximation of the Brillouin precursor over the initial space–time domain  2 .1; 1 / in a double resonance Lorentz model dielectric as

422

13 Evolution of the Precursor Fields

s

  2   2    1=2 ı0 b 0 b0 ı2 b22 b22  0 c 2 Ab .z; t /  C 4 C C 4 3 0 . /  2ı0 . / 2z !04 !2 !04 !2 n o i < i e uQ .!SPnC  !c /  z .2ı . /  3 0 .// 0    exp 3c h 2   2  i  2 ı0 b0 ı2 b22 b0 b22 .2ı ./3 0 .// 0 ./ 3 ı0 ./ C C !4 C !4 C !4 30 2 !4 0

2

0

2

(13.79) as z ! 1 uniformly for 1 <   1  " with arbitrarily small " > 0. The dynamical behavior described by this equation is the same as that described in the single resonance case.

13.3.1.2

Case 2:  D 1

At the space–time point  D 1 D ct1 =z, the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour of integration C is deformed to the path P .1 / D Pd .1 / C PnC .1 / C PdC .1 /, where PnC .1 / is an Olver-type path with respect to the near saddle point SPn , located at [see (12.236)] 2ı (13.80) !SPn .1 / i ; 3˛ with ˛ D 1  ı 2 .4!12 C b 2 /=.3!02 !12 / 1. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the second-order near saddle point SPn when the original contour of in  .1 /CPm2 .1 /CPnC .1 /C tegration is deformed to the path P .1 / D Pd .1 /CPm1 C C C Pm1 .1 / C Pm2 .1 / C Pd .1 /, where PnC .1 / is an Olver-type path with respect to the second-order near saddle point SPn whose location is also given by (13.80) with   1 ı D ı0 [cf. (12.281)] and with ˛ D 1 C b22 !04 =b02 !24 1 C .ı2 =ı0 /.b22 !04 =b02 !24 / . Because both of the functions uQ .!  !c / and .!;  / appearing in the integrand of (13.10) are analytic about this second-order near saddle point, then (10.3) and (10.4) are Taylor series expansions about that point with  D 3 and  D 1. Because  D 3, the argument presented in Sect. 10.1.2 leading up to (10.18), which is valid only for the case when  D 2, must now be appropriately modified. The second coefficient in the Taylor series expansion in (10.3) of the complex phase function .!; 1 / about this second-order near saddle point is given by p0 .!SPn ; 1 / D

1 000  .!SPn ; 1 / 3Š

(13.81)

and the first coefficient in the Taylor series expansion in (10.4) of the initial pulse envelope spectrum is given by

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

423

    q0 .!SPn 1 / D uQ !SPn .1 /  !c ;

(13.82)

the specific form of which depends upon the particular initial pulse envelope function u.t / under consideration. Notice that for both the single and double resonance cases,  000 .!SPn ; 1 / ' i j 000 .!SPn ;1 /j.  The proper value of ˛N 0 arg   000 .!SPn ; 1 / ' arg .i / must now be determined according to the convention defined in Olver’s method [see (10.7)]. For simplicity, the Olver-type path Pn .1 / through the second-order near saddle point SPn is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.7. The contour integral for the wavefield A.z; t / along this path may be written as A.z; t1 / D Ab .z; t1 / D I C  I  ; where I ˙ denote the contour integrals taken in opposite directions leading away from the saddle point along that corresponding half of the Olver-type path Pn .1 /, as indicated in Fig. 13.7. The angle of slope of the two steepest descent paths leaving the saddle point are given by [see (12.238) and Fig. 12.41] ˛N C D =6 and ˛N  D 5=6, as indicated in the figure. Consequently, because  arg .z/ D 0, the proper values of ˛N 0˙ for these two paths, as determined by the inequality given in (10.7), is ˛N 0C D =2 for I C and ˛N 0 D 5=2 for I  . The phase difference between the two coefficients a0˙ .!SPn ; 1 / then results in the factor C

e i ˛N 0

=3

 =3

 e i ˛N 0

D e i=6  e i5=6 D 2 cos

  6

D

p

3;

where a0C .!SPn ; 1 /

1=3   3Š 1  D uQ !SPn .1 /  !c e i=6 : 3 j 000 .!SPn ; 1 /j

(13.83)

'' I−

_

I+

Pn

' _ SPn

Fig. 13.7 Illustration of the steepest descent choice of the Olver-type path Pn .1 / through the second-order near saddle point SPn at  D 1 . The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!; 1 / <  .!SPn ; 1 / is satisfied

424

13 Evolution of the Precursor Fields

The asymptotic expansion of the second or Brillouin precursor field Ab .z; t / at the fixed space–time point  D 1 D ct1 =z is then given by [17–19] Ab .z; t / D

1=3   . 13 / 6c p 2 3 j 000 .!SPn ; 1 /jz n  z

 o  < i e i uQ !SPn .1 /  !c e c .!SPn ;1 / 1 C O z1=3 (13.84)

as z ! 1.

The Single Resonance Lorentz Model Dielectric From the set of relations preceeding (13.74), one finds that 2ı .!SPn ; 1 /

3˛  000 .!SPn ; 1 / 3i



4ı 2 b 2 0  1 C 9˛0 !04

˛b 2 : 0 !04

 ;

(13.85) (13.86)

Substitution of these results in (13.84) then yields the asymptotic approximation Ab .z; t / 

  20 !0 c 1=3 ˚ i  < i e uQ !SPn .1 /  !c 2 ˛b z    2ız 4ı 2 b 2 0  1 C  exp (13.87) 3˛c 9˛0 !04

 . 13 / p !0 2 3



as z ! 1 at the fixed space–time point  D 1 D ct1 =z. From (12.225), it is seen that 0  1 C

4ı 2 b 2 2ı 2 b 2

 ; 4 9˛0 !0 9˛0 !04

and hence, as was expected, the second precursor field attenuates exponentially with increasing propagation distance z > 0 at the fixed space–time point  D 1 . In the special (limiting) case when ı D 0, it then follows that 1 D 0 exactly and the two first-order near saddle points coalesce into a single second-order saddle point at the origin. The peak amplitude in the Brillouin precursor then decays with the propagation distance z > 0 only as z1=3 [as described by (13.87)] instead of the usual z1=2 behavior [see (13.76)] observed when ı > 0 [26].

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

425

The Double Resonance Lorentz Model Dielectric From (12.280) and (12.281) for 1 and !SPn .1 / in a double resonance Lorentz model dielectric and the set of relations preceeding (13.77), one finds that 3  4 ı0 b02 !24 C ı2 b22 !04 .!SPn ; 1 /   2 ; 270 !04 !24 b02 !24 C b22 !04  000 .!SPn ; 1 / 3i

b02 !24 C b22 !04 : 0 !04 !24

(13.88) (13.89)

Substitution of these approximate expressions into (13.84) then yields the asymptotic approximation !1=3 ˚ 20 !04 !24 c   2 4 < i e i uQ .!SPn .1 /  !c / Ab .z; t1 /  p 2 4 2 3 z b0 !2 C b2 !0 " 3 #  4z ı0 b02 !24 C ı2 b22 !04  exp  (13.90) 2  270 !04 !24 c b02 !24 C b22 !04 

1 3

as z ! 1 at the fixed space–time point  D 1 D ct1 =z.

13.3.1.3

Case 3:  > 1

For space–time values  in the domain  > 1 , the conditions of Olver’s theorem are satisfied at the symmetric pair of first-order near saddle points SPn˙ when the original contour of integration C is deformed to the path P . / D Pd ./CPn . /C PnC . /CPdC . /, where Pn . / is an Olver-type path with respect to the near saddle point SPn and PnC . / is an Olver-type path with respect to the near saddle point SPnC , as illustrated in Fig. 12.66. For a double resonance Lorentz model dielectric that satisfies the inequality p < 0 , the conditions of Olver’s theorem are satisfied at the pair of near saddle points SPn˙ when the original contour of integration is   ./ C Pm2 ./ C Pn ./ C PnC ./ C deformed to the path P . / D Pd ./ C Pm1 C C C Pm1 . / C Pm2 . / C Pd . /. In either case, (10.18) applies for each of the two near saddle points SPn˙ with (10.3) and (10.4) taken as Taylor series expansions about this first-order saddle point, in which case  D 2. Furthermore, because uQ .!  !c / is analytic at each of these saddle points, then  D 1. Hence, from Olver’s theorem (Theorem 2) and the results of Sect. 12.4, the contour integral in (13.10) taken over the two Olver-type paths Pn . / and PnC ./ yields the asymptotic expansion of the conclusion of the space–time behavior of the second or Brillouin precursor field Ab .z; t / given by [17–19]

426

13 Evolution of the Precursor Fields

 Ab .z; t / D

c z

1=2

( < i e i

a0 .!SPnC /e

.z=c/.!

Ca0 .!SPn /e

C ;/ SPn



  1 C O z1

.z=c/.!SPn ;/

  1 C O z1

)

(13.91) as z ! 1 uniformly for all   1 C " with arbitrarily small " > 0. To evaluate the pair of coefficients a0 .!SP ˙ / appearing in (13.91), the first two n coefficients in the Taylor series expansion (10.3) of the complex phase function .!;  / as well as the first coefficient in the Taylor series expansion (10.4) of the initial envelope spectrum uQ .!  !c / about each saddle point SPn˙ must now be determined. The latter quantity is given by q0 .!SP ˙ .// D uQ .!SP ˙ ./  !c /; n

n

(13.92)

the specific form of which depends upon the particular pulse envelope function u.t /. Because the second coefficient in the Taylor series expansion (10.3) of the complex phase function is given by p0 .!SP ˙ ;  / D  00 .!SP ˙ ;  /=2Š, the coefficient n n a0 .!SP ˙ ; / is found to be given by [see (10.9)] n

uQ .!SP ˙ ./  !c / n a0 .!SP ˙ ;  / D h i1=2 : n 00 2 .!SP ˙ ;  /

(13.93)

n

With this substitution, the nonuniform asymptotic expansion (13.91) of the Brillouin precursor becomes ("

Ab .z; t /  < i e

i

#1=2 c=z .z=c/.! C ;/ SPn uQ .!SPnC ./  !c /e  00 2 .!SPnC ;  / ) 1=2  c=z  C  uQ .!SPn ./  !c /e .z=c/.!SPn ;/ 2 00 .!SPn ;  / (13.94)

as z ! 1 uniformly for all   1 C " with arbitrarilysmall " > 0. The proper value of ˛N 0˙ arg   00 .!SP ˙ ;  / must now be determined n according to the convention defined in Olver’s method [see (10.7)]. Consider first C the value for the near saddle point SPn in the right half of the complex !-plane. For simplicity, the Olver-type path PnC ./ through this first-order saddle point is taken to locally lie along the path of steepest descent through that saddle point, as illustrated in Fig. 13.8. From (12.241), the angle of slope of the contour at the sadin the diagram. Because  arg .z/ D 0, dle point is ˛N C D =4, as indicated   the proper value of ˛N 0C arg   00 .!SPnC ;  / , as determined by the inequality in

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

Pn

_

427

'

SPn

Fig. 13.8 Illustration of the steepest descent choice of the Olver-type path PnC . / through the first-order near saddle point SPnC in the right half of the complex !-plane for  > 1 . The shaded area indicates the local region of the complex !-plane about the saddle point where the inequality  .!;  / < .!SPnC ;  / is satisfied

(10.7), is ˛N 0C ' =4. By a similar argument for the near saddle SPn in the  point  00 left half of the complex !-plane, the proper value of ˛N 0 arg   .!SPn ;  / is ˛N 0 ' =4. The Single Resonance Lorentz Model Dielectric From the set of relations preceeding (13.74), the complex phase function and its second derivative at the two first-order near saddle point locations ! D !SP ˙ . / D n ˙ . /  i 23 ı ./ are found to be given by (

  2 b2  1  ˛ . / 2 . /

. /.  0 / C 4 3 0 !0  )  4 2 2 1 ˛ . /  1 C ı . / 9 3 (  )   4 2 b2 2 ı . / 2  ˛ . / C ˛ . / ; ˙i ./ 0   C 20 !04 3

.!SP ˙ ; / ı n

(13.95)  00 .!SP ˙ ; /

n

  b2  2ı ˛ . /  1 ˙ 3i ˛ . / : 4 0 !0

(13.96)

One then has that #1=2 " h i1=2 2 !  0   00 .!SP ˙ ; /

0   n b 2ı ˛ . /  1 ˙ 3i ˛ . / s !02 0 e ˙i=4 ;

b 3˛ ./

(13.97)

428

13 Evolution of the Precursor Fields

where the finalapproximation is valid for all  > 1 such that the inequality  3˛ ./  2ı ˛ . /  1 is satisfied. Notice that this inequality will be satisfied provided that  is not too close to 1 , a requirement that isn’t overly restrictive in the nonuniform asymptotic description (when  approaches 1 from above, the nonuniform description must be replaced by the uniform asymptotic description). Substitution of these approximate expressions in (13.94) then gives the asymptotic approximation of the second or Brillouin precursor field as [3, 4, 17] s

(  z 2 0 c exp  ı

./.  0 / 6 ˛ ./z c 3 )    ˛ b2 4 2 2 2

. /  1 .1  ˛ .// ./ C ı ./ C 9 3 0 !04 " ( ( z i < i e . / 0   uQ .!SPnC ./  !c / exp i c )  #   4 2 b2  2 C ı . / 2  ˛ . / C ˛ . / Ci 4 20 !04 3 " ( z . / 0   CQu.!SPn ./  !c / exp  i c ))  #   4 2 b2  2 ı . / 2  ˛ . / C ˛ ./ C i 4 20 !04 3

!2 Ab .z; t /  0 b

(13.98) as z ! 1 for  > 1 . The second precursor field is then seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as  increases above 1 , the attenuation factor increasing with increasing  . Notice that the quantity is a fixed phase factor associated with the initial pulse carrier wave [where D 0 corresponds to a sine-wave carrier and D =2 corresponds to a cosine-wave carrier, as described in (11.34)], whereas the function . / describes the space– time dependence of the real part of the near saddle point location for  > 1 , as described by (13.66).

The Double Resonance Lorentz Model Dielectric With the substitution ! D !SP ˙ ./ D ˙ ./  i 23 ı0 . / in the set of relations n preceeding (13.77), one obtains

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

2 62 .!SP ˙ ; / ı0 4 . /.  0 / C n 3

 b02 1 C

429

 4

ı2 b22 !0 ı0 b02 !24 0 !04



3  4 7 2 ./  ı02 2 . / 5 9

    ı2 b22 !04 4ı 2 b 2

. / ; ˙i ./ 0   C 0 40 1 C 0 !0 ı0 b02 !24

(13.99)

and     b22 !04 ı2 b22 !04 1 C 2 4 . /  2 1 C b0 !2 ı0 b02 !24   b2!4 3b02 1 C 22 04 ./ ˙i 4 20 !0 b0 !2   3b02 b22 !04

˙i 1C 2 4 ./; (13.100) 20 !04 b0 !2

ı0 b02  .!SP ˙ ; /

n 0 !04 00

where the final approximation is valid for all  > 1 such that the inequality 3 ./  2ı0 . /  .b02 !24 C .ı2 =ı0 /b22 !04 /=.b02 !24 C b22 !04 / is satisfied. When this inequality is not satisfied, the nonuniform expansion is becoming invalid and must then be replaced by the uniform expansion. One then has that h i1=2 !2  00 .!SP ˙ ; /

0 n b0

s

20 b02 !24  e ˙i=4 : 3 b02 !24 C b22 !04 . / 

(13.101)

Substitution of these approximate expressions in (13.94) then gives the asymptotic approximation of the Brillouin precursor in a double resonance Lorentz model dielectric as s 0 b 2 ! 4 c !02  2 4 0 22 4  Ab .z; t /  b0 6 b0 !2 C b2 !0 ./z   z 2 .ı0 b02 !24 Cı2 b22 !04 /. 2 ./ 49 ı02 2 .//

./.   / C  exp ı0 0 0 ı0 !04 !24 c 3 ( < i e i

uQ .!SPnC ./  !c / (

zh  exp i ./ 0   C c CQu.!SPn . /  !c / (  exp

4ı02 b02 0 !04

zh  i ./ 0   C c



1C

4ı02 b02 0 !04

ı2 b22 !04 ı0 b02 !24

 1C



)

i

. / C i

ı2 b22 !04 ı0 b02 !24



i

 4

))

. /  i

 4

(13.102)

430

13 Evolution of the Precursor Fields

as z ! 1 for  > 1 . As in the single resonance case, the Brillouin precursor field is seen to be oscillatory and increasingly attenuated with the propagation distance z > 0 as  increases above 1 , the attenuation factor increasing with increasing  .

13.3.2 The Uniform Approximation The set of relations given in (13.76), (13.84), and (13.98) for the single resonance Lorentz model and in (13.79), (13.90), and (13.102) for the double resonance Lorentz model represent the nonuniform asymptotic approximation of the second or Brillouin precursor field for space–time values  ct =z in the successive domains 1 <  < 1 ,  D 1 , and  > 1 , respectively. The results are discontinuous at the critical transition point at  D 1 at which the two first-order near saddle points have coalesced into a single second-order saddle point. To obtain a continuous transition in the behavior of the Brillouin precursor field as  is allowed to vary across the critical space–time point  D 1 , the uniform asymptotic expansion due to Chester, Friedman, and Ursell [25] is employed (see Theorem 4 of Sect. 10.3). The required uniform asymptotic approximation is obtained by direct application of Theorem 4. Because the behavior of the near saddle points and the path of integration in the present case is the same as in the example treated in Sect. 10.3.2, it follows from the discussion in that example that the path of integration L appearing in (10.52) is an L21 contour (see Fig. 10.6) so that the function C. / appearing in (10.51)  is given  by (10.72) with indices i D 2; j D 1, so that C. / D e i2=3 Ai e i2=3 , where Ai ./ is the Airy function. Although Theorem 4 is directly applicable to the present problem over the entire space–time domain  > 1, it is still necessary to treat the two cases 1 <   1 and   1 separately because the approximate expressions for the near saddle point locations differ in the two cases. Nonetheless, the results for these two cases combined are continuous at the critical space–time point at  D 1 and therefore constitute an asymptotic approximation of the Brillouin precursor field Ab .z; t / that is uniformly valid for all  > 1. Consider first the uniform asymptotic behavior of the Brillouin precursor field over the initial space–time domain 1 <  < 1 . In that case, the two near saddle point locations are given by (13.65) as  !SP ˙ . / D i ˙ n

 2 ./  ı . / ; 0 3

(13.103)

where 02 . / D  2 . / with ./ and . / given in (13.66) and (13.67) for a single resonance Lorentz model dielectric and by (13.68) and (13.69) for the double resonance case. The analysis presented here will focus on the single resonance case, the double resonance case being left as an exercise. Application of Theorem 4 yields the asymptotic expansion [4, 17, 20, 27]

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

431

 1=3 z c < e i e c ˛0 ./ z h  i  uQ .!SPnC  !c /h1 ./ C uQ .!SPn  !c /h2 ./ C O z1

1 Ab .z; t / D  2 (

e i C

2 3

  2 Ai ˛1 ./e i 3 .z=c/2=3

.c=z/1=3 h uQ .!SPnC  !c /h1 . /  uQ .!SPn  !c /h2 . / 1=2 ˛1 . / )   1  i i 4 0  2 i 2=3 e 3 Ai ˛1 ./e 3 .z=c/ CO z (13.104)

as z ! 1 uniformly for all  2 .1; 1 . From (10.54)–(10.56) with substitution from (13.74) and (13.75), the coefficients appearing in this uniform expansion are found to be given by  1 .!SPnC ; / C .!SPn ;  / 2 2

 ı . /.  0 / 3      4 2 2 1 ıb 2 2 ˛ . /  1 ;  0 . / ˛ . /  1 C ı ./ 9 3 0 !04 (13.105)  1=3  3 1=2 ˛1 . / .!SPnC ; /  .!SPn ;  / 4 "   #1=3 3 3 b2 1=3 2 2 2 2

0 . / .  0 / C ; ˛ . / C ˛ı ./  2ı . / 2 0 !04 4 0 ˛0 . /

(13.106) and "

#1=2 1=2 2˛1 . / h1;2 . / 00  .!SP ˙ ; / n " #1=2 2 !0 1=6 20  

. / b 0 3˛ 0 ./ ˙ 2ı 1  ˛ . / "   #1=6 3 3 b2 2 2 2 2 ˛ ./ C ˛ı . /  2ı . /  .  0 / C ; 2 0 !04 4 0 (13.107)

432

13 Evolution of the Precursor Fields

for  2 .1; 1 . Notice that the upper sign choice in (13.107) corresponds to h1 . / and the lower sign choice to h2 ./. In the limit as  approaches the critical value 1 from below, this expression reduces to [see (10.57)]  h.1 / lim h1;2 ./ D  !1

1=3  20 !04

 ; 3i ˛b 2

2

1=3

 000 .!SPn ; 1 / (13.108)

where the final approximation is obtained by substitution from (13.86). Analogous expressions are obtained for the double resonance case. The proper values of the multivalued functions appearing in (13.106)–(13.108) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h1;2 . / is specified by (10.60) as (in the notation of the present chapter)   lim arg h1;2 ./ D ˛N C ;

!1

(13.109)

where ˛N C is the angle of slope of the steepest descent path leaving the secondorder saddle point SPn at  D 1 . From Fig. 13.7 it is seen that ˛N C D =6. Hence, (13.109) shows that the argument of h.1 / is =6. Moreover, because the sixth power of the quantity appearing on the right-hand side of (13.107) is real and negative for all  2 .1; 1 , the argument  of h1;2 . / is independent of  over that space–time domain. Hence arg h1;2 ./ D =6 and (13.107) may be rewritten as ˇ1=2 ˇ ˇ ˇ1=6 ˇˇ !02 ˇˇ 2 ˇ 0 ˇ ˇ   ˇ e i=6 h1;2 . /

0 . / ˇ 3˛ 0 ./ ˙ 2ı 1  ˛ . / ˇ b ˇ  ˇ1=6 ˇ3 ˇ 3 b2 2 2 2 2 ˇ ˛ 0 ./ C ˛ı . /  2ı . / ˇˇ ;  ˇ .  0 / C 4 2 0 !0 4 (13.110) for all  2 .1; 1 . 1=2 The proper value of the phase of the quantity ˛1 ./ is determined from (10.63) with n D 0, so that (in the notation of the present chapter)   1=2 lim arg ˛1 ./ D ˛N 12  ˛N C ;

!1

(13.111)

where ˛N 12 is the angle of slope of the vector from the saddle point SPn to the saddle point SPnC . Because ˛N 12 D =2 (see Fig. 12.40) and ˛N C D =6 , then    1=2 lim arg ˛1 ./ D : !1 3

(13.112)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

433

Moreover, because the cube of the quantity appearing on the right-hand side of 1=2 (13.106) is real and negative for all  2 .1; 1 , the argument of ˛1 ./ is indepen 1=2  dent of  over that space–time domain. Hence, arg ˛1 . / D =3 and (13.106) may be rewritten as ˇ 1=2 ˛1 . / ˇ

0 . /

ˇ1=3 i=3 ˇ e

ˇ  ˇ3 3 b2 ˇ ˛ ˇ .  0 / C ˇ2 0 !04 4

 ˇˇ1=3 ˇ 2 2 2 2 ; 0 ./ C ˛ı ./  2ı . / ˇ ˇ (13.113)

for all  2 .1; 1 .  Because arg ˛1 . / D 2=3, the argument of the Airy function Ai . / and its first derivative A0i . / appearing in the uniform asymptotic expansion in (13.104) is real and nonnegative for all  2 .1; 1 . With the above results for the arguments of the quantitites h1;2 . / and ˛1 ./, one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27] Ab .z; t / 

 1=3 z c e c ˛0 ./ z ( h ˇ ˇ ˇ ˇi i uQ .!SPnC  !c /ˇh1 ./ˇ C uQ .!SPn  !c /ˇh2 . /ˇ < i e 1 2

ˇ ˇ  Ai ˇ˛1 ./ˇ.z=c/2=3 ˇ ˇ ˇ ˇi .c=z/1=3 h ˇ ˇ1=2 uQ .!SPnC  !c /ˇh1 . /ˇ  uQ .!SPn  !c /ˇh2 . /ˇ ˇ˛1 . /ˇ ) ˇ ˇ  0 ˇ 2=3 ˇ A ˛1 ./ .z=c/ (13.114) i

as z ! 1 uniformly for all  2 .1; 1 . As  approaches the critical value 1 from below, the argument of the Airy function and its first derivative in (13.114) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of (13.114) in this limit is found from the limiting forms given in (10.57)–(10.59) and (13.108) as Ab .z; t1 / 

  20 !0 c 1=3 ˚ i  < i e uQ !SPn .1 /  !c 2 ˛b z    2ız 4ı 2 b 2 0  1 C (13.115)  exp 3˛c 9˛0 !04

 . 13 / p !0 2 3



as z ! 1 with  D 1 D ct1 =z. This result is identical with that given in (13.87) using Olver’s saddle point method.

434

13 Evolution of the Precursor Fields

Consider now the uniform asymptotic behavior of the Brillouin precursor field for   1 . In that case the near saddle points form a symmetric pair with locations given by (13.65) as 2 (13.116) !SP ˙ ./ D ˙ ./  i ı ./; n 3 the approximate complex phase behavioir at these points being given by (13.95) and (13.96) in the single resonance case, the double resonance case being left as an exercise. Application of Theorem 4 then yields the uniform asymptotic expansion [4, 17, 20, 27]  1=3 z c < e i e c ˛0 ./ z h  i  uQ .!SPnC  !c /h1 ./ C uQ .!SPn  !c /h2 ./ C O z1

1 Ab .z; t / D  2 (

e i C

2 3

  2 Ai ˛1 ./e i 3 .z=c/2=3

.c=z/1=3 h uQ .!SPnC  !c /h1 . /  uQ .!SPn  !c /h2 . / 1=2 ˛1 . / )   1  i i 4 0  2 i 2=3 e 3 Ai ˛1 ./e 3 .z=c/ CO z (13.117)

as z ! 1 uniformly for all   1 . From (10.54)–(10.56) with substitution from (13.95) and (13.96), the coefficients appearing in this expression are found to be given by  1 .!SPnC ; / C .!SPn ;  / 2 ( 2

ı . /.  0 / 3   b2  1  ˛ . / C 4 0 !0

˛0 . /

1=2



˛1 . / (

2

)  4 2 2 1 ./ C ı . / ˛ . /  1 9 3 (13.118)

  1=3 3 .!SPnC ; /  .!SPn ;  / 4

 i 32

h . /   0 

b2 20 !04

4

  ı 2 . / 2  ˛ . / C ˛ 3

2

./

i

) 1=3 ;

(13.119)

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

435

and " h˙ . /

1=2

2˛ . /

00 1  .!SP ˙ ;  /

#1=2

n

#1=2  1=6 " 3 20    i ./

i b 2 3˛ ./ 2i ı ˛ . /  1 "   #1=6   4 2 b2 2 ı . / 2  ˛ . / C ˛ ./    0  ; 20 !04 3  1=6  1=2 !2 3 20 i

0  i ./ b 2 3˛ ./ "   #1=6   4 2 b2 2    0  ; ı . / 2  ˛ . / C ˛ ./ 20 !04 3 !02

(13.120)   where the final approximation is valid provided that 3˛ ./  2ı ˛ . /  1 , which is found to be satisfied for all   1 . In the limit as  approaches 1 from above, this expression is replaced by its limiting form [see (10.57)]  h.1 / lim h˙ ./ D  !1C

2

1=3

 000 .!SPn ; 1 /

1=3  20 !04

 ; 3i ˛b 2

(13.121)

which is the same as that given in (13.108). The proper values of the multivalued functions appearing in (13.119)–(13.121) are determined by the conditions presented in Sect. 10.3.2. In particular, the phase of h˙ . / is specified by (10.60) as   lim arg h˙ ./ D ˛N C ;

!1C

(13.122)

where ˛N C is the angle of slope of the steepest descent path leaving the second-order saddle point SPn at  D 1 . From Fig. 13.7 it is seen that ˛N C D =6. Hence, (13.122) shows that the argument of h.1 / is =6. Moreover, because the sixth power of the approximate quantity appearing on the right-hand side of (13.120) is real and negative for all   1 , the argument of h˙ . / is approximately independent of  over that space–time domain. Hence arg h˙ ./ =6 and (13.120) may be rewritten as

436

h˙ . /

13 Evolution of the Precursor Fields

ˇ ˇ ˇ1=6 ˇ ˇ ˇ 20 ˇ1=2 i=6 !02 ˇˇ 3 ˇ ˇ ˇ . / ˇ ˇ 3˛ ./ ˇ e b ˇ2 ˇ  ˇ   4 2 b2 ˇ ˇ  0  ı . / 2  ˛ . / C ˛ 4 ˇ 20 !0 3

2

 ˇˇ1=6 ˇ ./ ˇ ˇ

(13.123)

  for all   1 , the accuracy of the approximation arg h˙ . / =6 decreasing as  increases above 1 . 1=2 The proper value of the phase of the quantity ˛1 ./ is determined from (10.63) with n D 0, so that   1=2 lim arg ˛1 ./ D ˛N 12  ˛N C ; (13.124) !1

where ˛N 12 is the angle of slope of the vector from the saddle point SPn to the saddle point SPnC . Because ˛N 12 D 0 (see Fig. 12.42) and ˛N C D =6 , then    1=2 (13.125) lim arg ˛1 ./ D  : C 6 !1 Moreover, because the cube of the quantity appearing on the right-hand side of 1=2 (13.119) is negative imaginary for all   1 , the argument of ˛1 ./ is indepen 1=2  dent of  over that space–time domain. Hence, arg ˛1 . / D =6 and (13.119) may be rewritten as ˇ ˇ ˇ h i ˇ1=3    2 ˇ3 ˇ 1=2 ˛1 . / ˇ 2 . /   0  2b ! 4 43 ı 2 . / 2  ˛ . / C ˛ 2 . / ˇ e i=6 0 0 ˇ ˇ (13.126) for all   1 . Because arg .˛1 . / D =3, the argument of the Airy function Ai . / and its first derivative A0i . / appearing in (13.117) is real and nonpositive for all   1 . With these results for the arguments of the quantities h˙ ./ and ˛1 . /, one finally obtains the uniform asymptotic approximation of the Brillouin precursor [4, 17, 20, 27]   1 c 1=3 z ˛0 ./ ec Ab .z; t /  2 z ( h ˇ ˇ ˇ ˇi i uQ .! C  !c /ˇhC ./ˇ C uQ .!SP   !c /ˇh . /ˇ < i e SPn

n

ˇ  ˇ  Ai ˇ˛1 ./ˇ.z=c/2=3 ˇ C ˇ ˇ  ˇi .c=z/1=3 h ˇ ˇ1=2 uQ .!SPnC  !c /ˇh ./ˇ  uQ .!SPn  !c /ˇh . /ˇ ˇ˛1 . /ˇ ) ˇ  ˇ  0 2=3 A ˇ˛1 ./ˇ.z=c/ (13.127) i

as z ! 1 uniformly for all   1 .

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

437

As  approaches the critical value 1 from above, the argument of the Airy function and its first derivative in (13.127) tends to zero as the amplitude coefficients in that equation tend to indeterminate forms. The determinate form of (13.127) in this limit is found from the limiting forms given in (10.57)–(10.59) and (13.121) to be precisely that given in (13.87) and (13.115). Taken together, the set of expressions given in (13.114), (13.115), and (13.127) constitute the uniform asymptotic approximation of the Brillouin precursor field in a single resonance Lorentz model dielectric that is uniformly valid for all  > 1. Because the argument of the Airy function and its first derivative in (13.127) is real and negative for all  > 1 , the Brillouin precursor is oscillatory over this space–time domain. In a similar manner, because the argument of the Airy function and its first derivative in (13.114) is real and positive for  2 .1; 1 , one may conclude that the Brillouin precursor is nonoscillatory over this initial space–time domain; however, this does not mean that the Brillouin precursor begins as a static field as it rapidly builds to its peak amplitude point near  D 0 over this initial space–time domain. Finally, for values of  bounded away from unity, the uniform asymptotic approximation in (13.114) simplifies to the nonuniform approximation given in (13.76), and the uniform asymptotic approximation given in (13.127) simplifies to the nonuniform approximation given in (13.94) as z ! 1.

13.3.3 The Instantaneous Oscillation Frequency The instantaneous angular frequency of oscillation of the Brillouin precursor field Ab .z; t / is defined [6, 7] as minus the time derivative of the oscillatory phase, where the minus sign is included in order to obtain a positive-valued angular frequency. Notice that the oscillatory phase terms appearing in both the nonuniform (13.94) and uniform (13.127) asymptotic approximations of this second precursor field are identical for  > 1 . Because d=dt D c=z for all z > 0 and (with ˛ 1) 0 2 2 d . / 3c d @   0 C D dt 2z d  2   2 C 0 2 !02

2b 2 !02 3b 2 !02

 2    02

1 A

3b 2 c  !02 z  2  02 C

0

 2  02 C

2b 2 !02

3b 2 !02

2 ;

12 31=2 A 7 5

d . / c d 6 D  ı2 @ 4 2 dt z d  2   2 C 3b22  2  02 C 3b 0 !0 !02    3b 2 ı2 2 2  2  2  02 C 3    C 2 2 2 0 cb  !0 !0

  2 3 z . /  2  02 C 3b 2 !

2b 2 !02

 ;

0

then the instantaneous angular frequency of oscillation of the second or Brillouin precursor field is given by

438

13 Evolution of the Precursor Fields

i d hz j˛. /j dt c 

    z d 4 2 b2  2 . /   0  ı '

. / 2 

. / C ./ C c dt 4 20 !04 3     2 2 2 " 3b ı 2b 2 2 2 2 3b 4  3   0 C !02  2 !02   0 C !02

. / 1    2 3 20 !04  2  02 C 3b 2 !0 # 2 4 1  . / 4ı b     2 2 0 !06  2  02 C 3b 2 !0     3b 2 ı2 2b 2 2 2 2 2 b 2  3   0 C !02  2 !02   0 C !02 C   2 3 . /  2  02 C 3b !02   2 2   2ı b    0 

. / 2  . / : 30 !04

!b . / 

  2ı 2 b 2 From (12.225) it is seen that   0  3 4 . / 2  . /   1 for all  > 1 . 0 !0 With this substitution and a bit of algebra, the above expression simplifies to 2

!b . /



b4 . / 41  0 !04

9 8ı 2 2 C !2 0

   3 2 2 2  2 02 C 3b2  15ı2  2 02 C 2b2 !0 !0 !0 5  3 3b 2  2 02 C 2 !0

   3b 2 ı2 2 2  2  2  02 C   C 3  2 2 0 b  !0 !0 C .  1 /   2 3 . /  2  02 C 3b 2 ! 2

2b 2 !02



0

(13.128) for  > 1 , which may be approximated as ˚ !b . / < !SPnC ./ D

. /:

(13.129)

The approximation given in (13.129) also holds in the double resonance case. Although an approximation, the identification of the instantaneous angular frequency of oscillation of the Brillouin prescursor as being given by the real part of the near saddle point location !SPnC ./ in the right half of the complex !-plane for  > 1 is intuitively pleasing and complements the analogous result given in (13.44) for the Sommerfeld precursor. As in that case, the notion of an instantaneous oscillation frequency is only a heuristic mathematical identififcation which, in certain circumstances, may yield completely erroneous or misleading results [22]. Although this is not the case for the Brillouin precursor whose instantaneous oscillation frequency monotonically increases with increasing  > 1 , approaching either the

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

439

q limiting value .1/ D !02  ı 2 for a single resonance Lorentz model dielectric q or the limiting value .1/ D !02  ı02 for a double resonance Lorentz model dielectric as  ! 1, the zero oscillation frequency it predicts for the Brillouin precursor over the initial space–time domain  2 .1; 1 is misleading, leading some to erroneously conclude that the Brillouin precursor is a static field over this space– time domain. It is not for all finite z > 0. A physically meaningful effective oscillation frequency may be defined over the initial space–time domain  2 .1; 1 by the temporal width of the build-up to the peak amplitude point that occurs between the space–time points  D SB and  D 0 . From (12.257),   between these two space–time points is given  the difference by  D 0  SB 4ı 2 b 2 = 30 !04 . This difference corresponds to the effective half-period Teff D 2=!eff of the field over this space–time domain through the relation  D .c=z/.Teff =2/, so that !eff .0 /

30 !04 c : 4ı 2 b 2 z

(13.130)

Notice that this effective angular oscillation frequency of the Brillouin precursor at  D 0 asymptotically approaches zero as z ! 1, in agreement with the limiting behavior as  ! 1C of the asymptotic result given in (13.129). The efficacy of this description is considered in Sect. 13.3.5 for the Heaviside step function signal.

13.3.4 The Delta Function Pulse Brillouin Precursor For an input delta function pulse at time t D 0 the corresponding initial envelope D 0. For space–time values  2 spectrum is given by uQ .!  !c / D i with .1; 1 , the uniform asymptotic approximation given in (13.114) becomes [4, 17, 20]

Aıb .z; t / 

  !02 c 1=3 z ˛0 ./ ec 2b z ( " 

3˛

" 

3˛

20

0 ./C2ı



1˛ ./

1=2

 C

3˛

20

0 ./2ı



1˛ ./

ˇ1=4 ˇ ˇ ˇ  ˇ˛1 ./ˇ Ai ˇ˛1 ./ˇ. cz /1=3 1=2  20 20  

0 ./C2ı

1˛ ./

3˛

0 ./2ı

1=2 #



1=2 #

1˛ ./

ˇ z 1=3  .c=z/1=3 0 ˇˇ ˇ ˇ1=4 Ai ˛1 . /ˇ. c / ˇ˛1 ./ˇ

)

(13.131)

440

13 Evolution of the Precursor Fields

as z ! 1 for all  2 .1; 1 . Accurate approximations of the functions ˛0 ./ and ˛1 . / appearing in this expression are given in (13.106) and (13.113), respectively. At the critical space–time point  D 1 the asymptotic field value is obtained from (13.115) as Aıb .z; t1 / 

 . 13 / p !0 2 3



20 !0 c ˛b 2 z

1=3

 exp

2ız 3˛c

  4ı 2 b 2 0  1 C (13.132) 9˛0 !04

as z ! 1 with  D 1 D ct1 =z. Finally, for space–time values   1 , the uniform asymptotic approximation given in (13.127) becomes [4, 17, 20] 1=2  1=3 ˇ ˇ ˇ  ˇ  c ˇ˛1 ./ˇ1=4 e cz ˛0 ./ Ai ˇ˛1 ./ˇ. z /2=3 c z (13.133) as z ! 1 uniformly for all   1 . A transitional asymptotic approximation (see Sect. 10.3.3) of the Brillouin precursor for the delta function pulse has also been given [28] in order to numerically bridge the small  -interval about the critical space–time point at  D 1 where the the coefficient ˛1 . / may become numerically indeterminate due to a lack of numerical accuracy. This problem is now completely eliminated through the use of accurate numerically determined saddle point locations [27]. Nevertheless, the transitional expansion is useful when analytic approximations for the saddle point locations and the complex phase behavior at them are used (see Problem 13.6). !2 Aıb .z; t /  0 b



20 3˛ ./

13.3.5 The Heaviside Step Function Pulse Brillouin Precursor For a Heaviside unit step function modulated signal with fixed angular carrier frequency !c > 0, the spectrum of the envelope function is given by (11.56) so that 

uQ H !SP ˙ . /  !c n



  ˙ 13 3 0 ./ 2ı . /  i !c i D D  2 (13.134) !SP ˙ ./  !c !c2 C 19 3 0 . / 2ı . / n

for 1 <   1 , and   uQ H !SP ˙ . /  !c D n

   23 ı . / C i ˙ . /  !c i (13.135) D 2 !SP ˙ ./  !c ˙ . /  !c C 49 ı 2 2 . / n

for   1 . At  D 1 , both of these equations simplify to the approximate expression 2ı    i !c uQ H !SPn .1 /  !c '  3˛ : (13.136) 4ı 2 !c2 C 9˛ 2 Analogous expressions hold in the double resonance Lorentz model case.

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

441

For space–time values  2 .1; 1 , the uniform asymptotic approximation given in (13.114) becomes [4, 17, 20] AH b .z; t / 

!02 !c 2b (" 

 1=3 z c e c ˛0 ./ z

1=2 20   3˛ 0 ./ C 2ı 1  ˛ . / 0 ./  2ı ./ 1=2 #  20 1   C  2 3˛ 0 . /  2ı 1  ˛ . / !c2 C 19 3 0 ./ C 2ı ./ ˇ ˇ1=4 ˇ ˇ  ˇ˛1 ./ˇ Ai ˇ˛1 ./ˇ. cz /1=3 " 1=2  20 1     2 3˛ 0 ./ C 2ı 1  ˛ . / !c2 C 19 3 0 ./  2ı ./  1=2 # 20 1     2 3˛ 0 ./  2ı 1  ˛ . / !c2 C 19 3 0 ./ C 2ı ./ ) ˇ z 1=3  .c=z/1=3 0 ˇˇ ˇ ˇ A ˛1 ./ˇ. c / ˇ˛1 ./ˇ1=4 i (13.137)  !c2 C 19 3

1



2

as z ! 1 for all  2 .1; 1 . Accurate approximations of the functions ˛0 ./ and ˛1 . / appearing in this expression are given in (13.106) and (13.113), respectively. At the critical space–time point  D 1 the asymptotic field value is obtained from (13.115) as   4ı 2 b 2 0  1 C 9˛0 !04 (13.138) as z ! 1 with  D 1 D ct1 =z. Finally, for space–time values   1 , the uniform asymptotic approximation in (13.127) becomes [4, 17, 20] AH b .z; t1 / 

 . 13 / !0 !c p 4ı 2 2 3 !c2 C 9˛ 2



20 !0 c ˛b 2 z

1=3



exp

2ız 3˛c

 1=2  1=3 z !2 0 c AH b .z; t /   0 e c ˛0 ./ b 6˛ . / z ( " ˇ1=4 ˇ 2 1  ı ./ˇ˛1 ./ˇ  2 4 3 ./ C !c C 9 ı 2 2 ./ # ˇ  ˇ  1 Ai ˇ˛1 ./ˇ. cz /1=3  2 4 . /  !c C 9 ı 2 2 ./

442

13 Evolution of the Precursor Fields

"

.c=z/1=3 ./  !c ˇ ˇ  2 ˇ˛1 . /ˇ1=4 ./  !c C 49 ı 2 2 ./ ) # ˇ z 1=3   ˇ ./ C !c 0 ˇ ˇ  Ai  ˛1 ./ . c / 2 . / C !c C 49 ı 2 2 ./ (13.139) as z ! 1 uniformly for all   1 . Accurate approximations of the functions ˛0 . / and ˛1 . / appearing in this expression are given in (13.118) and (13.126), respectively. Taken together, (13.137)–(13.139) constitute the uniform asymptotic approximation of the Brillouin precursor for an input Heaviside unit step function modulated signal. For space–time values  2 .1; 1 / with  bounded away from 1 , substitution of the expression given in (13.134) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in (13.76) results in [3]

AH b .z; t / 

!02 !c

"

0 c=.z/   4ı 1  ˛ . / C 6˛

h  2 i b !c2 C 19 3 0 ./  2ı ./ (   z  2ı ./  3 0 ./ 0    exp 3c b2  C 2ı ./  3 180 !04

 h  0 ./ 2ı 3  ˛ . /  3˛

#1=2 0 ./

0 ./

) i

(13.140) as z ! 1. The same nonuniform result is obtained from the uniform asymptotic approximation given in (13.137) with substitution of the dominant term in the large argument asymptotic expansion of both the Airy function and its first derivative [see the pair of expressions preceeding (10.75)]. For  0 , this nonuniform asymptotic approximation simplifies somewhat to AH b .z; t / 

 1=2 !02 !c 0 c i h   2ı 2 6 ˛./z b !c2 C ./  3˛ (   z 2ı  ./ 0    exp c 3˛    ) 2ı b2 4 C  ./ ˛./ C ı (13.141) 3 20 !04 3˛

13.3 The Brillouin Precursor Field in Lorentz Model Dielectrics

443

as z ! 1, where  . /

20 !04 4ı 2 C .0   / 9˛ 2 3˛b 2

1=2 :

The nonuniform asymptotic approximation of the second precursor field for the unit step function modulated signal given in (13.141) is the classical result for the second forerunner obtained by Brillouin [6] for  2 .1; 1 /; see also page 66 of [7]. That result is then seen to be accurate only for values of  near 0 , becoming invalid as  approaches 1 . At the space–time point  D 0 D ct0 =z, at which 0 .0 / D .0 / D 2ı=3˛, both (13.140) and (13.141) simplify to the result !2 AH b .z; t0 /  0 b!c



0 c 4ız

1=2 ;

(13.142)

at which point the Brillouin precursor varies with the propagation distance z > 0 only as z1=2 as z ! 1, making this space–time point in the field evolution entirely unique. The same result is obtained for the uniform asymptotic approximation given in (13.139) for sufficiently large values of z > 0. For space–time values  > 1 with  bounded away from 1 , substitution of the expression given in (13.135) for the input signal envelope spectrum evaluated at the near saddle point locations into the nonuniform asymptotic approximation given in (13.98) results in [3]

AH b .z; t / 

q

20 c 3 ˛ ./z

i h i 2 2 . /  !c C 49 ı 2 2 ./ ./ C !c C 49 ı 2 2 ./ ( " z 2 .  0 / ./  exp  ı c 3 b

h

!02 !c

 b 2  C 1  ˛ . / 4 0 !0

2

 ˛ 4

. /  1 ./ C ı 2 2 ./ 9 3

(  4 2 2 2 2  !c C ı ./  ./ 9   h 4 ./.2˛ .// ı2 b2    C C  cos ./z 0 4 c 3 2 ! 0 0

4 C ı . / . / 3 h   sin ./z 0   C c

ı2 b2 20 !04



4 ./.2˛ .// 3

C

˛

2 ./

˛



ı2

2 ./

ı2



#)

C 4

C 4

i

) i

(13.143)

444

13 Evolution of the Precursor Fields

as z ! 1. This nonuniform asymptotic approximation ˇ ˇreduces to the classical result given by Brillouin [6, 7] if ./ is replaced by ˇ. /ˇ and . / is replaced by ˛ 1 ' 1. However, these two replacements are valid only for space–time points not too distant from 0 . Hence, Brillouin’s classical expression for the second precursor field over the space–time domain  > 1 is an approximation, valid for  near 1 , of an expression that becomes invalid as  approaches 1 from above. As a result, Brillouin’s expression for the asymptotic behavior of the second precursor field (or second forerunner) for the unit step function modulated signal over the space–time domain  > 1 is not applicable. The temporal evolution of the second or Brillouin precursor field AH b .z; t / for a unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s at one absorption depth in a single resonance Lorentz model dielectric with Brillouin’s choice of the model parameters is represented by the solid curve in Fig. 13.9 when numerical saddle point locations are used in the uniform asymptotic approximation given in (13.137)–(13.139). In that case, 0 D 1:5 and 1 1:50275. As evident in Fig. 13.9, the Brillouin precursor field amplitude builds up rapidly as  increases to 0 and then decays with increasing  > 0 . The instantaneous oscillation frequency !b ./ of the Brillouin precursor is also seen to monotonically increase with increasing  > 1. Comparison of this

0.4

0.3

AHb(z,t)

0.2

0.1

0 −0.1 −0.2

1.4

1.5

1.6

1.7

1.8

Fig. 13.9 Temporal evoution of the Brillouin precursor field AH b .z; t / at one absorption depth model dielectric with Brillouin’s choice of z D zd ˛ 1 .!c / in a single resonance Lorentz p the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s. The solid curve results from numerically determined saddle point locations and the dashed curve from the second approximate expressions

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

445

behavior with that depicted in Figs. 13.4 and 13.5 for the Sommerfeld precursor at the same propagation distance in the same medium shows that the peak amplitude of the Brillouin precursor is approximately two orders of magnitude larger than that for the Sommerfeld precursor in this below resonance frequency case. As !c is increased above the angular resonance frequency !0 , the peak amplitude of the Brillouin precursor will diminish and the peak amplitude of the Sommerfeld precursor will increase at any fixed propagation distance z > 0. However, because of its unique z1=2 peak amplitude decay, the Brillouin precursor will eventually dominate the Sommerfeld precursor for sufficiently large observation distances provided that the input pulse spectral energy is nonvanishing in the spectral domain j!j < !0 below resonance. Previously published research [4, 20] exhibited a discontinuity in the uniform asymptotic behavior of the Brillouin precursor about the critical space–time point  D 1 in a Lorentz medium, originally thought to be due to numerical instabilities in the limiting behavior of the coefficients ˛11 ./, h1;2 . /, and h˙ . / for  near 1 .4 To bridge this small space–time neighborhood about  D 1 , a transitional asymptotic approximation (see Sect. 10.3.3) was then used [28] with complete success for the delta function pulse Brillouin precursor (see Problem 13.6) but only with limited success for the Heaviside step function signal case. However, as was pointed out by Cartwright [27, 29], this instability is actually due to the use of an unnecessary approximation of the lower near saddle point location !SPn . / for  2 .1; 1 /. The dashed curve in Fig. 13.9 describes the transitional asymptotic behavior of the Brillouin precursor when the second approximate expressions (13.65)–(13.67) for the near saddle point locations are used, resulting in a discontinuity in the field behavior just prior to the critical space–time point at  D 1 . This discontinuous behavior in the Brillouin precursor evolution is completely eliminated when accurate, numerical saddle point locations are used in the uniform asymptotic expressions without any further approximation, as seen in Fig. 13.9.

13.4 The Brillouin Precursor Field in Debye Model Dielectrics For a single relaxation time Rocard–Powles–Debye model dielectric with complex index of refraction given in (12.125), there is a single near saddle point with location given approximately in (12.305) as [30] # " r 3 !SPn . / i 1  1 C 2 .  0 3 4

(13.144)

At that time (circa 1975), numerical computations of the precursor fields using the second approximate saddle point locations with the appropriate, approximate expressions of the complex phase .!;  / behavior at them were performed in FORTRAN IV using double precision complex arithmetic on the University of Rochester’s IBM 370 computer. Any observed discontinuous behavior in the computed field evolution was then thought to be due to numerical instabilities caused by this limited numerical accuracy.

446

13 Evolution of the Precursor Fields

p for all   0  2 =3 with 0 n.0/ D s , where [cf. (12.306) and

 (12.307)] a0 p =.20 / and .a0 m2 =.20 // p2 .1 C 3s /=.4s m2 /  1 , with p 0 C f 0 and m2 0 f 0 . Numerical results presented in Figs. 12.25–12.30 show that this saddle point moves down the imaginary axis as  increases from the value p 1 1 , crossing the origin at  D 0 and then approaching the upper branch point singularity !p2 D i=0 as  ! 1. The accuracy of this approximation is illustrated in Fig. 12.60. A direct application of Olver’s theorem to the contour integral in (13.10) taken over the contour that results when C is deformed to an Olver-type path through the single near first-order saddle point SPn results in the asymptotic description of the Brillouin precursor in a Debye-type dielectric given by [30] r Ab .z; t / 

) ( c Q .!SPn ./  !c / z .!SPn ;/ i u c e < e 2z . 00 .!SPn ;  //1=2

(13.145)

p as z ! 1 with  > 1 . Unlike that for a Lorentz model dielectric where there are two neighboring near saddle points that coalesce into a single second-order saddle point at some critical space–time point, thereby requiring the application of a uniform asymptotic expansion technique, the asymptotic expression given in (13.145) p is uniformly valid for all finite space–time points  > 1 provided that any pole singularities of the spectral envelope function uQ .!SPn . /  !c / are sufficiently well removed from the near saddle point location lying along the imaginary axis. The dynamical structure of the Brillouin precursor in a single relaxation time Rocard–Powles–Debye model dielectric when the input pulse is a Heaviside unitstep-function signal with fc D 1 GHz carrier frequency at three absorption depths (z D 3zd ) into the simple Rocard–Powles–Debye model of triply distilled water whose frequency dispersion is depicted in Fig. 12.21, where zd ˛ 1 .!c /, is illustrated in Fig. 13.10. The solid curve describes the asymptotic solution given in (13.145) with numerically determined near saddle point locations and the dashed curve describes the asymptotic solution with the approximate saddle point location given in (13.144). Notice that this Brillouin precursor, which is characteristic of Debye-type dielectrics (see Case 2 in Sect. 12.1.1), appears as a single positive pulse with peak amplitude occurring at the space–time point  D 0 D n.0/. This peak amplitude point then propagates with the velocity v0 D c=0 D c=n.0/ through the dispersive material [31]. Since n.0/ > nr .!/ for all real ! > 0, the peak amplitude velocity is the minimum phase velocity for a pulse in the dispersive dielectric. Since !SPn .0 / D 0 and .!SPn .0 /; 0 / D .0; 0 / D 0, (13.145) then shows that [30, 31] ( AB .z; t0 /  < e

i



c uQ .!c / i 4 n0 .0/z

1=2 ) ;

(13.146)

13.4 The Brillouin Precursor Field in Debye Model Dielectrics

447

0.16 0.14

AHb(z,t)

0.12 0.1 0.08 0.06 0.04 0.02 0

8

8.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Fig. 13.10 Temporal evoution of the Brillouin precursor field AH b .z; t / at three absorption depths z D 3zd with zd ˛ 1 .!c / in the simple Rocard–Powles–Debye model of triply distilled water for a Heaviside unit step function modulated signal with fc D 1 GHz carrier frequency. The solid curve describes the temporal behavior when numerically determined saddle point locations are used in the asymptotic approximation and the dashed curve when the approximate near saddle point solution is used

as z ! 1 with t0 D 0 z=c, and the peak amplitude point in the Brillouin precursor p only decays algebraically as 1= z . The instantaneous oscillation frequency of the Brillouin precursor at the peak amplitude point is identically zero; however, this does not mean that the Brillouin precursor is a static field. In fact, the instantaneous oscillation frequency can be quite misleading [22] so that the effective oscillation frequency of the Brillouin precursor needs to be carefully examined. One frequency measure that is physically meaningful is determined by the e 1 points of the exponential function expŒ.z=c/.!SPn . /;  / when !c  0, which are given by the solutions of the equation .!N ./; / D c= z:

(13.147)

Since these points occur about the origin where the peak value in the Brillouin precursor occurs, the complex phase function may be approximated by the first few terms in its Maclaurin series expansion as 1 .!;  / Š .0; / C  0 .0; /! C  00 .0;  /! 2 ; 2

(13.148)

448

13 Evolution of the Precursor Fields

where .0; / D 0,  0 .0; / D i.  0 /, and  00 .0;  / D 2i n0 .0/. As a first approximation, .!;  / i. 0 /!, in which case (13.145) with (13.144) yields the solution pair   a0 .0 C f 0 /c 1=2 ˙ 0 ˙ (13.149) 0 z for z > 0. The temporal width of the Brillouin precursor is then given by TB

  a0 .0 C f 0 / 1=2 z .C   / 2 z ; c 0 c

(13.150)

as z ! 1. This then corresponds to the effective oscillation frequency [30] fB

1 1

2 TB 4



0 c a0 .0 C f 0 /z

1=2 (13.151)

of the Brillouin precursor as z ! 1. These two results then show that the temporal width and oscillation frequency of the Brillouin precursor are set by the material parameters independent of the input pulse for sufficiently large propagation distances z > 0. Notice that the effective oscillation frequency of the Brillouin precursor approaches zero as the propagation distance increases to infinity, in which limit the Brillouin precursor becomes a static field (with zero amplitude), but is nonzero for finite propagation distances.

13.5 The Middle Precursor Field The asymptotic behavior of the middle precursor field in a double resonance Lorentz model dielectric is determined by the phase behavior about the middle saddle points ˙ , j D 1; 2, whose dynamical evolution in the right half of the complex !-plane SPmj C (i.e., the -evolution of SPmj ), illustrated in the sequence of graphs in Figs. 12.12– 12.19, is summarized here in Fig. 13.11. Because of the inherent symmetry of the problem about the imaginary axis, as expressed in the equivalent field representations [from (11.12) and (11.13), and (11.28) and (11.29) with (11.45)] Z iaC1 z 1 fQ.!/e c .!;/ d! A.z; t / D <  ia Z iaC1 z 1 D fQ.!/e c .!;/ d! 2 ia1

(13.152) (13.153)

C for all z  0, just the middle saddle point pair SPmj , j D 1; 2, in the right half plane needs to be considered. The asymptotic description of the middle precursor field Am .z; t / is then obtained from the uniform asymptotic expansion of the contour

13.5 The Middle Precursor Field

449

''

a

+ SPm1

+

SPn

' (0)

SPd+

(1) (2)

(3)

+ SPm2

P( )

''

b

+ SPm1

+

SPn

(0)

' + SPm2

(1)

SPd+ (2)

(3)

P( )

c

''

' SPn+

+ SPm1 (0)

(1)

+

SPm2

SPd+ (2)

(3)

P( )

Fig. 13.11 Illustration of the saddle point evolution in a double resonance Lorentz model dielectric for (a) 1 <  < N1 , (b)  D N1 , and (c)  > N1 in the right half of the complex !-plane. The hatched areas in each plot indicate the local region about each saddle point where the real part  .!;  / of the complex phase function .!;  / is less than that at that saddle point. The shaded region in part b illustrates the local behavior about the effective second-order middle saddle point at C C  D N1 when the two first-order middle saddle points SPm1 and SPm2 come into closest proximity to each other

450

13 Evolution of the Precursor Fields

C integral appearing in (13.152) about the middle saddle point pair SPmj for all  > 1. Because the space–time evolution of this middle saddle point pair is analogous to that for the near saddle point pair, coming into closest proximity to each other at  D N1 , the analysis of their asymptotic contribution is analogous to that given in Sect. 13.3.2 for the Brillouin precursor. Although this middle saddle point pair remains separated at this critical space–time point, they do approach close enough that an effective second-order saddle point is produced at  D N1 , as indicated in part (b) of Fig. 13.11 (also see Fig. 12.16), reinforcing this analogy with the near saddle point behavior. The angle of slope of the steepest descent path P .Ns / through the upper midC as it leaves the effective second-order saddle point is seen dle saddle point SPm1 in Fig. 13.11b to be ˛N s D =6. In addition, with the change ! appearing in (10.74) taken to lie along the portion of the path P .Ns / approaching the effective saddle point from the left in Fig. 13.11b, then arg.f !g/ D 5=6, and (10.74) then states that the argument of v lying along the corresponding portion of the transformed contour [under the cubic change of variable defined in (10.49)] is given by arg.f vg/ D =6 C 5=6 D 2=3, showing that the transformed contour originates in Region 2 of Fig. 10.6. If the change ! is taken to lie along the portion of the path P .Ns / leaving this effective saddle point toward the right in Fig. 13.11b, then arg.f !g/ D =6 so that arg.f vg/ D 0, showing that the transformed contour terminates in Region 1 of Fig. 10.6. Consequently, the contour of integration P . / is transformed into an L21 path so that the function C. / appearing in the uniform expansion given in (10.51) of Theorem 4 is given by

  C. / D e i2=3 Ai e i2=3 where Ai . / denotes the Airy function. For early space–time values  2 .1; N1 , the deformed contour of integration is taken along a set of Olver-type paths passing through the upper near saddle point C , and the distant saddle point SPdC in the SPnC , the upper middle saddle point SPm1 right-half plane, as depicted in Fig. 13.11a. For later space–time values  > N1 , the deformed contour of integration is taken along a set of Olver-type paths passing C C and SPm2 , through the near saddle point SPnC , the pair of middle saddle point SPm1 C and the distant saddle point SPd in the right-half plane, as depicted in Fig. 13.11c. At the critical space–time value  D N1 when the two first-order middle saddle points are in closest proximity to each other, the deformed contour passing through the C is chosen to pass through the effective secondupper middle saddle point SPm1 order saddle point residing between the middle saddle point pair, as depicted in Fig. 13.11b. The analysis leading to (13.104) and (13.117) then applies, so that, beginning ˚ with (13.152) with fQ.!/ D < iei uQ .!  !c / , one obtains

13.5 The Middle Precursor Field

451

 1=3 z c Am .z; t / D  < e i e c ˛N 0 ./ z ( h      i  uQ !SP C  !c hN 1 ./ C uQ !SP C  !c hN 2 ./ C O z1 m1

m2

  e i2=3 Ai ˛N 1 ./e i2=3 .z=c/2=3    .c=z/1=3 h  uQ !SP C  !c hN 1 . /  uQ !SP C  !c hN 2 ./ C 1=2 m1 m2 ˛N 1 . / )   1 i i4=3 0  i2=3 2=3 e Ai ˛N 1 ./e .z=c/ CO z (13.154) as z ! 1 for  > 1, where    1   !SP C C  !SP C ; m1 m2 2        1=3 3 1=2  !SP C   !SP C ˛N 1 . / ; m1 m2 4 2 31=2 1=2 ./ 2 ˛ N 5 ; hN 1;2 . / 4 00  1  !SP C ;  ˛N 0 . /

(13.155) (13.156)

(13.157)

m1;2

the upper sign corresponding to hN 1 ./ and the lower sign to hN 2 . /. The proper values of the multivalued functions appearing in (13.156) and (13.157) are determined by the conditions presented in Sect. 10.3.2. Consider first obtaining these values when  2 .1; N1 /. The phase of hN 1;2 ./ is determined by the limiting behavior given in (10.60) as   lim arg hN 1;2 ./ D ˛N C ;

!N1

(13.158)

where ˛N C D =6 is the angle of slope of the steepest descent path leaving the effective second-order saddle point at  D N1 . However, notice that different values are obtained at the actual middle saddle point locations, as depicted in Fig. 13.12. In particular, the angle of slope of the steepest descent path leaving the upper middle saddle point at  D N1 is equal to =6, as seen in Fig. 13.11b. This then translates into the effective value ˛N C D =6 for the steepest descent path leaving the effective second-order saddle point depicted in Fig. 13.11b, as stated above. The phase of the 1=2 quantity ˛N 1 . / is then obtained from (10.63) with n D 0 as  1=2  lim arg ˛N 1 ./ D ˛N 12  ˛N C ;

!N1

(13.159)

452

13 Evolution of the Precursor Fields

-

a

-

-

arg {h1,2 ( )}

b

-

arg {h1}

-

-

arg {h2}

arg { 1 ( )}

c

Fig. 13.12 Depiction of the  -dependence of (a) the angle of slope ˛N 12 of the vector from the 1=2 C C middle saddle point SPm2 to SPm1 and the arguments of the complex quantities (b) ˛N 1 . /, and N (c) h1;2 . / for the middle saddle point pair in a double resonance Lorentz model dielectric

where ˛N 12 is the angle of slope of the vector from the lower middle saddle point C C to the upper middle saddle point SPm1 . Numerical calculations show that ˛N 12 SPm2 increases from  =2 at  D 1 to   as  increases above N1 , passing through the value  3=4 at  D N1 , as depicted in Fig. 13.12a. The limiting relation stated C then shows that the in (13.159) as applied to the upper middle saddle point SPm1 1=2 proper branch of ˛N 1 . / passes through the value 11=12, increasing from  2=3 at  D 1 to  5=6 as  increases above N1 , as depicted in Fig. 13.12c. However, for the effective middle saddle point, ˛N 12 D =2 and ˛N C D =6 so that  1=2   lim arg ˛N 1 ./ eff D :  N 3 !1

(13.160)

13.5 The Middle Precursor Field

453

The dotted curves in each part of Fig. 13.12 describe this effective behavior. With these results, (13.154) yields the uniform asymptotic approximation of the middle precursor field as Am .z; t / 

 1=3 z c < e i e c ˛N 0 ./ z ( h  ˇ ˇi  ˇ ˇ  uQ !SP C  !c ˇhN 1 ./ˇ C uQ !SP C  !c ˇhN 2 . /ˇ m1

m2

ˇ ˇ  Ai ˇ˛N 1 ./ˇ.z=c/2=3 ˇ ˇi  ˇ ˇ .c=z/1=3 h  ˇN ˇ Q ! C  !c ˇhN 2 . /ˇ ˇ C  !c h1 . /  u ˇ1=2 uQ !SPm1 SPm2 ˇ˛N 1 . /ˇ ) ˇ ˇ  0 ˇ 2=3 ˇ A ˛N 1 ./ .z=c/ ; (13.161) i

as z ! 1 uniformly for all  2 .1; N1 /. In a similar manner, the effective limiting behavior as  ! N1C that is depicted in Fig. 13.12 results in  1=2  5 : lim arg ˛N 1 ./ eff D C 6 !N1

(13.162)

The uniform asymptotic approximation of the middle precursor is then obtained from (13.154) as  1=3 z c Am .z; t /  < e i e c ˛N 0 ./ z ( h  ˇ ˇi  ˇ ˇ  uQ !SP C  !c ˇhN 1 ./ˇ C uQ !SP C  !c ˇhN 2 . /ˇ m1

m2

ˇ  ˇ  Ai ˇ˛N 1 ./ˇ.z=c/2=3 ˇ ˇi  ˇ ˇ .c=z/1=3 h  ˇN ˇ Q ! C  !c ˇhN 2 . /ˇ ˇ C  !c h1 . /  u ˇ1=2 uQ !SPm1 SPm2 ˇ˛N 1 . /ˇ ) ˇ  ˇ  A0 ˇ˛N 1 ./ˇ.z=c/2=3 ; (13.163) i

as z ! 1 uniformly for all   N1 . The temporal evolution of the middle precursor field AH m .z; t / for a Heaviside unit step function modulated signal with angular carrier frequency !c D 2  1016 r=s at five absorption depths in the passband between the two absorption bands of a double resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, p 0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and b0 D

454

13 Evolution of the Precursor Fields 0.015

0.01

AHm(z,t)

0.005 _

0

−0.005

−0.01

−0.015 1.15

1.2

1.25

1.3

1.35

1.4

1.45

Fig. 13.13 Temporal evoution of the middle precursor field Am .z; t / at five absorption depths z D 5zd inp a double resonance Lorentz model dielectric with medium parameters !0 D 1  1016 r=s, b0 D p0:6  1016 r=s, ı0 D 0:1  1016 r=s for the lower resonance line and !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s for the upper resonance line for a Heaviside unit step function modulated signal with angular carrier frequency !c D 2  1016 r=s

p !2 D 7  1016 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s for the upper resonance line is illustrated in Fig. 13.13. Notice that this middle precursor reaches its peak amplitude (which is nearly twice the value e 5 0:0067 for the signal at the applied carrier frequency) near the critical space–time point  D N1 . In addition, notice that the instantaneous oscillation frequency chirps up as  increases to N1 and then chirps down as  increases above this critical critical space–time point with angular frequency that is essentially contained within the passband. Finally, notice the interference between the two middle saddle points for  > N1 .

13.6 Impulse Response of Causally Dispersive Materials A canonical pulse type of fundamental mathematical interest in the description of dispersive pulse propagation phenomena is the Dirac delta function pulse (see Sect. 11.2.1) (13.164) Aı .0; t / D ı.t / whose dynamical evolution yields the impulse response of the dispersive medium. Because the temporal frequency spectrum of this initial pulse function is unity for all !, the propagated plane wave pulse field, given by

13.6 Impulse Response of Causally Dispersive Materials

Aı .z; t / D

1 2

Z

455

e .z=c/.!;/ d!

(13.165)

C

for all z  0, exhibits the pure asymptotic contributions from the saddle points of the complex phase function .!; / at sufficiently large z > 0. The impulse response is then comprised of the precursor fields that are a characteristic of the material dispersion. For a single resonance Lorentz model dielectric the asymptotic behavior of the impulse response is given by [3] Aı .z; t /  Aıs .z; t / C Aıb .z; t /

(13.166)

as z ! 1 for all   1, the propagated wavefield identically vanishing over the entire superluminal space–time domain  < 1. As illustrated in Fig. 13.14, the Sommerfeld precursor Aıs .z; t / arrives with infinite frequency at the speed of light point  D 1 (see Sect. 13.2.5). As  increases above unity, the distant sad0 in from ˙1, approaching the outer branch points !˙ as dle points SPd˙ move ˚  ! 1, so that < ˙ !SP ˙ . / monotonically decreases toward the limiting value d q ˚ !12  ı 2 . At the same time, the attenuative part  .!SP ˙ ;  / < .!SP ˙ ;  / d d of the complex phase function at the distant saddle points monotonically decreases from zero as  increases above unity, resulting in an increase in the wave amplitude attenuation as  increases and the wavefield evolves, as evident in Fig. 13.14. At

2

x 104

Ad (z,t)

1

0

+ SB

−1

−2

1

1.2

1.4

1.6

1.8

2

Fig. 13.14 Numerically determined impulse response of a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) at z D 1 m

456

13 Evolution of the Precursor Fields

the space–time point  D SB , the saddle point dominance changes from the distant to the near saddle points and the Brillouin precursor Aıb .z; t / then dominates the impulse response for all  > SB . As  increases over the space–time interval .SB ; 0 /, the value of  .!SPnC ;  / monotonically increases to zero, vanishes at  D 0 , and then monotonically decreases as  increases above 0 . The field amplitude then experiences zero exponential decay at  D 0 , the amplitude varying with propagation distance z > 0 only as z1=2 at this space–time point, the effective angular oscillation frequency being given by (13.130) over this space–time domain. The instantaneous angular frequency of oscillation of the Brillouin precursor then monotonically increases with increasingq > 1 from the effective value !eff .0 / at

 D 0 , approaching the limiting value !02  ı 2 as  ! 1, the amplitude attenuation also increasing monotonically with increasing  > 0 , as seen in Fig. 13.14. Similar behavior is obtained for a double resonance Lorentz model dielectric if the resonance frequencies !0 and !2 are sufficiently close that the middle saddle ˙ , j D 1; 2, are never the dominant saddle points because p > 0 [see points SPmj (12.116) and (12.117)]. This is illustrated in Fig. 13.15 which depicts the impulse response of a double-resonance medium with model parameters 16 !0 D 1 p 10 r=s; b0 D 0:6  1016 r=s; ı0 D 0:1  1016 r=s;

!2 D 4p 1016 r=s; b2 D 12  1016 r=s; ı2 D 0:1  1016 r=s;

Ad (z,t)

5000

0

−5000 1

SB

1.2

1.4

1.6

1.8

2

Fig. 13.15 Numerically determined impulse response at z D 5 m in a double resonance Lorentz model dielectric when the inequality p > 0 is satisfied

13.6 Impulse Response of Causally Dispersive Materials

457

at the propagation distance z D 5 m. In that case, the asymptotic behavior is described by (13.166) with the transition between the Sommerfeld precursor field dominance and the Brillouin precursor field dominance occuring at the space–time point  D SB ' 1:370. This impulse response is seen to be essentially indistinguishable from that of an equivalent single resonance medium. If the upper resonance frequency in this double resonance medium example is increased to !2 D 7  1016 r=s, then p < 0 and the middle saddle points become the dominant saddle points over the space–time interval  2 .SM ; MB / following the distant saddle point dominance and preceeding the near saddle point dominance, where SM ' 1:201 and MB ' 1:279 for this set of model material parameters. In that case the asymptotic behavior of the impulse response is given by [32] Aı .z; t /  Aıs .z; t / C Aım .z; t / C Aıb .z; t /

(13.167)

as z ! 1 for all   1, the propagated wavefield identically vanishing over the entire superluminal space–time domain  < 1. The impulse response of the medium then contains a middle precursor the evolves between the Sommerfeld and Brillouin precursors, as illustrated in Fig. 13.16. The instantaneous angular frequency of oscillation ofqthis middle precursor first increases as  increases to N1 and then decreases to ! 2  ı 2 as  increases above N1 , whereas its rate of exponential at1

0

tenuation with propagation distance z > 0 first decreases and then increases with increasing  .

Ad (z,t)

5000

0 SM

−5000 1

1.2

MB

1.4

1.6

1.8

2

Fig. 13.16 Numerically determined impulse response at z D 5 m in a double resonance Lorentz model dielectric when the inequality p < 0 is satisfied

458

13 Evolution of the Precursor Fields

A dispersive material of central interest to current research in ultrawideband electromagnetics, particularly to bioelectromagnetics, is water, as this substance is pervasive. The dielectric permittivity of triply distilled water may be expressed as (see Sect. 4.4.5 of Vol. 1) .!/ D or .!/ C res .!/  1;

(13.168)

where or .!/ describes the frequency dependence due to orientational polarization phenomena as described, for example, by the Rocard–Powles extension [33] of the Debye model [34], and where res .!/ describes the frequency dependence due to resonance polarization phenomena as described, for example, by the Lorentz model [35]. The 1 term appearing in this equation is introduced to compensate for the fact that each component model includes a C1 term to describe the vacuum. The multiple resonance Lorentz model description of the dominant resonance features appearing in the measured frequency dispersion of triply distilled water (see Figs. 4.2 and 4.3 of Vol. 1) is given by [see (4.214)] res .!/=0 D 1 

8 X

bj2

.jeve n/

! 2  !j2 C 2i ıj !

(13.169)

j D0

with (rms best fit) parameter values (see Table 4.2) !0 !2 !4 !6 !8

D 2:08  1013 r=s; D 1:05  1014 r=s; D 3:27  1014 r=s; D 6:19  1014 r=s; D 2:30  1016 r=s;

b0 b2 b4 b6 b8

D 4:98  1012 r=s; D 8:48  1013 r=s; D 1:98  1013 r=s; D 1:65  1014 r=s; D 2:01  1016 r=s;

ı0 ı2 ı4 ı6 ı8

D 3:56  1012 r=s; D 4:51  1013 r=s; D 3:08  1012 r=s; D 2:86  1013 r=s; D 5:82  1015 r=s:

The angular frequency dispersion described by this multiple resonance Lorentz model of triply distilled water is given in Fig. 13.17, where the solid curve describes the real part  0 .!/=0 and the dashed curve the imaginary part  00 .!/=0 of the relative dielectric permittivity. The numerically determined impulse response at z D 1 m in this multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water is illustrated in Fig. 13.18, where 0 D

s X 1C .bj =!j /2 ;

(13.170)

j

in this pure resonance case. Notice that this impulse response is almost entirely comprised of a high-frequency Sommerfeld precursor field component followed by a low-frequency Brillouin precursor field component, as described by (13.166). Although the middle precursor never completely appears as the dominant field behavior over some space–time domain in the dynamical field evolution in water, the

13.6 Impulse Response of Causally Dispersive Materials

459

101

Real & Imaginary Parts of

0

'

0

0

10

''

0

−1

10

10−2 1012

0

1013

4

1014

6

8

1015

1016

-r/s

1017

1018

Fig. 13.17 Angular frequency dependence of the real  0 .!/=0 (solid curve) and imaginary  00 .!/=0 (dashed curve) parts of the relative dielectric permittivity as described by the dominant Lorentz model resonance features of triply distilled H2 O

5

x 105

4 3 2

Ad (z,t)

1 0 −1 −2 −3 −4 −5

1

1.2

1.4

1.6

1.8

2

Fig. 13.18 Numerically determined impulse response at z D 1 m in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wavefield amplitude is in arbitrary units

460

13 Evolution of the Precursor Fields

middle saddle points do show their influence on the Brillouin precursor by quenching the oscillatory relaxation of this field component, as seen through a comparison of the field evolution illustrated in Fig. 13.18 with that in Fig. 13.14. This is due to interference between the asymptotic contribution from the near saddle points SPn˙ ˙ and the upper middle saddle points SPm1 for  > 1 whose exponential attenuation is only slightly greater than that for the near saddle points for all  > 1 [36]. As the propagation distance z > 0 increases and the dispersive behavior matures,5 the space–time point at which the peak amplitude in the Brillouin precursor occurs shifts upward to the critical space–time point at  D 0 1:5963 where the wavefield amplitude A.z; t0 /, with t0 D .c=z/0 , only decays with the propagation distance as z1=2 . This is illustrated in Fig. 13.19 which gives the impulse response following the Sommerfeld precursor evolution at the propagation distance z D 1 mm in the multiple resonance Lorentz model description of the frequency dispersion of triply distilled water. The Cole–Cole extension of the multiple relaxation time Rocard–Powles–Debye model of the orientational polarization component in the dielectric permittivity of triply distilled water is given by [see (4.214)]

140 120 100

Ad (z,t)

80 60 40 20 0 −20 1.4

1.6

1.8

2

Fig. 13.19 Numerically determined impulse response at z D 1 mm in the multiple resonance Lorentz model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.17. Notice that the horizontal axis for the wavefield amplitude is in the same units as that in Fig. 13.18

5

In the mature dispersion regime [37], “the field is dominated by a single real frequency at each space–time point. That frequency !E is the frequency of the time-harmonic wave with the least attenuation that has energy velocity equal to z=t .” A more detailed description is given in Chap.16.

13.6 Impulse Response of Causally Dispersive Materials

or .!/=0 D 1 C

461

2 X

aj  1j  : 1  i !fj j D1 1  i !j

(13.171)

With 1 D 0 and 2 D 1=2, the rms best fit parameter values appearing in this model are given by (see Table 4.1) a1 D 74:65; a2 D 2:988;

1 D 8:30  1012 s; 2 D 5:91  1014 s;

f 1 D 1:09  1013 s; f 2 D 8:34  1015 s:

The angular frequency dispersion of the relative dielectric permittivity described by the full Cole–Cole extension of the composite Rocard–Powles–Debye–Lorentz model of triply distilled water, obtained by combining (13.169) and (13.171) in (13.168), is illustrated in Fig. 13.20. Comparison of this behavior with that presented in Fig. 13.17 reveals the influence of the low-frequency orientational polarization response on the high-frequency resonance polarization response of the material dispersion. The numerically determined impulse response at z D 100 m in this composite dispersion model of the dielectric frequency dispersion of triply distilled water is illustrated in Fig. 13.21 over a space–time domain immediately following the Sommerfeld precursor evolution. In this case

2

Real & Imaginary Parts of

0

10

101 '

0

100 ''

0

10−1

−2

10

1010

1012

1014

1016

1018

- r/s Fig. 13.20 Angular frequency dependence of the real  0 .!/=0 (solid curve) and imaginary  00 .!/=0 (dashed curve) parts of the relative dielectric permittivity as described by the Cole–Cole extension of the composite Rocard–Powles–Debye–Lorentz model of triply distilled H2 O

462

13 Evolution of the Precursor Fields 4

1

x 10

0.8 0.6 0.4

Ad (z,t)

0.2 0

−0.2 −0.4 −0.6 −0.8 −1 1.5

2.5

3.5

4.5

5.5

Fig. 13.21 Numerically determined impulse response at z D 100 m in the Cole–Cole extension of the Rocard–Powles–Debye model description of the dielectric frequency dispersion of triply distilled water depicted in Fig. 13.20. Notice that the horizontal axis for the wavefield amplitude is in arbitrary units

0 D

s X X 1C aj C .bj =!j /2 ; j

(13.172)

j

resulting in the approximate value 0 8:9547. The detailed temporal field evolution illustrated in Fig. 13.21 displays the interference between the middle and Brillouin precursor field contributions to the total wavefield evolution. As the propagation distance z > 0 increases and the dispersive wavefield behavior matures, the space–time point at which the peak amplitude in the Brillouin precursor occurs shifts upward to the critical space–time point at  D 0 where the wavefield amplitude A.z; t0 /, with t0 D .c=z/0 , only decays with the propagation distance as z1=2 . The dynamical evolution of the impulse response Aı .z; t / becomes increasingly dominated by the Sommerfeld and Brillouin precursor fields as the propagation distance increases to infinity (i.e., as z ! 1) through the mature dispersion regime.

References ¨ 1. A. Sommerfeld, “Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig), vol. 28, pp. 665–737, 1909. ¨ 2. A. Sommerfeld, “Uber die fortpflanzung des lichtes in disperdierenden medien,” Ann. Phys., vol. 44, pp. 177–202, 1914.

References

463

3. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 4. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 5. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1. ¨ 6. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 7. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 8. P. Debye, “N¨aherungsformeln f¨ur die zylinderfunktionen f¨ur grosse werte des arguments und unbeschr¨ankt verander liche werte des index,” Math. Ann., vol. 67, pp. 535–558, 1909. 9. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Sect. 6.52. 10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 6.222. 11. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. Sect. 6.1. 12. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. New York: MacMillan, 1943. Sect. 5.2. 13. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 14. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 15. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 16. R. A. Handelsman and N. Bleistein, “Uniform asymptotic expansions of integrals that arise in the analysis of precursors,” Arch. Rat. Mech. Anal., vol. 35, pp. 267–283, 1969. 17. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 18. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic theory of pulse propagation in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (Stanford University), pp. 34–36, 1977. 19. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 69, no. 10, p. 1448A, 1979. 20. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 21. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, vol. 55 of Applied Mathematics Series. Washington, D.C.: National Bureau of Standards, 1964. 22. L. Mandel, “Interpretation of instantaneous frequencies,” Am. J. Phys., vol. 42, no. 10, pp. 840– 846, 1974. 23. S. He and S. Str¨om, “Time-domain wave splitting and propagation in dispersive media,” J. Opt. Soc. Am. A, vol. 13, no. 11, pp. 2200–2207, 1996. 24. A. Karlsson and S. Rikte, “Time-domain theory of forerunners,” J. Opt. Soc. Am. A, vol. 15, no. 2, pp. 487–502, 1998. 25. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Phil. Soc., vol. 53, pp. 599–611, 1957. 26. K. E. Oughstun and N. A. Cartwright, “Ultrashort electromagnetic pulse dynamics in the singular and weak dispersion limits,” in Progress in Electromagnetics Research Symposium, (Prague, Czech Republic), 2007. 27. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Rev., vol. 49, no. 4, pp. 628–648, 2007. 28. J. A. Solhaug, K. E. Oughstun, J. J. Stamnes, and P. Smith, “Uniform asymptotic description of the Brillouin precursor in a single-resonance Lorentz model dielectric,” Pure Appl. Opt., vol. 7, no. 3, pp. 575–602, 1998.

464

13 Evolution of the Precursor Fields

29. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 30. K. E. Oughstun, “Dynamical evolution of the Brillouin precursor in Rocard-Powles-Debye model dielectrics,” IEEE Trans. Ant. Prop., vol. 53, no. 5, pp. 1582–1590, 2005. 31. K. E. Oughstun, “Dynamical evolution of the precursor fields in linear dispersive pulse propagation in lossy dielectrics,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 257–272, New York: Plenum Press, 1994. 32. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 33. J. McConnel, Rotational Brownian Motion and Dielectric Theory. London: Academic Press, 1980. 34. P. Debye, Polar Molecules. New York: Dover Publications, 1929. 35. H. A. Lorentz, The Theory of Electrons. Leipzig: Teubner, 1906. Chap. IV. 36. J. E. K. Laurens and K. E. Oughstun, “Electromagnetic impulse response of triply-distilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 243–264, New York: Plenum Press, 1999. 37. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981.

Problems 13.1. Derive the second order approximations given in (13.21) and (13.22) of the complex phase behavior at the distant saddle points of a double resonance Lorentz model dielectric. 13.2. Derive the uniform asymptotic approximation of the Sommerfeld precursor for a Heaviside unit step function signal in a double resonance Lorentz model dielectric using the second approximate expressions for the distant saddle point locations. 13.3. Derive the general expression for the uniform asymptotic expansion of the Sommerfeld precursor in a single resonance Lorentz model dielectric when the input pulse is given by A.0; t / D f .t / [see (12.1)]. 13.4. Derive the asymptotic approximation given in (13.45) of the delta function pulse Sommerfeld precursor Aıs .z; t / in either a single or double resonance Lorentz model dielectric. 13.5. Derive (13.88) and (13.89) for the approximate phase behavior at the secondorder near saddle point in a double resonance Lorentz model dielectric when  D 1 . 13.6. Obtain the uniform asymptotic approximation for the Brillouin precursor in a double resonance Lorentz model dielectric in the separate space–time domains  2 .1; 1 and   1 . Show that each of these uniformly valid expressions reduces to the asymptotic approximation given in (13.90) at the critical space–time point  D 1 when the two first-order near saddle points have coalesced into a single second-order saddle point.

Problems

465

13.7. Apply the asymptotic method presented in Sect. 10.3.3 to obtain the transitional asymptotic approximation of the delta function pulse Brillouin precursor Aıb .z; t / in a single resonance Lorentz model dielectric. 13.8. Derive the asymptotic description given in (13.145) of the Brillouin precursor field Ab .z; t / in a single relaxation time Rocard-Powles-Debye model dielectric.

Chapter 14

Evolution of the Signal

The contribution Ac .z; t / to the asymptotic behavior of the propagated plane wavefield A.z; t / that is due to the presence of any simple pole singularities of the spectral function uQ .!  !c /, where A.0; t / D u.t / sin .!c t C / with fixed angular carrier frequency !c  0, is now condidered in some detail, with primary attention given to the oscillatory case when !c > 0. As discussed in Sect. 12.4, the field component Ac .z; t / is associated with any long-term signal that is being propagated through the dispersive material. The velocity of propagation of the signal and the transition from the total precursor field to the signal field are determined by the relative asymptotic dominance of the component fields As .z; t /, Ab .z; t /, Am .z; t /, and Ac .z; t /. Consequently, discussion of these topics is deferred to Chap. 15 where the asymptotic description of the dynamical evolution of the total propagated wavefield A.z; t / is considered by combining the results of this chapter with those of Chap. 13. The first section of this chapter presents the nonuniform asymptotic analysis based on the direct application of Olver’s method [1] and the Cauchy residue theorem [2, 3]. Even though the nonuniform expression exhibits a discontinuous change in behavior as the space–time parameter  ct =z varies, it is a useful approximation for the pole contribution Ac .z; t / in the final expression for the total wavefield A.z; t / for all  > 1 provided that the dominant saddle point remains isolated from the pole, because in that case, Ac .z; t / is asymptotically negligible in comparison to the precursor field at the space–time point when the discontinuity occurs. This is precisely the situation for Debye-type dielectrics as the only saddle point in that case is the near saddle point SPn that moves down the imaginary axis as  increases, crossing the origin at  D 0 n.0/. The nonuniform approximation may then be applied in this case provided that the pole is sufficiently removed from the origin. When applied to the unit step function modulated signal in a single-resonance Lorentz model dielectric, the nonuniform expression for the pole contribution Ac .z; t / is the same as that obtained by Brillouin [4, 5] except for the space–time value s at which the discontinuous change occurs. For those cases in which the nonuniform result is useful, the difference in values of the space–time point at which the discontinuity occurs is of no consequence in the final expression for the total wavefield A.z; t / because Ac .z; t / is asymptotically negligible during a space–time interval that includes both values. Although Brillouin attached physical K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 14, 

467

468

14 Evolution of the Signal

significance to the space–time value  D s at which this discontinuous change occurs, calling ts D .z=c/s the time of arrival of the signal, it is shown in Chap. 15 that the time of this discontinuous change has no physical significance and that the signal arrives at a later time. The nonuniform expression for the pole contribution Ac .z; t / is not useful in those cases in which the dominant saddle point passes near the pole singularity because Ac .z; t / is then not negligible at the space–time point s at which the discontinuity occurs. For such cases, it is necessary to apply the uniform asymptotic expansion technique due to Bleistein [6, 7] and Felsen and Marcuvitz [8], as summarized in Theorem 5 of Sect. 10.4, in order to obtain an asymptotic expression for the pole contribution Ac .z; t / that is uniformly valid in  . That analysis, as extended by Cartwright [9, 10], is presented in Sect. 14.3. The remaining sections apply this uniform asymptotic description to the analysis of the simple pole contribution for a Heaviside unit step function modulated signal in both Lorentz-type dielectrics and conducting media.

14.1 The Nonuniform Asymptotic Approximation This section obtains the nonuniform asymptotic approximation of the field component Ac .z; t / due to the contribution of any pole singularities appearing in the integrand of the Fourier–Laplace integral representation of the propagated plane wavefield [see (11.48)] A.z; t / D



Z 1 < i e i uQ .!  !c /e .z=c/.!;/ d! 2 C

(14.1)

as z ! 1. For that purpose, let !p denote a simple pole singularity of the spectral function uQ .!  !c /e i with residue p given by

 p D lim .!  !p /Qu.!  !c /e i :

(14.2)

!!!p

In accordance with the asymptotic procedure presented in Sect. 12.4, each of the functions Cd˙ .z; t /, Cm˙ .z; t /, and Cn˙ .z; t / appearing in (12.353) and (12.354) is zero in the nonuniform asymptotic approximation, so that ˚ Ac .z; t / D < 2 i./ ;

(14.3)

where . / D

X p

Res ! D !p



i uQ .!  !c /e i e .z=c/.!;/ 2

X i p e .z=c/.!p ;/ D 2 p

(14.4)

14.1 The Nonuniform Asymptotic Approximation

469

is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P ./ that is specified in Sect. 12.4 (see Fig. 12.66). For reasons of simplicity, it is assumed here that the deformed contour of integration P . / is near only one pole at a time, attention then being restricted to obtaining the nonuniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for multiple pole contributions. The contribution of the simple pole singularity at ! D !p occurs when the original contour of integration C , extending along the straight line path from i a1 to i a C 1 in the upper half of the complex !-plane, is deformed across the pole to P . /. More specifically, assume that there is only one pole and let the original contour C and the deformed contour P ./ lie on the same side of the pole for  < s and on opposite sides for  > s . Then, from (14.4), one has that . / D 0; z i .s / D p e c .!p ;s / ; 4 z i . / D p e c .!p ;s / ; 2

for  < s ; for  D s ;

(14.5)

for  > s :

Upon substitution of this set of results into (14.3), one immediately obtains the nonuniform asymptotic approximation for the simple pole contribution at ! D !p as

Ac .z; t /  0;  < s ; h    i 1 z Ac .z; t /  e c .!p ;s / p0 cos cz $ .!p ; s /  p00 sin cz $ .!p ; s / ;  D s ; 2 h    i z Ac .z; t /  e c .!p ;/ p0 cos cz $ .!p ;  /  p00 sin cz $ .!p ;  / ;  > s ; (14.6) as z ! 1, where p0 s ; (14.12)

as z ! 1 for real-valued !p > 0. It is then seen that, in this special case, the simple pole contribution results in a field contribution that is oscillating with fixed angular frequency !p and with an amplitude that is attenuated with propagation distance z > 0 at a constant, time-independent rate given by the attenuation coefficient ˛.!p / [compare this result with the general expression given in (14.6)]. The exact integral representation of the propagated plane wavefield given in (14.1) is a continuous function of the space–time parameter  ct =z for all z  0, and, in particular, is continuous at the space–time point  D s . However, the resulting asymptotic approximation of A.z; t / is a discontinuous function of  at  D s when the pole contribution Ac .z; t / is nonvanishing (i.e., when p ¤ 0) and is given by the nonuniform asymptotic approximation in (14.6).1 The discontinuity is of no consequence for fixed values of the propagation distance z larger than some positive constant Z, however, because the contribution to the propagated wavefield from the dominant saddle point at !SP varies exponentially as e .z=c/.!SP ;/ which dominates2 the exponential behavior of the pole contribution given in the second part of (14.6) at  D s . Hence, at the space–time point  D s when the discontinuity in the asymptotic behavior of the propagated wavefield occurs, the pole contribution is asymptotically negligible in comparison to the saddle point contribution, and, as a consequence, the discontinuous behavior is itself asymptotically negligible. For that reason, the particular value of s at which the pole crossing occurs is of little or no importance to the asymptotic behavior of the total propagated wavefield A.z; t /. 1

The value of s depends upon which Olver-type path is chosen for P . /. If that path is taken to lie along the path of steepest descent that passes through the saddle point nearest the pole, then the value of s is specified by the relation $ .!SP ; s / D $ .!p ; s /. 2 The asymptotic dominance of the saddle point contribution over the pole contribution at  D s is guaranteed by the fact that P .s / is an Olver-type path.

14.2 Rocard–Powles–Debye Model Dielectrics

471

14.2 Rocard–Powles–Debye Model Dielectrics Because the Debye-type dielectric has just the single near saddle point SPn that p moves down the imaginary axis as  increases from 1 D 1 , crossing the origin p at  D 0 s and then approaching the branch point !p1 D i=0 as  ! 1, as depicted in Fig. 14.1 (see also Figs. 12.25–12.30), the pole contribution at the real angular signal frequency !c > 0 will remain isolated from this saddle point for all   1 provided that !c is not too small. In that case, the pole contribution is given by (14.12) with s determined by the steepest descent path P . / passing through the near saddle point SPn into the right half of the complex !-plane. Because this steepest descent path is parallel to the real ! 0 -axis as it leaves the saddle point, s ' 0 ;

(14.13)

the accuracy of this approximation improving as the value of !c > 0 decreases. Notice further that (14.14) s  0 ; as evident from the sequence of illustrations in Fig. 14.1. For a Heaviside unit step function modulated signal with fixed angular carrier frequency !c > 0, the initial pulse spectrum is given by [from (11.56)] uQ H .!  !c / D i=.!  !c / which possesses a simple pole singularity at !p D !c with residue   i D i: (14.15)  D lim .!  !c / !!!c .!  !c / With this substitution, (14.12) becomes AHc .z; t /  0;

 < s ;

  1 AHc .z; t /   e z˛.!c / sin ˇ.!c /z  !c ts ;  D s ; 2   AHc .z; t /  e z˛.!c / sin ˇ.!c /z  !c t ;  > s ;

(14.16)

as z ! 1. The accuracy of this asymptotic approximation of the pole contribution may be assessed by first numerically computing the propagated wavefield at a fixed distance z > 0 into a particular Rocard–Powles–Debye model dielectric and then subtracting the asymptotic behavior of the Brillouin precursor [using (13.145) with numerically determined near saddle point locations] at that same propagation distance in the same dielectric, resulting in the numerical estimate of the pole contribution AnHc .z; t / AH .z; t /  AH b .z; t /;

(14.17)

which can then be compared with that described by the sequence of expressions in (14.16). An example of this calculation is presented in Figs. 14.2 and 14.3 for a fc D 1 GHz Heaviside unit step function signal at one absorption depth [z=zd D 1 with zd ˛ 1 .!c /] in the single relaxation time Rocard–Powles–Debye model of

472

14 Evolution of the Signal

a

ω“

SP

ω‘ P(θ)

branch cut

ωa

b

ω“

SP

ω‘

branch cut

P(θ)

c

ω“

SP

ω‘

branch cut

P(θ)

Fig. 14.1 Evolution of the steepest descent path P . / through the near saddle point in a single relaxation-time Debye-model dielectric. In part (a) 1 <  < 0 , (b)  D 0 , and (c)  > 0 . The short dashed curves are isotimic contours of  .!;  / below the value  .!SPn ;  / at the saddle point and the alternating long and short dashed curves are isotimic contours above that value. The shaded area in each part indicates the region of the complex !-plane where the inequality  .!;  / < .!SPn ;  / is satisfied

14.2 Rocard–Powles–Debye Model Dielectrics

473

0.5 AH(z,t)

0 8

A(z,t)

AHb(z,t)

−0.5

RP

2

4

6

8

10

12

14

16

18

20

Fig. 14.2 Numerically determined Heaviside step function AH .z; t / evolution with 1GH z carrier frequency (solid curve) and the asymptotic behavior of the associated Brillouin precursor AH b .z; t / (dashed curve) at one absorption depth in H2 O

0

−0.5

8

AHc(z,t)

0.5

2

RP

4

6

8

10

12

14

16

18

20

Fig. 14.3 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at one absorption depth in H2 O

474

14 Evolution of the Signal

triply distilled water described by (12.300). Both the numerically determined wavefield A.z; t / and the asymptotic description of the Brillouin precursor field Ab .z; t / are presented in Fig. 14.2, the former by the solid curve and the latter by the dashed curve. Notice that the leading edge peak in the total wavefield A.z; t / is primarily due to the Brillouin precursor field.3 The resultant estimation of the pole contribution, given by the difference between these two wavefields as expressed by (14.17), is presented in Fig. 14.3. This result is in keeping with the nonuniform asymptotic approximation of the signal contribution Ac .z; t / given in (14.16). In particular, the pole contribution is seen to occur at the space–time point  D s ' 0 , as stated in (14.13). Similar results are obtained as the propagation distance increases, as illustrated in Fig. 14.4 when z D 3zd and in Fig. 14.5 when z D 5zd . Notice that the relative peak amplitude of the leading edge of the estimated signal (or pole) contribution AHc .z; t / increases with increasing propagation distance. Unfortunately, this numerically observed phenomenon is not described by the nonuniform asymptotic description given in (14.16). Nevertheless, the accuracy of the asymptotic description of the total wavefield evolution increases as z ! 1.

14.3 The Uniform Asymptotic Approximation If the saddle point at !sp . / approaches close to the pole singularity at !p when  D s so that the quantity j!sp .s /  !p j becomes small, then the quantity j.!sp .s /; s /  .!p ; s /j also becomes small. In that case the positive constant Z introduced at the end of Sect. 14.1 becomes increasingly large. As a result, it becomes impractical to take z > Z in order to make the pole contribution Ac .z; t / asymptotically negligible at  D s . To avoid the discontinuous behavior of Ac .z; t / at the space–time point  D s when the saddle point is near the pole, the uniform asymptotic approximation stated in Theorem 5 (see Sect. 10.4) due to Felsen and Marcuvitz [8, 11] and Bleistein [6, 7, 12], and later extended by Cartwright [9, 10] must be applied. That asymptotic technique is employed in this section in order to obtain the uniform asymptotic approximation of the pole contribution Ac .z; t / in both Lorentz model dielectrics and Drude model conductors. From (12.353) and (12.354) in Sect. 12.4, the pole contribution in a double resonance Lorentz model dielectric is given by Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C CnC .z; t /;

3

(14.18)

The numerically determined wavefield A.z; t / in Fig. 14.2, as well as that used in obtaining the results presented in Figs. 14.3–14.5, has been slightly shifted in space–time by a fixed amount (  D 0:060) that is determined by the requirement that the peak amplitude point in A.z; t / for a sufficiently large propagation distance occurs at the same space–time point ( D 0 ) as that described by the asymptotic description of the Brillouin precursor.

14.3 The Uniform Asymptotic Approximation

475

AHc(z,t)

0.1

0

−0.1

8

9

10

11

12

Fig. 14.4 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at three absorption depths (z=zd D 3) in a single relaxation-time Debye model of H2 O

0.02

AHc(z,t)

0.01

0

−0.01

8

9

10

11

12

Fig. 14.5 Estimated signal contribution AHc .z; t / D AH .z; t /  AH b .z; t / for a 1GH z Heaviside step function signal at five absorption depths (z=zd D 5) in a single relaxation-time Debye model of H2 O

476

14 Evolution of the Signal

for 1    1 , and Ac .z; t / D < f2 i./g C Cd .z; t / C CdC .z; t / CCm .z; t / C CmC .z; t / C Cn .z; t / C CnC .z; t /;

(14.19)

for  > 1 , where . /, which is given by (14.4), is the sum of the residues of the poles that are crossed when the original contour of integration C is deformed to the path P . /. For either a single resonance Lorentz model dielectric or a Drude model conductor, the terms Cm˙ .z; t / are set equal to zero in these two expressions. For reasons of simplicity, it is again assumed that the deformed contour of integration P . / is near only one pole at a time (i.e., that the poles are isolated), and attention is restricted to obtaining the uniform asymptotic contribution due to that single pole alone. The results obtained are easily generalized to account for several individual pole contributions. By Theorem 5 in Sect. 10.4, the C -functions appearing in (14.18) and (14.19) are given by ( "  p  z 1 < i  ˙ i erfc i . / z=c e c .!p ;/ C.z; t / D 2 #) p c=z z .!sp ;/ ec C ; . /

˚ > = . / < 0;

(14.20)   p z z 1 < i  i erfc i . / z=c e c .!p ;/  i e c .!p ;/ C.z; t / D 2 #) p ˚ c=z z .!sp ;/ C ; = . / D 0; ./ ¤ 0; ec . / (

"

(14.21)  1=2 2c 1 C.z; t / D  < i   00 2 z .!sp ;  / )    000 .!sp ;  / z 1 .! ;/ sp ec  C 00 ; ./ D 0; !sp  !c 6 .!sp ;  / (

(14.22) as z ! 1, where !sp denotes the location of the interacting saddle point. Here erfc. / D 1  erf. / denotes the complementary error function defined in Sect. 10.4.1. The particular form of the C -function to be employed depends upon the sign of the imaginary part of the quantity . /, which is defined by [cf. (10.89)]

1=2 ./ .!sp ;  /  .!p ;  / :

(14.23)

14.3 The Uniform Asymptotic Approximation

477

The proper argument of this square root expression is determined by the limiting relation given in (10.93) [see also (10.90)] as lim

!p !!sp ./

 arg . / D ˛N c  ˛N sd C 2n;

(14.24)

where ˛N c is the angle of slope of the vector from !sp to !p in the complex !-plane, ˛N sd is the angle of slope of the tangent vector to the path of steepest descent at the interacting saddle point, and where n is an integer value which is chosen such that the argument of ı. / lies within the principal domain .;  for all   1. Consider first the case in which either one of the distant saddle points SPd˙ approaches the pole singularity at ! D !p that is located in a region of the complex !-plane bounded away from the limiting values ˙1  2ıi approached by !SP ˙ . / d

as  ! 1C , respectively. Specifically, let SPdC approach the point ! D !p which is assumed here to be the only pole singularity of the spectral function uQ .! !c /. Then CdC .z; t / is given by (14.20)–(14.22) with !SP C ./ substituted for !sp throughout, d and the remaining C -functions appearing in (14.18) and (14.19) are asymptotically negligible in comparison to both CdC .z; t / and the residue contribution. Similarly, if the distant saddle point SPd approaches the simple pole singularity at ! D !p , then Cd .z; t / is given by (14.20)–(14.22) with !SPd ./ substituted for !sp throughout, and the remaining C -functions appearing in (14.18) and (14.19) are asymptotically negligible in comparison to both Cd .z; t / and the residue contribution. Consider next the case in which one of the near saddle points SPnC for  2 Œ1; 1

or either SPn˙ for  > 1 approaches the pole singularity at ! D !p . Specifically, let SPnC approach the point ! D !p , which is again assumed to be the only pole singularity of the spectral function uQ .!  !c /. Then CnC .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.18) are asymptotically negligible in comparison to both CnC .z; t / and the residue contribution. If the near saddle point SPnC approaches the simple pole singularity at ! D !p when  > 1 , then CnC .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.19) are asymptotically negligible in comparison to both CnC .z; t / and the residue contribution. Similarly, if the near saddle point SPn approaches the simple pole singularity at ! D !p when  > 1 , then Cn .z; t / is given by (14.20)–(14.22) with !SPnC . / substituted for !sp throughout, and the remaining C -functions appearing in (14.19) are asymptotically negligible in comparison to both Cn .z; t / and the residue contribution. ˙ approaches Consider now the case when one of the middle saddle points SPmj C the pole singularity at ! D !p . Specifically, let SPm1 approach the point ! D !p , which is again assumed to be the only pole singularity of the spectral function uQ .!  !c /. Then CmC .z; t / is given by (14.20)–(14.22) with !SP C . / substituted m1 for !sp throughout, and the remaining C -functions appearing in (14.18) are asymptotically negligible in comparison to both CmC .z; t / and the residue contribution. If C approaches the point ! D !p , then CmC .z; t / is given the middle saddle point SPm2

478

14 Evolution of the Signal

by (14.20)–(14.22) with !SP C ./ substituted for !sp throughout, and the remainm2 ing C -functions appearing in (14.18) are asymptotically negligible in comparison to both CmC .z; t / and the residue contribution. Analogous results hold for the middle  in the left-half plane. saddle points SPmj Finally, for the case in which none of the saddle points approaches close to the pole singularity at ! D !p , then the two nearest saddle points to the pole through which the deformed contour of integration passes must be considered through application of Corollary 2 in Sect. 10.4.3. This situation may occur, for example, when the simple pole at ! D !p is situated near a branch cut of the complex phase 0 function .!;  /. Specifically, let !p be situated just above the branch cut !C !C associated with the resonance frequency !0 in a single resonance Lorentz model dielectric. The deformed contour of integration P ./ passing through both the distant saddle point SPdC and the near saddle point SPnC then interacts with this pole with both of these saddle points remaining isolated from it for all   1. In that case, both CdC .z; t / and CnC .z; t / are given by (14.20)–(14.22) with !SP C ./ and !SPnC . / d substituted for !sp throughout, respectively, the remaining C -functions appearing in (14.18) being asymptotically negligible in comparison to both CdC .z; t /, CnC .z; t /, and the residue contribution. Analogous results hold for Drude model conductors as well as for each branch cut in a multiple resonance Lorentz model dielectric. A case of special interest is that for which the pole singularity !p of the spectral function uQ .!  !c / is real and positive. The complex phase behavior .!;  /  i ! n.!/   at the simple pole singularity at ! D !p is then given by   .!p ; / D !p ni .!p / C i !p nr .!p /   ;

(14.25)

˚  .!p / < .!p ;  / D !p ni .!p /;  ˚  $ .!p ; / = .!p ;  / D !p nr .!p /   ;

(14.26) (14.27)

so that

where nr .!p / and ni .!p / denote the real and imaginary parts of the complex index of refraction, respectively [see, for example, (12.74) and (12.75)]. The saddle points in the left half of the complex !-plane then do not interact with the pole at ! D !p , and hence, their contributions Cj , j D d; m; n, are negligible. Attention is now focused on the pole contribution in a single resonance Lorentz model dielectric.

14.4 Single Resonance Lorentz Model Dielectrics Because the real coordinate location of the near saddle point SPnC in the right q half of the complex !-plane lies within the below resonance domain 0  ! 0 

!02  ı 2

and the real coordinate location of the distant saddle point SPdC in the right-half

14.4 Single Resonance Lorentz Model Dielectrics

479

q plane lies within the above absorption band domain ! 0  !12  ı 2 for all   1, the uniform asymptotic description of the pole contribution in a single resonance Lorentz model dielectric separates naturally into three cases. For real-valued q angular frequency values !p in the below absorption band domain 0  !p 

!02  ı 2 ,

SPnC

the near saddle point will interact with the simple pole singularity at ! D !p . In that case, the function CnC .z; t / appearing in (14.18) and (14.19) may be significant, the remaining C -functions being asymptotically negligible q by comparison. For values of !p in the above absorption band domain !p 

!12  ı 2 , the dis-

tant saddle point SPdC interacts with the simple pole singularity at ! D !p . In that case, the function CdC .z; t / appearing in (14.18) and (14.19) may be significant, the remaining C -functions being asymptotically negligible by comparison. q q Finally, for values of !p within the absorption band, so that !02  ı 2 < !p < !12  ı 2 , neither the near nor distant saddle points paases in close proximity to the simple pole singularity at ! D !p . In that case, both of the functions CdC .z; t / and CnC .z; t / appearing in (14.19) may be significant, the remaining C -functions being asymptotically negligible by comparison. The uniform asymptotic description of the pole contribution in each of these three cases is now treated in detail.

14.4.1 Frequencies below the Absorption Band For real-valued q angular frequency values !p in the below absorption band domain

0  !p  !02  ı 2 , it is the near saddle point SPnC in the right half of the complex !-plane that interacts with the simple pole singularity at ! D !p . The set of uniform asymptotic expressions given in (14.20)–(14.22) then apply to CnC .z; t / with !sp denoting the near saddle point location !SPnC ./ for all  > 1. Furthermore, the quantity ./ is given by (14.23) with either numerically determined near saddle point locations or with the second approximate expressions [from (13.74), (13.85), and (13.95)]  1 2ı . /  3 0 ./ .0  / 3  2  b2  2ı ./  3 0 ./ 2ı 3  ˛ . / C 3˛ C 4 540 !0 1 <  < 1 ;  2 2  2ı 4ı b 0  1 C ;  D 1 ; .!SPn ; 1 /

3˛ 9˛0 !04

.!SPnC ; /

0 ./

 ;

(14.28) (14.29)

480

14 Evolution of the Signal

(

  2 b2 

. /.  0 / C 1  ˛ . / 2 . / 4 3 0 !0  )  4 2 2 1 ˛ . /  1 C ı ./ 9 3 (  )   4 2 b2 2 Ci ./ 0   C ı . / 2  ˛ . / C ˛ . / ; 20 !04 3

.!SPnC ; / ı

 > 1 :

(14.30)

The argument of ./ must now be determined by the limiting expression given in (14.24), taken in the limit as !p approaches the saddle point location, with the integer n chosen such that this argument lies within the principal domain .; /. A sequential depiction of the near saddle point SPnC interaction with the simple pole singularity at ! D !p with !p bounded away from the origin along the positive real ! 0 -axis is given in Fig. 14.6. It is then seen that =2  ˛N c <  for all positive, real values of !p for all  > 1, where ˛N c increases monotonically with increasing  > 1. Furthermore, as described in the derivation of the uniform asymptotic description of the Brillouin precursor in Sect. 13.3.2, the angle of slope ˛N sd of the path of steepest

a

b

''

sd

SPn

c

''

P

=

P

P

c

'

p

''

SPn

sd = =0 c

p

'

'

p c

sd =

SPn

d

e

''

f

''

P

P s

c

SPn

p

'

p sd

=

P

'

p

c= sd =

SPn s

''

s

SPnC

' c

sd

=

SPn s

Fig. 14.6 Interaction of the near saddle point with the simple pole singularity at ! D !p located along the real ! 0 -axis when !p is near the upper end of the below absorption band domain q 0  !p  !02  ı 2 . The shaded area in each diagram of this  -sequence indicates the region of the complex !-plane where the inequality  .!SPnC ;  / > .!;  / is satisfied

14.4 Single Resonance Lorentz Model Dielectrics

481

descent through the near saddle point SPnC is equal to 0 as  increases from unity to 1 , ˛N sd D =6 at  D 1 , and ˛N sd D =4 as  increases above 1 . Notice that, although the value of ˛N sd changes abruptly at the critical space–time point  D 1 , the path of steepest descent through this near saddle point varies in a continuous  fashion  with  for all  > 1. Substitution of these results in (14.24) then yields arg . / D     =2 for 1 <  < s , arg .s / D 0, and arg . / D 3=4 for  > s . Notice   q that s > 0 for all !p 2 0; !02  ı 2 and that s D 0 for !p D 0. Upon application of (14.18)–(14.21), the uniform  contribution of the  qasymptotic simple pole singularity at ! D !p with !p 2 0; For  < s , =f ./g < 0 and (14.20) gives

!02  ı 2 is given as follows.

q #) c q   z z z 1 .! C ;/ c .! ;/ z SPn Ac .z; t /  ; < i   i erfc i . / c e c p C e 2 . / (

"

 < s ;

(14.31)

as z ! 1. At the space–time point  D s D cts =z, =f . /g D 0 and (14.21) gives q #) c q   z z z 1 .! ; / s C Ac .z; ts /  < i  i erfc i .s / cz e c .!p ;s / C e c SPn 2 .s / n z o C< e c .!p ;s / ;  D s ; (14.32) (

"

as z ! 1 with !p ¤ 0. Because the argument of the complementary error function is pure imaginary at  D s ,R this equation can be expressed in terms of Dawson’s  integral FD ./ exp .2 / 0 exp . 2 /d [see (10.97)] as q #) ( " c q   z p z 1 .! ; / C s c z SP n Ac .z; ts /  < i e  2 FD j .s /j c C 2 .s / o n z 1  D s ; (14.33) C < e c .!p ;s / ; 2 as z ! 1 with !p ¤ 0. For  > s , =f . /g > 0 and (14.20) gives q #) c q  z z z 1 c .!SP C ;/ .!p ;/ z c n Ac .z; t /  C ; < i  i erfc i . / c e e 2 . / o n z  > s ; (14.34) C< e c .!p ;/ ; (

as z ! 1.

"



482

14 Evolution of the Signal

For the special case when !p D 0, the upper near saddle point SPnC crosses the simple pole singularity at ! D !p when  D 0 , so that s D 0 and .0 / D 0. For  < 0 , =f ./g < 0 and the uniform asymptotic behavior of the pole contribution is given by (14.31). For  > 0 , =f . /g > 0 and the uniform asymptotic behavior of the pole contribution is given by (14.34). At  D s D 0 , (14.22) applies for CnC .z; t / appearing in (14.18). Because the path P .0 / crosses over the pole at !p D 0, then z 1 1 i i e c .0;0 / D ; (14.35) .0 / D 2 2 4 where .0; 0 / D 0. Hence, in this special case the simple pole contribution at  D s D 0 is given by 8 " #1=2 " #9  000 .!SPnC ; 0 / = 2c 1 < 1 Ac .z; t0 /  < i C 00 2 : z 00 .!SPnC ; 0 / !SPnC .0 / 6 .!SPnC ; 0 / ; 1 C s , =f . /g > 0 and (14.20) gives q #) c q  z .! C ;/ z z 1 c .! ;/ z SP d < i  i erfc i . / c e c p C e Ac .z; t /  ; 2 . / o n z  > s ; (14.43) C< e c .!p ;/ ; (

"



as z ! 1. Taken together, (14.40)–(14.43) constitute the uniform asymptotic approximation of q the pole contribution Ac .z; t / due to the simple pole at ! D !p with

!12  ı 2 . For space–time values  < s and sufficiently large propagap tion distances z > 0 such that the quantity j . /j z=c  1 is sufficiently large, the dominant term in the asymptotic expansion of the complementary error function  p  erfc i . / z=c may be substituted into (14.40) with the result that the first and second terms in that equation identically cancel. Hence, for space–time values  > 1 that are sufficiently less than s , there is no contribution to the asymptotic behavior of the total wavefield A.z; t / from the simple pole singularity. On the other hand, for space–time values  > p s and sufficiently large propagation distances z > 0 term in such that the quantity j ./j z=c  1 is sufficiently large, the dominant  p  the asymptotic expansion of the complementary error function erfc i . / z=c may be substituted into (14.43) with the result !p 

o n z Ac .z; t /  < e c .!p ;/

     e ˛.!p /z  0 cos ˇ.!p /z  !p t   00 sin ˇ.!p /z  !p t

(14.44)

as z ! 1 with  > s bounded away from s . These two limiting results are in agreement with the nonuniform asymptotic approximation given in the first and third parts of (14.12), where the amplitude attenuation coefficient ˛.!p / is given in (14.9) and the phase propagation factor ˇ.!p / is given in (14.11).

14.4.3 Frequencies within the Absorption Band For real-valued angular frequency q values !p within q the absorption band of the 2 2 Lorentz model dielectric, so that !0  ı < !p < !12  ı 2 , both the near SPnC

486

14 Evolution of the Signal

and distant SPdC saddle points interact with the simple pole singularity at ! D !p . It is then important to determine which saddle point’s steepest descent path crosses the pole and thereby determines the space–time value  D s > 1 at which this crossing occurs. Because the steepest descent path through a given saddle point is detemined by the imaginary part $ .!;  / =f.!; /g of the complex phase function, the space–time point s at which this crossing occurs must satisfy the relation $ .!sp ; s / D $ .!p ; s /;

(14.45)

where !sp denotes the relevant saddle point. The relevant saddle point whose steepest descent path sets the value of s has been shown [9] to be determined by the value of $ .!p ; / at the initial space–time point  D 1. This can be seen by considering of the imaginary part of the complex phase function .!; /   the behavior  i ! n.!/   along the positive real ! 0 -axis, where $ .! 0 ;  / D nr .! 0 /   ! 0 . For a single resonance Lorentz model dielectric, there exists a real angular frequency value !$ within the absorption band such that nr .! 0 / > 1 for 0  ! 0 < !$ and nr .! 0 / < 1 for finite ! 0 > !$ , where5 nr .!$ / 1;

q q !02  ı 2 < !$ < !12  ı 2 :

(14.46)

It is then seen that $ .!p ; 1/ > 0 for 0  ! 0 < !$ and that $ .!p ; 1/ < 0 for finite ! 0 > !$ . Because of this behavior, if $ .!p ; 1/ > 0, then the steepest descent path passing through the near saddle point SPnC and crossing the positive real ! 0 -axis determines the value of s through (14.45). If $ .!p ; 1/ < 0, then the steepest descent path passing through the distant saddle point SPdC and crossing the positive real ! 0 -axis determines the value of s through (14.45). Finally, if $ .!p ; 1/ D 0, then s D 1. The resultant uniform asymptotic approximation of the pole contribution is then determined through a direct application of Corollary 2 due to Cartwright [9,10]; see Sect. 10.4.3. The uniform asymptotic description of the pole contribution at ! D !p when $ .!p ; 1/  0 is then given by 2

3 q c z .! ;/   p z z 1 C c 6 7 SPd < i  4i erfc i d ./ cz e c .!p ;/ C e Ac .z; t /  5 2 d ./ (

2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C

q

c z

n . /

3 e

z c .!SP C ;/ n

 < s ; 5

7 5

)

(14.47)

As an example, consider the real part of the complex index of refraction for a single resonance Lorentz model dielectric illustrated in Fig. 12.2 for Brillouin’s choice of the medium parameters. Equation (14.46) is then found to be satisfied when !$ ' 4:2925  1016 r=s.

14.4 Single Resonance Lorentz Model Dielectrics

487

2

3

q

( c z  p z  z .!p ;/ z 1 c .!SP C ;/ 7 6 c d Ac .z; t /  C < i  4i erfc i d ./ c e e 5 2 d . / 2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C

q

n . /

n z o C< e c .!p ;/ as z ! 1 with

3

c z

e

z c .!SP C ;/ n

  s ;

7 5

)

(14.48)

q !02  ı 2 < !p  !$ . Here

1=2 j . / .!SP C ;  /  .!p ;  /

(14.49)

j

for j D d; n. The uniform asymptotic description of the pole contribution at ! D !p when $ .!p ; 1/  0 is given by 2

(

Ac .z; t / 

q

3

c z

 p  z 1 6 < i  4i erfc i d ./ cz e c .!p ;/ C e 2 d ./ 2 

6 Ci  4i erfc i n ./

pz c

q e

z c .!p ;/

C

z c .!SP C ;/ d

3

c z

n . /

e

z c .!SP C ;/ n

 < s ; 2

(

Ac .z; t / 

q

c z

 p  z 1 6 < i  4i erfc i d ./ cz e c .!p ;/ C e 2 d . / 2  p  z 6 Ci  4i erfc i n ./ cz e c .!p ;/ C o n z C< e c .!p ;/

q

c z

n . /

)

3 z c .!SP C ;/ d

z c .!SP C ;/ n

  s ;

7 5

(14.50)

3 e

7 5

7 5

7 5

)

(14.51)

q as z ! 1 with !$  !p < !12  ı 2 , where j . / for j D d; n is given by (14.49). The set of expressions given in (14.47) and (14.48), and (14.50) and (14.51) constitute the uniform asymptotic approximation of the pole contribution Ac .z; t / due

488

14 Evolution of the Signal

to the simple pole singularity at ! D !p when !p is situated within the absorption band of a single resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either (14.38) or (14.39) for zero for  < s , for sufficiently large propagaspace–time values  > s , and yield p tion distances z > 0 such that j . /j z=c  1.

14.4.4 The Heaviside Unit Step Function Signal For a Heaviside unit step function signal f .t / D uH .t / sin .!c t /, the initial pulse envelope spectrum [see (11.56)] uQ H .!  !c / D i=.!  !c / possesses a simple pole singularity at the applied signal (or carrier) angular frequency !c  0, with residue   D lim

!!!c

i .!  !c / !  !c

 D i:

(14.52)

Substitution of this result into (14.31)–(14.34) with !p set equal to !c then yields the uniform asymptotic descriptionqof the pole contribution for below resonance 

angular signal frequencies !c 2 0; !02  ı 2 as 9 q c > = q  z z z 1 .! C ;/ AHc .z; t /  ; < i erfc i . / cz e c .!c ;/  e c SPn > 2 ˆ . / ; : 8 ˆ
< = q   z z z 1 .! C ;/ c .! ;/ z c SP n < i erfc i . / c e c e ;  AHc .z; t /  > 2 ˆ . / : ;

AHc .z; ts / 

  e ˛.!c /z sin ˇ.!c /z  !c t ;

 > s ;

(14.55)

1=2 as z ! 1 with !c ¤ 0, where . / .!SPnC ;  /  .!p ;  / . If !c D 0, in which case s D 0 , (14.54) must be replaced by !2 AHc .z; t0 /  0 2b

s

1 0 c ; ız !SPnC .0 /

 D s D 0 ;

(14.56)

14.4 Single Resonance Lorentz Model Dielectrics

489

0.4

AHc(z,t)

0.2

0 s

c

−0.2

−0.4 1.4

1.6

1.8

2

2.2

Fig. 14.8 Temporal evoution of the pole contribution AHc .z; t / at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz p model dielectric with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) for a Heaviside unit step function modulated signal with below resonance angular carrier frequency !c D 1  1016 r=s

as z ! 1 with fixed 0 D ct0 =z. Again, notice that even though the term 1=!SPnC .0 / is singular because !SPnC .0 / D 0, this expression for the pole contribution Ac .z; t0 /, when combined with the asymptotic approximation of the Brillouin precursor field (which is also singular at  D 0 , but with the opposite sign to the singularity appearing in the above expression), yields a uniform asymptotic asymptotic approximation of the total wavefield A.z; t / that is well behaved at  D 0 . The dynamical evolution of the pole contribution AHc .z; t / as described by (14.53)–(14.55) for a Heaviside unit step function signal is illustrated in Fig. 14.8 at one absorption depth z D zd ˛ 1 .!c / in a single resonance Lorentz model dielectric choice of the medium parameters (!0 D 4  1016 r=s, p with Brillouin’s 16 b D 20  10 r=s, ı D 0:28  1016 r=s) with below resonance angular carrier frequency !c D 1  1016 r=s. This field componment was calculated using numerically determined near saddle point locations. Comparison of this field component with that given in Figs. 13.4 and 13.9 shows that, in this particular case, the peak amplitude in the pole contribution is two orders of magnitude greater than that for the Sommerefeld precursor field and is the same order of magnitude as that for the Brillouin precursor field. Both precursor fields diminish in amplitude for smaller propagation distances as the pole contribution increases to that of the initial wavefield A.0; t / D uH .t / sin .!c t /. However, as the propagation distance increases above a single absorption depth, the peak amplitudes of both precursor fields will decrease at a slower rate with z than that for the pole contribution so that either one or both of them dominate the propagated field evolution.

490

14 Evolution of the Signal

The uniform asymptotic description of the pole contribution for a Heaviside unit step function signal with finite, above absorption band angular carrier frequency q !c  !12  ı 2 is obtained from (14.40)–(14.43) with substitution from (14.52) as 8 ˆ
= q  z z z 1 c .!SP C ;/ .! ;/ z c d < i erfc i . / c e c e AHc .z; t /  C ; > 2 ˆ . / : ; 8 ˆ
=

q  z  z z 1 c .!SP C ;s / d AHc .z; ts /  < i erfc i .s / cz e c .!c ;s / C e > 2 ˆ .s / : ;

  e ˛.!c /z sin ˇ.!c /z  !c t ;  D s ; (14.58) 8 9 q c ˆ > < = q   z z z 1 c .!SP C ;/ d < i erfc i . / cz e c .!c ;/ C e ; AHc .z; t /  > 2 ˆ . / : ;   e ˛.!c /z sin ˇ.!c /z  !c t ;  > s ; (14.59) as z ! 1. The dynamical evolution of the pole contribution AHc .z; t / as described by (14.57)–(14.59) for a Heaviside unit step function signal is illustrated in Fig. 14.9 at three absorption depths z D 3zd in a single resonance Lorentz model dielectric choice of the medium parameters (!0 D 4  1016 r=s, p with Brillouin’s 16 b D 20  10 r=s, ı D 0:28  1016 r=s) with above absorption band angular carrier frequency !c D 1  1017 r=s. This propagated wavefield componment was calculated using numerically determined near saddle point locations. Because the distant saddle point SPdC passes within close proximity to the simple pole singularity at ! D !c when  D s in this above absorption band case, the magnitude of the quantity ./ becomes small about this space–time point, resulting in the appearance of resonance peak in the pole contribution, as seen in Fig. 14.9. A similar resonance peak also appears in the Sommerfeld precursor field for this case, illustrated in Fig. 14.10, but with the opposite sign. When added together to construct the total wavefield, these two resonance peaks destructively interfere and cancel each other out, as described in Sect. 15.2. Consider finally the case when the carrier frequency !c of the Heaviside unit step within the absorption function signal A.0; t / D uH .t / sin .!c t / is situated q q band of

the single resonance Lorentz model dielectric, so that !02  ı 2 < !c < !12  ı 2 . The uniform asymptotic description then separates into two cases that are dependent upon the sign of the quantity $ .!c ; 1/. From (14.47) and (14.48) with the substitution  D i , the uniform asymptotic description of the pole contribution at ! D !c when $ .!c ; 1/  0 is given by [9, 10]

14.4 Single Resonance Lorentz Model Dielectrics

491

0.06

0.04

AHc(z,t)

0.02

0 s

c2

c1

−0.02

−0.04

−0.06 1

1.1

1.2

1.3

1.4

Fig. 14.9 Temporal evoution of the pole contribution AHc .z; t / at z D 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band angular carrier frequency !c D 1  1017 r=s

0.06

0.04

AHS(z,t)

0.02

0

−0.02

−0.04

−0.06 1

1.1

1.2

1.3

1.4

Fig. 14.10 Temporal evoution of the Sommerfeld precursor AHS .z; t / at z D 3zd in a single resonance Lorentz model dielectric for a Heaviside unit step function modulated signal with above absorption band carrier frequency !c D 1  1017 r=s

492

14 Evolution of the Signal

q

( c z  p z  z .!c ;/ z 1 c .!SP C ;/ c d AHc .z; t /  C < i erfc i d ./ c e e 2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn e Ci erfc i n ./ c e c C n . /  < s ; (14.60) q

(

c

z  p  z z 1 c .!SP C ;/ d AHc .z; t /  < i erfc i d ./ cz e c .!c ;/ C e 2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn erfc i n ./ c e c C e n ./   e ˛.!c /z sin ˇ.!c /z  !c t ;   s ; (14.61)

as z ! 1 with

q !02  ı 2 < !c  !$ . Here

1=2 j . / .!SP C ;  /  .!c ;  /

(14.62)

j

for j D d; n. The uniform asymptotic description of the pole contribution at ! D !c when $ .!c ; 1/  0 is obtained from (14.50) and (14.51) as [9, 10] (

q

c

z  p  z z 1 c .!SP C ;/ d < i erfc i d ./ cz e c .!c ;/ C e AHc .z; t /  2 d . / q ) c z  p z  z .!c ;/ z .! C ;/ c SPn e i erfc i n ./ c e c C n . /

(

AHc .z; t / 

q

 < s ; (14.63) c z

z  p  z 1 c .!SP C ;/ d <  i erfc i d ./ cz e c .!c ;/ C e 2 d ./ q ) c z  p z  z .!c ;/ z .! C ;/ c SP n e i erfc i n ./ c e c C n . /   e ˛.!c /z sin ˇ.!c /z  !c t ;   s ; (14.64)

as z ! 1 with !$  !c < (14.62).

q !12  ı 2 , where j ./ for j D d; n is given by

14.4 Single Resonance Lorentz Model Dielectrics

493

The accuracy of this uniform asymptotic description of the pole contribution has been thoroughly investigated beginning with the earlier work of Smith et al. [16,17] and culminating in the recent work by Cartwright et al. [9, 10]. An accurate numerical estmate of the pole contribution can be determined by first computing the total field evolution AH .z; t / at a fixed propagation distance z > 0 and then subtracting the uniform asymptotic approximations of the Sommerfeld and Brillouin precursor fields from it, resulting in the numerical estimate AnHc .z; t / D AH .z; t /  AH s .z; t /  AH b .z; t /. For greatest accuracy, numerically determined saddle point locations are used in each of the uniform asymptotic field descriptions. The above and below absorption band cases all yield expected results, the rms error between the asymptotic approximation and numerical estimate of the pole contribution decreasing monotonically with increasing propagation distance z  zd , all in keeping with the asymptotic sense of Poincar´e’s definition (see Definition 5 of Appendix F). For the intra-absorption band case, consider first the on-resonance carrier frequency !c D !0 example, which satisfies the condition that $ .!c ; 1/ > 0. A comparison of the asymptotic (dashed curve) and numerical estimate (solid curve) of the pole contribution at the fixed propagation distance z D 21:3zd in a single resonance Lorentz model dielectric p with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, ı D 0:28  1016 r=s) is presented in Fig. 14.11, where the open circle in the figure indicates the critical space–time point  D s when the steepest descent path P ./ through the near saddle point SPnC

6

x 10

−3

4

AHc(z,t)

2

0

−2

−4

−6

1

2

3

4

5

6

7

8

Fig. 14.11 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with resonant angular carrier frequency !c D !0 at z  21:3zd (from Fig. 7.9 of Cartwright [9]). In this case, $ .!c ; 1/ > 0

494

14 Evolution of the Signal −4

10

x 10

8

RMS error

6

4

2

0 0

2

4 z (m)

6

8 −8

x 10

Fig. 14.12 RMS error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with resonant angular carrier frequency !c D !0 as a function of the propagation distance z (from Fig. 7.10 of Cartwright [9])

crosses the pole at ! D !c . Notice that the high-frequency ripple in the asymptotic result (the dashed curve in the figure) is due to the contribution from the distant saddle point SPdC in (14.60) and (14.61), as can easily be ascertained by eliminating this contribution from this uniform asymptotic description [9]. As seen in Fig. 14.12, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z  zd . Similar remarks apply to the intra-absorption band example !c D 1:25!0 illustrated in Fig. 14.13, which satisfies the condition $ .!c ; 1/ < 0. As seen in Fig. 14.14, the rms error between this asymptotic approximation and the numerical estimate of the pole contribution decreases monotonically with increasing propagation distance z  zd .

14.5 Multiple Resonance Lorentz Model Dielectrics The asymptotic analysis of the pole contribution at ! D !p in a double resonance Lorentz model dielectric naturally separates into five separate frequency domains the low-frequency, normally dispersive below resowhen !p  0 is real-valued: 

q 2 nance domain !p 2 0; !0  ı02 , the high-frequency, normally dispersive above q resonance domain !p > !32  ı22 , the intermediate frequency, normally dispersive

14.5 Multiple Resonance Lorentz Model Dielectrics

495

0.04

AHc(z,t)

0.02

0

−0.02

−0.04

1

2

3

4

5

q

6

7

8

9

10

Fig. 14.13 Comparison of the numerical (solid curve) and uniform asymptotic (dashed curve) pole contributions for a Heaviside step function signal with above resonance angular carrier frequency !c D 1:25!0 at z  21:3zd (from Fig. 7.11 of Cartwright [9]). In this case, $ .!c ; 1/ < 0

10

x 10−4

RMS error

8

6

4

2

0 0

0.2

0.4

0.6 z (m)

0.8

1

1.2 x 10−7

Fig. 14.14 RMS error between the uniform asymptotic and numerical pole contributions for a Heaviside step function signal with above resonance angular carrier frequency !c D 1:25!0 as a function of the propagation distance z (from Fig. 7.12 of Cartwright [9])

496

14 Evolution of the Signal

q  q 2 passband !p 2 !1  ı02 ; !22  ı22 , and the two anomalously dispersive abq q  

q 2

q 2 !0  ı02 ; !12  ı02 and !p 2 !2  ı22 ; !32  ı22 . sorption bands !p 2 The uniform asymptotic approximation ofqthe simple pole contribution Ac .z; t /   in the below resonance domain !p 2 0; !02  ı02 is given by (14.31)–(14.34) with (14.36) substituted for (14.33) in the special case when !p D 0. The uniform asymptotic q approximation of the pole contribution in the above resonance domain

!p > !32  ı22 is given by (14.40)–(14.43). In the lower absorption band !p 2 q 

q 2 !0  ı02 ; !12  ı02 , the uniform asymptotic approximation of the pole contribution is given by either (14.47)–(14.48) when $ .!p ; 1/ > 0 or (14.50) and (14.51) when $ .!p ; 1/ < 0 with the distant saddle point SPdC replaced by the middle saddle q 

q 2 C point SPm1 throughout. In the upper absorption band !p 2 !2  ı22 ; !32  ı22 , the uniform asymptotic approximation of the pole contribution is given by either (14.47)–(14.48) when $ .!p ; 1/ > 0 or (14.50) and (15.51) when $ .!p ; 1/ < 0 C throughwith the near saddle point SPnC replaced by the middle saddle point SPm2 out. q q   !12  ı02 ; !22  ı22 between the Attention is now focused on the passband

C two absorption bands. Because the middle saddle point SPm1 crosses the real ! 0 axis at the angular frequency value ! D !˛mi n where the absorption is a minimum within that passband, this crossing occuring at the space–time point  D ˛mi n , the uniform asymptotic description of the pole contribution at ! D !p includes the case where .˛mi n / D 0 when !p D !˛mi n . As an illustration, for the double resonance Lorentz model dielectric example considered in Sect. 12.3.2, it is seen in Fig. 12.56 that !˛mi n 2:6  1016 r=s, which is in good agreement with the numerically determined angular frequency value !˛mi n ' 2:6283  1016 r=s. C C and SPm2 in the right-half plane then interact with Both middle saddle points SPm1 the simple pole at ! D !p . The resultant uniform asymptotic description of this pole contribution is then determined through a direct application of Corollary 2 (see  q 2 Sect. 10.4.3) to each of the three cases !p 2 !1  ı02 ; !˛mi n , !p D !˛mi n , and q   !p 2 !˛mi n ; !22  ı22 . Theq uniform asymptotic description of the pole contribution at ! D !p when   !p 2 !12  ı02 ; !˛mi n is given by

14.5 Multiple Resonance Lorentz Model Dielectrics

497

2

q

2

q

3

( c z  p z  z .!p ;/ z 1 6 c .!SP C ;/ 7 c m2 Ac .z; t /  C < i  4i erfc i 2 ./ c e e 5 2 2 . /  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

c z

1 ./

3 e

z c .!SP C ;/ m1

 < s ; 2

(

Ac .z; t / 

q

7 5

)

(14.65) 3

c z

z  p  z 1 6 c .!SP C ;/ 7 m2 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 5 2 2 . /

2  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

q

c z

1 ./

o n z C< e c .!p ;/

3 e

z c .!SP C ;/ m1

  s ;

7 5

)

(14.66)

as z ! 1, where

1=2 j . / .!SP C ;  /  .!p ;  / mj

(14.67)

for j D 1; 2. The uniform asymptotic description of the pole contribution at ! D !p when !p D !˛mi n is given by (14.65) and (14.66) when  ¤ s . When  D s , where s D ˛mi n , the pole contribution is obtained from (14.22) as 2

3 q c z   .! ;/ p z z 1 C 6 7 c SPm2 e Ac .z; ts / D  < i  4i erfc i 2 ./ cz e c .!p ;/  5 2 2 ./ (

"

#1=2 2c Ci   00 z .!SP C ; s / m1 " ) # z  000 .!SP C ; s / .! C ;s / 1 c m1 SPm1  C 00 ; e !SP C  !˛mi n 6 .!SP C ; s / m1 m1 o 1 n z (14.68) C < e c .!˛mi n ;s / 2

498

14 Evolution of the Signal

as z ! 1 with fixed s D ˛mi n D cts =z. Notice that even though the term .!SP C  m1

!˛mi n /1 is singular at  D s , this expression for the pole contribution Ac .z; ts /, when combined with the asymptotic approximation of the middle precursor field, yields a uniform asymptotic approximation of the total wavefield A.z; t / that is well behaved at  D s . The uniform description of the pole contribution at ! D !p when q asymptotic   !p 2 !˛mi n ; !22  ı22 is given by 2

(

Ac .z; t / 

q

3

c z

 p  z 1 6 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 2 2 . / 2  p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C

q

z c .!SP C ;/ m2

3

c z

1 ./

e

z c .!SP C ;/ m1

 < s ; 2

(

Ac .z; t / 

q

c z

 p  z 6 Ci  4i erfc i 1 ./ cz e c .!p ;/ C o n z C< e c .!p ;/

q

c z

1 ./

)

(14.69)

z c .!SP C ;/ m2

3 e

7 5

3

 p  z 1 6 < i  4i erfc i 2 ./ cz e c .!p ;/ C e 2 2 . / 2

7 5

z c .!SP C ;/ m1

  s ;

7 5

7 5

)

(14.70)

as z ! 1, where j . / for j D 1; 2 is given by (14.67). The set of expressions given in (14.65), (14.66), and (14.68)–(14.70) constitute the uniform asymptotic approximation of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is situated within the passband of a double resonance Lorentz model dielectric. These uniform asymptotic expressions reduce to the nonuniform result given in either (14.38) or (14.39) for space–time values  > s , and yield zero p for  < s , for sufficiently large propagation distances z > 0 such that j ./j z=c  1. A similar analysis holds in each additional passband of a multiple resonance Lorentz model dielectric. Numerical illustrations are given in the next chapter where the uniform asymptotic description of the total propagated wavefield behavior is constructed. With reference to the middle saddle point dynamics depicted in Figs. 12.56 and 13.11, as well as to (14.23) and (14.24), the argument of j ./ is determined in the following manner. The angle of slope ˛N sd1 of the tangent vector to the path of C is found to vary over the domain steepest descent at the middle saddle point SPm1

14.6 Drude Model Conductors

499

˛N sd1  ! =6 as  D 1 ! N1 and then over the domain ˛N sd1 =6 ! =4 as  D N1 ! 1. Because the angle of slope ˛N c1 of the vector from !SP C to m1 !p is seen to vary over the domain ˛N c1 =2 ! =2 as  D 1 ! 1, it then follows from (14.23) with n D 0 that argŒ 1 ./ D 3=2 ! 3=4 as  D 1 ! 1. On the other hand, the angle of slope ˛N sd2 of the tangent vector to the path of C is found to vary over the domain steepest descent at the middle saddle point SPm2 ˛N sd2 =2 ! =3 as  D 1 ! N1 and then over the domain ˛N sd2 =3 ! =4 as  D N1 ! 1. Because the angle of slope ˛N c2 of the vector from !SP C to !p is m2 seen to vary over the domain ˛N c2 =2 !  as  D 1 ! 1, it then follows from (14.23) with n D 0 that argŒ 2 ./ D 0 ! 3=4 as  D 1 ! 1. Finally, notice that when !p D !˛mi n , lim!˛mi n ˛N c1 D =2 whereas lim!˛C ˛N c1 D C=2. mi n

14.6 Drude Model Conductors   The complex phase function .!; / D i ! n.!/   for a Drude model conductor with complex index of refraction [see (12.153)] !1=2 !p2 (14.71) n.!/ D 1  !.! C i  / possesses a pair of distant saddle points SPd˙ given by [see (12.309)–(12.311)] !SP ˙ ./ D ˙./  i d

  1 C . / 2

(14.72)

that beginqat ˙1  i  at  D 1 and move into the respective branch point zeros !z˙ D ˙ !p2  .=2/2  i =2 as  ! 1 and a single near saddle point SPn that moves down the positive imaginary axis, approaching the branch point singularity !pC D 0 as  ! 1, as described by the approximate expression given in (12.317). The Sommerfeld precursor in a Drude model conductor is then very similar to that for a Lorentz model dielectric whereas the Brillouin precursor is nonoscillatory and hence, more like that in a Debye model dielectric but with a long exponential tail due to the asymptotic approach of the near saddle point to the origin as  ! 1 [18,19]. Let the deformed contour of integration P ./ through the near and distant saddle points be composed of the set of Olver-type paths with respect to each saddle point such that, within a neighborhood about each saddle point, the Olver-type path is taken along the path of steepest descent through that saddle point. The space–time point  D s when the contour P ./ crosses the pole at ! D !p along the positive real ! 0 -axis is then defined by the equation $ .!sp ; s / D $ .!p ; s /;

(14.73)

500

14 Evolution of the Signal

where $ .!;  / =f.!;  /g, which may then be used to determine which saddle point interacts with the pole. First of all, because the near saddle point SPn is situated along the imaginary axis for all   1, then $ .!SPn ;  / D 0 for all   1. At the distant saddle point SPdC in the right-half plane, $ .!SP C ;  /  0 is equal d to zero at  D 1 and then decreases monotonically with increasing  > 1. Because  $ .!p ; / D !p nr .!p /   when !p is real-valued [cf. (14.27)], it follows that the near saddle point SPn interacts with the pole when !p 2 .0; !$ / whereas the distant saddle point SPdC interacts with the pole when !p  !$ , where the finite, real-valued angular frequency value !$ is defined by the relation (see Fig. 12.34) nr .!$ / D 1:

(14.74)

The uniform asymptotic description of the pole contribution at ! D !p is then given by q #) c q   z .! C ;/ z z 1 c .! ;/ z SP d < i  ˙ i erfc ˙i . / c e c p C e Ac .z; t /  ; 2 . / (

"

(

"

q

 < s ;

(14.75) #)

c q  z  z z 1 c .!SP C ;s / d < i  i erfc i .s / cz e c .!p ;s / C e 2 .s / n z o C< e c .!p ;s / ;  D s D cts =z; (14.76) q ( " #) c q  z  z .! C ;/ z 1 c .! ;/ SPd < i  i erfc i . / cz e c p C e ; Ac .z; t /  2 . / o n z  > s ; (14.77) C< e c .!p ;/ ;

Ac .z; ts / 

as z ! 1, where

1=2 ./ .!sp ;  /  .!c ;  //

(14.78)

with !sp D !SPn . / when 0 < !p < !$ and !sp D !SP C . / when !p  !$ . d Numerical illustrations are given in the next chapter where the uniform asymptotic behavior of the total propagated wavefield is constructed.

References 1. F. W. J. Olver, “Why steepest descents?,” SIAM Rev., vol. 12, no. 2, pp. 228–247, 1970. 2. E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable. London: Oxford University Press, 1935. p. 110. 3. E. C. Titchmarsh, The Theory of Functions. London: Oxford University Press, 1937. Sect. 10.5.

Problems

501

¨ 4. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 5. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 6. N. Bleistein, “Uniform asymptotic expansions of integrals with stationary point near algebraic singularity,” Com. Pure Appl. Math., vol. XIX, no. 4, pp. 353–370, 1966. 7. N. Bleistein, “Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities,” J. Math. Mech., vol. 17, no. 6, pp. 533–559, 1967. 8. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973. 9. N. A. Cartwright, Uniform Asymptotic Description of the Unit Step Function Modulated Sinusoidal Signal. PhD thesis, College of Engineering & Mathematical Sciences, University of Vermont, 2004. 10. N. A. Cartwright and K. E. Oughstun, “Uniform asymptotics applied to ultrawideband pulse propagation,” SIAM Review., vol. 49, no. 4, pp. 628–648, 2007. 11. L. B. Felsen and N. Marcuvitz, “Modal analysis and synthesis of electromagnetic fields,” Polytechnic Inst. Brooklyn, Microwave Res. Inst. Rep., 1959. 12. N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston, 1975. 13. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 14. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 15. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 16. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 17. K. E. Oughstun and P. D. Smith, “On the accuracy of asymptotic approximations in ultrawideband signal, short pulse, time-domain electromagnetics,” in Proceedings of the 2000 IEEE International Symposium on Antennas and Propagation, (Salt Lake City), pp. 685–688, 2000. 18. S. Dvorak and D. Dudley, “Propagation of ultra-wide-band electromagnetic pulses through dispersive media,” IEEE Trans. Elec. Comp., vol. 37, no. 2, pp. 192–200, 1995. 19. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in an isotropic collisionless plasma,” in 2007 CNC/USNC North American Radio Science Meeting, 2007.

Problems 14.1. Show that the uniform asymptotic expressions given in (14.31)–(14.34) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated below the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result p given in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1. 14.2. Show that the uniform asymptotic expressions given in (14.40)–(14.43) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated above the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result p given in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1.

502

Problems

14.3. Show that the uniform asymptotic expressions given in (14.47) and (14.48), and (14.50) and (14.51) of the pole contribution Ac .z; t / due to the simple pole singularity at ! D !p when !p is real-valued and situated within the absorption band of a single resonance Lorentz model dielectric reduce to the nonuniform result given p in (14.12) for sufficiently large propagation distances z > 0 such that j ./j z=c  1. 14.4. Show that the uniform asymptotic expressions given in (14.65) and (14.66), and (14.69) and (14.70) of the pole contribution Ac .z; t / due to the simple pole singularity atq! D !p when !p is real-valued and situated within the passband q 2  !1  ı02 ; !22  ı22 between the two absorption bands of a double resonance Lorentz model dielectric reduce to the nonuniform result given p in (14.12) for sufficiently large propagation distances z > 0 such that j . /j z=c  1. 14.5. Derive an approximate expression for (a) the real angular frequency value q 

q 2 2 2 2 !$ 0 2 !0  ı0 ; !1  ı0 at which $ .!$ 0 ; 1/ D 0, and (b) the real angular q 

q 2 !2  ı22 ; !32  ı22 at which $ .!$ 2 ; 1/ D 0 in a double frequency value !$ 2 2 resonance Lorentz model dielectric. 14.6. Derive an approximate expression for the finite, real-valued angular frequency value !$ which satisfies (14.74) for a Drude model conductor.

Chapter 15

Continuous Evolution of the Total Field

This chapter combines the results of the preceding two chapters in order to obtain the uniform asymptotic description of the total pulsed wavefield evolution in a given causally dispersive material. From the discussion given in Sect. 12.4, the propagated plane wavefield in either a single resonance Lorentz model dielectric [see (12.352)] or a Drude model conductor [see (12.356)] may be expressed either in the form A.z; t / D As .z; t / C Ab .z; t / C Ac .z; t /

(15.1)

for all subluminal space–time points  D ct=z  1, or as a linear superposition of fields that are each expressible in this form. The field components Ab .z; t / and Ac .z; t / are both negligible when the first (or Sommerfeld) precursor field As .z; t / is predominant, As .z; t / and Ac .z; t / are both negligible when the second (or Brillouin) precursor field Ab .z; t / is predominant, and the field components Ab .z; t / and Ac .z; t / are both negligible when the pole contribution Ac .z; t / is predominant. Two or three field components become important at the same time during transition periods, giving a continuous asymptotic description of the space–time evolution of the total wavefield A.z; t / for sufficiently large z > 0 for all   1. Analogous results hold for the asymptotic description of the propagated plane wavefield in a double resonance Lorentz model dielectric, which may be expressed either in the form A.z; t / D As .z; t / C Ab .z; t / C Am .z; t / C Ac .z; t /

(15.2)

for all  D ct =z  1, or as a linear superposition of fields that are each expressible in this form, where the dominance of the field component Am .z; t / describing the middle precursor over a finite space–time interval is dependent upon whether or not the necessary condition that p < 0 is satisfied for that particular medium [see (12.116) and (12.117)]. Each additional resonance feature appearing in the material dispersion then introduces the possibility of an additional middle precursor appearing in the dynamical field evolution. Finally, the asymptotic description of the propagated plane wavefield in a Rocard–Powles–Debye model dielectric may be expressed either in the form A.z; t / D Ab .z; t / C Ac .z; t / K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 15, 

(15.3) 503

504

15 Continuous Evolution of the Total Field

for all  D ct =z  1, or as a linear superposition of fields that are each expressible in this form. Composite models then naturally combine the features of each separate model. If the initial pulse A.0; t / at the plane z D 0 identically vanishes for all t < 0, then application of Cauchy’s residue theorem showed that A.z; t / D 0 for all superluminal space–time points  0, the second precursor field Ab .z; t / is asymptotically negligible in comparison to the first precursor field As .z; t / when 1   < SB , and the first precursor field As .z; t / is asymptotically negligible in

15.1 The Total Precursor Field

505

comparison to the second precursor field Ab .z; t / when  > SB . Both field components As .z; t / and Ab .z; t / are important in the transition region between the first and second precursors that lies in a small neighborhood of the space–time point  D SB . When asymptotic approximations of As .z; t / and Ab .z; t /, each uniformly valid for  in a specific space–time domain, are applied to (15.4), it follows from Corollary 1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap .z; t / that is uniformly valid over the same space–time domain. Hence, application in (15.4) of the uniform asymptotic approximations of As .z; t / and Ab .z; t / obtained in Sect. 13.2.2 and Sect. 13.3.2, respectively, provides an asymptotic approximation of the total precursor field Ap .z; t / in a single resonance Lorentz model dielectric that is valid uniformly for all   1. Based upon these results and the results of Chap. 13, the general dynamic behavior of the total precursor field Ap .z; t / in a single resonance Lorentz model dielectric is as follows.1 At  D 1 the front of the first (or Sommerfeld) precursor arrives, and this first precursor is dominant for all  2 Œ1; SB /. For space–time values  soon after the luminal space–time point  D 1, the peak amplitude in the first precursor occurs, and for all later space–time values the amplitude of the field envelope decays exponentially. Furthermore, the instantaneous angular frequency of oscillation !s . / of the first precursor field, which is initially infinite at  D 1, monotonically q

decreases with increasing  and approaches the limiting value !s ./ ! !12  ı 2 from above as  ! 1. For space–time values close to SB , the transition from the first to the second precursor field takes place. Because 1 < SB < 1 and because the second precursor is nonoscillatory (but not static!) for all  2 Œ1; SB , the total precursor field behavior for space–time values about the point  D SB is that of the superposition of an exponentially decaying, down-chirped oscillatory wavefield with an increasing nonoscillatory field. Finally, for all  > SB , the second (or Brillouin) precursor is dominant. The magnitude of the second precursor amplitude increases as  approaches 0 from below. At the space–time point  D 0 , the second precursor experiences zero exponential decay and decays with the propagation distance z > 0 only as z1=2 . The field behavior at this critical space–time point is then unique in all of dispersive pulse propagation phenomena with far-reaching implications (from a biological perspective) and applications (from an imaging perspective). Finally, for  > 0 , the exponential decay increases with increasing  and, for  > 1 , the second precursor becomes oscillatory with instantaneous angular frequency !b . / monotonically increasing q from zero with increasing   1 , approaching the limiting value !b ./ ! !12  ı 2 from below as  ! 1. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.14. Similar behavior is obtained for the total precursor field in a Drude model conductor, except that the Brillouin precursor remains nonoscillatory for all  > 1.

1

This description does not include the nonphysical phenomena of a so-called resonance peak which, if it appears, is exactly cancelled by the pole contribution.

506

15 Continuous Evolution of the Total Field

The total precursor field for a double resonance Lorentz model dielectric is given by (15.5) Ap .z; t / D As .z; t / C Am .z; t / C Ab .z; t / when the inequality p < 0 is satisfied [see (12.116) and (12.117)]; if the opposite inequality is statisfied (i.e., if p  0 ), then the total precursor field is given by (15.4) and the preceding asymptotic description for a single resonance Lorentz model dielectric applies. For space–time points   1 bounded away from both SM and MB , it follows from the results of Sect. 12.3.2 that for sufficiently large propagation distances z > 0, both the middle precursor field Am .z; t / and second precursor field Ab .z; t / are asymptotically negligible in comparison to the first precursor field As .z; t / when 1   < SM , both the first precursor field As .z; t / and second precursor field Ab .z; t / are asymptotically negligible in comparison to the middle precursor field As .z; t / when SM   < MB , and both the first precursor field As .z; t / and middle precursor Am .z; t / are asymptotically negligible in comparison to the second precursor field Ab .z; t / when  > MB . Both field components As .z; t / and Am .z; t / are important in the transition region between the first and middle precursors that lies in a small neighborhood of the space–time point  D SM , and both field components Am .z; t / and Ab .z; t / are important in the transition region between the middle and second precursors that lies in a small neighborhood of the space–time point  D MB . When asymptotic approximations of As .z; t /, Am .z; t /, and Ab .z; t /, each uniformly valid for  in a specific space–time domain, are applied to (15.5), it follows from Corollary 1 of Sect. 10.3.1 that the result is an asymptotic approximation of Ap .z; t / that is uniformly valid over the same space– time domain. Hence, application in (15.5) of the uniform asymptotic approximations of As .z; t /, Ab .z; t / and Am .z; t / obtained in Sects. 13.2.2, 13.3.2 and 13.4, respectively, provides an asymptotic approximation of the total precursor field Ap .z; t / in a double resonance Lorentz model dielectric that is valid uniformly for all   1. For each additional resonance feature included in a multiple resonance Lorentz model dielectric, the possibility of an additional middle precursor field is introduced subject to the condition given in (12.117). The general dynamic behavior of the total precursor field in a double resonance Lorentz model dielectric is the same as that in a single resonance medium when p > 0 . However, when p < 0 , the middle precursor becomes the dominant precursor field over the space–time domain  2 .SM ; MB / between the Sommerfeld precursor evolution and the Brillouin precursor evolution. For space–time values close to SM , the transition from the first to the middle precursor field takes place, and for space–time values close to MB , the transition from the middle to the second precursor field takes place. An illustration of this total precursor field evolution for a delta function pulse is given in Fig. 13.15. The total precursor field for a Rocard–Powles–Debye model dielectric is given by Ap .z; t / D Ab .z; t /

(15.6)

and is comprised of just the Brillouin precursor whose dynamical evolution is illustrated in Fig. 13.10. Although this precursor is not oscillatory in the usual

15.2 Resonance Peaks of the Precursors and the Signal Contribution

507

time-harmonic sense, its effective frequency of oscillation has been shown in Sect. 13.4 [see (13.151)] to depend upon the material parameters alone. When combined with either the Lorentz or Drude models, the resulatant composite model of the material dispersion yields a total precursor field that possesses the salient features of each individual model.

15.2 Resonance Peaks of the Precursors and the Signal Contribution Examination of the asymptotic expressions for the Sommerfeld, Brillouin, and middle precursor fields shows that each of them exhibits a resonance peak as  varies if the relevant saddle point passes near a first-order pole of the input pulse (or pulse envelope) spectrum [3]. Indeed, it is apparent from the general expression given in (10.18) for the asymptotic contribution of a first-order saddle point, viz., Z I.z; t / D

q.!/e P

zp.!;/



2 d!  q.!sp /  00 zp .!sp ;  /

1=2

e zp.!sp ;/

(15.7)

as z ! 1, that I.z; t / becomes large if the saddle point !sp . / approaches a pole !p of the spectral function q.!/ as  varies. It is the sole purpose of this short section to show that such a resonance peak is not exhibited by the total wavefield A.z; t /. That resonance peak is cancelled by an identical resonance peak with opposite sign in the term C.z; t / appearing in the uniform asymptotic expression for the pole contribution Ac .z; t /. That is to say, these resonance peaks are an artifact of the separation of the asymptotic behavior of the integral representation of the pulse into the various component wavefield contributions of precursor and pole contribution. Under the conditions that lead to the appearance of a resonance peak in the saddle point contribution to the wavefield behavior, the saddle point passes near a firstorder pole. As a result, the uniform asymptotic expression for the pole contribution Ac .z; t / must be included in the asymptotic approximation of the total wavefield A.z; t /. The result can be written as  A.z; t /  q.!sp / 

2 zp 00 .!sp ;  /

1=2

e zp.!sp ;/ C Ac .z; t /

(15.8)

as z ! 1. From (14.20) and (14.21), if the simple pole and saddle point do not coalesce, the uniform asymptotic approximation of Ac .z; t / can be written as Ac .z; t / 

 . /

r

 zp.!sp ;// e C f0 .!p / z

(15.9)

508

15 Continuous Evolution of the Total Field

as z ! 1, where f0 .!p / is an analytic function of complex !p and where  is the residue of the pole of the spectral function q.!/. Because !sp is a first-order saddle point of the complex phase function p.!;  /, this phase function evaluted at ! D !p can be expanded in a Taylor series about !sp in the form  2 1 p.!p ; / D p.!sp ; / C p 00 .!sp ;  / !p  !sp . / C    : 2

(15.10)

As a consequence [see (14.23)],

1=2 ./ p.!sp ;  /  p.!p ;  /

1=2   !p  !sp . / C f1 .!p / D  12 p 00 .!sp ;  /

(15.11)

for !p sufficiently close to !p , where f1 .!p / is an analytic function of !p that goes to zero as !sp . / ! !p . As a result, (15.9) can be written as  Ac .z; t / 

   1=2  12 p 00 .!sp ; / !p  !sp ./

r

 zp.!sp ;// e C f2 .!p / z

(15.12)

as z ! 1, where f2 .!p / is an analytic function of !p . Similarly, because q.!/ has a first-order pole at ! D !p with residue  , the first term on the right-hand side of (15.8) can be written as  q.!sp / 

2 00 zp .!sp ; /

1=2 e zp.!sp ;/ D

1=2  2   00 e zp.!sp ;/ !sp ./  !p zp .!sp ;  / Cf3 .!p /; (15.13)

where f3 .!p / is an analytic function of !p . Substitution of (15.12) and (15.13) into (15.8) then yields A.z; t /  f2 .!p / C f3 .!p /:

(15.14)

Hence, A.z; t / is an analytic function of !p in a neighborhood of the saddle point !sp . /, and therefore cannot have a singularity at !p D !sp . Consequently, the resonance peak appearing in the precursor field is exactly cancelled by an identical resonance peak appearing in the pole contribution. The total propagated wavefield A.z; t / then does not exhibit the resonance peaks exhibited by its component subfields Ap .z; t / and Ac .z; t /.

15.3 The Signal Arrival and the Signal Velocity

509

15.3 The Signal Arrival and the Signal Velocity Attention is now given to to the detailed description of the arrival of the signal due to the contribution of any simple pole singularity appearing in the spectral function uQ .!  !c / in the integrand of the propagated plane wavefield given in (14.1), viz.,

Z 1 i .z=c/.!;/ A.z; t / D uQ .!  !c /e d! < ie 2 C

(15.15)

for z  0. The transition of the total propagated wavefield A.z; t / from the precursor field Ap .z; t / to the signal Ac .z; t / then defines the signal velocity of the pulse in the dispersive medium. As in Chap. 14, the pole !p is taken to lie along the positive real ! 0 -axis of the complex !-plane. Because of its historical significance, the analysis focuses on the signal velocity in a single resonance Lorentz medium. The extension of these results to more complicated dispersive model media is also included.

15.3.1 Transition from the Precursor Field to the Signal From the results of Chap. 14, the contribution of the simple pole singularity at ! D !p occurs when the original contour of integration C , which extends along the straight line from i a1 to i aC1 in the upper half of the complex !-plane, lies on the opposite side of the pole singularity than does the Olver-type path P . / through the accessible saddle points. That is, P ./ and the original integration contour C lie on the same side of the pole when  < s and lie on opposite sides when  > s [see (14.5)]. Consequently, for  < s the pole is not crossed when the original contour is deformed to P . / and there is no residue contribution, whereas for  > s the pole is crossed in deforming the contour C to P ./ and there is a residue contribution to the asymptotic behavior of the propagated wavefield. The value of s depends upon which Olver-type path is chosen for P ./. If that path is taken to lie along the path of steepest descent through the saddle point nearest the pole, then, because $ .!;  / =f.!;  /g is constant along the path of steepest descent, it follows that the value of s is defined by the expression [1, 2] $ .!sp ; s / D $ .!p ; s /;

(15.16)

where !sp D !sp . / denotes the saddle point which interacts with the pole singularity. At  D s , however, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to A.z; t / because P . / is an Olvertype path with respect to that saddle point. Consequently, the particular value of s at which the pole contribution occurs is of little or no significance to the asymptotic behavior of the propagated wavefield A.z; t /. An example of such an Olver-type path at a fixed space–time point  > 1 when the two near saddle points SP˙ n

510

15 Continuous Evolution of the Total Field

''

c1

c2

c6

c4

' −

SPn

+

SPn

'

c3

c5

'

SPd-

+

SPd

P(q)

Fig. 15.1 A deformed contour of integration P . / passing through both the near and distant saddle points for a fixed space–time value  > 0 . This contour is an Olver-type path with respect to the near saddle point SPC n in the right half of the complex !-plane, and is an Olver-type path with respect to the near saddle point SP n in the left-half of the complex !-plane. The lighter shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP˙ ;  / n is satisfied and the darker shaded area indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SP˙ ;  / is satisfied d

are dominant over the two distant saddle points SP˙ d in a single resonance Lorentz model dielectric is depicted in Fig. 15.1. The path P . / through the pair of near C saddle points SP n and SPn can lie anywhere within the shaded region of the figure. With the path P . / shown in this figure,  < s if the pole !p lies in the angular frequency interval !c2 < !p < !c4 ,  > s if !p lies within either of the angular frequency intervals 0  !p < !c2 or !p > !c4 , and  D s if either !p D !c2 or !p D !c4 . If the path P . / was chosen to be completely in the lower half of the complex !-plane (such an Olver-type path is possible for the situation illustrated in Fig. 15.1), then  > s for all !p  0. The pole contribution at ! D !p is the dominant contribution to the asymptotic behavior of the propagated wavefield when  > c > s , where c is defined as the space–time value that satisfies the relation [3–6]  .!sp ; c / D  .!p /;

(15.17)

15.3 The Signal Arrival and the Signal Velocity

511

where !sp D !sp .c / denotes the dominant saddle point at  D c . Notice that  .!p / is independent of the value of  when !p is real-valued, as is assumed here. For space–time values  < c such that the inequality  .!sp ;  / >  .!p / is satisfied, the saddle point is the dominant contribution to the asymptotic behavior of the propagated wavefield and the pole contribution is asymptotically negligible by comparison. For space–time values  > c such that the inequality  .!sp ;  / <  .!p / is satisfied, however, the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wavefield and the saddle point contribution is asymptotically negligible by comparison. For example, for the space–time point depicted in Fig. 15.1, (15.17) is satisfied if either !p D !c1 or !p D !c6 and the value of  is then c for either of these two pole locations. Furthermore, for the space–time ;  / >  .!p / is value  depicted in Fig. 15.1,  < c and the inequality  .!SPC n ;  / <  .!p / is satisfied if !c1 < !p < !c6 , and  > c and the inequality  .!SPC n satisfied if either 0  !p < !c1 or !p > !c6 . Consequently, the pole contribution is asymptotically negligible in comparison to the saddle point contribution to the propagated wavefield at the space–time value depicted in Fig. 15.1 if !c1 < !p < !c6 , whereas the pole contribution is the dominant contribution to the asymptotic behavior of the propagated wavefield and the saddle point contribution is asymptotically negligible in comparison to it if either 0  !p < !c1 or !p > !c6 . Based upon these results, it is seen that the signal arrival for a fixed value of !p occurs at the space–time point  D c satisfying (15.17). Notice that this definition of the signal arrival yields a signal velocity [3–6] vc .!p /

c c .!p /

(15.18)

that is independent of the initial pulse envelope function and the propagation distance z, depending only upon the dispersive medium properties and the value of !p . Notice that this pulse velocity measure always satisfies relativistic causality; that is, vc .!p /  c 8 !p . A general overview of the arrival of the signal and its interaction with the Sommerfeld and Brillouin precursor fields is now presented for the case of a single resonance Lorentz model dielectric. This description is based upon (15.17) and the numerical results describing the topography of  .!;  / in the complex !-plane as a function of  presented in Figs. 12.4–12.9 of Sect. 12.2 for Brillouin’s choice of the medium parameters. A similar description may be given for the other models of the material dispersion considered here. Consider first the topography of  .!;  / in the right half of the complex !-plane over the space–time domain  2 Œ1; SB in which the distant saddle points SP˙ d are and are of equal dominance initially dominant over the upper near saddle point SPC n at  D SB , as illustrated in the sequence of illustrations given in Figs. 12.4–12.6. It is seen from these three diagrams that for high angular frequency values !p  !SB , where !SB is defined in (12.258), the signal arrival will occur during the evolution of the first (or Sommerfeld) precursor field, as determined by the space–time point when the isotimic contour  .!;  / D  .!SPC ;  / through the distant saddle point d

512

15 Continuous Evolution of the Total Field

0 SPC d crosses the pole at ! D !p . At the luminal space–time point  D 1 (Fig. 12.4), the symmetric pair of distant saddle points SP˙ d are located at ˙1  2i ı [see (12.204)] and no signal can have arrived (except for the nonphysical signal with infinite angular frequency !p ). At  D 1:25 (Fig. 12.5) it is seen that for values of !p  9:4  1016 r=s the signal has already arrived and is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. Finally, at  D SB ' 1:33425 (Fig. 12.6), the pair of distant saddle points SP˙ d and the upper near saddle point SPC n are of equal exponential importance in their individual contributions to the asymptotic behavior of the total wavefield A.z; t /. In that case, for values of !p > !SB , where !SB ' 8:6  1016 r=s, the signal has already arrived and is the dominant contribution to the total wavefield A.z; t /. For values !p < !SB , the signal (or pole contribution) has yet to arrive at  D SB and the precursor field Ap .z; t / is the dominant contribution to the total wavefield A.z; t /. For all space–time points  > SB , first the upper near saddle point SPC n for SB <   1 and then the two near saddle points SP˙ n for all  > 1 are dominant over the distant saddle point pair. The asymptotic contribution of these near saddle points to the field A.z; t / yields the second (or Brillouin) precursor. For space–time values  2 .SB ; 0 , during which the upper near saddle point SPC n , lying along the positive ! 00 -axis, is the dominant saddle point, a careful consideration of the iso;  / (inclined at a positive angle of =4 radians timic contour  .!;  / D  .!SPC n to the positive ! 0 -axis) through that saddle point, reveals that the signal due to any pole singularity at !p > !SB loses its asymptotic dominance in the total wavefield evolution because of the decreasing exponential decay of the evolving Brillouin precursor. That is, as  increases over the space–time domain .SB ; 0 , the isotimic ;  / at the angle =4 through the dominant upper near contour  .!;  / D  .!SPC n saddle point recrosses any pole singularity at !p > !SB that had previously been crossed by the isotimic contour  .!;  / D  .!SPC ;  / through the distant saddle d

point SPC d when  varied over the initial space–time interval .1; SB /. Note, however, that the pole contribution at !p > !SB has not been cancelled or negated by this occurrence, but rather has only become less dominant than the evolving sec; 0 / D 0 and the isotimic contour ond precursor field [4]. At  D 0 ,  .!SPC n 0  .!;  / D  .!SPC ;  / intersects the real ! -axis at the origin (where the near 0 n C saddle point SPn happens to be) and at infinity, and remains above the ! 0 -axis for all other positive values of ! 0 . Consequently, at the space–time point  D 0 , the second precursor field (which experiences zero exponential attenuation at this point) is exponentially dominant over all other contributions to the asymptotic behavior of the propagated wavefield A.z; t /. Consider finally the remaining three plots depicting the topography of  .!;  / in the right half of the complex !-plane when  > 0 . At the space–time point  ' 1 ' 1:501 (Fig. 12.7), it is seen that the pole contribution at ! 0 D !p is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / in either the low-frequency domain 0  !p < 0:005  1016 r=s or in the highfrequency domain !p > 20  1016 r=s, whereas for values in the interval 0:005  1016 r=s < !p < 20  1016 r=s, the second precursor is the dominant contribution.

15.3 The Signal Arrival and the Signal Velocity

513

At  D 1:65 (Fig. 12.8), it is seen that the pole contribution at ! 0 D !p is the dominant contribution in either the low-frequency domain 0  !p < 1:27  1016 r=s or in the high-frequency domain !p > 15:6  1016 r=s, whereas for values in the interval 1:27  1016 r=s < !p < 15:6  1016 r=s, the second precursor is the dominant contribution. At  D 5:0 (Fig. 12.9), it is seen that the pole contribution at ! 0 D !p is the dominant contribution in either the low-frequency domain 0  !p < 3:2  1016 r=s or in the high-frequency domain !p > 6:35  1016 r=s, whereas for values in the interval 3:2  1016 r=s < !p < 6:35  1016 r=s, the second precursor is the dominant contribution. Because the near saddle points are dominant over the distant saddle points for all  > SB , and because  .!sp / at the near saddle points monotonically decreases with increasing  > 0 , a critical space–time point  D m will finally be reached at which the relation ; m / D  .!min /  .!SPC n

(15.19)

is satisfied. Here !min is that value of ! 0 along the positive real axis at which  .! 0 / attains its minimum value [see (12.80) and (12.83)]. At this space–time point, the ; m / lies entirely in the lower half of the complex isotimic contour  .!/ D  .!SPC n !-plane with the exception of the two points ! 0 D ˙!min where it just touches the real ! 0 -axis, as illustrated in Fig. 15.2. Consequently, if !p D !min , the signal arrival is at c D m which is larger than any other value of c . That is, the signal velocity vc .!min / D c=m is the absolute minimum signal velocity in any given single resonance Lorentz model dielectric. For all later space–time points  > m , the signal contribution at any real frequency value !p has already arrived and is the dominant contribution to the propagated wavefield A.z; t /. In summary, the signal arrival in a single resonance Lorentz model dielectric separates naturally into two distinct cases dependent upon the value of the real angular frequency !p of the pole in comparison to the critical angular frequency value !SB defined in (12.258). For values of !p in the angular frequency interval 0  !p  !SB , the signal arrival is due to the crossing of the isotimic contour  .!/ D  .!sp / with the simple pole singularity at ! D !p , where !sp denotes the location of the upper near saddle point SPC n for 1 <  < 1 , the second-order near saddle point SPn at  D 1 , and the near saddle point SPC n for all  > 1 . For such values of !p , the signal due to the pole contribution at ! D !p is preceded by the first and second precursor fields, and the signal evolves essentially undisturbed as  increases above c . For pole values !p > !SB , however, the signal arrival first occurs due to the crossing of the isotimic contour  .!/ D  .!SPC ;  / through the d

distant saddle point SPC d with the simple pole singularity. This first arrival occurs at some space–time point  2 .1; SB / for finite !p . At some later space–time point  2 .SB ; 0 /, this pole is again crossed, but in the opposite direction, by the isotimic ; / through the upper near saddle point SPC contour  .!/ D  .!SPC n , rendering n the pole contribution asymptotically less dominant than the second precursor field. Finally, for some still later space–time point  > 0 , the pole is again recrossed ;  / through the in the original direction by the isotimic contour  .!/ D  .!SPC n

514

15 Continuous Evolution of the Total Field

''

O r i g i n a l

C o n t o u r

o f

I n t e g r a t i o n

min

SPd−

Branch Cut

min

SPn−

SPn+

Branch Cut

'

SPd+

P(qm )

Fig. 15.2 A deformed contour of integration P .m / passing through both the near and distant saddle points at the fixed space–time point  D m when the condition  .!SP˙ ; m / D  .!min / n

is satisfied. This contour is an Olver-type path with respect to the near saddle point SPC n in the right half of the complex !-plane, and is an Olver-type path with respect to the near saddle point SP n in the left half of the complex !-plane. The lighter shaded area indicates the region of the complex !-plane wherein the inequality  .!; m / < .!SP˙ ; m / is satisfied and the darker shaded area n indicates the region of the complex !-plane wherein the inequality  .!; m / <  .!SP˙ ; m / is d satisfied

near saddle point SPC n so that it finally becomes asymptotically dominant over all other contributions to the propagated wavefield A.z; t / for all remaining space–time values. Consequently, for pulsed sources with !p > !SB there is the existence of a so-called prepulse [3] due to the interuption of the signal evolution by the second precursor field which becomes dominant over the pole contribution for some short space–time interval. This prepulse formation is seen to be an integral part of the dynamic evolution of the second precursor field superimposed upon the evolution of the signal contribution. The space–time evolution of the signal contribution when !p > !SB may then be considered to be separated into three parts: the so-called prepulse which is preceeded by the first precursor and then followed by the second precursor field superimposed upon the signal contribution, which is then finally followed by the signal which remains dominant for all later space–time points. It is important to keep in mind that the prepulse is not independent of the signal evolution. Indeed, the prepulse formation is simply a consequence of the superposition of the signal (or pole) contribution with the second precursor field which becomes dominant over the signal for a finite space–time interval. As a final point regarding the signal arrival, it is of interest to notice that the uniform asymptotic approximation of the pole contribution at ! D !p takes on a

15.3 The Signal Arrival and the Signal Velocity

515

particularly useful form for numerical calculations at the critical q space–time point  D c . For example, for the below resonance case 0 < !p  !02  ı 2 of a single resonance Lorentz model dielectric, the uniform asymptotic approximation of the pole contribution at the space–time point  D c > s is given by (14.34), where ./ and . / is given by (14.23). At !sp denotes the near saddle point location !SPC h p i n  D c , ˛N sd D =4, arg i .c / z=c D =4, and the complementary error function appearing in (14.34) may be replaced by the right-hand side of (10.98) [see also (10.113)], with the result [3, 7, 8] (     q  q   p 1 i 4 2z 2z < i  i 2e C j .c /j c C i S j .c /j c Ac .z; tc /  2 q ) c o z z z 3 n z .! C ;c / c .! ; / SPn e c p c C C < e c .!p ;c / e .c / 2 (15.20) as z ! 1 with fixed  D c D ctc =z. Here C. / and S. / are the cosine and sine Fresnel integrals, respectively [see (10.99) and (10.100)]. This expression is reminiscent of the special form the Lommel function expression for the diffracted wavefield takes at the geometric shadow boundary (see of Born and Wolf [9]). Analogous results hold for multiple resonance Lorentz model dielectrics as well as for Drude model conductors.

15.3.2 The Signal Velocity The analysis presented in Chaps. 13 and 14 furnishes a complete uniform asymptotic description of the propagated ultrawideband pulsed wavefield in both single and multiple Lorentz model dielectrics, Debye model dielectrics, and Drude model conductors. The signal velocity in each of these causal medium models is now considered in some detail.

15.3.2.1

Signal Velocity in Single Resonance Lorentz Model Dielectrics

The first (or Sommerfeld) precursor field As .z; t / in a single resonance Lorentz model dielectric arrives at the luminal space–time point  D 1, rapidly building to a peak amplitude value immediately following its arrival, the amplitude then decreasing2 as it experiences increasing exponential attenuation as  continues to increase 2

The possible appearance of any resonance peak in the individual precursor evolution is not considered here as it does not appear in the total wavefield evolution, as discussed in Sect. 15.2.

516

15 Continuous Evolution of the Total Field

above unity. At the space–time point  D SB > 1, the second (or Brillouin) precursor field Ab .z; t / becomes dominant over the first precursor field and remains so for all  > SB . The amplitude of this field component rapidly builds up to a peak amplitude value around the space–time point  D 0 > SB , after which its amplitude experiences increasing exponential attenuation as  increases above 0 . If the spectral envelope function uQ .!  !c / for the initial pulse has a pole singularity at ! D !p , then the final contribution to the propagated wavefield A.z; t / arises from the contribution due to that pole. It is assumed here that !p  0 is real-valued. From the results of Sect. 14.4, this pole contribution, when it is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /, is given by [see, for example, (14.38) and (14.44)]

    Ac .z; t /  e ˛.!p /z  0 cos ˇ.!p /z  !p t   00 sin ˇ.!p /z  !p t

(15.21)

as z ! 1 with  > s bounded away from s , where  0 1 . For all space–time points  > c , the pole contribution given in (15.21) is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / for all !p  0. For angular frequency values !p > !SB , the pole contribution given in (15.21) is the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t / for space–time points in the interval  2 .c1 ; c2 /, where 1 < c1 < SB and where SB < c2 < c for finite !p . The space–time point  D c1 is defined by the relation (15.23)  .!SPC ; c1 / D  .!p /I 1 < c1 < SB ; !p > !SB ; d

at which point the pole contribution is of equal dominance with the first precursor field, this pole contribution remaining dominant over the first precursor for all  > c1 . However, at the space–time point  D c2 defined by the relation ; c2 / D  .!p /I  .!SPC n

SB < c2 < 0 ; !p > !SB ;

(15.24)

15.3 The Signal Arrival and the Signal Velocity

517

the second precursor is of equal dominance with the pole contribution, and over the subsequent space–time interval  2 .c2 ; c /, the second precursor field is dominant over the pole contribution. Finally, at the space–time point  D c defined in (15.22), these two contributions are again of equal dominance, and for all later space–time points  > c , the pole contribution remains as the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. Physically, the first (or Sommerfeld) precursor field is due to the high-frequency (above absorption band) energy present in the frequency spectrum of the initial pulse as filtered by the material dispersion, whereas the second (or Brillouin) precursor field is due to the low-frequency (below resonance) energy present in the initial pulse spectrum as filtered by the material dispersion. The pole contribution given in (15.21) is physically due to the frequency component in the initial pulse spectrum at the angular frequency !p , as can readily be seen from that equation. For the majority of canonical pulse types considered in this book, the pole occurs at !p D !c , the applied carrier or signal (radian) frequency of the initial plane wave pulse at z D 0. A well-defined signal velocity for these canonical pulse types is now given. The main signal arrival is defined to occur at that space–time value  D c satisfying (15.22) at which the pole contribution given in (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wavefield A.z; t /. The velocity at which this space–time point in the wavefield propagates through the dispersive medium is defined as the main signal velocity, given by [3–6] vc .!c /

c ; c .!c /

(15.25)

where c is the vacuum speed of light. Furthermore, for angular frequency values !c > !SB , there is the appearance of a so-called pre-pulse whose front arrives at the space–time point  D c1 satisfying (15.23) when the pole contribution given in (15.21) becomes the dominant contribution to the asymptotic behavior of the propagated wavefield, and whose back arrives at the space–time point  D c2 satisfying (15.24) when the second precursor field becomes the dominant contribution to the asymptotic behavior of the propagated wavefield. The velocity at which the front of this prepulse propagates through the dispersive medium is called the anterior pre-signal velocity, given by vc1 .!c /

c I c1 .!c /

!c > !SB ;

(15.26)

and the velocity at which the back of this prepulse propagates through the dispersive medium is called the posterior pre-signal velocity, given by vc2 .!c /

c I c2 .!c /

!c > !SB :

(15.27)

518

15 Continuous Evolution of the Total Field

From the inequalities given in (15.22)–(15.24), these three pulse velocities are seen to satisfy the inequality vc .!c / 

c c < vc2 .!c / < < vc1 .!c / < c: 0 SB

(15.28)

Notice that the main signal,anterior pre-signal, and posterior pre-signal velocities depend only upon the value of !c and the medium parameters. The angular frequency dependence of these three signal velocities is presented in Fig. 15.3 for a single-resonance Lorentz model dielectric characterized p by Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, !p D 20  1016 r=s, ı D 0:28  1016 r=s). The numerical values for these signal velocity graphs are obtained [3] using (15.22)–(15.24) to determine accurate numerical estimates of the values of c .!c /, c1 .!c /, and c2 .!c / through a comparison of the numerically determined behavior of  .!;  / at either the near or distant saddle points with the value of  .!/ at the angular frequency value !c . As evident from these numerical results presented in Fig. 15.3, the main signal velocity vc .!c / attains a minimum value near the resonance frequency of the medium. The actual minimum occurs at

1

vc 1 / c 0.8

vc 2 / c 0.6

v/c

vE /c vc / c

0.4

0.2

0

0

0

5 c

(r/s)

SB

10

15 x 1016

Fig. 15.3 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c /, relative anterior pre-signal velocity vc1 .!c /=c D 1=c1 .!c /, and relative posterior pre-signal velocity vc2 .!c /=c D 1=c2 .!c / in a single-resonance Lorentz model dielectric characterized by Brillouin’s choice of the medium parameters. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium

15.3 The Signal Arrival and the Signal Velocity

519

p the value !c D !min !0 1 C 2ı=!0 where  .!c / attains its minimum value along the positive real axis [see (12.83)]. Consequently, the signal velocity does not peak to the vacuum speed of light c near resonance, as indicated by Brillouin [1, 2], but rather attains a minimum value near resonance.3 Approximate expressions for the critical space–time points c .!c /, c1 .!c /, and c2 .!c /, whose inverses give the relative main signal, anterior preisignal, and posterior pre-signal velocities, respectively, may be obtained from the defining expressions given in (15.22)–(15.24), repsectively, through the application of appropriate approximations for  .!;  /. For the main signal velocity one obtains the approximate relationship !2

 .!c / ' ı.c  0 /

c2  02 C 2 !p2 0

c2



02

!p2

(15.29)

C 3 !2 0

for c  0 . For values of c close to 0 (i.e., when either 0  !c  !0 or !c  !1 ), this relation yields the solution c .!c / ' 0 

3 .!c / ; 2ı

(15.30)

whereas for large values of c (i.e., when !0  !c  !1 ), this relation yields the solution  .!c / : (15.31) c .!c / ' 0  ı Substitution of these expressions into (15.25) then yields the desired analytic approximation for the main signal velocity. For the anterior pre-signal velocity, one finds the approximation  .!c / (15.32) c1 .!c / ' 1  2ı for 1  c1 < SB when !c > !SB , from which the approximate behavior of the anterior pre-signal velocity vc1 .c / may be determined. Finally, for the posterior pre-signal velocity, one obtains the approximation  .!c / C

0 !04 02 !0 2 .   /  .0  c2 /3 ' 0 0 c2 4ı!p2 16ı 3 !p4

(15.33)

for SB < c2 < 0 when !c > !SB . The proper solution to this cubic equation corresponds to the one that has the limiting values c2 .1/ D 0 and c2 .!SB / D !SB . 3

Brillouin [1, 2] incorrectly interpreted the signal arrival to occur when the simple pole singularity at the signal frequency was crossed in deforming the original contour of integration to the path of steepest descent through the relevant near and distant saddle points. This result was partially corrected by Baerwald [10] in 1930.

520

15 Continuous Evolution of the Total Field

15.3.2.2

Signal Velocity in Multiple Resonance Lorentz Model Dielectrics

The main signal arrival in a double resonance Lorentz model dielectric is defined to occur at the space–time point  D c satisfying the relation [cf. (15.22)] ; c / D  .!c /;  .!SPC n

c  0

(15.34)

at which the pole contribution becomes the dominant contribution to the asymptotic behavior of A.z; t / and remains so for all  > c . The main signal velocity describes the rate at which this space–time point travels through the dispersive medium and so is given by [cf. (15.25)] c vc .!c / D : (15.35) c .!c / Because of the double resonance character of the dispersive medium, this velocity possesses a local minimum near each resonance frequency as well as attaining a local maximum c=p at some frequency value !c D !p in the passband between the two resonance frequencies, as illustrated in the graph presented in Fig. 15.4. This

1

vc1 /c 0.8

vc2 /c

1/q 0 0.6

vE /c

v/c

vE /c vc /c

vc /c

0.4

0.2

0

0

0

2

6 c (r/s)

SB

8

10 x 1016

Fig. 15.4 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c / and the relative pre-signal velocities vc1 .!c /=c D 1=c1 .!c / and vc2 .!c /=c D 1=c2 .!c / 16 in a double p 1  10 r=s, p resonance Lorentz model dielectric with medium parameters !0 D 16 16 16 b0 D 0:6  10 r=s, ı0 D 0:1  10 r=s and !2 D 4  10 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v.0/=c D 1=0

15.3 The Signal Arrival and the Signal Velocity

521

local maximum value cannot exceed the value c=0 D c=n.0/, which the main signal velocity reaches at both zero and infinite frequency values, as seen in the figure. Notice that this local maximum is significantly less than the peak value attained by the energy velocity vE .!c / in this passband, as described by the dashed curve in the figure, where (see Sect. 5.2.6 of Vol. 1) vE .!/ D c=E .!/ with 1 E .!/ D nr .!/ C nr .!/

"

# b02 ! 2 b22 ! 2 C ;  2 2 ! 2  !02 C 4ı02 ! 2 ! 2  !22 C 4ı22 ! 2 (15.36)

from (5.212) of Vol. 1. It is from this result that the necessary condition given in (12.117) for the appearance of the middle precursor in a double resonaqnce Lorentz model dielectric was obtained [11]. Separate attention must then be given to these two individual cases. If p > 0 , then the middle saddle points never become the dominant saddle points (see Fig. 12.59) and the middle precursor doesn’t appear in the dynamical field evolution. The signal arrival and velocity is then similar to that for a single resonance Lorentz model dielectric, being described by a main signal velocity and anterior and posterior pre-signal velocities for !c > !SB , the only additional feature being the appearance of a local maximum in the main signal velocity in the passband between the two absorption bands, as illustrated in Fig. 15.4. The main signal arrival then occurs at the space–time point  D c satisfying (15.22) and, for angular frequency values !c > !SB , the anterior pre-signal arrival and posterior pre-signal departure occur at the successive space–time points  D cj , j D 1; 2, satisfying (15.23) and (15.24), respectively. This particular branching character of the signal velocity dispersion is a direct consequence of the asymptotic dominance of the Brillouin precursor over the space–time domain  2 .c2 ; c / between the main signal and posterior presignal velocities when !c > !SB and is the same as that obtained for a single resonance Lorentz model dielectric (see Fig. 15.3). The impulse response for this double resonance Lorentz model medium is given in Fig. 13.15. This branching character of the signal velocity is complicated further when the middle saddle points SP˙ m1 become the dominant saddle points over some nonzero space–time interval. In this case the prepulse signal velocity when !c > !SM > !SB is interrupted by the dominance of the upper middle saddle point pair SP˙ m1 over the space–time interval  2 .SM ; MB /, as seen in Figs. 12.57–12.58. The impulse response for this double-resonance Lorentz model medium is given in Fig. 13.16. The space–time description of the prepulse arrival and departure points then separates into two angular frequency domains (see Fig. 12.57). For !c > !MB , where the real-valued angular frequency value !MB is defined by the relation  .!SPC ; MB / D  .!MB /; m1

(15.37)

with  .!SPC ; MB / D  .!SPC ; MB / [see (12.121)] the prepulse arrival and deparn m1 ture is described by the pair of relations [cf. (15.23) and (15.24) with SB replaced by SM in (15.23) and with SB replaced by MB in (15.24)]

522

15 Continuous Evolution of the Total Field

 .!SPC ; c1 / D  .!c /I

1 < c1 < SM ; !c > !MB ;

(15.38)

 .!SPC ; c2 / D  .!c /I n

MB < c2 < 0 ; !c > !MB :

(15.39)

d

For !c 2 .!SM ; !MB /, where the real-valued angular frequency value !SM is defined by the relation (15.40)  .!SPC ; SM / D  .!SM /; m1

with  .!SPC ; SM / D  .!SPC ; SM / [see (12.119)], the set of expressions in m1 d (15.38) and (15.39) is augmented by the prepulse arrival–departure branch  .!SPC ; cm / D  .!c /I

SM < cm < MB :

m1

(15.41)

The corresponding signal velocity branches vc1 .!c / c=c1 .!c /, vcm .!c / c=cm .!c /, vc2 .!c / c=c2 .!c /, and vc2 .!c / c=c2 .!c /, illustrated in Fig. 15.5, are then seen to satisfty the inequality

1

vc1 /c vcm /c

0.8

vcm /c vE /c

vc2 /c

1/q 0

vE /c

vc /c

vc /c

v/c

0.6

vc2 /c

0.4

0.2

0

0

1

0

c

(r/s)

2 x 1017

Fig. 15.5 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c / and the relative pre-signal velocities vc1 .!c /=c D 1=c1 .!c /, vc2 .!c /=c D 1=c2 .!c /, and vcm .!c /=c D 1=cm .!c / in a double-resonance Lorentz model dielectric with medium pap 16 16 16 16 rameters p !0 D 1  10 r=s, b0 D 0:6  10 r=s, ı0 D 0:1  10 r=s and !2 D 7  10 r=s, b2 D 12  1016 r=s, ı2 D 0:1  1016 r=s. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium, and the dotted line indicates the zero frequency velocity value v.0/=c D 1=0

15.3 The Signal Arrival and the Signal Velocity

0 < vc .!c /
!12  ı 2 for finite values of   1, such a resonance peak may only occur when q !c > !12  ı 2 . However, even for such a high applied signal frequency !c above the medium absorption band, this resonance peak may not appear for the case of a highly absorptive medium (i.e., large ı) because of the larger distance between the distant saddle point SPC d and !c at  D r . The dynamical evolution of the first (or Sommerfeld) precursor field is illustrated in Figs. 15.10 and 15.11 for these two different possibilities. The first precursor wave evolution illustrated in Fig. 15.10 is typical of that observed in either a highly absorptive Lorentz model dielectric or when the applied signal frequency !c  q !12  ı 2 ; the resonance peak is absent in either of these two situations. The first precursor wave evolution illustrated in Fig. 15.11, on the q other hand, is typical of that

observed in a weakly absorbing medium when !c > !12  ı 2 ; the resonance peak is present in this situation. As shown in Sect. 15.2, a resonance peak of similar form but opposite sign also appears at  D r in the pole contribution to the propagated wavefield such that this resonance phenomena does not appear in the total wavefield evolution, as illustrated in Fig. 15.12.

15.5 The Heaviside Step Function Modulated Signal

535

AHs(z,t)

0.05 0 −0.05 qr

AHs(z,t) + AHc(z,t)

AHc(z,t)

0.05 0 −0.05 0.05 0 −0.05 1

1.1

1.2

1.3

1.4

1.5

q

Fig. 15.12 Superposition of the Sommerfeld precursor field AHs .z; t / and the pole contribution AHc .z; t / resulting in the cancellation of the resonance peaks in each component field

The instantaneous frequency of oscillation of the Sommerfeld precursor, given in (13.43), is approximately equal to the real part of the distant saddle point location in the right half of the complex !-plane, so that [see (13.44)] n o !s . / < !SPC ./ D ./:

(15.73)

d

This instantaneous angular frequency is at first infinite (when  D 1 and the field amplitude vanishes), and then rapidly decreases as  initially increases away from unity, the rate of this decrease decreasing as  continues to increase and the distant 0 saddle point SPC d asymptotically approaches the outer branch point !C . Consider next the spatiotemporal field structure of the second (or Brillouin) precursor field AHb .z; t / whose dynamical evolution is due to the -evolution of first the upper near saddle point SPC n for 1 <  < 1 and then the pair of near saddle points SP˙ n for   1 . As  increases away from unity and approaches 0 from below, the amplitude of the Brillouin precursor steadily increases as the attenuation monotonically decreases, the attenuation vanishing at  D 0 . Then, as  increases above 0 , the attenuation increases monotonically and the amplitude steadily decreases, but remains larger than the corresponding amplitude of the first precursor field which, for all  > SB , possesses a larger exponential decay rate with propagation distance z > 0 than does the second precursor field. However, due to the presence of the . /  !c /1 as a factor in the asymptotic approximation of resonance term .!SPC n the second precursor field [see (13.139)], a resonance peak in the wavefield may now occur when the near saddle point SPC n approaches close to the applied signal

536

15 Continuous Evolution of the Total Field 0.12 0.1 0.08 0.06 AHb(z,t)

0.04 0.02 0 −0.02

qSB

q0

−0.04 qr

−0.06 −0.08 1.4

1.5

1.6

1.7

Fig. 15.13 Brillouin precursor field evolution when the applied signal frequency !c satisfies 0  q !c
0, because critical aspects of the wavefield evolution (e.g., the critical space–time points SB , 0 , c , and cj , j D 1; 2) are then independent of the propagation distance z. The dynamical signal evolution depicted in Figs. 15.19–15.21 illustrates the q behavior in the below absorption band domain !c 2 .1; !02  ı 2 /. In each case, the dynamical field evolution begins with the Sommerfeld (or first) precursor over the space–time domain  2 .1; SB /, followed by the Brillouin (or second) precursor

0.05 0 z = zd

AH(z,t)

−0.05

0.1 z = 3zd 0

0.05

z = 5zd

0 1

qSB

2

q0 qc

2.5

q

Fig. 15.19 Dynamical signal evolution with below resonance angular carrier frequency !c D 1  1016 r=s at one, three, and five absorption depths

544

15 Continuous Evolution of the Total Field

0.1 0

z = zd

AH(z,t)

−0.1

0.2 z = 3zd 0

0.1

z = 5zd

0 qc 1

qSB q 0

2

3 q

4

5

Fig. 15.20 Dynamical signal evolution with below resonance angular carrier frequency !c D 2  1016 r=s at one, three, and five absorption depths

0.1 0 z = zd

AH(z,t)

−0.1

0.2 z = 3zd 0

0.1

z = 5zd

0 −0.1

1

qSB q 0

2

3 q

qc 4

5

Fig. 15.21 Dynamical signal evolution with below resonance angular carrier frequency !c D 3  1016 r=s at one, three, and five absorption depths

15.5 The Heaviside Step Function Modulated Signal x 1016

4

Instantaneous Oscillation Frequency (r/s)

545

c

2

1

0

q0

1

2

3 q

qc 4

5

Fig. 15.22 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 3  1016 r=s propagated signal wavefields in Fig. 15.21

1

0.5

AH(z,t)

0

z = zd

0.5 z = 3zd 0

0.25

z = 5zd

0 −0.25

q0 0 qSB

10

20

q

30 q c

40

50

Fig. 15.23 Dynamical signal evolution with on resonance angular carrier frequency !c D 4  1016 r=s at one, three, and five absorption depths

546

15 Continuous Evolution of the Total Field

0.1 0 z = 10zd

AH(z,t)

−0.1

0.2 z = 30zd 0

0.1

z = 50zd

0 q0

−0.1

1 qSB

3

5 q

7

9

Fig. 15.24 Dynamical signal evolution with on resonance angular carrier frequency !c D 4  1016 r=s at 10, 30, and 50 absorption depths

1

z = zd

0.5

AH(z,t)

0

0.5 z = 3zd 0

0.25

z = 5zd

0 q0 0 qSB

10

20 q c

30

40

50

q

Fig. 15.25 Dynamical signal evolution with intra-absorption band angular carrier frequency !c D 5  1016 r=s at one, three, and five absorption depths

15.5 The Heaviside Step Function Modulated Signal

547

0.2 z = 10zd

0.1

AH(z,t)

0

0.1 z = 30zd 0 −0.1 0.1 z = 50zd 0 −0.1

q0 1q SB

3

5

7

9

q

Fig. 15.26 Dynamical signal evolution with intra-absorption band angular carrier frequency !c D 5  1016 r=s at 10, 30, and 50 absorption depths

Instantaneous Oscillation Frequency (r/s)

10

x 1016

8

6 c

4

2

0

5

10

q

15

20

qc

25

Fig. 15.27 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 51016 r=s propagated signal wavefield at 5 (illustrated in Fig. 15.25), 10 (illustrated in Fig. 15.26), and 15 absorption depths

548

15 Continuous Evolution of the Total Field 0.6 0.4 0.2

AH(z,t)

0

z = zd

0.2

z = 3zd

0 0.2 z = 5zd

0 q0 1 q SB

3

5

q

7

qc

9

Fig. 15.28 Dynamical signal evolution with (barely) above absorption band angular carrier frequency !c D 6  1016 r=s at one, three, and five absorption depths

field, which evolves over the space–time domain  2 .SB ; c /, followed by the main signal evolution for all  > c . Because of the low carrier frequency in Fig. 15.19, the Sommerfeld precursor is barely visible in comparison to both the Brillouin precursor and main signal. However, as the carrier frequency !c is increased, the amplitude j!  !c j1 of the initial pulse spectrum increases in the high-frequency domain above the absorption band, resulting in an increased amplitude of the Sommerfeld precursor, as seen in Figs. 15.20 and 15.21. Finally, notice that both the first and second precursor fields are well defined at one absorption depth (z=zd D 1) in both the !c D 1  1016 r=s (Fig. 15.19) and !c D 2  1016 r=s (Fig. 15.20) cases, but not so in the !c D 3  1016 r=s (Fig. 15.21). This is due to the fact that the absorption depth decreases as the carrier frequency !c approaches the absorption band and the propagation distance must be further increased to reach the so-called mature dispersion regime where the precursor fields are well defined and described by the asymptotic theory. The numerically determined instantaneous angular frequency of oscillation of each of the propapagated signal structures presented in Fig. 15.21 is presented in Fig. 15.22, where the  symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Each numerically determined sampled value of the instantaneous angular oscillation frequency is computed from the expression c ; !j .Nj / D z j

(15.78)

15.5 The Heaviside Step Function Modulated Signal

549

where j D j C1  j represents the absolute difference between the space–time values j at successive zero crossings in the computed wavefield evolution at a fixed value of the propagation distance z > 0. The space–time point at which the numerically determined angular frequency value !j is then assigned to the midpoint value Nj of the space–time interval .j ; j C1 / between two adjacent zeros. Because the near saddle point interacts with the below absorption band pole at ! D !c and the total field evolution is dominated by the Brillouin precursor after the Sommerfeld precursor dies out and before the main signal arrival, the instantaneous oscillation frequency of the total field in this below absorption case is found to increase monotonically to the carrier frequency !c as  increases above SB , and is approximately equal to !c at  D c and remains so for all larger  > c , as seen in Fig. 15.22. The dynamical signal and instantaneous oscillation frequency evolution depicted q in Figs. 15.23–15.27 illustrates the behavior in the absorption band domain q

!c 2 Œ !02  ı 2 ; !12  ı 2 . For applied signal frequencies at the lower end of the absorption band (!c !0 ) the dominant field structure is dominated by the Brillouin precursor, as seen in Figs. 15.23 and 15.24 for the on-resonance angular carrier frequency case (!c D !0 4  1016 r=s), but as !c increases up through the absorption band, the Sommerfeld precursor becomes increasingly important in the total propagated field structure, as seen in Figs. 15.25 and 15.26 for the !c D 5  1016 r=s angular carrier frequency case (which is approximately midway through the absorption band). Because the absorption reaches a maximum so that the absorption depth zd D ˛ 1 .!c / reaches a minimum in the absorption band, it can take several absorption distances to reach the mature dispersion regime in the absorption band. The propagated signal field structures presented in Figs. 15.23 and 15.25 at one, three, and five absorption depths are seen to be in the immature dispersion regime, whereas those presented in Figs. 15.24 and 15.26 are seen to be in the mature dispersion regime, the transition from immature to mature dispersion occuring at approximately ten absorption depths (this is to be contrasted with a single absorption depth in the below absorption band frequency domain). Notice that, although the main signal at !c has largely attenuated away when z > 10zd , the peak amplitudes in the Sommerfeld and Brillouin precursors have not (for example, at ten absorption depths, the main signal amplitude has been attenuated from unity to the value e 10 ' 0:0000454, while the peak amplitude of the interfering Sommerfeld and Brillouin precursors is approximately 0:2, over three orders of magnitude larger). The dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 5  1016 r=s propagated signal wavefield at 5 ( symbols), 10 (C symbols), and 15 ( symbols) absorption depths is illustrated in Fig. 15.27. The effects of interference between the high-frequency Sommerfeld precursor and the low-frequency Brillouin precursor are evident as the oscillation frequency approaches the signal carrier frequency !c from below as  approaches c from below. This dynamical field behavior continues just above the upper edge of the absorption band, as illustrated in Figs. 15.28 and 15.29 when !c D 6  1016 r=s. It is evident that an experimental measurement of the propagated field structure presented in either Figs. 15.24, 15.26, or 15.29 would detect only the interfering

550

15 Continuous Evolution of the Total Field

0.1 z = 10zd

AH(z,t)

0 0.1 z = 30zd 0

0.1 z = 50zd 0

1

q SB q 0

2

3

4

q Fig. 15.29 Dynamical signal evolution with (barely) above absorption band angular carrier frequency !c D 6  1016 r=s at 10, 30, and 50 absorption depths

precursor field structure as the main signal has attenuated away, and consequently would measure a “signal velocity” that is associated with the pronounced peak in the precursor field. This measured velocity value would then be close to the vacuum speed of light c, which is much greater than the actual signal velocity (see, for example, the signal velocity measurement reported in [20] and its criticism in [21]). As the applied angular signal frequency !c is increased above the medium absorption band, the Sommerfeld (or first) precursor field becomes more pronounced in the total field evolution, as is readily evident in Fig. 15.30 when !c D 7  1016 r=s and Fig. 15.31 when !c D 8  1016 r=s. In both cases, the angular carrier frequency is sufficiently small that !c < !SB so that there is no prepulse formation. Notice that, in both cases, the propagated field structure at a single absorption distance (z D zd ) is in the immature dispersion regime wherein the precursor field structure is not yet sufficiently well defined, whereas at five absorption depths (z D 5zd ) the propagation distance is large enough to be in the mature dispersion regime wherein the precursor field structure is well defined. The transition between these two dispersion regimes occurs at approximately three absorption depths for both the !c D 7  1016 r=s (Fig. 15.30) and !c D 8  1016 r=s (Fig. 15.31) angular carrier frequency cases. The interference between the trailing tail of the Sommerfeld precursor with the entire Brillouin precursor evolution is clearly evident in these two field structures. Notice that as the carrier frequency !c increases above the medium absorption band and the material attenuation ˛.!c / decreases monotonically, it is still much larger than that over much of the precursor field evolution so that, for the

15.5 The Heaviside Step Function Modulated Signal

551

0.5 z = zd

0.1 0

AH(z,t)

−0.1 0.2 z = 3zd 0 0.1

z = 5zd

0

q0 1 qSB

3 qc

5

7

9

q Fig. 15.30 Dynamical signal evolution with above absorption band angular carrier frequency !c D 7  1016 r=s at 10, 30, and 50 absorption depths

0.5 z = zd

0.1

AH(z,t)

0

0.2 z = 3zd 0 0.1

z = 5zd

0 1

qSB q 0

2

qc

3

q Fig. 15.31 Dynamical signal evolution with above absorption band angular carrier frequency !c D 8  1016 r=s at 10, 30, and 50 absorption depths

552

15 Continuous Evolution of the Total Field 0.4 z = zd 0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

q c1 q c2 1

qSB

q0

2

q

qc

2.5

Fig. 15.32 Dynamical signal evolution with above absorption band angular carrier frequency !c D 9  1016 r=s at 10, 30, and 50 absorption depths

largest propagation distance considered (z D 5zd ) in these two figures, the amplitude of the main signal evolution is becoming negligible in comparison to the peak amplitude of either precursor field. The dynamical field evolution of the propagated signal wavefield when !c > !SB is illustrated in Figs. 15.32 (for !c D 9  1016 r=s) and 15.33 (for !c D 10  1016 r=s) at one, three, and five absorption depths. The corresponding numerically determined values of the instantaneous angular oscillation frequency for the three signal evolutions described in Fig. 15.33 is given in Fig. 15.34, where the symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. This instantaneous angular oscillation frequency is then seen to first reach the signal carrier frequency !c from above at  D c1 , remains equal to !c over the space–time interval c1 <  < c2 , then scatters about !c as  increases above c2 (because of interference with the Brillouin precursor field evolution), and finally stabilizes at !c at  D c and remains at that value for all  > c . The prepulse formation predicted by the asymptotic theory is therefore clearly obtained when !c > !SB . The numerically determined values of c1 , c2 , and c at each selected value of !c may then be used to calculate the relative signal velocity values vc1 =c D 1=c1 , vc2 =c D 1=c2 , and vc =c D 1=c , which may then be compared to the signal velocity values predicted by the asymptotic theory for the same Lorentz model dielectric. This has been done by Oughstun, Wyns, and Foty [22] for a single resonance Lorentz model dielectric with model parameters similar to that used by Brillouin except that the value of the phenomenological damping constant ı has been halved. The results are presented in Fig. 15.35. As can be seen, excellent agreement between the

15.5 The Heaviside Step Function Modulated Signal 0.4

553

z = zd

0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

qSB 1

q c1

q c2

q0

q

qc 2

2.5

Fig. 15.33 Dynamical signal evolution with above absorption band angular carrier frequency !c D 10  1016 r=s at 10, 30, and 50 absorption depths

Instantaneous Oscillation Frequency (r/s)

2

x 1017

1.8 1.6 1.4 1.2 wc 0.8 0.6 0.4 0.2 0

1

q c1

q c2 1.5

q

qc 2

2.5

Fig. 15.34 Dynamical evolution of the instantaneous angular frequency of oscillation of the !c D 10  1016 r=s propagated signal wavefields in Fig. 15.33

554

15 Continuous Evolution of the Total Field 1.0

v c 1 /c 0.8

v c2 /c

1/q0

vE/c

0.6

v /c

v c /c vc/c

0.4

0.2

0 0

2

0

6

8

SB

10

12

14

(X 1016 r/s)

Fig. 15.35 Angular frequency dependence of the relative main signal velocity vc .!c /=c D 1=c .!c /, relative anterior pre-signal velocity vc1 .!c /=c D 1=c1 .!c /, and relative posterior pre-signal velocity vc2 .!c /=c D 1=c2 .!c / in a single-resonance Lorentz model dielectric characp terized by the medium parameters !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:14  1016 r=s as described by the asymptotic theory (solid curves) and the numerically measured results (data points with error bars) of Ref. [22]. The dashed curve describes the frequency dependence of the relative energy transport velocity vE .!c /=c in the dispersive medium

numerical experimental results (indicated by the data points with error bars) and the description afforded by the modern asymptotic theory is maintained over the entire angular frequency domain considered. In addition to demonstrating the accuracy of the modern asymptotic theory in describing the complete evolution of a Heaviside step function modulated signal in a single resonance Lorentz model dielectric, these results also provide a physical measure of the signal velocity that is based solely on the measurable instantaneous angular frequency of oscillation of the propagated field in the mature dispersion regime.

15.5.2 Signal Propagation in a Double Resonance Lorentz Model Dielectric Because the distant saddle points in a double resonance Lorentz model dielectric evolve in the angular frequency domain above the upper absorption band, the dynamical evolution of the Sommerfeld precursor is similar to that in an equivalent singleq resonance Lorentz model dielectric with angular resonance frequency !N2 D !2 .b02 C b22 /=.b22 C b02 !22 =!02 /, plasma frequency b 2 b02 C b22 , and

15.5 The Heaviside Step Function Modulated Signal

555

damping constant ı .ı0 C ı2 /=2 with ı0 ı2 [see (13.11) and compare (12.202) and (12.203) with (12.272) and (12.273)]. The description of the Heaviside step function Sommerfeld precursor AHs .z; t / presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the total propagated signal. In like fashion, because the near saddle points in a double resonance Lorentz model dielectric evolve in the angular frequency domain below the lower absorption band, the dynamical evolution of the Brillouin precursor is similar to that for an equivalent singleq resonance Lorentz model dielectric with angular resonance frequency !N 0 D !0 .b02 C b22 /=.b02 C b22 !02 =!22 /,

plasma frequency b 2 b02 .1 C b22 !04 =.b02 !24 //, and damping constant ı ı0 ı2 [see (13.65) and compare (13.66) and (13.67) with (13.68) and (13.69)]. The description of the Heaviside step function Brillouin precursor AHb .z; t / presented in Sect. 15.5.1 for a single resonance Lorentz model dielectric then applies here, including the description of the resonance phenomena and its cancellation by the pole contribution in the construction of the asymptotic behavior of the total propagated signal. As a consequence, the analysis presented here q can focus on q the propagated   2 2 !1  ı0 ; !22  ı22 in the signal behavior for angular carrier frequencies !c 2 passband between the two absorption bands of the double resonance Lorentz model dielectric. The uniform asymptotic behavior of the individual Sommerfeld, middle, and Brillouin precursor fields at five absorption depths (z D 5zd ) in a double resonance Lorentz model dielectric with p < 0 is illustrated in Fig. 15.36 when the angular q carrier frequency !c is near the lower end of the medium pass band q 2  2 !1  ı0 ; !22  ı22 . The superposition of these three precursor fields then produces the resulatant total precursor field evolution illustrated in Fig. 15.37. At this propagation distance the main signal contribution has been reduced in amplitude to the value e z=zd D e 5 ' 0:006738, which is almost an order of magnitude smaller than the peak amlitude values of both the middle and Brillouin precursor fields. Similar results are obtained when the angular carrier frequency !c is shifted upward toward the upper end of the medium pass band, as illustrated in Figs. 15.38 and 15.39. Notice the increase in the relative amplitude of the Sommerfeld precursor AHs .z; t / and the decrease in the relative amplitudes of both the middle AHm .z; t / and Brillouin AHb .z; t / precursor fields as the carrier frequency is increased through the pass band. As a result, the main signal amplitude is now only about a third of the peak amplitude in the precursor field evolution. Nevertheless, as the propagation distance increases further, the main signal amplitude will decrease at a faster rate than the precursor fields, resulting in a propagated wavefield structure that is completely dominated by the total precursor field AHp .z; t / D AHs .z; t / C AHm .z; t / C AHb .z; t /;

(15.79)

556

15 Continuous Evolution of the Total Field

AHs(z,t)

0.02 0

−0.02

AHm(z,t)

0.02 0

−0.02

AHb(z,t)

0.02 0 _

−0.02

1

q1 1.2

qSB

q1 1.4 q0

1.6

1.8

2

q

Fig. 15.36 Superposition of the Sommerfeld AHs .z; t /, middle AHm .z; t /, and Brillouin AHb .z; t / precursors to produce the total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the lower end of the pass band of a double resonance Lorentz model dielectric Middle Precursor Evolution

0.04

AHp(z,t)

0.02 Second Precursor Evolution

First Precursor Evolution

0

q1

qSB

q0

−0.02

−0.04

_ q1 1

1.2

1.4

q

1.6

1.8

2

Fig. 15.37 Total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the lower end of the pass band of a double resonance Lorentz model dielectric

for z > 0 when the inequality p > 0 is satisfied. When the opposite inequality is satisfied, so that p < 0 , the total precursor field is given by AHp .z; t / D AHs .z; t / C AHb .z; t /; for z > 0, the middle precursor field now being absent.

(15.80)

15.5 The Heaviside Step Function Modulated Signal

557

AHs(z,t)

0.01

0

AHm(z,t)

0.01

0

AHb(z,t)

0.01

0

−0.01

1

_ q1 1.2 qSB

q1 q 1.4 0

1.6

1.8

2

q

Fig. 15.38 Superposition of the Sommerfeld AHs .z; t /, middle AHm .z; t /, and Brillouin AHb .z; t / precursors to produce the total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the upper end of the pass band of a double resonance Lorentz model dielectric 0.02

0.01

AHp(z,t)

Middle Precursor Evolution First Precursor Evolution

Second Precursor Evolution

q1

0 qsb

q0 _ q1

−0.01

1

1.2

1.4

1.6

1.8

2

q Fig. 15.39 Total precursor field for the propagated Heaviside step function signal wavefield AH .z; t / with angular carrier frequency !c near the upper end of the pass band of a double resonance Lorentz model dielectric

The accuracy of this uniform asymptotic description is clearly evident in the associated figure pairs presented in Figs. 15.40 and 15.41, and Figs. 15.42 and 15.43. The first figure in each figure pair presents three successive propagated signal waveforms

558

15 Continuous Evolution of the Total Field 0.4 0.2

z = zd

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd

0 |

1

|

qSB q0

|

q

2 qc

3

Fig. 15.40 Dynamical signal evolution with intra-passband angular carrier frequency !c 2 .!1 ; !2 / at one, three, and five absorption depths when p > 0

Instantaneous Oscillation Frequency (r/s)

3

x 1016

2.5

c

1.5

1

0.5

0

1

2

3

q

Fig. 15.41 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wavefields in Fig. 15.40

at three successive propagation distances at the same fixed angular carrier frequency !c that is chosen to be in the medium pass band between the two absorption bands. These numerical results were computed using a 222 -point FFT simulation of the exact Fourier–Laplace integral representation of the propagated plane-wave pulse given in (12.1) with !max D   1019 r=s. The results are displayed most conveniently as a function of the dimensionless space–time parameter  D ct=z with

15.5 The Heaviside Step Function Modulated Signal

559

0.4 0.2

z = zd

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd

0

|

1

1.2

|

q 01.4

q c1.6

1.8

q Fig. 15.42 Dynamical signal evolution with intra-passband angular carrier frequency !c 2 .!1 ; !2 / at one, three, and five absorption depths when p < 0

Instantaneous Oscillation Frequency (r/s)

5

x 1016

4.5 4 3.5 ωc 2.5 2 1.5 1 0.5 0

|

1.2

qc 1.6

1.4

1.8

q Fig. 15.43 Dynamical evolution of the instantaneous angular frequency of oscillation of the propagated signal wavefields in Fig. 15.42

fixed z > 0, because critical aspects of the wavefield evolution (e.g., the critical space–time points SB , 0 , c , cm , and cj , j D 1; 2) are then independent of the propagation distance z. The sequence of numerically determined wavefield plots presented in Fig. 15.40 describe the dynamical evolution of a Heaviside unit step function signal AH .z; t /

560

15 Continuous Evolution of the Total Field

with fixed angular carrier frequency !c D 2  1016 r=s in the pass band of a douD 1  1016 r=s, ble resonance Lorentz model dielectric with medium parameters !0 p p b0 D 0:61016 r=s, ı0 D 0:11016 r=s, and !2 D 41016 r=s, b2 D 0:21016 r=s, ı2 D 0:1  1016 r=s. In that case, p > 0 , the middle saddle points never become the dominant saddle points and the middle precursor doesn’t appear in both the total precursor field evolution [see (15.80)] and the dynamical field evolution (see Fig. 13.15 for the impulse response of this particular medium and Fig. 15.4 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.40 is presented in Fig. 15.41, where the  symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Notice that in this case when p > 0 , the instantaneous angular oscillation frequency settles to the signal frequency !c after the complete evolution of the precursor fields. The sequence of numerically determined wavefield plots presented in Fig. 15.42 describe the dynamical evolution of a Heaviside unit step function signal AH .z; t / with fixed angular carrier frequency !c D 3  1016 r=s in the pass band of a douD 1  1016 r=s, ble resonance Lorentz model dielectric with medium parameters !0 p p 16 16 16 b0 D 0:610 r=s, ı0 D 0:110 r=s, and !2 D 710 r=s, b2 D 0:21016 r=s, ı2 D 0:1  1016 r=s. In that case p < 0 , the middle saddle points become the dominant saddle points over the short space–time interval  2 .SM ; MB /, and the middle precursor appears between the Sommerfeld and Brillouin precursors in both the total precursor field evolution [see (15.79)] and the dynamical field evolution (see Fig. 13.16 for the impulse response of this particular medium and Fig. 15.5 for the frequency dispersion of the signal velocity). The corresponding numerically determined instantaneous angular oscillation frequency for each signal pattern illustrated in Fig. 15.42 is presented in Fig. 15.43, where the symbols are for the single absorption depth wavefield, the C symbols are for the three absorption depth wavefield, and the  symbols are for the five absorption depth wavefield. Notice that in this case when p < 0 , the instantaneous angular oscillation frequency first settles to the signal frequency !c at the space–time point  D cm after the evolution of the Sommerfeld and middle precursor fields, but then diverges away from this value over the space–time interval  2 .c2 ; c / because of interference with the Brillouin precursor evolution, after which it resettles to !c and remains at that value for all  > c . These numerical results then confirm the prepulse formation in the pass band of a double resonance Lorentz model dielectric when p < 0 . Comparison of these results with the description afforded by the group velocity approximation in Sect. 15.5.2 reveals the importance of the precursor field evolution in correctly describing the observed signal distortion (see Fig. 11.29). This middle precursor evolution in a double resonance Lorentz model dielectric has also been observed by Karlsson and Rikte [23] using the dispersive wave splitting approach introduced by He and Str˘om [24] in 1996.

15.5 The Heaviside Step Function Modulated Signal

561

15.5.3 Signal Propagation in a Drude Model Conductor The complex index of refraction of a Drude model conductor [25] is given by [see (12.153)] !1=2 !p2 n.!/ D 1  ; (15.81) !.! C i  / where !p is the plasma frequency and  is the damping constant which, for a plasma, is given by the effective collision frequency. Estimates of these model parameters for the E-layer of the ionosphere, given by [26] !p   107 r=s;    105 r=s; are used in this section to describe the transient wavefield phenomena associated with ionospheric signal propagation. The results presented here are based on the asymptotic results presented by Cartwright et al. [27, 28]. From the analysis presented in Sect. 12.3.4, the Drude model possesses a pair of distant saddle points [cf. (12.309)–(12.311)] !SP˙ . / D ˙./  i d

  1 C . / 2

(15.82)

for   1, with the second approximate expressions s ./

. /

!p2  2 2  1

!p2  2 1

C



2 ; 4

2 .27/.4/

./

;

(15.83)

(15.84)

which begin at !SP˙ .1/ D ˙1  i  and move into the outer branch point zeros d q !z˙ D ˙ !p2  .=2/2  i =2 (see 12.31) as  ! 1, and a single near saddle point [cf. (12.317)] 8 3 !p2

1  3 !p2   2 2 A 2 @ C 4 2 2 0

!SPn . / i  9 !p2   8 2 31=2 9  2 ˆ > ˆ > 2 2 < 9 !p   6 7 = 7   1   16 4 ˆ > 3.!p2  2 / 5 ˆ > : ; 16 4  2 C 4 2

(15.85)

562

15 Continuous Evolution of the Total Field

that moves down the positive imaginary axis as   1 increases, asymptotically approaching the origin as  ! 1 (see Figs. 12.36–12.38). The uniform asymptotic description of the Sommerfeld precursor AHs .z; t / for the unit Heaviside step function signal in a Drude model conductor is then given by (13.61) for all   1. The resultant first precursor field evolution of a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the Drude model of the E-layer of the ionosphere is presented in Fig. 15.44. This transient field structure is characteristic of the Sommerfeld precursor in Lorentz-type dielectrics, with amplitude that begins at zero at the luminal space–time point  D 1, rapidly building to its peak value soon after this point, and then decreasing monotonically to zero as  ! 1. In addition, the instantaneous angular frequency of oscillation !s . / of this Sommerfeld precursor (see Sect. 13.2.4) begins at infinity (when the field amplitude vanishes) and then chirps down qwith increasing  > 1, asymptotically approaching the limiting value C 1, with the result [27] (

c=z  2 00 .!SPn ;  /

AHb .z; t /  
0. Most noticeable, however, is the very long tail exhibited by the Brillouin precursor in a conducting medium. This characteristic is not observed in a dispersive medium with zero conductivity and may have important health and safety implications. The uniform asymptotic description of the signal contribution AHc .z; t / due to the simple pole singularity at ! D !c is given in (14.75)–(14.78) and is illustrated in Fig. 15.46 with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths into the Drude model of the E-layer of the ionosphere. The resultant total signal evolution, given by the superposition of the Sommerfeld precursor field, the Brillouin precursor field, and the signal contribution as AH .z; t / D AHs .z; t / C AHb .z; t / C AHc .z; t /;

(15.87)

is then given by the sum of the field plots in Figs. 15.44–15.46, with result given by the solid curve in Fig. 15.47. For comparison, the numerically determined Heaviside unit step function signal wavefield with the same carrier frequency (!c D 1105 r=s) propagated the same distance (z D 5zd ) in the same Drude model medium using a 222 -point FFT simulation of the dispersive propagation problem with maximum

564

15 Continuous Evolution of the Total Field 0.01

AHc(z,t)

Fig. 15.46 Dynamical evolution of the signal contribution AHc .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27])

0

−0.01

0

1000

3000

2000

4000

0.03

AH(z,t)

0.02

0.01

0

−0.01 0

1000

2000 q

3000

4000

Fig. 15.47 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1105 r=s at five absorption depths (z D 5zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

sampling frequency fmax D 5  108 r=s sampled at the Nyquist rate is described by the dashed curve in Fig. 15.47. Although this is not sufficiently large to adequately describe the Sommerfeld precursor component,5 it is more than sufficient to provide an accurate description of the Brillouin precursor and signal contributions to the total field evolution, as evident in Fig. 15.47. The result is a high-frequency ripple riding on the low-frequency Brillouin precursor, as illustrated in Fig. 15.47 at 5 5

Complementary hybrid numerical-asymptotic techniques have been developed by both Dvorak et al. [29,30] and Hong et al. [31] to properly handle this problem by extracting the high-frequency behavior in order to treat it analytically, leaving the lower frequency behavior to be dealt with numerically.

15.5 The Heaviside Step Function Modulated Signal

565

0.006

AH(z,t)

0.004

0.002

0

0

2000

4000 q

6000

8000

Fig. 15.48 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at ten absorption depths (z D 10zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

AH(z,t)

0.00005

0

−0.00005

0

10000

20000

30000

40000

50000

q

Fig. 15.49 Dynamical evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1  105 r=s at 200 absorption depths (z D 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

absortpion depths, Fig. 15.48 at ten absorption depths, and in Fig. 15.49 at 200 absorption depths. At this final propagation distance (z D 200zd ), the high-frequency Sommerfeld precursor is the dominant contribution to the total propagated wavefield structure. However, because of its (comparatively) very short space–time lifetime

566

15 Continuous Evolution of the Total Field

AH(z,t)

0.00005

0

−0.00005

0

1.5

2.0

2.5

Fig. 15.50 Early space–time evolution of the total propagated signal wavefield AH .z; t / due to a Heaviside unit step function modulated signal with VLF angular carrier frequency !c D 1105 r=s at 200 absorption depths (z D 200zd ) into the E-layer of the ionosphere (from Cartwright and Oughstun [27]). The solid curve describes the uniform asymptotic behavior and the dashed curve the numerically determined behavior

at the front of the signal, it is not discernable in Fig. 15.49, and so is depicted in Fig. 15.50 with the same amplitude scale used in Fig. 15.49. When a Heaviside step function signal propagates through a pure (i.e., zero conductivity) dispersive dielectric, such as that described by either the Lorentz or Debye models, the Brillouin precursor is found to decay only algebraically as z1=2 due to the fact that the near saddle point crosses the origin at  D 0 n.0/. In a purely conducting medium such as that described by the Drude model, however, the complex dielectric permittivity c .!/ D .!/ C k4ki  .!/=! possesses a simple pole at the origin which prevents the near saddle point from reaching that critical point for finite  . As a consequence, the Brillouin precursor in a conductor will decay at a faster rate than the algebraic z1=2 rate found in a pure dielectric. Nevertheless, detailed numerical studies [28] show that the algebraic decay rate of the peak amplitude point of the Sommerfeld precursor approaches z3=4 as z ! 1, while that of the Brillouin precursor approaches z2 as z ! 1.

15.5.4 Signal Propagation in a Rocard–Powles–Debye Model Dielectric The asymptotic description of dispersive signal propagation in a Debye-type dielectric results in the representation [see (15.3)] AH .z; t / D AHb .z; t / C AHc .z; t /;

(15.88)

15.5 The Heaviside Step Function Modulated Signal

567

where the asymptotic behavior of the Brillouin precursor is described in Sect. 13.4 and the asymptotic behavior of the signal contribution is described in Sect. 14.2. The numerically determined signal propagation in a double relaxation time Rocard-Powles-Debye model of triply-distilled water, whose angular frequency dispersion is illustrated in Fig. 15.51, is depicted in Fig. 15.52 at one (z D zd ), three (z D 3zd ), and five (z D 5zd ) absorption depths zd ˛ 1 .!c / when the angular carrier frequency is !c D 2fc with fc D 1 GHz. This carrier frequency is indicated by the plus sign in Fig. 15.51 on both the curves for the real and imaginary parts of .!/. The numerical results presented in Fig. 15.52 were obtained using a 221 -point FFT simulation of the dispersive propagation problem with maximum sampling frequency fmax D 1  1012 s sampled at the Nyquist rate, where the minimum, nonzero sampled frequency value is then given by fmin D fmax =.2N / with N D 221 ; the radian equivalent values of both of these frequency values are indicated by the -symbol on both of the curves in Fig. 15.51. A similar set of calculations is presented in Figs. 15.53 and 15.54 when fc D 200 GHz with N D 221 and fmax D 11013 s. These numerical results show that the observed signal distortion is primarily due to the Brillouin precursor, as described by the asymptotic theory [32], whose peak amplitude at  D 0 decays algebraically with propagation distance z > 0 as z1=2 . Comparison of these two sets of numerical results shows that, as the carrier frequency increases through the absorption peak at ! 2=0 , where 0 D 8:30  1012 s for triply distilled water, the relative time scale between the Brillouin precursor and the signal contribution changes as the latter becomes more of a ripple riding on the (relatively) slowly decaying tail of the Brillouin precursor.

10

r ( c)

Real and Imaginary Parts of the Relative Dielectric Permittivity

8 r( )

6

4

2 i( ) i ( c)

0

105

1010 (r/s)

1015

Fig. 15.51 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply distilled H2 O. The values of r .!c / and i .!c / at !c D 2fc with fc D 1 GHz are marked on each curve

568

15 Continuous Evolution of the Total Field z = zd 0.4 0.2 0

AH(z,t)

−0.2 0.2 z = 3zd 0 0.2 z = 5zd 0 q0 5

7

9

11

q

13

15

Fig. 15.52 Dynamical signal evolution with a 1 GHz carrier frequency at one, three, and five absorption depths in triply distilled H2 O

10

r( )

Real and Imaginary Parts of the Relative Dielectric Permittivity

8

6

4 r ( c)

i( )

2

i ( c)

0 105

1010 (r/s)

1015

Fig. 15.53 Frequency dispersion of the real and imaginary parts of the relative dielectric permittivity of triply distilled H2 O. The values of r .!c / and i .!c / at !c D 2fc with fc D 100 GHz are marked on each curve

15.5 The Heaviside Step Function Modulated Signal

569

0.6 z = zd

0.4 0.2

AH(z,t)

0 −0.2 0.2 z = 3zd 0 z = 5zd 0

q0 0

10

20

30

40

50

q

Fig. 15.54 Dynamical signal evolution with a 100 GHz carrier frequency at one, three, and five absorption depths in triply distilled H2 O

15.5.5 Signal Propagation along a Dispersive Transmission Line Although it was not identified as such at the time, the Brillouin precursor has also been observed by Veghte and Balanis [33] in their analysis of transient signal propagation along a dispersive microstrip transmission line. The cross-sectional geometry of the microstrip waveguide is depicted in Fig. 15.55, the space between the metallic conductors being filled with a dispersive dielectric material with relative permittivity .!/ and constant magnetic permeability  D 0 . The voltage signal along the dispersive line is given by the Fourier–Laplace integral representation [33] 1 V .z; t / D 2

Z

1

Q VQ .z0 ; !/e i .k.!/ z!t / d!

(15.89)

1

for all z z  z0  0, where VQ .z0 ; !/ D

Z

1

V .z0 ; t /e i!t dt

(15.90)

1

is the Fourier spectrum of the initial voltage pulse at z D z0 . Here ! Q k.!/ ˇ.!/ C i ˛.!/ D  1=2 .!/ c

(15.91)

570

15 Continuous Evolution of the Total Field

Fig. 15.55 Cross-sectional geometry of a microstrip transmission line

w t sub

h

is the complex wavenumber that is a characteristic of the transmission line properties. Veghte and Balanis [33] considered a lossless, dispersive transmission line using the effective microstrip dispersion model developed by Pramanick and Bhartia [34], with real relative dielectric permittivity given by 0 .f / D sub  eff

sub  eff .0/   ;  .0/ f 2 1 C eff sub ft

(15.92)

Z0 ; 20 h

(15.93)

0 where eff .0/ D eff .0/, with

ft

where sub is the relative dielectric permittivity of the substrate (itself a dispersive medium) of thickness h, and Z0 is the characteristic impedance of the microstrip line. Typical values for these parameters for a microstrip line operating over the frequency domain from 1 GHz to 1  104 GHz are sub D 10:2, eff .0/ D 6:76, h D 0:0635 cm, and Z0 D 53:6 (, with width to height ratio w= h < 1 (see Fig. 15.55), so that ft D 33:585 GHz. Comparison of this dispersion model with the real part of the Debye model [see (4.179) in Vol. 1] shows that (15.92) can readily be generalized to a causal model as eff .!/ D sub C

eff .0/  sub ; 1 C i eff !

(15.94)

where the plus sign is now used in the denominator in order to make the model attenuative. The effective relaxation time in this model is defined here as p eff .0/=sub eff (15.95) !t with !t 2ft . The corresponding parameter values for the above microstrip line example are then eff D 3:8578  1012 s with !t D 2:1102  1011 r=s. The resultant angular frequency dispersion is illustrated in Fig. 15.56 (note that the imaginary part of the relative effective dielectric permittivity has been magnified ten times in the figure). Notice that, unlike the Debye model, the real part of the (effective) dielectric permittivity now increases as ! increases above 2=eff . The numerically determined voltage signal propagation along this dispersive microstrip transmission line using an N D 220 FFT simulation of the Fourier integral representation in

15.5 The Heaviside Step Function Modulated Signal

571 r(

)

3

Real and Imaginary Parts of the Relative Dielectric Permittivity

r ( c)

2 10 i (

c)

10 i ( )

1

0 105

1010 (r/s)

1015

Fig. 15.56 Frequency dispersion of the real and (ten times the) imaginary parts of the relative effective dielectric permittivity of a microstrip transmission line with sub D 10:2, eff .0/ D 6:76, Z0 D 53:6 (, and h D 0:0635 cm. The values of r .!c / and i .!c / at !c D 2fc with fc D 10 GHz are marked on each curve 0.6 z = zd

0.4

VH(z,t) - volts

0.2 0

− 0.2 0.2 z = 3zd

0 0.2 z = 5zd

0 1

2

q0

3 q

4

5

Fig. 15.57 Dynamical signal evolution with a 10 GHz carrier frequency at one, three, and five absorption depths along the microstrip transmission line

(15.89) with fmax D 1  1013 Hz sampled at the Nyquist rate (the radian equivalent values of both fmax and fmin D fmax =2N are indicated by the  symbols on the graphs in Fig. 15.56) is illustrated by the sequence of voltage plots in Fig. 15.57 at

572

15 Continuous Evolution of the Total Field

one, three, and five absorption depths when the input signal is a 1 V heaviside step function signal with fc D 10 GHz carrier frequency. These results clearly show the development of a Brillouin precursor at the leading edge of the propagated voltage signal that is strikingly similar to that for a Rocard–Powles–Debye model dielectric [cf. Fig. 15.52].

15.6 The Rectangular Pulse Envelope Modulated Signal From the results of Sect. 11.2.4, the propagated wavefield due to an initial rectangular envelope modulated sinusoidal wave with fixed angular frequency !c and initial time duration T > 0 (assumed to be an integral number of periods Tc D 2=!c of the carrier wave) may be represented as the difference between two Heaviside unit step function modulated signals separated in time by the initial pulse width T , so that [see (11.64)] AT .z; t / D AH .z; t; 0/  AH .z; t; T / Z

Z z z 1 1 1 z c 0 .!;/ d!  e i!c T z c T .!;T / d! D < 2 C !  !c C !  !c (15.96) for all z  0, where AH .z; t; T / denotes the propagated signal wavefield due to an input Heaviside unit step function signal that turns on (i.e., jumps discontinuously from zero to one) at time t D T , with   0 .!;  / .!; / D i ! n.!/   ; ct D ; z

(15.97) (15.98)

and   T .!; / i ! n.!/  T ; c T .t  T /: z

(15.99) (15.100)

Consequently, the results of Sect. 15.5 may be directly applied to obtain a detailed understanding of the asymptotic behavior of any given rectangular envelope pulse as z ! 1. Specifically, the uniform asymptotic behavior of the front AH .z; t; 0/ of the pulse is completely described by the results presented in Sect. 15.5, and the uniform asymptotic behavior of the back AH .z; t; T / of the pulse is also described by the results in Sect. 15.5 with  replaced by the retarded space–time parameter T and the constant phase factor e i!c T multiplying the final result. The total uniform asymptotic behavior of the propagated rectangular envelope modulated pulse is then given by the difference between these two component wavefields.

15.6 The Rectangular Pulse Envelope Modulated Signal

573

Consider first the asymptotic behavior of the rectangular pulse envelope wavefield AT .z; T / given in (15.96) for space–time values   T c , where T c is that value of T at which the pole contribution to AH .z; t; T / becomes the dominant contribution to that field component. Because T c > c for all T > 0 and finite z > 0, so that the pole contribution to AH .z; t; 0/ is also the dominant contribution to that field component, the asymptotic behavior of the total wavefield is then given by   !c nr .!c /    

 Ce ˛.!c /z sin cz !c nr .!c /  T  !c T

AT .z; t /  e ˛.!c /z sin

z

c

D0

(15.101)

as z ! 1 with   T c . That is, the pole contribution from the back of the pulse identically cancels the pole contribution from the front of the pulse for all   T c . Because T D   .c=z/T , it is then seen that the space–time evolution of the back of the pulse is retarded in  by the amount .c=z/T from the corresponding  -evolution of the front of the pulse. The critical space–time value  D T c at which the pole contribution from the back of the pulse arrives is then given by T c D c C .c=z/T . Hence, the -duration of the propagated rectangular envelope pulse AT .z; t / between the front and back pole contribution arrivals is c D

c T; z

(15.102)

and the corresponding temporal duration of the propagated pulse between these two space–time points is z (15.103) tc D c D T: c Consequently, any pulse broadening and envelope degradation due to dispersion is caused primarily by the precursor fields associated with the front and back of the pulse. If the precursor fields are neglected and only the pole contributions are considered by themselves (i.e., without any interference from the interacting saddle points), then the propagated wavefield becomes AT .z; t /  e ˛.!c /z sin

z c

  !c nr .!c /  

(15.104)

for all  2 Œc ; T c , and is zero for all other values of . Consequently, by neglecting the precursor phenomena associated with the front and back edges of the pulse, as well as any interference from the interacting saddle points with the pole contributions, the propagated pulse retains its rectangular envelope shape and temporal width T , and is simply attenuated with the propagation distance z > 0 by the amplitude factor e ˛.!c /z .

574

15 Continuous Evolution of the Total Field

15.6.1 Rectangular Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric To obtain a complete understanding of the propagation of a rectangular envelope modulated signal through a causally dispersive medium, the detailed interaction of the precursor fields associated with the front and back edges of the pulse with both the pole contributions and each other must be fully taken into account, and that requires a specific model for the material dispersion. Attention is first given here to the single resonance Lorentz model dielectric. As described in Sect. 15.5.1, the propagation characteristics of both the front and back edges of the pulse depend upon the value of the applied angular carrier frequency !c , the asymptotic description separating into the normally dispersive below absorption band do

q 2 main !c 2 0; !0  ı 2 , the normally dispersive above absorption band domain

q 2  !1  ı 2 ; 1 , and the anomalously dispersive absorption band domain !c 2 q  q 2 !0  ı 2 ; !12  ı 2 . As a further complication, the propagated wavefield !c 2 behavior also depends strongly upon both the initial pulse duration T > 0 and the propagation distance z > 0 at which the dynamical field behavior is observed, as characterized by the following sequence of space–time domains:  Minimal Distortion Domain: For a sufficiently small propagation distance z > 0

and=or a sufficiently long initial pulse duration T such that the inequality c  1 < .c=z/T is satisfied, the first and second precursor fields associated with the leading edge of the pulse will evolve undisturbed by the precursor field contributions associated with the trailing edge of the pulse, and the first and second precursor fields associated with the back of the pulse will only interfere with the pole contribution associated with the front of the pulse.  Intermediate Distortion Domain: For either larger propagation distances z or shorter initial pulse durations T such that SB  1 < .c=z/T < c  1, the first precursor field associated with the leading edge of the pulse will still evolve undisturbed, but the leading edge second precursor field will overlap and interfere with the trailing edge first precursor field, and the trailing edge second precursor field will interfere with the pole contribution associated with the leading edge of the pulse.  Maximal Distortion Domain: Finally, for even larger propagation distances z or even shorter initial pulse widths T such that the inequality 0 < .c=z/T < SB  1 is satisfied, the first and second precursor fields associated with either the leading or trailing edges of the pulse will overlap and interfere with each other. Consequently, for any given initial pulse duration T > 0, its  -value .c=z/T progressively falls into each one of the above space–time domains as the propagation distance z increases, so that for a sufficiently large value of z, the inequality 1 < .c=z/T < SB is eventually satisfied. For positive values of z and T in the initial, minimal distortion domain c  1 < .c=z/T , the envelope degradation and temporal spread of the propagated pulse is

15.6 The Rectangular Pulse Envelope Modulated Signal

575

minimal. As the propagation distance z increases, the envelope degradation and temporal width of the pulse increase as the precursor fields associated with the leading and trailing edges of the pulse increasingly interfere with each other and the two pole contributions come closer together in -space. This transitional behavior marks the intermediate distortion domain. For values of z and T in the final, maximal distortion domain where 0 < .c=z/T < SB  1, the pulse envelope degradation and temporal spread of the propagated pulse are severe. Pulses with short initial pulse widths T then degrade much faster with propagation distance than those with longer initial pulse widths. For two initial rectangular envelope pulses with the same angular carrier frequency !c but with different initial pulse widths T1 and T2 D mT1 , m > 0, the propagation distances in the same dispersive medium at which their propagated field structure in the -domain is identical are related by z2 D mz1 . Before full consideration can be given to a detailed description of the dynamical wavefield evolution due to an input rectangular envelope modulated pulse with constant angular carrier frequency !c , the signal arrival and associated signal velocity must first be carefully described. This description is afforded by the detailed asymptotic analysis presented in [3, 8, 35] which forms the basis of the presentation given here. For an input rectangular envelope modulated signal AT .z; t / D AH .z; t; 0/  AH .z; t; T / of initial time duration T > 0 and fixed angular carrier frequency !c 2 Œ0; !SB /, the signal arrival occurs at the space–time point  D c and the propagated wavefield ceases to oscillate at !c when T   cT=z D c . Both of these transition points propagate with the signal velocity vc .!c / D c=c .!c /. The main body of the propagated pulse that is oscillating at ! D !c then evolves over the space–time interval from  D c to  D c C cT=z with -width c D cT=z and corresponding temporal width tc D

z c D T: c

(15.105)

Consequently, any temporal pulse broadening and envelope degradation of the initial rectangular envelope pulse when !c 2 Œ0; !SB / is due primarily to the precursor field structure of the propagated wavefield that arises from the leading and trailing edges of the pulse. This interference of the signal contribution with the precursor field structure tends to shorten the space–time domain over which the propagated wavefield oscillates predominantly at the input angular carrier frequency !c . Strictly speaking, the temporal width of the rectangular envelope pulse signal is then found to decrease with increasing propagation distance z  0. The commonly observed phenomenon [14, 36–44] of dispersive pulse spreading is obtained only when the propagated signal is redefined to include a range of frequencies about !c , and this, in turn, implies the incorporation of some portion of the leading and trailing edge precursor fields in the definition of the main body of the pulse. For !c > !SB , the signal arrival first occurs at  D c1 when the simple pole at ! D !c is first crossed, and the propagated wavefield finally ceases to oscillate predominantly at !c when T   cT=z D c1 and the pole contribution is subtracted out. Both of these transition points propagate with the presignal velocity vc1 .!c / D c=c1 .!c /. The main body of the propagated pulse that is oscillating at

576

15 Continuous Evolution of the Total Field

! D !c then evolves over the space–time interval from  D c1 to  D c1 C cT=z with -width c D cT=z and corresponding temporal width tc D

z c D T: c

(15.106)

Between these two space–time points that define the main body of the pulse there are, at most, two other distinct transition points at  D c2 and  D c at which the propagated wavefield either ceases to oscillate predominantly at !c or begins again to oscillate predominantly at ! D !c due to the asymptotic dominance of the leading edge Brillouin precursor between these two space–time points. Because of this, the propagated wavefield due to such an input rectangular envelope pulse separates into, at most, two subpulses provided that cT=z > c  c1 , which eventually reduces to a single pulse at a sufficiently large propagation distance z such that cT=z < c  c1 . Apart from this pulse breakup, the only other source of envelope degradation and pulse broadening of the rectangular envelope pulse with carrier frequency !c > !SB is from the leading and trailing edge precursor fields. The resultant angular frequency dependence of the signal velocity for a finite duration rectangular envelope pulse when the inequality cT=z < c  c1 is satisfied is illustrated in Fig. 15.58. When the propagation distance is sufficiently small such that opposite inequality cT=z > c  c1 is satisfied, then the signal velocity is described by Fig. 15.3. As in that figure, the dashed curve describes the energy

1.0 vc 1 /c 0.8 1/q0 0.6

v/c

vE /c

0.4 vc /c 0.2

0 0

2

0

6

8

SB

10

12

14

(X 1016 r/s)

Fig. 15.58 Angular frequency dependence of the relative signal velocity for a rectangular envelope pulse with finite initial duration T > 0 and propagation distance z > 0 such that cT=zp< c c1 in a single resonance Lorentz model dielectric with parameters !0 D 41016 r=s, b D 201016 r=s, and ı D 0:14  1016 r=s. The long dashed curve describes the angular frequency dispersion of the energy transport velocity for a strictly monochromatic wavefield in the medium

15.6 The Rectangular Pulse Envelope Modulated Signal

577

transport velocity of a strictly monochromatic wavefield in the dispersive medium. The discontinuous jump in the signal velocity at ! D !SB is fundamentally due to the change in dominance of the precursor fields at  D SB . Below !SB the signal arrival occurs following the evolution of the leading edge Sommerfeld and Brillouin precursors, whereas above !SB the signal arrival occurs during the evolution of the leading edge Sommerfeld precursor field. When opposite inequality cT=z > c  c1 is satisfied (with z sufficiently large to be in the mature dispersion regime), the propagated wavefield is found to be separated into two pulse whose velocities are described in Fig. 15.3. The first is a prepulse with front velocity vc1 .!c / D c=c1 .!c / and back velocity vc2 .!c / D c=c2 .!c /, followed by a second subpulse with front velocity vc .!c / D c=c .!c / and back velocity vc1 .!c / D c=c1 .!c /. Because vc1 .!c / > vc .!c / [see (15.28)], the back of this second subpulse eventually catches up with the front and cancels it out, this occuring when cT=z D c  c1 . This then marks the transition from the signal velocity being described by Fig. 15.3 to that described in Fig. 15.58 when !c > !SB . From (15.96) and the uniform asymptotic representation of the Heaviside unit step function envelope signal given in (15.70), the uniform asymptotic description of the propagated wavefield due to an input rectangular envelope modulated signal with initial temporal duration T > 0 is given by [3, 8, 35] AT .z; t /  AHs .z; t; 0/ C AHb .z; t; 0/ C AHc .z; t; 0/ AHs .z; t; T /  AHb .z; t; T /  AHc .z; t; T /;

(15.107)

which is simply the difference between the propagated wavefields due to an input Heaviside unit step function modulated signal AH .z; t; 0/  AHs .z; t; 0/ C AHb .z; t; 0/ C AHc .z; t; 0/ that begins to oscillate harmonically at !c at time t D 0 in the z D 0 plane and the Heaviside unit step function modulated signal AH .z; t; T /  AHs .z; t; T / C AHb .z; t; T / C AHc .z; t; T / that begins to oscillate harmonically at !c at time t D T in the z D 0 plane. It is now shown that this asymptotic representation of the propagated rectangular envelope modulated wavefield provides a complete, accurate description of the dynamical pulse evolution in the mature dispersion regime for all !c 2 Œ0; 1/. Consider first the dynamical pulse evolution in the below absorption band domain   q 2 !c 2 0; !0  ı 2 . In this case, the peak amplitude of the Sommerfeld precursor is typically several orders of magnitude less than the peak amplitude of the Brillouin precursor so that the entire propagated wavefield structure in the mature dispersion regime is dominated by the Brillouin precursor and the signal contibution. In the minimal distortion domain cT=z > c  1, the precursor fields associated with the leading edge AH .z; t; 0/ of the pulse will completely evolve prior to the arrival of the precursor fields associated with the trailing edge AH .z; t; T /. Indeed, the trailing edge precursors will arrive only after the leading edge signal component has arrived (at  D c ) and is evolving. Hence, when this condition prevails the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is also minimal, as illustrated in Fig. 15.59. In the intermediate

578

15 Continuous Evolution of the Total Field

AH(z,t,0)

0

q 1

qSB

qc

AH(z,t,T) 0

q 1+cT/z qSB + cT/z

1

qc+cT/z

AT(z,t,) 0

q 1

qSB

qc

qc+cT/z

cT/z

Fig. 15.59 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration q T in the normally dispersive, below absorption band angular signal frequency domain   !c 2 0; !02  ı 2 with propagation distance z in the minimal distortion domain cT=z > c  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is minimal and the resultant pulse distortion is minimal

distortion domain c  1 > cT=z > SB  1, the leading edge first precursor field will still evolve undisturbed, but during the evolution of the leading edge Brillouin precursor, the arrival and evolution of the trailing edge precursors occurs. Hence, when this condition prevails there will be interference between the leading edge Brillouin precursor and the trailing edge Sommerfeld precursor, the trailing edge Brillouin precursor appearing soon after the signal arrival at  D c , as illustrated in Fig. 15.60, so that the resultant pulse distortion is found to be moderate. Finally, in the maximal distortion domain cT=z < SB  1 there will occur a nearly complete overlap of the leading and trailing edge precursors, as illustrated in Fig. 15.61, and the resultant pulse distortion is severe.

15.6 The Rectangular Pulse Envelope Modulated Signal

579

AH(z,t,0) q

0 1

qSB

qc

AH(z,t,T) q

0

1

1+cT/z

qSB + cT/z

qc+cT/z

AT(z,t,) q

0

1

qSB

qc

qc+cT/z

cT/z

Fig. 15.60 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T q in the normally dispersive, below absorption band angular signal frequency   domain !c 2 0; !02  ı 2 with propagation distance z in the intermediate distortion domain c  1cT=z > SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is moderate and the resultant pulse distortion is also moderate

In each below absorption band case, the pole contribution to the total wavefield evolution occurs at the space–time point  D c and is then subtracted out at  D c C cT=z so that the overall temporal width of the propagated signal contribution is equal to the initial pulse width T , as stated in (15.105). However, because of the asymptotic dominance of the trailing edge Brillouin precursor AHb .z; t; T /, this signal contribution is the dominant contribution to the total wavefield evolution only over the space–time domain from  D c to  ' 0 C cT=z. The corresponding temporal width of the propagated signal is then given by z tc ' T  .c  0 / c

(15.108)

580

15 Continuous Evolution of the Total Field

AH(z,t,0) 0

q 1

qSB

qc

AH(z,t,T) 0

q

qSB+cT/z qc+cT/z

1 1+cT/z

AT(z,t,) 0

q 1

qSB

qc

qc+cT/z

cT/z

Fig. 15.61 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration q T in the normally dispersive, below absorption band angular signal frequency domain   !c 2 0; !02  ı 2 with propagation distance z in the maximal distortion domain cT=z < SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

provided that 0 C cT=z > c , which is satisfied up through most of the intermediate distortion domain. When the opposite inequality 0 C cT=z < c is satisfied, the pulse distortion is severe and the total propagated wavefield is dominated by the leading and trailing edge precursor fields. Similar results are obtained in the intermediate signal frequency domain q !c 2

q 2 

q 2 2 2 !0  ı ; !SB which contains the medium absorption band !0 ı ; !12 ı 2 where the medium dispersion is anomalous and extends up to the critical angular frequency value !SB at which the signal velocity of the Heaviside step function modulated signal bifurcates into three branches (see Fig. 15.3). The major difference

15.6 The Rectangular Pulse Envelope Modulated Signal

581

from the below absorption band case is that the leading and trailing edge Sommerfeld precursor fields become more pronounced in the total wavefield evolution as the angular signal frequency moves up through this intermediate frequency domain. The construction of the dynamical space–time structure of the propagated rectandispersive, intra-absorption gular envelope wavefield AT .z; t / in the anomalously q 

q 2 2 band angular signal frequency domain !c 2 !0  ı 2 ; !1  ı 2 is illustrated in Fig. 15.62 when the propagation distance z is in the maximal distortion domain cT=z < SB  1. For values of the initial pulse width T > 0 and propagation distance z > 0 satisfying this inequality there is a nearly complete overlap of the two sets of precursor fields so that the propagated wavefield structure AT .z; t / is dominated by a pair of interfering leading and trailing edge Sommerfeld precursors, followed by a pair of interfering leading and trailing edge Brillouin precursors, which is then followed by the signal oscillating at ! D !c that evolves over the space–time interval from  D c to  D c C cT=z, as illustrated. Just prior to the signal arrival at  D c , the propagated wavefield is dominated by the interfering pair of Brillouin precursors whose instantaneous oscillation frequency is less than !q c.

 Near the lower end of the angular signal frequency domain !02  ı 2 ; !SB , the Sommerfeld precursor field is relatively insignificant in comparison to both the Brillouin precursor field and the signal contribution so that the temporal width of the propagated pulse is given by (15.108). On the other hand, near the upper end of this frequency domain the Sommerfeld precursor field is a dominant feature in the total propagated wavefield over the two space–time domains  2 .1; SB / and  2 .1CcT=z; SB CcT=z/. The propagated signal contribution, which arises from the pole contribution that evolves over the space–time domain extending from  D c to  D c C cT=z when cT=z > c  1, with temporal width z tc D T  .c  1/; c

(15.109)

and is then again the dominant contribution over a small space–time interval about the point  D SB C cT=z provided that cT=z > c  SB . When the inequality cT=z > c  1 is satisfied, the signal contribution is separated into two pulses, each oscillating at the input angular carrier frequency ! D !c . As the propagation distance increases such that the inequality c 1 > cT=z > c SB is satisfied, these two signal pulses reduce to a single pulse oscillating at ! D !c . Finally, when the propagation distance increases such that the inequality cT=z < c  SB is satisfied, the pulse distortion is severe and the total propagated wavefield is dominated by the leading and trailing edge precursor fields over the entire subluminal space–time domain   1. Consider finally the construction of the propagated wavefield in the high angular frequency domain !c > !SB where the propagated Heaviside step function signal separates into a prepulse that evolves over the space–time domain  2 .c1 ; c2 / and a main signal that evolves over the space–time domain  > c , these two signal

582

15 Continuous Evolution of the Total Field

AH(z,t,0) 0

q

1

qSB

qc

AH(z,t,T) 0

q

1

1+cT/z

qc+cT/z

AT(z,t,) q

0

1 1+cT/z qSB

qc

qc+cT/z

cT/z

Fig. 15.62 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T in the anomalously dispersive, intra-absorption band angular signal frequency doq

q 2  2 2 !0  ı ; !1  ı 2 with propagation distance z in the maximal distortion domain main !c 2 cT=z < SB  1. When this situation prevails, the interference between the leading and trailing edge precursor fields is nearly complete and the resultant pulse distortion is severe

components being separated by the Brillouin precursor field which is the asymptotically dominant field component over the space–time domain  2 .c2 ; c /, as described in Sect. 15.51. In this high-frequency domain the Sommerfeld precursor field is a dominant feature in the total wavefield evolution, evoling essentially undisturbed over the initial space–time domain  2 .1; c1 /. For a sufficiently long initial pulse width T and=or a sufficiently small propagation distance z such that cT=z > c  1, the precursor fields and prepulse associated with the leading edge AH .z; t; 0/ of the rectangular envelope pulse will completely evolve prior to the arrival and evolution of the precursor fields and prepulse associated with the railing edge AH .z; t; T / of the pulse, so that their interference is

15.6 The Rectangular Pulse Envelope Modulated Signal

583

minimal. When this condition is satisfied, the total propagated wavefield evolves in the following sequential manner: 1   < c1 c1    c2 c2 <  < c c    1 C cz T 1 C cz T   < c1 C cz T c1 C cz T   < SB C cz T SB C cz T < 

AT .z; t /  AHs .z; t; 0/ AT .z; t /  AHc .z; t; 0/ AT .z; t /  AHb .z; t; 0/ C AHc .z; t; 0/ AT .z; t /  AHc .z; t; 0/ AT .z; t /  AHs .z; t; T / C AHc .z; t; 0/ AT .z; t /  AHs .z; t; T / AT .z; t /  AHb .z; t; T /

where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space–time domain. The propagated rectangular envelope wavefield is thus separated into a prepulse that evolves over the space–time domain  2 Œc1 ; c2 with temporal width z tp D .c2  c1 / (15.110) c that increases linearly with the propagation distance z  0, and a main pulse that evolves over the space–time domain  2 Œc ; 1 C cT=z with temporal width z tc D T  .c  1/ c

(15.111)

that decreases to zero linearly with the propagation distance z over the propagation domain cT=z > c  1. The front and back of the prepulse then propagate with the anterior and posterior pre-signal velocities vc1 .!c / D c=c1 .!c / and vc2 .!c / D c=c2 .!c /, respectively, and the front of the main pulse propagates with the signal velocity vc .!c / D c=c .!c /, just as for the Heaviside step function envelope signal. A moments reflection on the limiting behavior of this asymptotic pulse structure as the propagation distance z becomes small (ignoring momentarily that these results are derived from asymptotic theory as z ! 1) shows that the prepulse width tp given in (15.110) approaches zero while the main pulse width tc given in (15.111) approaches the initial pulse width T as z ! 0. The main pulse is then clearly associated with the initial rectangular envelope pulse. As the propagation distance z increases so that the inequality cT=z < c  1 is satisfied, the main pulse vanishes from the propagated field structure and all that remains is the prepulse and the leading and trailing edge precursor fields. The prepulse remains intact, evolving essentially undisturbed over the space–time interval  2 Œc1 ; c2 until the inequality cT=z < c2  1 is satisfied. When this condition prevails, the prepulse becomes distorted as the trailing edge Sommerfeld precursor begins to evolve over this space–time interval. The construction of the propagated wavefield when c1  1 < cT=z < c2  1 is illustrated in Fig. 15.63. When this

15 Continuous Evolution of the Total Field

AH(z,t,0)

584

0

q

AH(z,t,T)

qc1

qc2

q

0 qc1+cT/z

1

AT(z,t,)

qc

qc2+cT/z

q

0

qc1 1

qc2

qc1+cT/z

qc2+cT/z

qc

1+cT/z

Fig. 15.63 Construction of the dynamical space–time structure of the propagated wavefield AT .z; t / D AH .z; t; 0/  AH .z; t; T / due to an input rectangular envelope pulse with temporal duration T in the normally dispersive, high angular signal frequency domain !c > !SB with propagation distance z in the space–time domain c1 1  cT=z < c2 1. When this situation prevails, the interference between the leading and trailing edge precursor fields is moderate to severe and the resultant pulse distortion is becoming severe

condition prevails, the total propagated wavefield evolves in the following sequential manner: 1   < c1 AT .z; t /  AHs .z; t; 0/ AT .z; t /  AHc .z; t; 0/ c1   < 1 C cz T AT .z; t /  AHs .z; t; T / C AHc .z; t; 0/ 1 C cT=z   < c2 AT .z; t /  AHb .z; t; 0/  AHs .z; t; T / C AHc .z; t; 0/ c2 <  < c1 C cz T AT .z; t /  AHb .z; t; 0/ c1 C cz T <  < c2 C cz T AT .z; t /  AHb .z; t; T / C AHb .z; t; 0/ c2 C cz T <  where the leading term in each asymptotic expression given here indicates that it is asymptotically dominant over any additional term included over that particular space–time domain. The propagated wavefield structure is then seen to be dominated by the leading and trailing edge precursor fields over all but the space–time interval  2 Œc1 ; 1 C cT=z . The temporal width of the prepulse is now given by z tp D T  .c1  1/; c

(15.112)

15.6 The Rectangular Pulse Envelope Modulated Signal

585

which decreases from its maximum value .z=c/.c2  c1 / when cT=z D c2  1 to zero when cT=z D c1  1 as the propagation distance z increases from z D cT =.c2  1/ to z D cT =.c2  c1 /. The validity of this uniform asymptotic description of rectangular envelope pulse propagation in a single resonance Lorentz model dielectric is completely borne out by comparison with detailed numerical calculations of the dynamical pulse evolution [8, 35]. The calculations presented here are for a strongly absorptive medium with Brillouin’s choice of the medium parameters [1, 2], viz., !0 D 4:0  1016 r=s; p b D 20  1016 r=s; ı D 0:28  1016 r=s: The dynamical evolution of the propagated rectangular envelope wavefield at several increasing values of the relative propagation distance z=zd is illustrated in Figs. 15.64–15.67 for the below resonance angular carrier frequency !c D 1:0  1016 r=s. The e 1 penetration depth at this signal frequency is given by zd .!c / ˛ 1 .!c / D 1:82  104 cm. The time origin in each sequence of propagated waveforms has been shifted by the amount tc D c z=c so that the signal arrival at time t D tc and the signal departure at time t D tc C T are aligned at each propagation distance; these time instances are indicated by the vertical dotted lines in each figure. The initial rectangular envelope pulse width in Fig. 15.64 is T D 0:6283 fs D 6:283  1016 s and corresponds to a single period of oscillation of the signal. In this case, the pulse distortion becomes severe (cT=z < SB  1) after only 1=3 of an absorption depth zd into the medium, after which the propagated pulse is dominated by the interfering leading and trailing edge Brillouin precursors. The initial pulse width is doubled to T D 1:257 fs in Fig. 15.65, corresponding to two periods of oscillation of the input signal. In this case the pulse distortion is minimal when z=zd D 0:055, moderate when z=zd D 0:55, and severe when z=zd  2:75. Each of these cases corresponds qualitively to the asymptotic constructions depicted in Figs. 15.59–15.61, respectively. The initial rectangular envelope pulse width is again doubled to T D 2:513 fs in Fig. 15.66. In this case, the pulse distortion is minimal when z=zd  0:7 and becomes severe when z=zd D 1:24, after which the propagated waveform is dominated by the leading and trailing edge Brillouin precursors. Finally, the initial pulse width is doubled once more to T D 5:026 fs in Fig. 15.67 which corresponds to eight oscillation periods of the initial signal. In this case the transition from minimal to moderate pulse distortion occurs when z=zd D 1:41 and the transition to severe pulse distortion occurs when z=zd D 2:48. By comparison, the transition to the severe pulse distortion regime for a picosecond pulse occurs when z=zd 500. Again, in the severe pulse distortion regime the propagated waveform is dominated by the interfering leading and trailing edge Brillouin precursors. Similar results for a single resonance Lorentz model dielectric have been obtained numerically by Barakat [45].

586

15 Continuous Evolution of the Total Field

Fig. 15.64 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 0:6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

T = 0.6283 fs 1 z=0 0

−1 0.5 z / zd = 0.055 0 −0.5

AT (z,t)

0.5 z/ zd = 0.55 0 −0.5 0.2 z/ zd = 2.75

0 −0.2 0.1

z / zd = 5.5

0 −0.1

−4

−2

0

2

t − qc z/c (fs)

Careful inspection of Figs. 15.64–15.67 shows that the propagated rectangular envelope pulse width given in (15.108) correctly describes the time duration over which the propagated waveform is dominated by the signal component oscillating at the input angular signal frequency !c . In particular, this pulse-signal width tc is seen to decrease with increasing propagation distance z from its initial value T to zero at the transition point to the maximal distortion domain. Nevertheless, the overall temporal width of the entire propagated pulse is seen to increase with the propagation distance z. Up into the maximal distortion domain, the propagated rectangular envelope pulse waveform is seen to be defined between the two space–time points  D 0 and  D c C cT=z, with corresponding temporal width z t D T C .c  0 /: c

(15.113)

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.65 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 1:257 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

587

T = 1.257 fs 1 z=0 0

−1 0.5

z/ zd = 0.055

0 −0.5

AT (z,t)

1 z/ zd = 0.55

0

−1 0.1

z/ zd = 2.75

0 −0.1 0.1

z / zd = 5.5

0 −0.1 −4

−2

0

2

t – qc z/c (fs)

Once into the maximal distortion domain, the propagated rectangular envelope wavefield structure becomes completely dominated by the leading and trailing edge Brillouin precursors whose peak amplitude points occur at the space–time points  D 0 and  D 0 C cT=z, and are thus separated in time by the initial pulse width T . Because these two points in the wavefield evolution experience no exponential decay, but rather decrease in amplitude with the propagation distance z > 0 only as z1=2 , they will remain the prominent feature in the propagated wavefield structure long after the signal contribution has attenuated away. This behavior applies throughout q the normally dispersive below absorption band angular frequency 

domain !c 2 0; !02  ı 2 and remains applicable up through most of the anomaq 

q 2 !0  ı 2 ; !12  ı 2 . Near the upper lously dispersive absorption band !c 2 q end of the absorption band and for signal frequencies !c > !12  ı 2 above the

588

15 Continuous Evolution of the Total Field

Fig. 15.66 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 2:513 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

T = 2.513 fs 1 z=0 0

−1 0.5

z / zd = 0.055

0 −0.5

AT (z,t)

0.5

z/ zd = 0.55

0 −0.5 0.2 z/ zd = 2.75

0 −0.2 0.1

z / zd = 5.5 0 −0.1

−6

−4

−2

0

2

4

t – qc z/c (fs)

absorption band, the leading and trailing edge Sommerfeld precursor fields become a dominant feature in the propagated wavefield AT .z; t / and must then be included in any description of its overall temporal width, as is now done. The dynamic evolution of the propagated rectangular envelope pulse wavefield AT .z; t / at several increasing values of the propagation distnace z is illustrated in Fig. 15.68 for the above absorption band signal frequency !c D 1:0  1017 r=s, where !c > !SB . The e 1 penetration depth at this signal frequency is zd ˛ 1 .!c / D 2:68  105 cm. The initial temporal pulse width in this example is T D 0:6283 fs D 6:283  1016 s, which corresponds to ten oscillation periods of the signal frequency. At the smallest propagation distance presented in Fig. 15.68, z=zd D 0:037 and at the intermediate propagation distance illustrated, z=zd D 0:37, so that both of these propagated pulse waveforms are in the immature dispersion regime. In both cases the inequality cT=z > c  1 is satisfied so that the pulse

15.6 The Rectangular Pulse Envelope Modulated Signal Fig. 15.67 Dynamical wavefield evolution due to an input rectangular envelope pulse with below resonance angular carrier frequency !c D 1:0  1016 r=s and initial pulse width T D 5:026 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

589

T = 5.026 fs 1 z=0 0

−1 0.5

z / zd = 0.055

0 −0.5

AT(z,t)

1 z / zd = 0.55 0

−1 0.1

z / zd = 2.75

0 −0.1

0.1

z / zd = 5.5

0 −0.1

−6

−4

−2

0

2

4

6

t – qc z/c (fs)

distortion is minimal. At the largest propagation distance illustrated in the figure, z=zd D 3:73 so that the propagated rectangular envelope waveform is in the mature dispersion regime. In this last case, cT=z c1  1, so that the prepulse is almost fully distorted due to interference with the trailing edge Sommerfeld precursor and the main pulse has almost completely disappeared, being replaced by the interfering leading and trailing edge Brillouin precursors. Notice that the time origin for each propagated waveform illustrated in Fig. 15.68 has been shifted by the amount c z=c, the vertical dotted lines in the figure depicting the location of the front and back of the initial, undistorted rectangular envelope pulse, both propagating at the main signal velocity vc .!c / D c=c .!c /.

590

15 Continuous Evolution of the Total Field T = 0.6283 fs 1.08

z/ zd = 0.037

0

−1.00

AT (z,t)

0.84

z/ zd = 0.373

0

−0.90 0.085

z/ zd = 3.73

0

−0.082 −3

−2

−1 t – qc z / c (fs)

0

1

Fig. 15.68 Dynamical wavefield evolution due to an input rectangular envelope pulse with above absorption band angular carrier frequency !c D 1:0  1017 r=s and initial pulse width T D 0:6283 fs in a single resonance Lorentz model dielectric at several increasing values of the relative propagation distance z=zd

The temporal width tc of the main pulse is seen to decrease from its initial value T to zero as the propagation distance increases from zero, as described by (15.111). In addition, the temporal width tp of the prepulse is seen to first increase with increasing propagation distance z  0, as described by (15.110), and then decrease with increasing propagation distance as the pulse distortion becomes severe, as described by (15.112). Nevertheless, the overall temporal width of the entire propagated waveform is seen to increase with the propagation distance z > 0. If just the high-frequency structure in the mature dispersion regime, which evolves

15.6 The Rectangular Pulse Envelope Modulated Signal

591

over the space–time domain from  D 1 to  0 , is included, the overall temporal pulse width becomes z t 0 ' .0  1/; (15.114) c provided that cT=z < c 1, whereas if both the low- and high-frequency structure is included, which evolves over the space–time domain from  D 1 to  ' 0 C cT=z, the overall temporal width is found to be given by z t ' T C .0  1/; c

(15.115)

again provided that cT=z < c  1. The relation given in (15.114) is the appropriate measure of the overall propagated rectangular envelope pulse width in the mature dispersion regime if only the high-frequency content of the wavefield is detected, whereas the relation given in (15.115) is the appropriate measure if all significant frequency components are included (i.e., detected).

15.6.2 Rectangular Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric Because the asymptotic description of a Heaviside step function signal in either a simple Debye model or a more accurate Rocard–Powles–Debye model dielectric is described by the superposition of the Debye-type Brillouin precursor (see Case 2 of Sect. 12.1.1) and the signal contribution, as described in (15.88), the dynamical evolution of a rectangular envelope pulse is then seen to be dominated by a pair of leading and trailing edge Brillouin precursors as the propagation distance z > 0 exceeds a single absorption depth zd D ˛ 1 .!c / at the initial pulse carrier frequency !c , as described in detail in [32]. The pulse evolution to this asymptotic behavior in the mature dispersion regime is illustrated in the sequence of graphs presented in Fig. 15.69 for an input ten-cycle rectangular envelope pulse with fc D 1 GHz carrier frequency and T D 10=fc D 10 ns initial pulse width at one, three, five, seven, and nine absorption depths in triply distilled water (see Fig. 15.53). Notice that the leading and trailing edge Brillouin precursors persist long after the 1 GHz signal has been significantly attenuated by the dispersive absorptive medium. Although these two Brillouin precursors penetrate very far into the material, they only carry a small fraction of the initial pulse energy in the particular case under consideration. The leading edge Brillouin precursor is essentially a remnant of the first half-cycle of the initial pulse and the trailing edge Brillouin precursor is a remnant of the last halfcycle. The input pulse energy available to the leading and trailing edge Brillouin precursors is then limited to that contained in a single cycle of the input rectangular envelope pulse. For a ten-cycle pulse as considered here and illustrated in Fig. 15.69, this means that, at most, only 10% of the input pulse energy is available to this precursor pair [46–48].

592

15 Continuous Evolution of the Total Field z/zd = 1

0.4 0.2 0

A(z,t)

−0.2 0.2 z/zd = 3

0 z/zd = 5

0.1 0 0.1 0 0.1 0 −0.1

z/zd = 7 z/zd = 9

1.24

1.26

1.28

1.3

1.32

t – q0z/c (s)

1.34

1.36

1.38 x 10−7

Fig. 15.69 Dynamical wavefield evolution of an input unit amplitude, ten-cycle rectangular envelope pulse with fc D 1:0 GHz carrier frequency at one, three, five, seven, and nine absorption depths in the simple Rocard–Powles–Debye model of triply distilled water

A more efficient way to generate a Brillouin precursor pair in a dispersive material is with a single cycle pulse because the input pulse energy available to this precursor pair then approaches 100% [32]. The pulse sequence presented in Fig. 15.70 illustrates the dynamical pulse evolution as a unit amplitude, rectangular envelope single-cycle pulse with fc D 1 GHz and T D 1=fc D 1 ns penetrates into triply distilled water at the successive pentration depths z=zd D 0; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. The evolution of the pulse into a pair of leading and trailing edge Brillouin precursors is clearly evident as the propagation distance exceeds a single absorption depth (z=zd > 1) and the peak amplitude attenuation transitions from exponential e z=zd to the z1=2 algebraic decay described in (13.146). This transition to nonexponential, algebraic decay is illustrated in Fig. 15.71 which presents a semilogarithmic graph of the numerically determined peak amplitude decay as a function of the relative propagation distance z=zd . The solid line in the figure describes the pure exponential decay function e z=zd . The numerically determined peak amplitude decay for three different initial single-cycle pulses with carrier frequency values fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz is presented in this figure by the , ı, and C symbols, respectively, each data set connected by a cubic spline fit. The temporal width of the leading edge Brillouin precursor as a function of the propagation distance z  0 is illustrated in Fig. 15.72. The ordinate in part (a) of the

15.6 The Rectangular Pulse Envelope Modulated Signal 1

593

z/ zd = 0

0 .8 0 .6 1

0 .4

2 3

A ( z,t)

0 .2

4

5

6

7

8

9

10

0 − 0 .2 − 0 .4 − 0 .6 − 0 .8 −1 1 .2

1 .3

1 .4

1.5

1 .6

1. 7

1 .8

1.9

2

x 10–7

t (s )

Fig. 15.70 Propagated pulse sequence of a unit amplitude, single-cycle rectangular envelope pulse with fc D 1:0 GHz carrier frequency in the simple Rocard–Powles–Debye model of triply distilled water 100

Peak Amplitude

10−1

10−2

10−3

10−4

10−5

0

2

4

6

8

10

z / zd Fig. 15.71 Peak amplitude attenuation as a function of the relative propagation distance for input unit amplitude single-cycle rectangular envelope pulses with carrier frequencies fc D 0:1 GHz ( symbols), fc D 1:0 GHz (ı symbols), and fc D 10 GHz (C symbols) in the simple Rocard– Powles–Debye model of triply distilled water. The solid curve describes the pure exponential attenuation experienced by the signal component oscillating at fc , given by e z=zd

594

15 Continuous Evolution of the Total Field

Temporal Width (s)

a 10−8

fc = 0.1GHz fc = 1GHz

10−10

fc = 10GHz

10−12

Temporal Width (s)

b

10−4

10−3

10−2

10−1 z(m)

100

101

102

10−6

10−8 fc = 0.1GHz

10−10

fc = 1GHz fc = 10GHz

10−12 −2 10

10−1

z / zd

100

101

Fig. 15.72 Temporal width of the leading-edge Brillouin precursor (in seconds) as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z=zd D ˛.!c /z for a single-cycle rectangular envelope pulse with fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10 GHz. The solid curves describe the limiting asymptotic behavior as z ! 1

figure is the absolute propagation distance z in meters and the ordinate in part (b) is the relative propagation distance z=zd . The dependence of the absorption depth zd ˛ 1 .!c / on the angular carrier frequency !c of the pulse is reflected in the individual curves appearing in Fig. 15.72b. The solid curves in the figure describe the asymptotic result given in (13.150), viz., TB

  a0 .0 C f 0 / 1=2 z .C   / 2 z ; c 0 c

(15.116)

as z ! 1, where ˙ describe the e 1 amplitude points in the Brillouin precursor [see (13.149)], and the three sets of data points represent numerical results obtained for the fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz single-cycle pulse cases, each data set connected by a cubic spline fit. Notice that TB ! 3=.8fc / as z ! 0 for each case and that the numerical results approach the asymptotic behavior described by (15.116) as z ! 1. The temporal width of either the leading or trailing edge Brillouin precursor is then seen to increase monotonically with increasing propagation distance z  0 from the value 3=.8fc / at z D 0, asymptotically approaching the curve described by (15.116) as z ! 1, the transition to the asymptotic behavior occurring when z=zd  1.

15.6 The Rectangular Pulse Envelope Modulated Signal

595

a 1011 1010 fc = 10GHz

Effective Oscillation Frequency (Hz)

109 fc = 1GHz

108 fc = 0.1GHz

107

10−4

10−3

10−2

10−1

100

101

102

z (m)

b 1012

1010

fc = 10GHz fc = 1GHz

108

fc = 0.1GHz

106 10−2

10−1

z/zd

100

101

Fig. 15.73 Effective oscillation frequency (in Hz) of a single cycle rectangular envelope pulse as a function of (a) the propagation distance z (in meters), and (b) the relative propagation distance z=zd D ˛.!c /z for a single-cycle rectangular envelope pulse with fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10 GHz. The solid curves describe the limiting asymptotic behavior as z ! 1

The effective oscillation frequency feff of the single-cycle pulse as a function of the propagation distance z  0 is illustrated in Fig. 15.73. The solid curve describes the asymptotic result [cf. (13.151)]  1=2 0 c 1 1

(15.117) feff ! fB 2 TB 4 a0 .0 C f 0 /z as z ! 1, and the three sets of data points present numerical results for the fc D 0:1 GHz, fc D 1:0 GHz, and fc D 10:0 GHz single-cycle pulse cases, each data set connected by a cubic spline fit. Each measured value of the effective oscillation frequency was determined from the temporal distance between the peak amplitude points in the leading and trailing edge half-cycles of the pulse as it propagated through the dispersive model of water given in (12.300). Notice that feff ! fc as z ! 0 for each case and that the numerical results for the effective oscillation frequency approach the asymptotic behavior given in (15.117) as z ! 1. The effective oscillation frequency of the Brillouin precursor in a Debye model dielectric is then seen to decrease monotonically with increasing propagation distance z  0 from the initial pulse carrier frequency fc at z D 0, asymptotically approaching the curve described by (15.117) as z ! 1, the transition to the asymptotic behavior occuring when z=zd  1.

596

15 Continuous Evolution of the Total Field z /zd = 0 100

80

|Ã(z, )|

1 60

2 3 40

10

20

0 105

106

107

108

109

1010

f (Hz) Fig. 15.74 Magnitude of the pulse spectra for the single-cycle rectangular envelope pulse sequence presented in Fig. 15.70

The magnitude of the spectra for the single-cycle rectangular envelope pulse sequence illustrated in Fig. 15.70 is presented in Fig. 15.74. Notice that the peak value of the spectrum for the input single-cycle pulse is slightly downshifted from the input pulse carrier frequency value fc D 1:0 GHz to the value fp ' 0:83 GHz. This peak amplitude point in the pulse spectrum then shifts to lower frequency values as the propagation distance z  0 increases, shifting from the value fp ' 0:54 GHz at z=zd D 1 to the value fp ' 0:21 GHz at z=zd D 10, in general agreement with the results presented in Fig. 15.73 describing the decrease in the effective pulse frequency with propagation distance. Notice further that the initial and propagated pulse spectra depicted in Fig. 15.74 are effectively contained above 1 MHz over the range of propagation distances considered here. When an ideal high-pass filter with cutoff frequency fmin D 1 MHz is applied to the initial pulse spectrum when computing the propagated pulse wavefield using an adequately sampled FFT simulation of the integral representation of the propagated plane wave pulse given in (12.1), the results presented here for the single-cycle pulse with fc D 1:0 GHz remain essentially unchanged from zero through at least 20 absorption depths [32]. In particular, the algebraic, nonexponential peak amplitude decay of the leading and trailing edge Brillouin precursors [see (13.146)] presented in Fig. 15.71 remains essentially unaltered by application of this ideal high-pass filter operation. It is then clear that this unique, distinguishing behavior of the Brillouin precursor is not a zero frequency phenomenon for finite propagation distances.

15.6 The Rectangular Pulse Envelope Modulated Signal

597

15.6.3 Rectangular Envelope Pulse Propagation in Triply Distilled Water The general dynamical characteristics for rectangular envelope pulse propagation in both Debye-model and Lorentz-model dielectrics presented in the previous subsections are now examined in greater detail for triply distilled water. These results are based on the published research by P. D. Smith [46] et al. [49]. The angular frequency dispersion of the relative dielectric permittivity for this complicated medium is described here by [cf. (4.214) of Vol. 1 as well as (13.168), (13.169), and (13.171)] .!/=0 D 1 C

2 X

X bj2 aj  ; .1  i !j /.1  i !fj / j D0;2;4;6 ! 2  !j2 C 2i ıj ! j D1 (15.118)

with parameter values (compare with those given in Tables 4.1 and 4.2 of Vol. 1) a0 D 74:1, 0 D 8:44  1012 s, f 0 D 4:93  1014 s and a2 D 2:90, 2 D 6:05  1014 s, f 2 D 8:59  1015 s for the orientational polarization part of the model (describing the angular frequency dispersion up through the microwave region of the spectrum), and with !0 =2 D 1:81013 Hz, b0 =2 D 1:21013 Hz, ı0 =2 D 4:3 1012 Hz, !2 =2 D 4:9  1013 Hz, b2 =2 D 6:8  1012 Hz, ı2 =2 D 8:4  1011 Hz, and !4 =2 D 1:0  1014 Hz, b4 =2 D 2:0  1013 Hz, ı4 =2 D 2:8  1012 Hz for the resonance polarization part of the model describing the angular frequency dispersion in the infrared region of the spectrum, and with !6 =2 D 3:7  1015 Hz, b6 =2 D 3:2  1015 Hz, ı6 =2 D 8:0  1014 Hz describing the angular frequency dispersion in the ultraviolet region of the electromagnetic spectrum. Although these model parameters are slightly different from the values given in Tables 4.1 and 4.2 of Vol. 1 with one less resonance line, the resulting frequency dispersion is remarkably similar over the spectral domain from zero through the infrared where this numerical study is focused. The numerically determined dynamical field evolution for each of the five frequency cases depicted in Fig. 15.75 is presented in Figs. 15.76–15.80 [46, 49]. For each frequency case the temporal location of the peak amplitude point in the leading edge Brillouin precursor is denoted by t0eff D

z 0 ; c eff

(15.119)

where 0eff denotes the space–time point whose value is given by the effective zero frequency limit of the index of refraction that is “seen” by the initial pulse spectrum. The temporal location of the peak amplitude point in the trailing edge Brillouin precursor is then given by t0eff C T . As the pulse carrier frequency fc moves up through the various spectral domains that are dominated by different aspects of the dispersion model, the value of 0eff changes as does the character of the leading and trailing edge Brillouin precurors.

598

15 Continuous Evolution of the Total Field c1

1

c2

c3

c4

c5

nr ( )

10

0

10

8

10

10

10

10

12

14

16

10

(r/s)

18

10

10

Fig. 15.75 Angular frequency dependence of the real part nr .!/ D !12  ı 2 in a single resonance Lorentz model dielectric. The contour C denotes the original contour of integration that extends along the horizontal line from i a  1 to i a C 1 with 0 < a < 2ˇT . The shaded area in each figure plot indicates the region of the complex !-plane wherein the inequality  .!;  / <  .!SPC ;  / is satisfied d

610

15 Continuous Evolution of the Total Field

Consider first the case where the inequality ˇT >

 ı 1 C .r / 2

(15.139)

is satisfied. The geometry of this situation in the complex !-plane is illustrated in part (a) of Fig. 15.84. In this case, the simple pole singularities in the lower half of the complex !-plane at !n D !c  i 2nˇT , n D 1; 2; 3; : : : , all lie below the distant point SPC d for all   1. Nevertheless, for a sufficiently small value of ˇT satisfying the inequality in (15.137), the simple pole singularity !1 D !c  i 2ˇT , together with the pole singularity !˙0 D !c , will interact with the path P . / that passes through the saddle point SPC d , and so will contribute to the asymptotic behavior of the signal contribution to Aht .z; t /. The approximate expression given in (15.137) for the complex phase behavior at the simple pole singularities shows that the behavior at the poles with n  2 possess a larger exponential decay than that at both the n D 0 and n D 1 poles, and hence are asymptotically negligible in this case. For smaller values of ˇT , such as for the two cases depicted in parts (b) and (c) of Fig. 15.84, the original contour of integration C will be deformed across at least one of the simple pole singularities !Cn in the upper-half plane. However, any such pole contribution to the asymptotic approximation of Aht .z; t / occurs only during a space–time interval when the exponential attenuation associated with it is much greater than that at the distant saddle point SPC d . In the space–time domain when the value of  .!Cn ; / at any of the pole singularities with n D 1; 2; 3; : : : is either comparable with or less than that at the distant saddle point SPC d , the original contour of integration C does not cross that pole singularity when it is deformed to the path P . / through SPC d . Consequently, for all values of the rise-time parameter ˇT on the order of ı or greater, the simple pole singularities in the upper-half of the complex !-plane do not significantly contribute to the asymptotic behavior of the q

propagated wavefield A.z; t / for all !c  !12  ı 2 . The propagated wavefield is then essentially unchanged from the limiting instantaneous rise-time case, the Sommerfeld and Brillouin precursor fields dominating the wavefield structure as z ! 1, in agreement with the simple argument given in connection with (15.130)–(15.133). For even smaller values of ˇT such that Tr  1=ˇT  1=ı, the pole singularities located at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , approach close to the real axis and the interaction of the deformed contour of integration with them becomes increasingly important. In that case, several of their contributions to the asymptotic behavior of the propagated wavefield Aht .z; t / are no longer negligible in comparison with the contribution that is due to the simple pole singularity at !˙0 D !c as well as the contributions that are due to several of the simple pole singularities at !n D !c  i 2nˇT , n D 1; 2; 3; : : : , in the lower-half plane. Each of these pole contributions oscillates at the angular signal frequency !c , so that the precursor fields, whose spectral amplitudes decrease as ˇT decreases, become negligible in comparison with the total signal contribution oscillating at !c . The hyperbolic tangent envelope wavefield is then quasimonochromatic.

15.7 Noninstantaneous Rise-Time Signals Fig. 15.85 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with above absorption band carrier frequency !c D 2:5!0 at the fixed propagation distance z D 5:84zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

611

0.002

0

b T = 0.5d

−0.002

0.002

Aht(z,t)

0

bT = d

−0.002

0.002 b T = 5d

0 −0.002

0.002 b T = 10d

0 −0.002

1.0

qc1

qc2

1.2

1.4 q

1.6

1.8

The numerically determined [54] dynamical wavefield evolution of the hyperbolic q tangent envelope signal in the very high-frequency domain !c > !SB > !12  ı 2 of a single resonance Lorentz model dielectric is illustrated in Fig. 15.85 for several values of the rise-time parameter ˇT when !c D 2:5!0 at the fixed propagation distance z D 5:84zd , where zd ˛ 1 .!c /. The propagated wavefield is seen to be quasimonochromatic for ˇT < ı. As the rise-time parameter ˇT is increased above the medium damping constant ı, the Sommerfeld precursor begins to dominate the early time wavefield evolution, and for ˇT  ı the Brillouin precursor appears in a short space–time interval about  D 0 , thereby bifurcating the steady-state signal evolution into a prepulse that evolves over the space–time interval c1    c2 and the main signal which oscillates undisturbed for all  > c . Finally, notice that the ratio of the peak amplitude of the Sommerfeld precursor to the steady-state amplitude of the signal monotonically increases as ˇT increases above ı.

612

15 Continuous Evolution of the Total Field

Case 2. Below Absorption Band Domain !c 

q !02  ı 2 .

In the normally dispersive below absorption band domain, the near saddle point SPC n is the interacting saddle point and the value of r is defined by .r / D !c ;

(15.140)

where the second-order approximation for ./, valid for all  > 1 , is given in (12.220). In addition, the imaginary part of this near saddle point location is given by [see (12.219)] n o 2 = !SPC ./ D  ı ./; (15.141) n 3 where the second-order approximation for . /, valid for all   1 , is given in (12.221). As in the above absorption band case, only the pole singularities on and below the real ! 0 -axis necessarily interact with the deformed contour of integration P . / that passes through the near saddle point SPC n for  > 1 . The approximate behavior of the complex phase function at these simple pole singularities is given by    b2

.n˛ˇT  ı/ !c2  4n2 ˇT2 C 2n˛ˇT !c2 4 0 !0

  b2  2 2 2 i !c   0  ˛ !c  4n ˇT  8nˇT .n˛ˇT  ı 20 !04 (15.142)

.!n ; / 2nˇT .  0 / C

for n D 0; 1; 2; 3; : : : . By comparison, the approximate behavior of .!;  / at the near saddle point SPC n for  > 1 is given by [see (13.95)] ( .!SPC ; / ı n

2

. /. 3

 0 /

)  2 1  b 2  4 2 2 1  ˛ . / C ./ C 9 ı . / 3 ˛ . /  1 0 !04

   b2 4 2 2 i ./   0  ı . / 2  ˛ . / C ˛ . / ; 20 !04 3 (15.143) where the frequency-independent factor ˛ 1 [not to be confused with the frequency-dependent absorption coefficient ˛.!/] is given by (12.218). Consider first the case when the inequality ˇT >

ı

.r / 3

(15.144)

15.7 Noninstantaneous Rise-Time Signals

613

0.004 0

b T = 0.2d

–0.004 0.004 b T = 0.5d

0 –0.004 0.01 Aht(z,t)

0

bT = d

–0.01 0.02 b T = 1.5d

0 –0.02 0.02

b T = 2d

0 –0.02 0.01

b T = oo

0 –0.01 1.4

qc 1.6

1.8

2.0

q

Fig. 15.86 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with below absorption band carrier frequency !c D 0:25!0 at the fixed propagation distance z D 5:495zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

is satisfied. The deformed integration contour P . / through the near saddle point SPC n for  > 1 then interacts with the simple pole singularities at !0 D !c and !1 D !c  i 2ˇT . The other pole singularities either do not enter into the asymptotic description of Aht .z; t / or are asymptotically negligible in comparison with these contributions. For increasingly large values of the rise-time parameter such that ˇT  ı .r /=3, the exponential decay associated with the contribution from the !1 pole singularity becomes much greater than that at !0 D !c and so is asymptotically negligible by comparison. In that limiting case the asymptotic behavior of the propagated wavefield Aht .z; t / approaches that for a Heaviside step function modulated signal, as seen in Fig. 15.86. In the opposite sense, as the risetime parameter ˇT approaches close to the value ı .r /=3, the contribution of the

614

15 Continuous Evolution of the Total Field

simple pole singularity at !1 D !c  i 2ˇT to the asymptotic behavior of the propagated wavefield Aht .z; t / must be taken into account. Furthermore, for values ˇT  ı .r /=3, the original contour of integration C will be deformed across at least one of the simple pole singularities at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , in the upper half of the complex !-plane. However, such a pole contribution occurs only during a space–time domain when the exponential attenuation associated with it is much greater than that associated with the near saddle point SPC n . Furthermore, ;  /, the original contour for those space–time values  when  .!Cn ;  /   .!SPC n of integration C will not cross that pole singularity when it is deformed to the path P . / that lies along the steepest descent path through the saddle point SPC n . Consequently, for all values of the rise-time parameter ˇT on the order of ı or greater, the simple pole singularities in the upper half of the complex !-plane do not contribute significantly to the asymptotic behavior of the propagated hyperbolic tangent envefrequency lope modulated wavefield Aht .z; t / for all values of the angular carrierq

in the normally dispersive, below absorption band domain 0 < !c  !02  ı 2 . In the quasimonochromatic limit of small ˇT  ı, however, the pole singularities !Cn approach close to the real axis and the interaction of the deformed contour of integration with them becomes important, as seen in Fig. 15.86. Similar results are obtained as the carrier frequency is shifted up to the medium resonance frequency, as illustrated in Fig. 15.87. Case 3. Intra-Absorption Band Domain

q q !02  ı 2 < !c < !12  ı 2 .

For applied signal frequencies in the intermediate angular frequency domain  q angularq 2 2 !c 2 !0  ı 2 ; !1  ı 2 , which contains the anomalously dispersive absorption band of the dielectric, neither the near nor distant saddle points come within close proximity of any of the simple pole singularities at !˙n D !c ˙ i 2nˇT , 0 n D 0; 1; 2; 3; : : : . Furthermore, because of the presence of the branch cut !C !C just below the region of anomalous dispersion in the lower half of the complex !plane (see Fig. 12.1), the pole singularities at !n D !c ˙i 2nˇT , n D 0; 1; 2; 3; : : : , do not contribute to the asymptotic behavior of the propagated hyperbolic tangent envelope signal Aht .z; t / unless ˇT < ı=2n. As in both the above and below absorption band cases, the contributions to the propagated wavefield that are due to the simple pole singularities located in the upper half of the complex !-plane at !Cn D !c C i 2nˇT , n D 1; 2; 3; : : : , are all asymptotically negligible by comparison unless ˇT  ı, in which case the quasimonochromatic limit is attained. The numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal Aht .z; t / with intra-absorption band carrier frequency is presented in Fig. 15.87 when !c D !0 (near the lower end of the absorption band) and in Fig. 15.88 when !c D 1:4375!0 (near the upper end of the absorption band) in a single resonance Lorentz model dielectric p with Brillouin’s choice of the medium parameters (!0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s). The propagated wavefield structure in the on-resonance case

15.7 Noninstantaneous Rise-Time Signals Fig. 15.87 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with on-resonance carrier frequency !c D !0 at the fixed propagation distance z D 2:67zd in a single resonance Lorentz model dielectric for several values of the initial rise-time parameter ˇT

615 0.08

b T = 0.2d

0

– 0.08 0.04 b T = 0.5d

0 – 0.04

Aht(z,t)

0.08 0

bT = d

– 0.08 0.4

b T = 5d

0 0.4

b T = 10d

0

– 40

0

q 0 40 q

80

120

depicted in Fig. 15.87 is similar to that in the below absorption band case illustrated in Fig. 15.86, the field being quasimonochromatic for ˇT < ı, whereas the Brillouin precursor dominates the field evolution when ˇT > ı. The dynamical wavefield evolution becomes more complicated when the carrier frequency is near the upper end of the absorption band, as illustrated in Fig. 15.88 when !c is just below !1 . This particular value of the carrier frequency corresponds to the angular frequency value at which the real-valued group velocity vg .!/ D .@ˇ.!/=@!/1 is equal to the speed of light in vacuum, viz., vg .!c / ' c. The propagated wavefield is clearly quasimonochromatic for ˇT  ı. As ˇT approaches ı from below, the wavefield begins to lose its quasimonochromatic character, as seen in part (a) of Fig. 15.88. As ˇT is increased through and above ı, the leading edge of the wavefield steepens and becomes increasing complicated as both the Sommerfeld and Brillouin precursors increase in relative amplitude. Finally, for ˇT  ı, these two precursor fields dominate the entire wavefield evolution, as seen in part (b) of Fig. 15.88.

616

15 Continuous Evolution of the Total Field

a

b T = 0.2d

b

.0002

0

0.02

−.0002

0

.0005

b T = 0.5d

−0.02 0.04

Aht (z,t)

0 Aht (z,t)

b T = 3d

b T = 5d

0

0.002 0

−0.04 0.1

−0.002

0

bT = d

b T = 10d

0.01 b T = 2d

0

−0.1 0

−0.01 0

1 q 02

4

q

6

8

1q 0 2

4

q

6

8

10

10

Fig. 15.88 Numerically determined dynamical wavefield evolution of a hyperbolic tangent envelope modulated signal with intra-absorption band carrier frequency !c D 1:4375!0 at the fixed propagation distance z D 8:26zd in a single resonance Lorentz model dielectric. (a) Small to moderate values for several values of the initial rise-time parameter ˇT and (b) large values of ˇT

In summary, the analysis presented here has shown that the Sommerfeld and Brillouin precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter ˇT for a hyperbolic tangent envelope signal that are on the order of ı or greater, where ı is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr  1=ˇT that are less than or equal to the characteristic relaxation time Tı  1=ı of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to ˇT =2 at the 1=2 amplitude point. This inequality requires that the maximum initial rise-time in a Lorentz model dielectric with Brillouin’s choice of the medium parameters is given by Tı  1=ı D 0:357 fs.

15.7 Noninstantaneous Rise-Time Signals

617

15.7.2 Raised Cosine Envelope Signal Propagation in a Single Resonance Lorentz Model Dielectric Similar results are obtained [54] for the raised cosine envelope signal with envelope function 8 9 0; t 0 <  =  (15.145) urc .t / 12 1  cos .ˇr t / ; 0  t  Tr : ; 1; Tr  t with rise-time parameter ˇr D =Tr , where Tr is the signal rise-time. Unlike the hyperbolic tangent envelope signal with envelope function given in (15.125), which is nonzero for finite times t  0 when ˇT is finite, the canonical envelope function defined above in (15.145) is identically zero for all t  0 and, moreover, fully attains its steady-state amplitude in the finite rise-time Tr . The temporal frequency spectrum of this envelope function is found to be given by    !  2 ! 1 i i 2ˇ 1 r  : cos 2ˇr  uQ rc .!/ D e 2 ! ! C ˇr !  ˇr

(15.146)

In the limit as ˇr ! 1, the initial signal rise-time goes to zero (Tr ! 0) and the envelope function urc .t / defined in (15.145) goes over to the Heaviside unit step function, viz., (15.147) lim urc .t / D uH .t /; ˇr !1

its spectrum having the appropriate limit given by lim uQ rc .!/ D

ˇr !1

i : !

(15.148)

The opposite limit as ˇr ! 0, however, is of no interest for the raised cosine envelope signal, as the entire wavefield then vanishes. The temporal frequency spectrum of the initial raised cosine envelope signal f .t / D urc .t / sin .!c t / with fixed angular carrier frequency !c is given by uQ rc .!  !c / D

  i i 2ˇ .!!c / e r cos 2ˇ r .!  !c / 2  1 1 2 ;    !  !c !  !c C ˇr !  !c  ˇr (15.149)

which  points at ! D !c and ! D !c ˙ ˇr . Because the factor  has critical cos 2ˇ r .!  !c / appearing in (15.149) vanishes at ! D !c ˙ˇr , these two critical points are removable singularities. Hence, the only pole contribution to the asymptotic behavior of the propagated wavefield Arc .z; t / that is due to an input raised

618

15 Continuous Evolution of the Total Field

cosine envelope signal is from the simple pole singularity at ! D !c . This contribution, when it is the dominant contribution to the asymptotic wavefield behavior, yields the steady-state signal evolution that oscillates harmonically in time with the input angular signal frequency !c . The ratio of the finite rise-time envelope spectrum uQ rc .! !c / to its instantaneous limit uQ H .!  !c / is given by    uQ rc .!  !c / D e i 2ˇr .!!c / cos 2ˇ r .!  !c / uQ H .!  !c /   !  !c !  !c  1  !  !c C ˇr !  !c  ˇr     .!  !c /2 i 2ˇr .!!c / 1 ; .!  !c / 1 C

e 2ˇr ˇr2

(15.150)

where the final approxiamtion is valid for sufficiently large values of the rise-time parameter ˇr and finite values of the quantity !  !c . Because the closest that either the near or distant saddle point can approach the point ! D !c > ı is given by ı, the behavior of the spectrum uQ rc .! !c / about ! D !c is essentially unaltered from its instantaneous rise-time behavior if the inequality ˇr > ı

(15.151)

is satisfied [54]. This is the same approximate inequality obtained for the hyperbolic tangent envelope signal [see (15.132)]. In terms of the characteristic relaxation time Tı  1=ı of the single resonance Lorentz model medium and the initial rise-time Tr  1=ˇr of the raised cosine envelope signal, this inequality becomes Tr < Tı :

(15.152)

Consequently, for finite angular carrier frequencies !c > 0, the precursor fields that are a characteristic of the dynamical field evolution in a single Lorentz model dielectric will persist nearly unchanged from their ideal behavior for a Heaviside step function envelope signal for values of the rise-time parameter ˇr for a raised cosine envelope signal that are on the order of ı or greater, where ı is the characteristic damping constant of the Lorentz model dielectric, or, equivalently, for values of the rise-time Tr  1=ˇr that are less than or equal to the characteristic relaxation time Tı  1=ı of the dispersive medium. Notice that this result can also be related to the slope of the envelope function which is equal to ˇr =2 at the 1=2 amplitude point. These results are completely borne out by precise numerical calculations of the propagated wavefield behavior in a single resonance Lorentz model dielectric [54].

15.7 Noninstantaneous Rise-Time Signals

619

15.7.3 Trapezoidal Envelope Pulse Propagation in a Rocard–Powles–Debye Model Dielectric A canonical pulse envelope shape of central importance to both radar and communications systems is the trapezoidal envelope pulse with envelope function [see (11.68)] 9 8 0; t  T0 > ˆ > ˆ > ˆ tT0 > ˆ > ˆ ; T  t  T C T 0 0 r > ˆ Tr > ˆ < 1; T0 C Tr  t  T0 C Tr C T = (15.153) utrap .t / r CT / ; T0 C Tr C T  t 1  t.T0 CT > ˆ > ˆ Tf > ˆ > ˆ ˆ  T0 C T r C T C Tf > > ˆ > ˆ ; : 0; T0 C Tr C T C Tf  t with initial rise-time Tr > 0, fall-time Tf > 0, and peak amplitude pulse duration T > 0. The total initial pulse duration is then given by T C Tr C Tf and the half-amplitude pulse width is T C .Tr C Tf /=2, as illustrated in Fig. 11.6. This initial pulse type may be described by the time-delayed difference between a pair of trapezoidal envelope signals with envelope functions given by 8 < 0;

9 t  Tj 0 = tTj 0 ; T  t  T C T utrapj .t / j0 j0 j : Tj ; 1; Tj 0 C Tj  t

(15.154)

for j D r; f . The temporal angular frequency spectrum of this trapezoidal signal envelope function is then given by [see (11.69)] Z

1

uQ j .!/ D

uj .t /e i!t dt

1

  i sinc !Tj =2 e i!.Tj 0 CTj =2/ ; (15.155) !   where sinc. / sin . /= . Notice that sinc !Tj =2 ! ı.!/ in the limit as Tj ! 1, in which case the initial signal envelope spectrum becomes uQ trapj .!/ ! .i=!/ı.!/e .i!.Tj 0 CTj =2// , where ı. / denotes the Dirac delta function. A monochroD

matic, time-harmonic signal isthen obtained in this limiting case. In the opposite  limit as Tj ! 0, sinc !Tj =2 ! 1 and the initial signal envelope spectrum becomes uQ trapj .!/ ! .i=!/e .i!Tj 0 / , which describes the ultrawideband spectrum for a step function envelope signal. In general, the spectrum uQ j .!  !c / described by (15.155) for a trapezoidal envelope signal with carrier frequency !c > 0 will be ultrawideband if the in> equality 2=Tj  !c is satisfied. In that case, the ultrawideband spectral factor 1 .!  !c / will remain essentially unchanged over the positive angular frequency

620

15 Continuous Evolution of the Total Field

domain Œ0; 2!c , as depicted in Fig. 11.7 and illustrated in Fig. 11.8. This inequality is equivalent to the inequality < (15.156) Tj  Tc ; for j D r; f , where Tc 1=fc D 2=!c is the oscillation period of the carrier signal that is modulated by the trapezoidal envelope function given in (15.154). The inequality given in (15.156) provides a necessary condition for the dominant appearance of the Brillouin precursor in the propagated field structure in a Rocard–Powles–Debye model dielectric. Notice that this condition is independent of the dispersive material, unlike that for either a hyperbolic-tangent modulated signal [54], a gaussian envelope pulse [55–58], or a van Bladel envelope pulse [59,60]. The reason for this is quite simple: the hyperbolic tangent, gaussian, and van Bladel envelope functions are each continuous with continuous partial derivatives while the trapezoidal envelope function is continuous with a discontinuous first derivative at two points in time (t D Tj 0 and t D Tj ). These two discontinuities in the envelope slope will always yield a precursor contribution in a dispersive material, but that precursor contribution will only be significant in comparison to the pole contribution when the above inequality in (15.156) is satisfied. Notice further that the Heaviside step function signal is discontinuous in both its value and its first derivative at time t D Tj 0 , so that the trapezoidal envelope signal retains just the latter feature, albeit displaced in time by the initial rise-time Tr . These results are completely borne out by detailed numerical calculations, as illustrated in Fig. 15.89 showing the propagated symmetric (i.e., equal rise- and

0.1 0.05 Tr = Tf = Tc / 2

Atrap (z,t)

0 0.05

Tr = Tf = Tc

0 0.05

Tr = Tf = 2Tc

0

2.6

2.65 t (s)

2.7 x 10

–7

Fig. 15.89 Propagated symmetric (Tr D Tf ) trapezoidal envelope pulse structure with fc D 3 GHz carrier frequency at five absorption depths (z D 5zd ) in triply distilled water for Tj < Tc , Tj D Tc , and Tj > Tc , j D r; f

15.8 Infinitely Smooth Envelope Pulses

621

fall-times Tr D Tf ) trapezoidal envelope pulse structure with fc D 3 GHz carrier frequency at five absorption depths [z D 5zd with zd ˛ 1 .!c /] in triply distilled water for the three cases Tj < Tc , Tj D Tc , and Tj > Tc , j D r; f . Because the carrier wave period is given by Tc D fc1 D 3:33  1010 s and the characteristic relaxation time for water is given by 1 D 8:3  1012 s, each trapezoidal envelope case depicted in Fig. 15.89 satisfies the inequality Tj  1 , j D r; f , demonstrating that the Brillouin precursor fidelity is indeed independent of this dispersive material factor, in spite of the fact that the material dispersion is an essential ingredient in the appearance of the Brillouin precursor; notice that the Brillouin precursor is still present in the Tr D Tf D 2Tc case depicted in Fig. 15.89, accounting for the leading- and trailing-edge pulse distortion, but stretched out in time with an amplitude approximately equal to the signal amplitude. Rather, the fidelity of the Brillouin precursor in the dispersive material is governed by the inequality given in (15.156). Because the trapezoidal envelope pulse is the canonical pulse type upon which both pulsed radar and digital wireless telecommunication systems are based [61, 62], this result poses a special challenge regarding public health and safety. To avoid the formation of the Brillouin precursor upon penetration into the human body, the rise and fall times (as well as any other rapid amplitude changes) of both radar and wireless digital communication systems (as well as any other pulsed electromagnetic radiation emitters) should strictly satisfy the inequality Tj > 1=fc ;

j D r; f

(15.157)

where fc is the characteristic oscillation frequency of the radiated pulse.

15.8 Infinitely Smooth Envelope Pulses The final canonical pulse type of interest here is the infinitely smooth envelope pulse. Because of this smoothness property, this pulse type, and in particular the gaussian envelope pulse, is a favorite among the group velocity adherents. In spite of this, the asymptotic theory has much to explain about its ultrashort behavior in a causal, linear dispersive system that the group velocity approximation fails to provide.

15.8.1 Gaussian Envelope Pulse Propagation in a Single Resonance Lorentz Model Dielectric Because of its central importance in optics as well as the central role it plays in the group velocity description, the correct description of gaussian pulse propagation in a dispersive medium provides a unique challenge to both theoreticians and experimentalists alike. An accurate description of gaussian pulse propagation in a

622

15 Continuous Evolution of the Total Field

dispersive medium begins with the 1970 analysis of Garrett and McCumber [63] who considered pulse propagation in the anomalous dispersion regime. Their main result showed that the peak amplitude point can, under certain conditions, propagate at the classical group velocity even when it exceeds the vacuum speed of light c. Crisp [64] then argued that the observed superluminal group velocity was due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light.

A decade later, Sherman and Oughstun [65] provided a detailed physical description of dispersive pulse dynamics based on Loudon’s [15] time-harmonic electromagnetic energy transport velocity in a Lorentz medium. Not only did this description explain each feature observed in the propagated pulse structure, it was also in complete keeping with relativistic causality. Shortly thereafter, Chu and Wong [66] presented experimental results for picosecond laser pulses propagating through thin samples of a linear dispersive dielectric whose peak absorption never exceeded six absorption depths, showing that the peak amplitude point of a gaussian light pulse travels in such a medium with the group velocity vg .!c / at the optical frequency even when vg .!c / > c, purporting to disprove the enrgy velocity description while verifying the group velocity description. An asymptotic description of the propagation of a gaussian wave packet in a Lorentz medium was then given by Tanaka, Fujiwara, and Ikegami [55] in 1986. They showed that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range

in agreement with the Sherman–Oughstun energy velocity description [65, 67]. In addition, Tanaka et al. concluded that [55] fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fouriercomponent waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long travelling distance.

Balictsis and Oughstun [55–58] then showed that the fast pulse component is nothing more than the Sommerfeld precursor and that the slow pulse component is just the Brillouin precursor. These final results provided a complete explanation of the apparent discrepancy between the energy and group velocity results for dispersive gaussian pulse propagation. Consider then an input gaussian envelope modulated harmonic wave f .t / D ug .t / sin .!c t C / with fixed angular carrier frequency !c > 0 and initial full pulse width 2T > 0 that is centered about the instant of time t0 at the plane z D 0, where [see (11.78)] 2 2 ug .t / D e .tt0 / =T ; (15.158)

15.8 Infinitely Smooth Envelope Pulses

623

which is propagating in the positive z-direction through a single resonance Lorentz model dielectric. The constant phase factor is used to adjust the location of the carrier wave with respect to the peak amplitude point in the gaussian envelope. Typically D =2 for an ultrashort pulse so that the carrier maximum coincides with the envelope maximum.

15.8.1.1

Classical Asymptotic Description

The exact, classical integral representation of the propagated gaussian pulse wavefield is given by [56] 1 < Ag .z; t / D 2

Z uQ g .!  !c /e

z 0 c .!; /

d!

(15.159)

C

for all z  0, with initial pulse spectrum uQ g .!/ D  1=2 T e T

2 ! 2 =4

e i.!c t0 C / ;

(15.160)

where

c (15.161)  0   t0 z is the shifted space–time parameter relative to the initial peak amplitude point of the gaussian envelope pulse. The contour of integration C is taken here as any contour in the complex !-plane that is homotopic to the real frequency axis extending from 1 to C1. Because this pulse envelope spectrum is an entire function of complex !, the propagated gaussian pulse wavefield has the representation [see (15.1)] Ag .z; t / D Ags .z; t / C Agb .z; t /

(15.162)

for all z  0, where the asymptotic behavior of the two component wavefields is given by [56, 57, 68]  Agj .z; t /  aj

c 2z

1=2

(

uQ .!SPj . 0 /  !c / z .!SP ; 0 / j < i

1=2 e c  00 .!SPj ;  0 /

) (15.163)

as z ! 1 for j D s; b. Here as D 2 and !SPs D !SPC . 0 / denotes the distant firstd

0 order saddle point SPC d of the complex phase function .!;  / in the right-half of 0 the complex !-plane for all  > 1, whereas ab D 1 for 1 <  0 < 1 and ab D 2 . 0 / denotes the near first-order saddle point SPC for  0 > 1 where !SPb D !SPC n n of the complex phase function .!;  0 / in the right-half of the complex !-plane. The nonuniform behavior exhibited in (15.163) in any small neighborhood of either the space–time point  0 D 1 or of the space–time point  0 D 1 may be corrected using the appropriate uniform asymptotic expansion procedure described in either

624

15 Continuous Evolution of the Total Field

Sect. 13.2.2 or Sect. 13.3.2, respectively. The gaussian pulse wavefield component Ags .z; t / is referred to as a gaussian Sommerfeld precursor field and the component Agb .z; t / is referred to as a gaussian Brillouin precursor field. Because of the form of the initial gaussian envelope spectrum uQ g .!/ given in (15.160), the asymptotic description of each gaussian pulse component Agj .z; t /, j D s; b, contains a gaussian amplitude factor of the form exp .T =2/2  2 i 0 with the ˚material attenuation that is given by the real phase behavior  .!SPj ;  0 / D < .!SPj ;  0 / at the relevant saddle point, and the instantaneous angular oscillation frequency of each component in the mature dispersion ˚ pulse regime is approximately given by < !SPj in the ultrashort pulse limit as T ! 0.   q 2 2 Consequently, for a below absorption band carrier frequency !c 2 0; !0  ı , the instantaneous angular oscillation frequency ofqthe gaussian Brillouin precursor

Agb .z; t / crosses !c as it chirps upward toward !02  ı 2 , whereas for an above q  resonance carrier frequency !c 2 !12  ı 2 ; 1 , the instantaneous angular oscillation frequency of the gaussian Sommerfeld precursor Ags .z; t / crosses !c as q 2 it chirps downward toward !1  ı 2 , in each case the gaussian amplitude factor ˚ peaking to unity when < !SPj D !c . For an intra-absorption band angular car q q !02  ı 2 ; !12  ı 2 the carrier frequency value is never rier frequency !c 2 attained by either pulse component [56]. If the input angular signal frequency !c is in  the medium absorption  band q q 2 2 2 2 !0  ı ; !1  ı , then where the dispersion is anomalous, so that !c 2 both gaussian pulse components Ags .z; t / and Agb .z; t / will be present in the propagated waveform in roughly equal proportion. The Brillouin precursor component q

Agb .z; t / becomes more pronounced as !c is decreased from !12  ı 2 !1 to q !02  ı 2 !0 and dominates the propagated gaussian pulse evolution as !c is decreased into the normally dispersive region below the medium resonance frequency, whereas the Sommerfeld precursor q q Ags .z; t / becomes more pronounced as

!c is increased from !02  ı 2 !0 to !12  ı 2 !1 and dominates the propagated gaussian pulse evolution as !c is increased into the normally dispersive region above the medium absorption band. As an illustration, the numerically determined dynamical wavefield evolution due to an input ultrashort gaussian envelope pulse with initial pulse width 2T D 0:2 fs and carrier frequency !c D 5:75  1016 r=s that is near the upper end of the absorption band of a single resonance Lorentz model dielectric with Brillouin’s medium parameters is illustrated in Fig. 15.90 for several values of the relative propagation distance z=zd D ˛.!c /z. This case is of particular interest because the group velocity vg .!/ D .@ˇ.!/=@!/1 at this carrier frequency

15.8 Infinitely Smooth Envelope Pulses X 10

−2

X 10−3 z /zd = 20.66

0

z/zd = 61.98

1 Ag(z,t)

Ag(z,t)

2.5

625

0 −1

−2.5 1

3

4

5

1 q

X 10−3

6

z /zd = 41.32

0

−5 1

X 10−4

3

q

4

5

z/zd = 82.64

0

−6

q0 2

2.8

2

q

Ag(z,t)

Ag(z,t)

5

q0 2

q0

1

q0 q

2

2.8

Fig. 15.90 Numerically determined dynamical wavefield evolution due to a 0:2 fs gaussian envelope pulse with intra-absorption band carrier frequency !c 2 .!0 ; !1 / satisfying vg .!c / ' c in a single resonance Lorentz model dielectric

in this medium is very nearly equal to the speed of light c in vacuum. The gaussian Sommerfeld precursor component is seen to first emerge from the propagated pulse structure as the propagation distance increases into the mature dispersion regime, its peak amplitude point traveling with a velocity just below c, as seen in the 21 and 41 absorption depth cases in the figure. As the propagation distance continues to increase, the gaussian Brillouin precursor component of the pulse emerges, its peak amplitude point traveling with a velocity that approaches the value c=0 D c=n.0/ from above as z ! 1. The propagated wavefield Ag .z; t / due to an ultrashort gaussian envelope pulse then separates (or bifurcates) into two distinct pulse components that propagate with different peak velocities, the faster pulse component being the high-frequency gaussian Sommerfeld precursor Ags .z; t / with instantaneous angular oscillation frequency !s . 0 / that chirps downward toward !1 as the space–time parameter  0 increases, followed by the slower, low-frequency gaussian Brillouin precursor Agb .z; t / with instantaneous angular oscillation frequency !b . 0 / that chirps upward toward !0 as  0 increases, in complete agreement with the asymptotic results presented by Tanaka, Fujiwara, and Ikegami [55]. Notice that this gaussian pulse bifurcation is a linear phenomenom and that, for a multiple resonance Lorentz model dielectric, the pulse can separate into as many subpulses as there are precursor fields; for a double resonance Lorentz model dielectric, the gaussian pulse can separate into three subpulses (a Sommerfeld, middle, and Brillouin

626

15 Continuous Evolution of the Total Field

precursor pulse) when the inequality given in (12.117) is satisfied. Each feature of this dynamical pulse evolution is properly described by the Sherman–Oughstun energy velocity description [65, 67]. As the initial pulse width 2T is increased, the asymptotic approximation of the Sommerfeld and Brillouin pulse components given in (15.163) for the propagated gaussian pulse wavefield representation given in (15.162) remains qualitatively correct while its quantitative accuracy decreases at any fixed, finite propagation distance z > 0. This nonuniform asymptotic description (as well as its uniform counterpart) will remain quantitatively accurate as the initial pulse width is increased provided that the propagation distance is also allowed to increase, in keeping with the definition of an asymptotic expansion in Poincar´e’s sense as z ! 1 (see Definition 5 of Appendix F). However, because the medium is attenuative, the usefulness of this description also decreases as 2T increases as the important features of the dynamical pulse evolution (particularly when compared to experimental observations) are typically measured at some fixed observation distance in the medium.

15.8.1.2

Modified Asymptotic Description

The classical integral representation of gaussian pulse propagation given in (15.159)–(15.161) may be rearranged so as to yield the modified integral representation [55, 58] Ag .z; t / D

Z z 1 0 e c ˚m .!; / d! < i UQ m 2 C

(15.164)

for all z  0, where UQ m  1=2 T e i.!c t0 C

/

(15.165)

is independent of the angular frequency !, and where ˚m .!;  0 / .!;  0 / 

cT 2 .!  !c /2 4z

(15.166)

is the modified complex phase function which depends not only on the dispersive properties of the host medium, but also on the initial pulse width 2T and carrier frequency !c as well as upon the propagation distance z > 0. In the ultrashort pulse limit as 2T ! 0, the modified phase function ˚m .!;  0 / reduces to the classical phase function .!;  0 / D i !.n.!/   0 / so that the asymptotic behavior of the modified integral representation given in (15.164) is determined by the behavior of the integrand about the saddle points of .!;  0 /, as in (15.162) and (15.163). This then establishes the following scaling law for gaussian pulse propagation [57]: if the classical asymptotic description given in (15.162) and (15.163) is valid (to some specifis degree of accuracy) for some given input pulse width 2T at a given propagation

15.8 Infinitely Smooth Envelope Pulses

627

distance z, then this description will remain equally valid (to that same degree of accuracy) as the initial pulse width is increased provided that z is also increased in such a manner that the ratio T 2 =z remains fixed.

The saddle point dynamics of the modified phase function ˚m .!;  0 / are now considered based on the analyses of Tanaka et al. [55] and Balictsis et al. [58, 69, 70]. Although the complex phase function .!;  0 / satisfies the symmetry relation   .!  ;  0 / D .!;  0 /, the modified phase function does not, viz., ˚m .!  ;  0 / ¤ ˚m .!;  0 /:

(15.167)

As a consequence, the modified complex phase behavior, as well as its saddle points, are not symmetric about the imaginary axis. Nevertheless, the branch cuts remain determined by the complex index of refraction and so are still the symmetric line 0 segments !0 ! and !C !C about the imaginary axis (see Fig. 12.1). Numerical results show that there are five first-order saddle points SPmk , k D 1; 2; : : : ; 5, for a single resonance Lorentz model dielectric with respective locations !SPmk . 0 / that remain isolated from each other over the entire space–time domain  0 2 .1; 1/. The dynamical evolution of these saddle points in the complex !-plane is illustrated in Fig. 15.91 as a function of the space–time parameter  0 for a 2T D 0:2 fs gaussian envelope pulse with D =2 and !c D 5:75  1016 r=s angular carrier frequency at 83:92 absorption depths of a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters.6 The space–time points rcmj at which these saddle points may cross the real ! 0 -axis are defined by the condition !SPmj .rcmj / D !rcmj ;

j D 1; 2; : : : ; 5;

(15.168)

−

0 −

−30 –15

SPm1

SPm2

SPm5

16

'' (10 r/s)

30

+ +

SPm4

SPm3

rcm5

rcm1

rcm2 c

0

15

16

' (10 r/s)

Fig. 15.91 Dynamical evolution of the five first-order saddle points SPmj , j D 1; 2; : : : ; 5, of the modified complex phase function ˚m .!;  0 / in the complex !-plane as a function of the space– time parameter  0 for a 2T D 0:2 fs gaussian envelope pulse with D =2 and !c D 5:75  1016 r=s angular carrier frequency at 83:92 p absorption depths of a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s 6

Notice that, for reasons of consistency, the j D 1 and j D 2 saddle points SPmj are interchanged from that used in [58, 69, 70].

628

15 Continuous Evolution of the Total Field

where the angular frequency value !rcmj is real-valued. As seen in Fig. 15.91, only the saddle points SPm1 , SPm2 , and SPm5 satisfy this condition, where [58] !rcm1 ' C9:0261  1016 r=s; !rcm2 ' C1:4220  1016 r=s; !rcm5 ' 7:3953  1016 r=s;

rcm1 ' 1:2831 rcm2 ' 1:6871 rcm5 ' 1:7036:

The integration contour C is then deformed into a new path P . 0 / that passes through all of the accessible saddle points of the modified phase function at any given space–time value  0 in such a manner that it may be partitioned into a continuous chain of component subpaths Pmj . 0 /, each an Olver-type path with respect to its corresponding saddle point SPmj . Under this transformation, the modified integral representation (15.164), of the propagated gaussian envelope pulse takes the form X Agj .z; t / (15.169) Ag .z; t / D j

with

( ) Z z 1 0/ ˚ .!; m Q < i Um ec d! Agj .z; t / D 2 Pmj . 0 /

(15.170)

for all z  0. Because the modified complex phase function ˚m .!;  0 / now explicitly depends upon the propagation distance z, the first condition of Olver’s theorem (see Sect. 10.1.1) is not satisfied. This condition serves to ensure that the phase function does not vanish as z ! 1. However, because lim ˚m .!;  0 / D .!;  0 /;

(15.171)

z!1

where the classical complex phase function .!;  0 / strictly satisfies all of the conditions of Olver’s theorem, then the first condition of Olver’s theorem may be relaxed for the modified complex phase function case considered here. Application of Olver’s theorem to each integral in (15.170) then gives  Agj .z; t / 

c 2z

(

1=2 <



i UQ m ˚m00 .!SPmj ;  0 /

1=2 e

z c ˚m .!SPmj

; 0 /

) (15.172)

as z ! 1. Detailed numerical results [58, 69, 70] show that only the two saddle points SPm1 and SPm2 , whose dynamical evolution lies in the right-half of the complex !-plane, become a dominant saddle point during the entire space–time domain of interest, each in its respective space–time interval mj , j D 1; 2. The asymptotic representation of the propagated gaussian envelope pulse is then given by Ag .z; t / D Ag1 .z; t / C Ag2 .z; t /;

(15.173)

15.8 Infinitely Smooth Envelope Pulses

629

where lim Ag1 .z; t / D Ags .z; t /;

(15.174)

lim Ag2 .z; t / D Agb .z; t /I

(15.175)

z!1 z!1

that is, the pulse component Ag1 .z; t / corresponds to the gaussian Sommerfeld precursor Ags .z; t / and the pulse component Ag2 .z; t / corresponds to the gaussian Brillouin precursor Agb .z; t /. Each pulse component Agj .z; t / contains a gaussian amplitude factor that is contained in the modified complex phase function ˚m .!SPmj ;  0 / appearing in the exponential factor of (15.172). To explicitly display this, (15.166) may be expressed as ˚m .!;  0 / D .!;  0 /  g .!/;

(15.176)

where

cT 2 .!  !c /2 (15.177) 4z accounts for the gaussian amplitude factor in each pulse component. The angular frequency dependence of the real part m .! 0 / D !c and travels at the classical group velocity vps D vg .!ps / ' 2:86c. As the propagation distance is increased to approximately 145 absorption depths, illustrated in the bottom field plot of Fig. 15.97, the instantaneous oscillation frequency at the space–time point where the peak in the pulse envelope occurs shifts to the higher angular frequency value !ps ' 5:35  1016 r=s and now travels at the classical group velocity vps D vg .!ps / ' 4:45c. As the propagation distance continues to increase, the low-frequency components that are present in the initial pulse spectrum are attenuated at a larger rate than are the highfrequency components, so that the propagated pulse spectrum becomes dominated by an increasingly higher frequency component, the peak in the envelope of the propagated pulse propagating with the classical group velocity at this frequency value. Again, as the propagation distance increases into the mature dispersion regime, the pulse dynamics evolve toward that described by the energy velocity description; however, the overall field amplitude also rapidly attenuates to zero in this case. Similar results are obtained for each of the subluminal group velocity cases depicted in Fig. 15.94. For example, the sequence of blue squares (cases 8, 2, and 9) show this shift away from the absorption band for a 2 fs gaussian envelope pulse with intra-absorption band carrier frequency !c D 5:751016 r=s where vg .!c / ' c. The dependence of the peak amplitude velocity on the initial gaussian envelope pulse width 2T for fixed carrier frequency !c D 5:75  1016 r=s and propagation distance z D 83:92zd D 1m is displayed by cases 1, 2, and 3 in Fig. 15.95, illustrating the manner in which the pulse dyamics change as the pulse becomes ultrawideband. Because of the small propagation distance of at most six absorption depths in their laboratory arrangement, the experimental results of Chu and Wong [66] are restricted to the small propagation distance limit below the mature dispersion regime. The modified asymptotic description introduced by Tanaka et al. [55] in 1986 and then fully developed by Balictsis et al. [58, 70] in the 1990s bridges the gap between these two regimes, being in agreement with the experimental results [66] at small propagation distances (i.e., in the immature dispersion regime) and reducing to the classical asymptotic description at sufficiently large propagation distances (i.e., in the mature dispersion regime) in the dispersive, attenuative medium [57]. Moreover, the modified asymptotic description provides a mathematically rigorous derivation of the correct group velocity description of gaussian pulse propagation in a dispersive, attenuative medium and clearly shows how that description evolves into the Sherman–Oughstun energy velocity description [65, 67] as the propagation distance increases into the mature dispersion regime.

15.8 Infinitely Smooth Envelope Pulses

637

Ag(z,t)

1

0

−1 −10fs 2X10

z=0

−5fs

0 t' = (t − t 0)

5fs

10fs

−25

Ag(z,t)

z = 58.05zd = 0.2mm

–2X10

0

−25

−10fs 35X10

tps = −0.4662fs −5fs

0 t' = (t − t 0)

z/c 5fs

10fs

−62

Ag(z,t)

z = 145.13zd = 0.2mm

–35X10

−62

−10fs

tps = −0.7505fs

z/c

−5fs

5fs

0 t' = (t − t 0)

10fs

Fig. 15.97 Numerically determined dynamical pulse evolution for a 2T D 10:0 fs, !c D 5:25  1016 r=s gaussian envelope pulse at (from top to bottom) z D 0, z D 58:05zdp , and z D 145:13zd in a single resonance Lorentz model dielectric with !0 D 4  1016 r=s, b D 20  1016 r=s, and ı D 0:28  1016 r=s, where zd ˛ 1 .!c /. The solid vertical line in the bottom two wavefield plots marks the retarded instant of time t 0 D z=c when the peak pulse amplitude would have arrived at that propagation distance if it had traveled with the speed of light c in vacuum, and the vertical dashed line marks the actual retarded instant of time t 0 D tps when the peak pulse amplitude point actually arrives at that propagation distance. The middle field plot corresponds to the negative velocity case 12 and the bottom diagram to the negative velocity case 13 in Fig. 15.95. (From Balictsis and Oughstun [58])

638

15 Continuous Evolution of the Total Field

15.8.2 Van Bladel Envelope Pulse Propagation in a Double Resonance Lorentz Model Dielectric The major difficulty with the gaussian pulse is that its envelope function is strictly nonzero for all finite time except in the vanishing pulse width limit 2T ! 0 when the initial pulse amplitude is inversely proportional to 2T . An important example of an infinitely smooth pulse envelope with compact temporal support is provided by the Van Bladel envelope pulse Avb .0; t / D uvb .t / sin .!c t C / with envelope function [see (11.76)] ( uvb .t /



e

2

1C 4t.t  /

0I



I

when 0 < t <  ; when either t  0 or t  

(15.187)

p with temporal duration  > 0 and full pulse width = 2 at the e 1 amplitude points in the envelope function, as illustrated in Fig. 11.3 for a two-cycle pulse ( D 2Tc ) and in Fig. 11.14 for a ten-cycle pulse ( D 10Tc ), with Tc 1=fc D 2=!c for a cosine carrier wave ( D =2). Because the envelope function uvb .t / vanishes identically outsde of the finite time interval .0;  /, its Fourier transform uQ vb .!/ is an entire function of complex !. Its resultant propagated wavefield in a double resonance Lorentz model dielectric is then given by Avb .z; t / D Avbs .z; t / C Avbm .z; t / C Avbb .z; t /

(15.188)

for all t  z=c with z > 0, the propagated wavefield identically vanishing for all t < z=c. The Sommerfeld precursor pulse component Avbs .z; t / is due to the distant saddle points SP˙ d , the middle precursor pulse component Avbm .z; t / is due to the middle saddle points SP˙ m1 , and the Brillouin precursor pulse component Avbb .z; t / is due to the near saddle points SP˙ n . The dynamical field evolution when the angular carrier frequency !c is set equal to the value !min at the minimum dispersion point in the passband between the two absorption bands of resonance Lorentz model dielectric is presented in Figs. 11.31–11.35.

15.8.3 Brillouin Pulse Propagation in a Rocard–Powles–Debye Model Dielectric; Optimal Pulse Penetration A problem of particular practical importance is the determination of the structural form of the input pulse that will best penetrate a finite distance into a given dispersive dielectric. The results presented in Figs. 15.70 and 15.71 indicate that the pulse that will provide near-optimal, if not indeed optimal, penetration is comprised of a pair of Brillouin precursor structures with the second precursor delayed in time and  phase shifted from the first. This so-called Brillouin pulse is obtained from

15.8 Infinitely Smooth Envelope Pulses

639

(13.145) with z D zd D ˛ 1=2 .!c / in the exponential, the other factors not appearing in the exponential set equal to unity, and is given by [32]    .!N .T /; T / .!N ./; /  exp ; fBP .t / D exp !c ni .!c / !c ni .!c / 

(15.189)

where T   cT=zd with T > 0 describing the fixed time delay between the leading and trailing-edge Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing-edges and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. With T D 1=.2fc / the input Brillouin pulse is approximately a single-cycle pulse with effective oscillation frequency equal to fc . The input Brillouin pulse when fc D 1 GHz is depicted in Fig. 15.98; part (a) of the figure shows the separate leading and trailing-edge Brillouin precursor structures and part (b) shows the final pulse obtained from the superposition of these two parts, as described by (15.189). The initial rise and fall time for this pulse is  0:6 ns. The dynamical evolution of this input Brillouin pulse in triply distilled water is illustrated by the pulse sequence given in Fig. 15.99 with z=zd D 0; 1; 2; : : : ; 10. Comparison with the pulse sequence 1

A1 & −A2

0.5 0 −0.5 −1 5

5.5

6

6.5

7

7.5

8

8.5 −9 x 10

7

7.5

8

8.5 x 10−9

t (s) 1

ABP1(z,t)

0.5 0 −0.5 −1

5

5.5

6

6.5

t (s)

Fig. 15.98 Temporal structure of the Brillouin pulse BP1 with time delay T D 1=.2fc / for fc D 1 GHz. The separate leading and trailing-edge precursor components are illustrated in (a) and their superposition is given in (b)

640

15 Continuous Evolution of the Total Field 1

z/zd = 0

0.8

1 2

0.6

3

4

0.4

5

6

7

8

9

ABP1(z,t)

0.2

10

0 −0.2 −0.4 −0.6 −0.8 −1 0

1

2

3

4 t (s)

5

6

7 −8 x 10

Fig. 15.99 Propagated pulse sequence for the Brillouin pulse BP1 with delay time T D 1=.2fc / for fc D 1 GHz in the simple Rocard-Powles-Debye model of triply-distilled water

depicted in Fig. 15.70 for a 1 GHz single-cycle rectangular envelope pulse shows that the Brillouin pulse decays much slower with propagation distance. Improved results are obtained when the delay is doubled to the value T D 1=fc . In this case there is a noticeable “dead-time” between the leading and trailing-edge Brillouin precursor structures which decreases the effects of destructive interference between these two components of the Brillouin pulse, resulting in improved penetration into the dispersive, absorptive material. However, this destructive interference can never be completely eliminated for all propagation distances as the time delay between the peak amplitude points for the leading and trailing-edge Brillouin precursors decreases with the inverse of the propagation distance (see Sect. 15.6.1). Nevertheless, it can be effectively eliminated over a given finite propagation distance by choosing the time delay T sufficiently large. The tradeoff in doing this is to decrease the effective oscillation frequency of the radiated pulse. The numerically determined peak amplitude decay with relative propagation distance z=zd is presented in Fig. 15.100. The lower solid curve depicts the exponential attenuation described by the function exp. z=zd /, and the lower dashed curve describes the peak amplitude decay for a single-cycle rectangular envelope pulse with fc D 1 GHz. Notice that the departure from pure exponential attenuation occurs when z=zd 0:5 as the leading and trailing-edge Brillouin precursors begin to emerge from the pulse. The dashed curve labeled BP1 describes the peak amplitude decay for the Brillouin pulse with T D 1=.2fc /, BP2 describes that for the Brillouin pulse with T D 1=fc , and BP3 describes that for T D 3=.2fc /. There

15.8 Infinitely Smooth Envelope Pulses

641

1 0.9 0.8

Peak Amplitude

0.7 0.6 0.5 0.4 BP3

0.3

BP2 BP1

0.2

Single Cycle Pulse exp(−Δ z /zd)

0.1 0

0

1

2

3

4

5

6

7

8

9

10

Δz/zd Fig. 15.100 Peak amplitude as a function of the relative propagation distance z=zd for the input unit amplitude single-cycle rectangular envelope pulse and the Brillouin pulses BP1 , BP2 , and BP3 with fc D 1 GHz. The solid curve describes the pure exponential decay given by exp. z=zd /

isn’t any noticeable improvement in the peak amplitude decay as the delay time T is increased beyond 3=.2fc / over the illustrated range of propagation distances. Notice that at ten absorption depths, exp. z=zd / D exp.10/ Š 4:54  105 , the peak amplitude of the single-cycle pulse is 0:0718, the peak amplitude of the Brillouin pulse BP1 is 0:2123, the peak amplitude of the Brillouin pulse BP2 is 0:2943, and the peak amplitude of the Brillouin pulse BP3 is 0:3015, over three orders of magnitude larger than that expected from simple exponential attenuation. The power associated with the observed peak amplitude decay presented in Fig. 15.100 may be accurately determined by plotting the base ten logarithm of the peak amplitude data vs. the base ten logarithm of the relative propagation distance. If the algebraic relationship between these two quantities is of the form Apeak D B. z=zd /p where B is a constant, then the value of the power p is given by the slope of the relation log.Apeak / D log.B/ C plog. z=zd /. The numerically determined average slope of the base ten logarithm of the numerical data presented in Fig. 15.100 is given in Fig. 15.101. The power factor p for the single-cycle pulse rapidly decreases to the value 1 as the propagation distance increases, this being due to destructive interference between the leading and trailing-edge Brillouin precursors. This destructive interference is somewhat reduced for the Brillouin pulse BP1 whose power factor p varies between 0 and 0:5 when the propagation distance increases up to 3 absorption depths. As the propagation distance increases further, the effects of destructive interference increase and p slowly decreases toward 1. This destructive interference is practically eliminated for the Brillouin

642

15 Continuous Evolution of the Total Field 0 – 0.1 – 0.2

Average Slope

– 0.3 – 0.4

BP3

– 0.5

BP2

– 0.6 BP1

– 0.7 – 0.8

Single Cycle Pulse – 0.9 –1

0

1

2

3

4

5

6

7

8

9

10

Δz/zd Fig. 15.101 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.100

pulse BP2 for propagation distances up to 4 absorption depths. Near optimal (if not indeed optimal) results are obtained for the Brillouin pulse BP3 which experiences negligible destructive interference for propagation distances through at least 20 absorption depths. At z=zd D20 the peak amplitude of this Brillouin pulse is 0:2154, eight orders of magnitude larger than that expected from exponential attenuation. At this penetration depth the power factor p has decreased to the value 0:485 as destructive interference between the leading and trailing-edge Brillouin precursors begins to take effect. The nonexponential, . z/1=2 algebraic decay of the Brillouin precursor makes it the ideal field structure for penetrating attenuative dielectric materials as well as for underwater communications. The fact that its temporal width and effective oscillation frequency depend upon the material parameters makes the Brillouin precursor ideally suited for remote sensing with direct application to foliage and ground penetrating radar as well as to biomedical imaging. However, this also means that the current IEEE/ANSI safety standards may need to be carefully examined for such ultrawideband pulses. Finally, if not indeed optimal, near optimal material penetration is obtained with the Brillouin pulse described by (15.189). If the initial pulse field is perturbed from that given in (15.189), the peak amplitude evolution is decreased from that described by BP1 , BP2 , or BP3 with the period T held fixed. By adjusting the time delay between the leading and trailing-edge Brillouin precursors in the initial pulse, near optimal (if not indeed optimal) pulse penetration can be obtained over a given finite propagation distance. However, as this delay time is increased, the effective oscillation frequency of the initial pulse is decreased.

15.9 The Pulse Centroid Velocity of the Poynting Vector

643

15.9 The Pulse Centroid Velocity of the Poynting Vector Many different definitions have been introduced for the sole purpose of describing the velocity of an electromagnetic pulse in a dispersive medium, the most prevalent of these being phase, group, and energy velocities. Although these different velocity measures provide comparable results in those frequency regions of the material dispersion where the loss is small, they disagree wherever the material loss is large. In fact, some definitions of the pulse velocity (e.g., the phase and group velocities) yield seemingly nonphysical results (superluminal or negative velocities), while others (e.g., the group velocity) apply only to certain pulse characteristics. In 1970, Smith [18] introduced the definition of the pulse centrovelocity in the hope of introducing a measurable pulse velocity which overcomes these shortcomings. This pulse centrovelocity is defined by the quantity ˇ Z ˇ ˇr ˇ

1

tE 2 .r; t /dt

.Z

1

1 1

ˇ1 ˇ E 2 .r; t /dt ˇˇ

(15.190)

where E.r; t / is the real-valued electric field intensity vector. Notice that this ratio is analogous to a center of mass calculation since it tracks the temporal center of gravity of the intensity of the pulse. Recently, Peatross et al. [71, 72] introduced a variant of Smith’s centrovelocity which tracks the temporal center of gravity of the real-valued Poynting vector  c    S.r; t / D   E.r; t /  H.r; t / 4

(15.191)

rather than that of the pulse intensity. Peatross and coworkers [71, 72] then proved that the delay between the initial and propagated temporal center of gravities of the Poynting vector can be expressed as the sum of two terms calculated in the frequency domain, which they call the group delay and the reshaping delay.

15.9.1 Mathematical Formulation For a plane-wave electromagnetic pulse traveling in the positive z-direction through a temporally dispersive HILL medium occupying the positive half-space z > 0 with initial field value specified at z D 0, the average centrovelocity vcv is defined by the equation z vcv (15.192) htz i  ht0 i where

R1 1O z  1 t S.z; t / dt htz i R1 S.z; t / dt 1O z  1

(15.193)

644

15 Continuous Evolution of the Total Field

is the arrival time of the temporal centroid of the Poynting vector at the plane z  0. Notice that calculation of the centrovelocity requires knowledge of the Poynting vector S.z; t / for the propagated wavefield which, in turn, requires expressions for E.z; t / and B.z; t / D 0 H.z; t /. As shown by Peatross et al. [71, 72], the difference between the propagated and initial temporal centers of gravity of the pulse Poynting vector can be expressed as the sum of two terms as (15.194) htz i  ht0 i D Gr C Rr0 ; where the centroid group delay Gr is defined by the expression R1 Gr

Q

@ 0, whereas the reshaping delay “represents a delay which arises solely from a reshaping of the spectrum through absorption (or amplification)” [71] and is calculated on the initial plane z D 0. In a typical experimental arrangement, the pulse (taken here to be traveling in the positive z-direction) is normally incident upon the material interface at the plane z D 0. The transmitted pulse is then calculated in the spectral domain through application of the normal incidence Fresnel transmission coefficients [9] EQ t .!/ 2n1 .!/ D n1 .!/ C n2 .!/ EQ i .!/ Q 2n2 .!/ Bt .!/ B .!/ D Q n .!/ C n2 .!/ Bi .!/ 1

E .!/

(15.197) (15.198)

where n1 .!/ D 1 for the case of vacuum in the negative half-space z < 0, n2 .!/ is the complex index of refraction of the dispersive material occupying the positive half-space z > 0, EQ t .!/ and EQ i .!/ are the transmitted and incident electric field spectra, and BQ t .!/ and BQ i .!/ are the transmitted and incident magnetic field spectra at the interface plane z D 0, respectively.

15.9 The Pulse Centroid Velocity of the Poynting Vector

645

15.9.2 Numerical Results The numerical method used to calculate the Poynting vector at any plane z  0 utilizes the fast Fourier transform (FFT) algorithm to determine the propagated plane wave electric and magnetic field components. For reasons of definiteness, MKS units are now used. To numerically determine the propagated electric field vector E.z; t / D 1O y E.z; t /, this code computes the FFT of the temporal evolution of the initial electric field vector E.0; t / D 1O y E.0; t /, propagates each monochroQ matic component by multiplication with the propagation factor expŒi k.!/z/ , where Q k.!/ D .!=c/n.!/, and then computes the inverse FFT of the propagated field spectrum, thereby constructing the temporal structure of the propagated electric field Z

1

E.z; t / D

Q Q !/e i.k.!/z!t/ E.0; d!

(15.199)

1

Q !/ is the Fourier spectrum of E.0; t /. Likewise, the propagated magwhere E.0; netic field vector B.z; t / D 1O x B.z; t / is numerically determine by performing the FFT of the initial electric field vector E.0; t /, multiplying the resulting spectrum by .1=c/n.!/, propagating each monochromatic component using the same propQ agation factor expŒik.!/z , and then computing the inverse FFT of the propagated spectrum, with the result B.z; t / D

1 c

Z

1

Q

Q !/e i.k.!/z!t/ d!: n.!/E.0;

(15.200)

1

The Poynting vector for the plane wave pulse is then directly calculated as S.z; t / D

1 E.z; t /B.z; t / 1O z 0

(15.201)

for any z  0. The accuracy of this numerical approach depends directly upon the highest frequency sampled at the Nyquist rate, the highest frequency necessary to accurately describe the initial pulse spectrum, and the highest frequency necessary to accurately describe the material dispersion. For a single resonance Lorentz model dielectric with Brillouin’s choice of the medium parameters, the maximum frequency sampled in the published calculations by Cartwright and Oughstun [73] is at least 2  1017 rad/s with at least 217 points sampled; a higher maximum frequency with more sample points is used when the applied signal frequency !c is set above the material absorption band. Because of its experimental importance, let the initial electric field vector of a plane wave pulse normally incident upon the dielectric interface at z D 0 be described by a single cycle gaussian envelope modulated cosine wave ( D =2) with fixed angular carrier frequency !c > 0. The transmitted electric and magnetic field vectors at z D 0C , and hence the transmitted Poynting vector, will then

646

15 Continuous Evolution of the Total Field

experience a small frequency chirp caused by the frequency dependence of the material refractive index appearing in the Fresnel transmission coefficients E .!/ and B .!/. An estimate of the average dispersive transmission induced frequency chirp in the transmitted Poynting vector is found [73] to be less than 5% of the doubled frequency value 2!c at the center of the Poynting vector for all cases considered.

15.9.2.1

Carrier Frequency Below the Absorption Band

When the carrier frequency of the pulse lies in the normal dispersion region below the region of anomalous dispersion, the amplitude of the gaussian Sommerfeld precursor pulse component Ags .z; t / is negligible compared to that of the gaussian Brillouin precursor pulse component Agb .z; t /. Hence, the centrovelocity will rapidly approach the limit vc D c=0 as z ! 1, which is the rate at which the peak amplitude point in the Brillouin precursor travels through the dispersive material. The average centrovelocity of the single cycle gaussian envelope modulated pulse was numerically calculated for each of the below resonance carrier frequency cases !c D 0:25!0 , !c D 0:5!0 , and !c D 0:75!0 over propagation distances from 0:1zd to 100zd , where zd D ˛ 1 .!c /. The results are presented in Fig. 15.102, where the circles, asterisks, and plus signs denote the data points for the !c D 0:25!0 , !c D 0:5!0 , and !c D 0:75!0 cases, respectively, and the solid curves are cubic 0.7

vcv /c

0.6

0.5

c = 0.25

0

c = 0.5

0

0.4 c = 0.75

0

0.3 10−1

100

z/zd

101

102

Fig. 15.102 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [73])

15.9 The Pulse Centroid Velocity of the Poynting Vector

647

spline fits through these data points. The approach to the limiting value .vc /=c D 1=n.0/ D 1=0 D 2=3 as z ! 1 is clearly evident in the figure, in agreement with the asymptotic theory. Similar behavior is obtained for a rectangular envelope pulse with below resonance signal frequency. In terms of the group and reshaping delays, the group delay begins and remains dominant over the reshaping delay for all propagation distances, in agreement with numerical results [73] obtained for a rectangular envelope modulated pulse with carrier frequency below the region of anomalous dispersion.

15.9.2.2

Carrier Frequency in the Absorption Band

The numerically determined average centrovelocity in the absorption band where the dispersion is anomalous is presented in Fig. 15.103 where the circles, asterisks and plus signs represent the data points for the !c D !0 , !c D 1:25!0 , and !c D 1:5!0 cases, respectively. As evident in the figure, the centrovelocity rapidly approaches the limiting value vc D c=0 D .2=3/c as z ! 1. Because the gaussian envelope pulse considered here effectively contains only one oscillation, it does not experience the strong phase delay effects that, for example, a ten-oscillation rectangular envelope pulse undergoes when the carrier frequency lies within the absorption band, as illustrated in Fig. 15.104. Because of this, the centrovelocity is expected to quickly ascend to the limit set by the peak

0.7

0.6

vcv /c

0.5

0.4

0.3

0.2 10−1

c = 1.5 0 c = 1.25 0

100

c= 0

z/zd

101

102

Fig. 15.103 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [73])

648

15 Continuous Evolution of the Total Field 1 c =1.25 0

0.8 0.6

c=

c =1.5 0

0

0.4

vcv /c

0.2

c =1.5 0

0

−0.2 −0.4

c =1.25 0

−0.6 −0.8 −1 10−1

100

z/zd

101

102

Fig. 15.104 Relative average centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the anomalous dispersion region in the medium absorption band. (From Cartwright and Oughstun [73])

amplitude point of the Brillouin precursor. However, when one considers a gaussian envelope pulse with five oscillations or more, the phase delay effects become increasingly significant for sufficiently small propagation distances and negative and superluminal centrovelocities are indeed observed. This extreme behavior is found [73] to be due primarily to pulse reshaping rather than to motion of the pulse itself. As in the case of a rectangular envelope modulated pulse with carrier frequency in the region of anomalous dispersion, the group and reshaping terms are of the same order of magnitude for small propagation distances, and hence, the reshaping term cannot then be ignored. 15.9.2.3

Carrier Frequency Above the Absorption Band

For carrier frequencies !c that lie in the normal dispersion region above the absorption band of the material, both the gaussian Sommerfeld and gaussian Brillouin precursor pulse components are evident during the pulse propagation. The large relative amount of spectral energy situated in the high-frequency domain above the absorption band implies that the gaussian Sommerfeld precursor pulse component is a significant contribution to (15.162) even for large propagation distances. The peak amplitude point of the gaussian Sommerfeld precursor component travels at a velocity just below c while the peak amplitude point of the gaussian Brillouin precursor

15.9 The Pulse Centroid Velocity of the Poynting Vector

649

0.9

c = 2.5 0

vcv /c

0.8

0.7

c=

2

0

0.6 10−1

100

z/zd

101

102

Fig. 15.105 Relative average centroid velocity for gaussian pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region above the medium absorption band. (From Cartwright and Oughstun [73])

travels at the velocity 0 c D .2=3/c. Thus, the value of the pulse centrovelocity will start above .2=3/c and slowly descend to this limit as the gaussian Sommerfeld precursor gradually decays in amplitude with increasing propagation distance. The relative average centrovelocity values for the gaussian envelope modulated pulse cases with angular carrier frequencies !c D 2!0 and !c D 2:5!0 , which correspond to frequencies in the normal dispersion region above the absorption band, are presented in Fig. 15.105 over propagation distances from 0:1zd to 100zd . The graph clearly shows that the centrovelocity starts above and then descends to the limiting value of .vc /=c D 2=3 in agreement with the asymptotic theory. As found for the corresponding rectangular envelope pulse case [73], the group delay is dominant over the reshaping delay when the propagation distance is greater than 101 absorption depths.

15.9.3 The Instantaneous Centroid Velocity The preceding numerical results are for an average centroid velocity of the Poynting vector that is determined by the initial and final centroid locations within the dispersive medium. This velocity measure, as it was originally introduced by Peatross et al. [71, 72], is appropriate for experimental measurements as one typically measures the input and ouput pulse shapes through a slab of dielectric material with

650

15 Continuous Evolution of the Total Field

given thickness. A more localized centroid velocity measure is given by the instantaneous centroid velocity of the Poynting vector that is defined as [73] vci .zj /

lim

zj C1 !zj

zj C1  zj ; htj C1 i  htj i

(15.202)

where htj i is the centroid of the Poynting vector of the pulse at the propagation distance zj . An accurate numerical estimate of this limiting expression may be obtained by selecting neighboring points .zj ; zj C1 / from the average centroid velocity data sets that are sufficiently close to each other. With the exception of the below resonance case for the rectangular envelope case presented in [73], the instantaneous centroid velocity results are are found to be qualitatively similar to the average centroid velocity results. In particular, the limiting value c (15.203) lim vci .z/ D z!1 0 is always obtained. For the case of a rectangular envelope modulated signal with below resonance carrier frequency, the instantaneous centroid velocity is found to peak to a maximum value at a propagation distance between z=zd D 2 and z=zd D 3, after which it approaches the limiting value c=0 from above as z ! 1, as seen in Fig. 15.106. 1.2

1

vci /c

0.8 c = 0.25

0.6

0.4

0.2 10−1

0

c = 0.50

0

c = 0.75

0

100 z/zd

101

Fig. 15.106 Relative instantaneous centroid velocity for rectangular envelope pulse propagation in a Lorentz model dielectric as a function of the relative propagation distance z=zd for several values of the initial carrier angular frequency !c in the normal dispersion region below the medium absorption band. (From Cartwright and Oughstun [73])

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

651

This peak value in the instantaneous centroid velocity increases as the initial pulse carrier frequency increases through the below resonance frequency domain and just becomes superluminal for angular signal frequency values !c 0:75!0 . The numerical results presented in Fig. 15.106 show that the pulse energy centroid initially “accelerates” until its instantaneous velocity reaches a peak value between two and three absorption depths, after which it “decelerates” toward the asymptotic value c=0 set by the velocity of the peak amplitude point of the Brillouin precursor in the dispersive medium.

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits The asymptotic description of the dynamical evolution of an ultrawideband electromagnetic pulse in a dispersive medium has established that the temporal pulse structure evolves into a set of precursor fields that are characteristic of the dispersive medium. Of particular interest is the evolution of the Brillouin precursor whose peak amplitude experiences zero exponential decay with propagation distance z > 0, decreasing algebraically as z1=2 in a dispersive, absorptive medium. The limiting behavior of this algebraic peak amplitude decay in both the zero damping limit as well as the zero density limit is now considered for a Lorentz model dielectric in order to establish whether or not this rather unique behavior persists in these two different limits. The weak dispersion limit is of particular interest as many optical systems are designed to possess minimal loss over the pulse bandwidth. The causal complex index of refraction is given here by  n.!/ D 1 

b2 ! 2  !02 C 2i ı!

1=2 ;

(15.204)

for a single resonance Lorentz model dielectric with undamped angular resonance frequency !0 and phenomenological damping constant ı > 0 with b 2 D k4=0 kN qe2 =m the square of the plasma frequency, where N denotes the number density of Lorentz oscillators in the medium. The material absorption then decreases when either ı ! 0 or when N ! 0. In the first limiting case, the material dispersion becomes increasingly localized about the resonance frequency as ı ! 0 and so is referred to here as the singular dispersion limit. In the second limiting case, the material absorption vanishes while the material dispersion approaches unity at all frequencies as N ! 0 and so is referred to here as the weak dispersion limit. These two limiting cases are fundamentally different in their effects upon ultrashort pulse propagation and are thus treated separately in the following two sections.

652

15 Continuous Evolution of the Total Field

15.10.1 The Singular Dispersion Limit ˙ The asymptotic theory shows that the two first-order near saddle points !SP ./ of n the complex phase function .!; / for a single resonance Lorentz model dielectric coalesce into a single second-order saddle point at [see (12.236)]

!SPn .1 / Š 

2ı i; 3˛

(15.205)

where [from (12.225)] 1 0 C

2ı 2 !p2 3˛0 !04

;

(15.206)

with 0 D n.0/ and ˛ 1. At the space–time point  D 0 D n.0/, the dominant . / crosses the origin [!SPC .0 / D 0] so that its contribution near saddle point !SPC n n to the asymptotic behavior of the propagated wavefield experiences zero exponential attenuation, viz., .0 /; 0 / D 0; (15.207) .!SPC n the peak amplitude point decaying only as z1=2 as z ! 1, while at the space–time point  D 1 this contribution to the asymptotic wavefield experiences a small (but nonzero) amount of exponential attenuation as well as a z1=3 algebraic decay as z ! 1, provided that ı > 0. In the singular dispersion limit as ı ! 0, however, the two near saddle points !SP˙ . / coalesce into a single second-order saddle point at the origin, resulting n in an asymptotic behavior whose peak amplitude experiences zero attenuation, the amplitude now decaying only as z1=3 . Notice that this limiting behavior is entirely consistent with the modern asymptotic theory. The numerically determined peak amplitude decay with relative propagation distance z=zd is presented in Fig. 15.107 for an input Heaviside unit step function modulated signal A.0; t / D f .t / D UH .t / sin .!c t / with below resonance carrier frequency !c D 3:0  1014 r=s in a single resonance Lorentz model dielectric with p angular resonance frequency !0 D 3:9  1014 r=s and plasma frequency b D 9:29  1014 r=s for several decreasing values of the phenomenological dampdepth in ing constant ı. Here zd ˛ 1 .!c / denotes the e 1 amplitude penetration n o Q the dispersive dielectric at the angular frequency !c , where ˛.!/ = k.!/ is the attenuation coefficient. The dashed line in the figure describes the pure exponential attenuation described by the function e z=zd . The peak amplitude used here is given by the measured amplitude of the first maximum in the temporal evolution of the propagated pulse at a fixed observation distance z  0. Notice that this “leadingedge” peak amplitude point initially attenuates more rapidly than that of the signal at ! D !c , but that as the mature dispersion regime is reached and the Brillouin precursor emerges, a transition is made from exponential attenuation to algebraic decay. Notice further that this transition occurs at a larger relative propagation distance z=zd as the phenomenological damping constant ı decreases and the medium

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

653

dispersion becomes increasingly localized about the medium resonance frequency !0 , and hence, more singular. As the material dispersion becomes more singular (i.e., as ı decreases), the number of sample points required to accurately model the material dispersion and resultant propagated field structure increases. At the smallest value of ı considered here, a 223 point FFT was required. The algebraic power associated with the measured peak amplitude decay presented in Fig. 15.107 may be accurately determined by plotting the base ten logarithm of the peak amplitude data vs. the base ten logarithm of the relative propagation distance z=zd , as described in Sect. 15.8.3. If the algebraic relationship between these two quantities is of the form Apeak D B.z=zd /p where B is a constant,then the value of the power p is given by the slope of the relation log .Apeak / D log .B/ C p log .z=zd /. The numerically determined average slope of the base ten logarithm of the data presented in Fig. 15.107 is given in Fig. 15.108 for each value of ı considered. These numerical results show that the power p increases from a value approaching 1=2 as z ! 1 to a value approaching 1=3 as z ! 1 when ı is decreased such that ı=!0  1, in complete agreement with the asymptotic theory. An example of the numerically computed dynamical field evolution in the singular dispersion limit is presented in Fig. 15.109. The initial wavefield at z D 0 is a Heaviside unit step function signal with below resonance angular carrier frequency !c D 3:0  1014 r=s. The propagated wavefield illustrated here was calculated at ten absorption depths into a single resonance Lorentz modelpdielectric with resonance frequency !0 D 3:9  1014 r=s, plasma frequency b D 9:29  1014 r=s, and phenomenological damping constant ı D 3:02  1010 r=s. Because ı=!0 D 7:74  105 , this case is well within the singular dispersion regime.

15.10.2 The Weak Dispersion Limit In the weak dispersion limit as N ! 0, the material dispersion approaches that for vacuum at all frequencies, i.e., n.!/ ! 1. This then introduces a rather curious difficulty into the numerical FFT simulation of pulse propagation in this weak dispersion limit as the number of sample points required to accurately model the propagated pulse behavior rapidly increases as the number density N decreases to zero. To circumvent this problem, an approximate equivalence relation may be used that allows one to compute the propagated field behavior in an equivalent dispersive medium that is strongly dispersive. This approximate equivalence relation, which becomes exact in the limit as N ! 0, directly follows from the integral representation of the propagated wavefield, given by (12.1) as Z 1 (15.208) fQ.!/e .z=c/.!;/ d!; A.z; t / D 2 C for z  0.

654

15 Continuous Evolution of the Total Field

Peak Amplitude of the Brillouin Precursor

10"0

10−1

X

r/s

10−2

10

−3

0

20

40

z/zd

60

80

100

Fig. 15.107 Numerically determined peak amplitude decay due to an input unit step function modulated signal with below resonance carrier frequency !c D p 3:0  1014 r=s in a single resonance Lorentz model dielectric with !0 D 3:9  1014 r=s and b D 9:29  1014 r=s as a function of the relative propagation distance z=zd for decreasing values of the phenomenological damping constant ı

Average Slope of the Logarithm of the Peak Amplitude Data

0

−1

r/s

X

−0.5

0

20

40

z/zd

60

80

100

Fig. 15.108 Average slope of the base ten logarithm of the numerical data presented in Fig. 15.107

15.10 Dispersive Pulse Propagation in the Singular and Weak Dispersion Limits

655

0.02 z/zd = 10

0.01

AH (z,t)

Brillouin Precursor

0 Sommerfeld Precursor

−0.01

−0.02 2

3

4 t (x10

5

6

–10

r/s)

Fig. 15.109 Propagated wavefield at ten absorption depths (z D 10zd ) due to an input Heaviside unit step function modulated signal with below resonance angular carrier frequency !c p D 3:0  1014 r=s in a single resonance Lorentz model dielectric with !0 D 3:9  1014 r=s, b D 9:29  1014 r=s, and ı D 3:02  1010 r=s

Two different propagation problems for the same input pulse A.0; t / D f .t / are identical provided that the relation kQ1 .!/z1  !t1 D kQ2 .!/z2  !t2

(15.209)

is satisfied for all !. Upon equating real and imaginary parts, there results the pair of relations ˇ1 .!/z1  !t1 D ˇ2 .!/z2  !t2 ; ˛1 .!/z1 D ˛2 .!/z2 ;

(15.210) (15.211)

n o Q both of which must be satisfied for all !, where ˇ.!/ < k.!/ and ˛.!/ n o Q = k.!/ . For the absorptive part, one obtains the equivalence relation z2 D

˛1 .!/ z1 ; ˛2 .!/

8 !:

(15.212)

If the two media differ only through p their densities, then because ˛.!/ D .!=c/ni .!/ for real ! and n.!/ D 1 C Ng.!/ ! 1 C 12 Ng.!/ as N ! 0, so that ni .!/ 12 Ng.!/, the above equivalence relation becomes z2

N1 z1 : N2

(15.213)

656

15 Continuous Evolution of the Total Field

The corresponding equivalence relation for the phase part then becomes ! c

    1 1 ! 1 C N1 g.!/ z1  !t1

1 C N2 g.!/ z2  !t2 2 c 2   1 ! N1 1 C N2 g.!/

z1  !t2 ; c 2 N2 (15.214)

so that

 t2 t1 C

 N1 z1 1 ; N2 c

(15.215)

which is the second part of the desired equivalence relation. For example, if N1 =N2 D 1  102 , then z2 D z1  102 and t2 t1  .0:33  108 s=m/z1 . In that case, the propagated wavefield structure illustrated in Fig. 15.109 also applies to the case when the plasma frequency b is reduced by the factor 10 and the propagation distance z is increased by the factor 100 provided that the time scale is adjusted according to the relation given in (15.215).

15.11 Comparison with Experimental Results The first experimental measurements of the precursor fields originally described by Sommerfeld [19] and Brillouin [1] in a single resonance Lorentz medium were published by Pleshko and Pal´ocz [74, 75] in 1969; it is apparent that they were the first to refer to the first and second precursors as the Sommerfeld and Brillouin precursors, respectively. As reported in Pleshko’s Ph.D. thesis [74], the transient responses for three different types of waveguiding structures were investigated: an air-filled rectangular cross-section metallic waveguide, a surface-waveguide, and a coaxial line that is filled with a longitudinally magnetized ferromagnetic material. Two types of baseband pulse generators were used in these experiments, each producing phased locked signals. The first is a so-called Bouncing Ball Pulse Generator (BBPG) [76] which produces a gaussian-type pulse in the X-band7 with 150200 ps pulse widths at the half-amplitude points, and the second was an HP 1105 A pulse generator with a tunnel diode mount which produced a 20 ps rise-time, 3 ¯s width pulse that could then be passed through shaping circuits in order to produce a variety of pulse waveforms, including a single-cycle pulse with ˙2:0 GHz bandwidth centered about a 4:0 GHz carrier frequency. The pulse waveforms were then measured with a 12:4 GHz sampling oscilloscope with 16 GHz maximum frequency response [74, pages 4–7]. Their principal experimental results are now briefly described. 7

The X-band denotes the microwave region of the electromagnetic spectrum extending from 7 to 12:5 GHz.

15.11 Comparison with Experimental Results

657

The dispersion relation for the fundamental mode of an air-filled rectangular waveguide oriented along the z-axis is given by ! kz D c

 1=2 !c2 1 2 ; !

(15.216)

where !c is the angular cutoff frequency. This dispersion relation then approximates the optic mode in a single resonance Lorentz model dielectric, obtained from (12.57) for j!j  !0 with !c replaced by the plasma frequency b. The transient wavefield is then comprised of just a Sommerfeld precursor with zero exponential attenuation, decaying algebraically as z1=2 as z ! 1, as described in (13.62) with ı D 0. This result was first verified experimentally by Pleshko [74] using three lengths of an air-filled rectangular cross-section waveguide with a BBPG source pulse waveform. The measured output waveforms are illustrated in Fig. 15.110 and the measured relative peak amplitude points (indicated by the arrows in Fig. 15.110) are compared in Fig. 15.111 with the theoretical z1=2 amplitude decay. The experimentally measured temporal evolution of the Sommerfeld precursor structure produced by a 20 ps rise-time step function is presented in Fig. 15.112. As then concluded by P. Pleshko [74, page 45], “it may be concluded that the agreement

Fig. 15.110 BBPG source pulse waveforms at increasing propagation distances in an air-filled rectangular waveguide. (Figure 15.17 from P. Pleshko [74])

658

15 Continuous Evolution of the Total Field 1

Relative Peak Amplitude

0.9

0.8

0.7

1/2

1/zrel

0.6

0.5

1

2

zrel

3

4

Fig. 15.111 Measured peak amplitude decay with relative propagation distance (open circles) of the Sommerfeld precursor dominated propagated waveforms in an air-filled rectangular waveguide presented in Fig. 15.109. The solid curve describes the z1=2 behavior predicted by the asymptotic theory. (Data from Fig. 15.18 of P. Pleshko [74])

Fig. 15.112 Step function response of an air-filled rectangular waveguide. (Figure 15.19 from P. Pleshko [74])

15.11 Comparison with Experimental Results

659

between the transient response calculated by the stationary phase method and transient response obtained experimentally is exceptionally good, with an accuracy well within expected measurement error.” Consider next the experimental results for a coaxial line filled with a longitudinally (i.e., along the waveguide axis) magnetized ferrimagnetic material.8 The dispersion relation may be approximated [78] by that given by Suhl and Walker [79] for the dominant TEM mode in a parallel plate waveguide filled with a lossless ferrite in the limit of small plate separation a as kz

 1=2 .!0 C !M /2  ! 2 !p r ; c !0 .!0 C !M /  ! 2

(15.217)

p provided that .!=c/ r a  1. Here r denotes the relative (real-valued) dielectric permittivity of the ferrite, and !0 D g Hi ; !M D k4kg Ms ;

(15.218) (15.219)

where g denotes the gyromagnetic ratio, Hi is the externally applied magnetic field intensity in the ferrite, and k4kMs is the saturation magnetization of the ferrite. The dispersion relation givenp in (15.217) has a low-frequency branch ! 2 Œ0; !R

p with asymptote k D . r =c/! 1 C .!M =!0 / as ! ! 0, where !R

p !0 .!0 C !M /;

(15.220)

p and a high-frequency branch !  !C with asymptote k D . r =c/! as ! ! 1, where (15.221) !C !0 C !M : As the externally applied magnetic field strength is increased, !0 increases and both !R and !C increase and approach each other as the two asymptotes merge, resulting p in a single, nondispersive dielectric mode with dispersion relation k D . r =c/!. This is illustrated in Fig. 15.113 which shows the measured line response to a narrow pulse excitation for increasing values of the applied magnetic field strength. As this is approximately the impulse response of the dispersive line, the resultant waveforms are dominated by the high-frequency branch Sommerfeld precursor followed by the low-frequency branch Brillouin precursor, as clearly evident in the figure. The experimentally measured Heaviside step function envelope modulated sine wave response of this garnet-filled coaxial line is presented in Fig. 15.114 for increasing values of the of the applied magnetic field strength. The input waveform had a 1 ns rise-time with a 625 MHz carrier frequency. as described by Pleshko [74]: Initially, with zero applied field, the sine wave pulse propagates on essentially a low dispersion portion of the high frequency mode. at 20 gauss, an initial high-frequency oscillation 8

A detailed description of ferromagnetic, antiferromagnetic, and ferrimagnetic materials may be found in Chap. 1 of [77].

660

15 Continuous Evolution of the Total Field

Fig. 15.113 Narrow pulse response of a garnet-filled coaxial line. (Figure 15.32 from P. Pleshko [74])

due to the high frequency components of the pulse is seen to be the first signal arriving, which is a Sommerfeld type of precursor, and since the cutoff frequency fC is low, it has appreciable amplitude. At this point also, the Brillouin type of precursor is also seen. There is no main signal at this field strength because the carrier frequency of the pulse lies in the stop band of the system. As the field strength is increased, the Sommerfeld type of precursor is no longer visible to the eye (masked by the noise of the oscilloscope) but the Brillouin precursor still has appreciable amplitude. The waveform obtained at an external field strength of 150 gauss . . . the carrier frequency is close to !R and thus the main body of the signal has very large dispersion and low amplitude. Thus, olnly the low frequency portion of the spectrum comes through with large amplitudes. With a field strength of 200 gauss, the Brillouin type precursor, and the main signal are seen with large amplitudes as Brillouin stated in his book (p. 128) but the Sommerfeld type precursor is not visible due to the fact that the noise of the oscilloscope is greater than the amplitude of the signal comprising the Sommerfeld type precursor.

Although their experiments were conducted in the microwave domain on waveguiding structures with dispersion characteristics that are similar to that described by either a Drude model conductor [compare (15.216) and (12.153)] or a single resonance Lorentz model dielectric [compare the low and high-frequency branches of (15.217) with the below resonance and above absorption band approximations of (12.57)], the results established the physical property of the now classical asymptotic theory developed by Sommerfeld and Brillouin. In particular, through several rather clever experimental arrangements, Pleshko [74] and Pal´ocz [75] were able

15.11 Comparison with Experimental Results

661

Fig. 15.114 Heaviside step function envelope modulated sine wave response of a garnet-filled coaxial line. (Figure 15.39 from P. Pleshko [74])

to isolate the dynamical evolution of the individual pulse components comprising the propagated signal representation given in (15.1). This “proof of principle” done, experimental verification in bulk (e.g., non-waveguiding) media then remained to be given. In an extension of this early experimental work, D. D. Stancil [80] measured magnetostatic precursory wave motion in thin ferrite films.9 These results showed the existence of three types of Brillouin-type precursors in an Yttrium iron garnet (YIG) film, with experimental observations for forward volume waves, backward volume waves, and surface waves. A complete, detailed description of this observed precursor-type phenomena in magnetostatic wave motion remains to be given. Precursor-type phenomena is also observed in fluids and acoustics. The signal velocity of sound in superfluid 3 He–B was measured by Avenel, Rouff, Varoquaux and Williams [82, 83] for moderate material damping. Their reported experimental results are in agreement with Brillouin’s original description [1] where the deformed contour of integration was constrained to entirely lie along the union of steepest descent paths through the distant and near saddle points, resulting in a signal 9

Magnetostatic waves (MSW), also called magnetic polarons or magnons, refer to oscillations in the magnetostatic properties of a magnetic material such as a ferrite. See, for example, the book on magnetostatic waves by Stancil [81].

662

15 Continuous Evolution of the Total Field

velocity that peaks to a maximum value near the material resonance frequency. However, their experiment did not use a step function modulated signal for which the signal velocity has been defined. Rather, they used a continuous envelope pulse for which the signal velocity is undefined. The observation of a “precursory” motion that is similar to the Sommerfeld (or first) precursor was later observed [83] by Varoquaux, Williams, and Avenel in superfluid 3 He–B. Detailed experiments measuring precursor phenomena on fluid surfaces have been presented by Falcon, Laroche, and Fauve [84]. The dispersion relation for the angular frequency !.k/ in terms of the wavenumber k (neglecting dissipation) for such surface waves is given by s   3 gk C k tanh .kh/; !.k/ D 

(15.222)

where g is the acceleration due to gravity,  is the fluid mass density,  is the surface tension, and h is the p depth of the liquid body. Associated with this fluid is the capillary length `c =.g/ and Bond number B0 .`c = h/2 . The wavenumber dispersion of the phase velocity vp .k/ !.k/=k is described by the dotted curve in Fig. 15.115 for mercury ( D 13:5  103 kg=m3 ,  D 1:5  103 Ns=m2 ,  D 0:4 N=m) with depth h D 3:7 mm. The solid curve in the figure describes the (numerically determined) exact wavenumber dependence of the group velocity vg .k/ @!.k/=@k. In either the long wavelength approximation or shallow fluid limit given by kh  1, the dispersion relation in (15.222) may be expanded and differentiated with respect to k to yield the approximate expression [84] vg .k/

i p h @!.k/ a4

gh 1  a2 .kh/2 C .kh/4 ; @k 4

(15.223)

where a2 13  B0 and a4 19  12 B0  13 B02 . The wavenumber dependence 90 of this approximation is described by the dashed curve in Fig. 15.115. Although this approximation is only valid for k  1= h  270, it does properly show that a minimum in the group velocity exists only when the Bond number satisfies the inequality 0  B0 < 13 , the value B0 D 13 corresponding to the critical depth p hc 3`c . A typical surface wave pattern generated by an impulsional excitation is illustrated in Fig. 15.116. The source is located to the right of the figure with the surface wave propagating to the left, led by a high-frequency Sommerfeld precursor SH followed by a low-frequency Brillouin precursor SL (referred to as a low-frequency Sommerfeld precursor in [84]). The observed pulse evolution can be described by the group velocity dispersion curve presented in Fig. 15.115 by first noting that as time increases at a fixed propagation distance z > 0, the horizontal line at vg .k/ D z=t moves p down the group velocity curve. For early times t < vg .0/, where vg .0/ D gh, there is just a single wavenumber solution ks to this equation [viz., vg .ks / z=t ] which decreases as t increases. For all values of the Bond

15.11 Comparison with Experimental Results

663

wave velocity (m/s)

0.22

0.2

vp(k) 0.18 vg(k)

0.16 0

100

200

300 400 wavenumber (r/m)

500

600

700

Fig. 15.115 Wavenumber dependence of the phase velocity vp .k/ D !=k (dotted curve) and the group velocity velocity vg .k/ D @!=@k (solid curve) for mercury with fluid depth h D 3:7 mm and Bond number B0 D 0:22. The dashed curve describes the approximate group velocity dispersion described by (15.223)

Fig. 15.116 Photograph of typical surface wave precursors for mercury with fluid depth h D 3:7 mm and Bond number B0 D 0:22. The full vertical scale corresponds to a 7 cm canal width. (Figure 15.1 from Falcon, Laroche, and Fauve [84])

number B0  13 , so that h  hc , there is just a single solution to this equation as the group velocity monotonically decreases to its static value. In that case, the propagated impulsive response is comprised of just a Sommerfeld-type precursor whose instantaneous oscillation frequency monotonically decreases to zero. On the other hand, when 0  B0 < 13 , so that h > hc , there are two solutions to (15.222) when

664

15 Continuous Evolution of the Total Field

p t > z= gh. In that case, the propagated impulsive response is comprised of two types of precursors: the fastest is the high-frequency Sommerfeld-type precursor SH (from the capillary branch k > kmin of the dispersion curve) characterised by an instantaneous oscillation that decreases with time, that is followed by the slower, low-frequency Brillouin-type precursor SL (from the gravity branch k < kmin of the dispersion curve) that is characterised by an instantaneous oscillation frequency that increases with time. Measurements of these precursor-type wave motions were performed by Falcon et al. [84] for a mercury fluid layer with heights h varying from 2:12 to 13:75 mm, so that 0:02  B0  0:67, the critical value B0 D 13 of the Bond number occuring at the critical fluid height h D hc 3 mm. The source was a horizontal impulsion and the resultant free-surface wave profiles were recorded at a fixed distance z from the source. Their results, using both optical (a and b) and inductive (c) measurement techniques, are presented in Fig. 15.117; the insets in parts (b) and (c) of the figure present a comparison of the two measurement techniques. In part (a) of the figure, h < hc and the dynamical wave evolution at z D 0:2 m is dominated by a Sommerfeld-type precursor whose oscillation frequency monotonically decreases to zero. In part (b) of the figure, h > hc and the high-frequency Sommerfeld-type precursor precedes the low-frequency Brillouin-type precursor at z D 0:2 m. The observation distance in (b) is increased to z D 0:6 m in part (c) of the figure. At this larger propagation distance the high-frequency Sommerfeld-type precursor has disappeared due to attenuation (from viscous dissipation) and only the low-frequency contribution from the gravity branch (the Brillouin-type precursor) is observed. The measured oscillation period of the high-frequency Sommerfeld-type and lowfrequency Brillouin-type precursors are presented p in Fig. 15.118 as a function of the dimensionless space–time parameter .z=t /= gh for a variety of experimental conditions. The solid curves in the figure describe the theoretical behavior obtained from the dispersion relation given in (15.222). Although the description provided by the stationary phase approximation with this real-valued dispersion relation is adequate in some respect, it fails to completely describe the dynamical evolution of the impulsive surface wave phenomena illustrated in Fig. 15.117. A more accurate description that properly (i.e., causally) includes the attenuative part of the dispersion relation remains to be given. The experimental observability of optical precursors has been proposed by Aaviksoo, Lippmaa, and Kuhl [85] in 1988 using the transient response of excitonic resonances to picosecond pulse excitation. The experimental observation of the Sommerfeld and Brillouin precursors was then reported by Aaviksoo, Kuhl, and Ploog [86] in 1991. In their experimental arrangement, an approximate double ex1 =˛2

400 fs ponential pulse [see (11.58)–(11.60)] with a steep rise-time of tr  ln˛1˛˛ 2 1 and a slow decay-time of td  ˛2 6:2 ps was transmitted through a 0:2 ¯m thick layer of GaAs crystal near the exciton resonance. The transmitted pulse was then measured through its cross-correlation with the incident pulse. The results agreed qualitatively (the amplitudes were off by a factor between 2 and 3) with theoretical cross-correlation results based on dispersive pulse propagation in a single resonance Lorentz model medium. Because these theoretical results are described

15.11 Comparison with Experimental Results

665

Fig. 15.117 Measured free-surface profiles of the impulsive wave response at the surface of merp cury as a function of the dimensionless time t =t0 , where t0 D z= gh using optical [(a) and (b)] and inductive [(c)] techiniques. The insets in (b) and (c) show a comparison of measurements made with both techniques. In (a) h < hc so that B0 > 1=3 and just the high-frequency Sommerfeld-type precursor SH is present and in (b) and (c) h > hc so that B0 < 1=3 and both the high-frequency Sommerfeld-type and low-frequency Brillouin-type precursors are present. (Figure 15.3 from Falcon, Laroche, and Fauve [84])

666

15 Continuous Evolution of the Total Field 300

Period (ms)

Low-Frequency Brillouin Precursors

200 High-Frequency Sommerfeld Precursors

100 Tmin

0

0.5

1

1.5

2

(z/t)/(gh)1/2

Fig. 15.118 Measured oscillation period of the high-frequency Sommerfeld and low-frequency p Brillouin precursors as a function of the dimensionless space–time parameter .z=t /= gh with 0:2 m  z  0:8 m for the fluid depth cases h D 2:12 mm (r) for depression pulses, h D 2:12 mm (4) for elevation pulses, h D 3:4 mm (), h D 5:6 mm (u t), h D 7:2 mm (ı), h D 10:4 mm (˘), and h D 13:75 mm (). (From Fig. 15.4 of Falcon, Laroche, and Fauve [84])

by overlapping Sommerfeld and Brillouin precursors (see, for example, Fig. 15.88), together with an exciton precursor that has been described by Birman and Frankel [87,88] their experimental results provide indirect evidence of these precursor fields. As stated by the authors in the conclusion of their paper [86]: we have experimentally demonstrated the existence of precursors for transient electromagnetic wave propagation through dispersive media in the optical frequency range. These precursors or forerunners appear in the transmitted optical pulse if long narrow-bandwidth pulses with steep fronts propagate near material resonances. Experiments on thin GaAs crystals with nearly exponential type pulses and a frequency tuned close to the free-exciton resonance have confirmed this prediction in a good agreement with corresponding theoretical calculations. The observation of separate Sommerfeld, Brillouin, and exciton precursors in the optical regime remains a challenge for future work.

Indeed, the observation of the separate Sommerfeld and Brillouin precursors in the optical domain has posed a special challenge to experimentalists. The experimental observation of the Brillouin precursor in bulk media using an ¨ ultrashort optical pulse has been reported by Choi and Osterberg [89] in 2004, but not without criticism [90] that is itself due criticism. In their reported obser¨ vation of a Brillouin precursor in deionized water, Choi and Osterberg state [89] that they “observe pulse breakup in a linear regime for 540 fs long pulses with a bandwidth of 60 nm. . . propagating through 700 mm of deionized water.” They “attribute the pulse breakup to the formation of optical precursors.” Their conclusions are “further supported by subexponential attenuation with distance for the new peak p as well as 1= z attenuation at distances exceeding 3:5 m.” Their experimental measurements of the peak amplitude decay as a function of propagation distance are

15.11 Comparison with Experimental Results

667

Peak Amplitude

100

e −z/zd

Tr f = T/10 Tr f = T

10−1

0

1

z/zd

2

3

Fig. 15.119 Experimentally measured peak amplitude decay (ı symbols) of an ultrashort optical pulse in deionized water. The dotted line describes pure exponential decay and the dashed curves describe the numerically determined peak amplitude decay with rise/fall time equal to the period of oscillation T D 1=f of the carrier wave (lower dashed curve) and to one-tenth of that period ¨ (upper dashed curve). (Experimental data provided by Choi and Osterberg [89])

presented in Fig. 15.119 by the open circles connected by solid line segments from a linear spline interpolation. Notice that the initial pulse spectrum in their experiments is centered about 700 nm, corresponding to a pulse frequency f ' 0:428 PHz with bandwidth f ' 0:0367 PHz. Because f =f ' 0:0857  1, this pulse is quasimonochromatic. ¨ In their published critique of the Choi and Osterberg results, Alfano et al. [90] state that “Sommerfeld precursors arise from the higher frequencies in the pulse while Brillouin precursors arise from low frequencies far away from resonance frequencies.” However, this statement is not entirely correct. What is necessary is that the pulse spectrum have sufficient energy either above resonance (for the Sommerfeld precursor) or below resonance (for the Brillouin precursor). Alfano et al. [90] further state that “the dispersion of water is almost flat in the region about 700 nm. No asymptotic behavior (saddle point) exists.” Although the real part of the dielectric permittivity of water exhibits relatively weak normal dispersion over the pulse bandwidth, the imaginary part does not (see Figs. 4.2 and 4.3 in Vol. 1). This does not eliminate the possibility of saddle point evolution and hence, the appearance of a precursor. Finally, they attribute the observed pulse breakup as being due to a vibrational overtone absorption band in water that is centered at 760 nm (0:394 PHz). This can also help explain the appearance of a Brillouin

668

15 Continuous Evolution of the Total Field

precursor in their measured attenuation data because small nonlinear effects have been shown [91] to enhance the Brillouin precursor in the dynamical field evolution. ¨ Choi and Osterberg’s experimental results [89], reproduced in Fig. 15.119, clearly exhibit nonexponential decay. The question as to whether or not this data p exhibit the 1= z amplitude attenuation characteristic of the Brillouin precursor is best addressed by determining the algebraic power law described by these values. If the relation between the peak amplitude Apeak and the relative propagation distance z=zd , where zd D ˛ 1 .!c / is the e 1 penetration depth at the pulse frequency, is given by Apeak D B.z=zd /p , where B is a constant, then the power p of the peak amplitude decay may be determined [32] from the slope of the base ten logarithm of the data. The results, presented in Fig. 15.120. show that the averaged experimental data varies between that for the the numerically determined peak amplitude decay for an ultrawideband pulse with rise/fall time equal to the period of oscillation T D 1=f of the carrier wave (lower dashed curve) and to one-tenth of that period (upper dashed curve) in a double resonance Lorentz model of the optical frequency dispersion of triply distilled water over the intermediate propagation distances between 0:25 and 1:75 absorption depths. The average slope values are presented here in order to help smooth the inherent variability in Choi and ¨ Osterberg’s experimental results, indicated by the C signs in Fig. 15.120. These averaged results are indicative of the appearance of a Brillouin precursor over this propagation domain. Further experimental results involving ultrashort optical pulse propagation in water have since been reported by Okawachi et al. [92]. However,

0 −0.2 −0.4 Averaged Experimental Results

Average Slope

−0.6 −0.8 −1

Trf = T/10

−1.2 −1.4

Trf = T

−1.6 −1.8

0

1

z/zd

2

3

Fig. 15.120 Averaged valuse (ı symbols) of the slope of the base ten logarithm of the experimental data (C symbols) obtained from Fig. 15.119 compared with that obtained from the two theoretical (dashed) curves in Fig. 15.119 for the T and T =10 rise/fall time cases

15.12 The Myth of Superluminal Pulse Propagation

669

their experiments were performed using a 540 fs gaussian envelope pulse whose spectrum in the ultraviolet region of the optical domain (using wavelengths of 800 and 1,530 nm) is clearly not ultrawideband, as reflected in their experimental results. Thier conclusion that [92] “we observe strictly monoexponential decay, confirming that propagation of femtosecond pulses in water obeys the Beer-lambert law” has no bearing in the ultrawideband domain. The asymptotic and numerical results p presented in this chapter show that the characteristic 1= z peak amplitude decay of the Brillouin precursor will not be observed in their experimental arrangement. More recent experimental observations [93] of both the Sommerfeld and Brillouin precursors in the optical domain when the input ultrashort pulse is in the region of anomalous dispersion has been reported by Jeong, Dawes, and Gauthier. In their experiment, a step function modulated signal with a 1:7 ns rise-time, which corresponds to a 206 MHz bandwidth with !c ' 3:9  1014 r=s, was transmitted through a dilute gas of potassium (39 K) atoms. The pulse frequency and bandwidth then essentially interact with a single resonance frequency at !0 ' 3:9  1014 r=s with frequency dispersion described by a single resonance Lorentz model dielectric with plasma frequency b ' 3:05  109 r=s and damping constant ı ' 3:02  107 r=s. This then corresponds to both the singular and weak dispersion limits described in Sect. 15.10. Because the pulse carrier frequency is in the anomalous dispersion region of this resonance and because the dispersion is both weak and singular, both the Sommerfeld and Brillouin precursors are observed superimposed on each other [see Fig. 15.88b]. Taken together, these experimental results provide an important (albeit partial) verification of the modern asymptotic theory in its description of ultrashort dispersive pulse dynamics. Additional experimental measurements are clearly needed, not just to verify the theoretical predictions, but also to apply the unique features of precursors to a variety of practical applications, including medical imaging, remote sensing, and communications in adverse environments. With the use of a temporal coherence synthesization scheme proposed by Park et al. [94], transform-limited pulses with fairly arbitrary envelope functions may be constructed in the optical domain.

15.12 The Myth of Superluminal Pulse Propagation There was a young chap named Devaney whose arguments went faster than electromagnetic energy. He published a paper in May, in an extremely noncausal way, with errata published the previous February! Clever limericks aside, the allure of superluminal pulse propagation in classical physics is practically irresistable (see the episode “The Lure of Light” from the 1953–1954 Flash Gordon television series). It certainly is more newsworthy than the fundamental restriction imposed by the special theory of relativity, as evidenced

670

15 Continuous Evolution of the Total Field

by the May 30, 2000 New York Times article “Faster Than Light, Maybe, But Not Back to the Future” as well as by the May 16, 2006 New York Times article “Impressive New Tricks of Light, All Within the Laws of Physics.” Such experiments are typically conducted with a gaussian envelope pulse as that particular pulse shape is ideally suited for the group velocity approximation. Therein lies the entire difficulty with these reported observations of superluminal pulse propagation. Sommerfeld’s relativistic causality theorem (see Theorem 6 in Sect. 13.1), first given in 1914, rigorously proves that information cannot be transmitted through a causal medium faster than the speed of light c in vacuum. In particular, notice that any initial plane wave pulse A.0; t / D f .t / at the plane z D 0 and propagating in the positive z-direction can be formally separated into two distinct parts as A.0; t / D f .t /uH .t0  t / C f .t /uH .t  t0 /;

(15.224)

where uH .t / denotes the Heaviside unit step function [defined here as uH .t / D 0 for t < 0, uH .t / D 12 for t D 0, and uH .t / D 1 for t > 0], and where t0 2 .1; C1/ is any finite, fixed instant of time. Sommerfeld’s theorem then rigorously shows that no part (i.e., information, energy, etc.) of the initial wavefield component A> .0; t / f .t /uH .t  t0 / can appear ahead of the luminal space–time point c.t  t0 /=z D 1 in the propagated wavefield for all z > 0; that is, A> .z; t / D 0

(15.225)

for all .z; t / with z > 0 such that cz .t  t0 / < 1. Notice that sufficiently slow parts of the initial wavefield component A< .0; t / f .t /uH .t0  t / can appear behind the luminal space–time point c.t  t0 /=z D 1 in the propagated wavefield. The observation of both superluminal and negative group velocities for gaussian pulse propagation in any causally dispersive system is then due to pulse reshaping. Because peak amplitude points are not causally related [95,96], there is no violation of special relativity. This simple fact was beautifully demonstrated with a simple electronic circuit by Prof. Kitano of Kyoto University at the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics (see Fig. 15.121 for a photograph of some of the participants at this workshop). All of the talks for this workshop can be found at the KITP Web site. The subtle effects of pulse reshaping on the group velocity of a gaussian envelope pulse in the anomalous dispersion region of a Lorentz model dielectric are illustrated in Figs. 15.95–15.97. Generalizations of the group velocity do not fare any better. In particular, the generalization of the group velocity to the pulse centroid velocity does not remove this fundamental difficulty. The detailed numerical study of the evolution of the pulse Poynting vector centrovelocity for both ultrawideband rectangular envelope and gaussian envelope plane-wave pulses traveling through a single resonance Lorentz model dielectric, presented here in Sect. 15.9 for the gaussian envelope pulse with more detailed results in [73] for both gaussian and rectangular envelope pulses, leads to the following set of conclusions:

15.12 The Myth of Superluminal Pulse Propagation

671

Fig. 15.121 Participants in the 2002 Quantum Optics Workshop on Slow and Fast Light at the Kavli Institute of Theoretical Physics, University of California at Santa Barbara. From left to right: First Row: Raymond Chiao, Daniel Gauthier, Lijun Wang, Aephraim Steinberg, Justin Peatross; Second Row: Masao Kitano, Michael Fleischhauer, Scott Galsgow, Peter Milonni, Ulf Leonhardt, A. Zee, Herbert Winful, Guenter Nimtz; Third Row: Curtis Broadbent, Joseph Eberly, Kurt Oughstun

 As z ! 1, both the average and instantaneous centrovelocity of an ultrawide-

band pulse tends toward the rate at which the peak of the Brillouin precursor travels through the medium, independent of the applied pulse frequency. This is precisely the result obtained from the asymptotic theory because the Brillouin precursor dominates the propagated ultrawideband pulse wavefield for sufficiently large propagation distances (typically greater than one absorption depth at the applied pulse frequency).  The reshaping delay [see (15.196)] may become significant [i.e., of the same order of magnitude as the group delay given in (15.195)] for small propagation distances when the carrier frequency of the pulse lies within the region of anomalous dispersion. In general, the relative significance of the reshaping delay is highly dependent on the dispersive material properties.  A phase delay between the electric and magnetic field vectors occurs when the carrier frequency of the input pulse lies within the region of anomalous dispersion of the medium. This phase delay primarily effects the trailing edge of a rectangular envelope pulse for small propagation distances and rapidly diminishes with increasing propagation distance. Because of this phase delay effect, the centroid of the propagated Poynting vector for a rectangular envelope pulse rapidly shifts to earlier times with small increases in the propagation distance, resulting in centrovelocity values that are initially negative, then become negatively infinite, jump discontinuously to positive infinity and then finally become subluminal for sufficiently large z, as illustrated in Fig. 15.104.

672

15 Continuous Evolution of the Total Field

 The effect of the phase delay on the centrovelocity is dependent upon the initial

time duration of the pulse. For example, for an input signal of one oscillation at a carrier frequency within the anamolous dispersion regime, both the rectangular and guassian modulated pulses are found to have centrovelocity values which are subluminal for all propagation distances z > 0 considered. However, when the input pulse consists of ten oscillations at a carrier frequency within the anamolous dispersion regime, both the rectangular and guassian modulated pulses experience superluminal centrovelocity values.  The centrovelocity may not accurately describe the pulse velocity through a given Lorentz model dielectric with regard to energy transport. For example, when the Sommerfeld precursor amplitude is of the same order as the Brillouin precursor amplitude, the centrovelocity will fall between the two precursors at a point where a negligible amount of pulse energy is located. It may then be concluded that superluminal pulse propagation is just an illusion. Neither electromagnetic energy nor information encoded in the electromagnetic field can travel faster than the speed of light in vacuum, all in keeping with Einstein’s [97] special theory of relativity.

References ¨ 1. L. Brillouin, “Uber die fortpflanzung des licht in disperdierenden medien,” Ann. Phys., vol. 44, pp. 204–240, 1914. 2. L. Brillouin, Wave Propagation and Group Velocity. New York: Academic, 1960. 3. K. E. Oughstun, Propagation of Optical Pulses in Dispersive Media. PhD thesis, The Institute of Optics, University of Rochester, 1978. 4. K. E. Oughstun and G. C. Sherman, “Optical pulse propagation in temporally dispersive Lorentz media,” J. Opt. Soc. Am., vol. 65, no. 10, p. 1224A, 1975. 5. K. E. Oughstun and G. C. Sherman, “Comparison of the signal velocity of a pulse with the energy velocity of a time-harmonic field in Lorentz media,” in Proceedings of the URSI Symposium on Electromagnetic Wave Theory, (M¨unchen), pp. C1–C5, 1980. 6. K. E. Oughstun and G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B, vol. 5, no. 4, pp. 817–849, 1988. 7. K. E. Oughstun and G. C. Sherman, “Uniform asymptotic description of electromagnetic pulse propagation in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. A, vol. 6, no. 9, pp. 1394–1420, 1989. 8. K. E. Oughstun and G. C. Sherman, Pulse Propagation in Causal Dielectrics. Berlin: SpringerVerlag, 1994. 9. M. Born and E. Wolf, Principles of Optics. Cambridge: Cambridge University Press, seventh (expanded) ed., 1999. ¨ 10. H. Baerwald, “Uber die fortpflanzung von signalen in disperdierenden medien,” Ann. Phys., vol. 7, pp. 731–760, 1930. 11. S. Shen and K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B, vol. 6, pp. 948–963, 1989. 12. K. E. Oughstun and N. A. Cartwright, “Dispersive pulse dynamics and associated pulse velocity measures,” Pure Appl. Opt., vol. 4, no. 5, pp. S125–S134, 2002. 13. H. M. Nussenzveig, Causality and Dispersion Relations. New York: Academic, 1972. Chap. 1.

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42. R. E. Haskell and C. T. Case, “Transient signal propagation in lossless, isotropic plasmas,” IEEE Trans. Antennas Prop., vol. 15, pp. 458–464, 1967. 43. L. E. Vogler, “An exact solution for waveform distortion of arbitrary signals in ideal wave guides,” Radio Sci., vol. 5, pp. 1469–1474, 1970. 44. J. R. Wait, “Electromagnetic-pulse propagation in a simple dispersive medium,” Elect. Lett., vol. 7, pp. 285–286, 1971. 45. R. Barakat, “Ultrashort optical pulse propagation in a dispersive medium,” J. Opt. Soc. Am. B, vol. 3, no. 11, pp. 1602–1604, 1986. 46. P. D. Smith, Energy Dissipation of Pulsed Electromagnetic Fields in Causally Dispersive Dielectrics. PhD thesis, University of Vermont, 1995. Reprinted in UVM Research Report CSEE/95/07-02 (July 18, 1995). 47. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation of ultra-wideband plane wave pulses in a causal, dispersive dielectric,” in Ultra-Wideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 285–295, New York: Plenum, 1995. 48. P. D. Smith and K. E. Oughstun, “Electromagnetic energy dissipation and propagation of an ultrawideband plane wave pulse in a causally dispersive dielectric,” Radio Sci., vol. 33, no. 6, pp. 1489–1504, 1998. 49. P. D. Smith and K. E. Oughstun, “Ultrawideband electromagnetic pulse propagation in triplydistilled water,” in Ultra-Wideband, Short-Pulse Electromagnetics 4 (E. Heyman, B. Mandelbaum, and J. Shiloh, eds.), pp. 265–276, New York: Plenum, 1999. 50. H. T. Banks, M. W. Buksas, and T. Lin, Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiers in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, 2000. 51. J. A. Fuller and J. R. Wait, “A pulsed dipole in the earth,” in Transient Electromagnetic Fields (L. B. Felsen, ed.), pp. 237–269, New York: Springer-Verlag, 1976. 52. R. W. P. King and T. T. Wu, “The propagation of a radar pulse in sea water,” J. Appl. Phys., vol. 73, no. 4, pp. 1581–1590, 1993. 53. N. A. Cartwright and K. E. Oughstun, “Ultrawideband pulse penetration in a Debye medium with static conductivity,” in Fourth IASTED International Conference on Antennas, Radar, and Propagation, 2007. 54. K. E. Oughstun, “Noninstantaneous, finite rise-time effects on the precursor field formation in linear dispersive pulse propagation,” J. Opt. Soc. Am. A, vol. 12, pp. 1715–1729, 1995. 55. M. Tanaka, M. Fujiwara, and H. Ikegami, “Propagation of a Gaussian wave packet in an absorbing medium,” Phys. Rev. A, vol. 34, pp. 4851–4858, 1986. 56. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of ultrashort Gaussian pulse propagation in a causal, dispersive dielectric,” Phys. Rev. E, vol. 47, no. 5, pp. 3645–3669, 1993. 57. K. E. Oughstun and C. M. Balictsis, “Gaussian pulse propagation in a dispersive, absorbing dielectric,” Phys. Rev. Lett., vol. 77, no. 11, pp. 2210–2213, 1996. 58. C. M. Balictsis and K. E. Oughstun, “Generalized asymptotic description of the propagated field dynamics in Gaussian pulse propagation in a linear, causally dispersive medium,” Phys. Rev. E, vol. 55, no. 2, pp. 1910–1921, 1997. 59. K. E. Oughstun and H. Xiao, “Failure of the quasimonochromatic approximation for ultrashort pulse propagation in a dispersive, attenuative medium,” Phys. Rev. Lett., vol. 78, no. 4, pp. 642– 645, 1997. 60. H. Xiao and K. E. Oughstun, “Failure of the group velocity description for ultrawideband pulse propagation in a double resonance Lorentz model dielectric,” J. Opt. Soc. Am. B, vol. 16, no. 10, pp. 1773–1785, 1999. 61. S. P. Sira, A. Papandreou-Suppappola, and D. Morrell, “Dynamic configuration of time-varying waveforms for agile sensing and tracking in clutter,” IEEE Trans. Signal Proc., vol. 55, no. 7, pp. 3207–3217, 2007. 62. V. Mitlin, Performance Optimization of Digital Communications Systems. Boca-Raton: Auerbach, 2006. Sect. 4.9. 63. C. G. B. Garrett and D. E. McCumber, “Propagation of a Gaussian light pulse through an anomalous dispersion medium,” Phys. Rev. A, vol. 1, pp. 305–313, 1970.

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64. M. D. Crisp, “Concept of group velocity in resonant pulse propagation,” Phys. Rev. A, vol. 4, no. 5, pp. 2104–2108, 1971. 65. G. C. Sherman and K. E. Oughstun, “Description of pulse dynamics in Lorentz media in terms of the energy velocity and attenuation of time-harmonic waves,” Phys. Rev. Lett., vol. 47, pp. 1451–1454, 1981. 66. S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Phys. Rev. Lett., vol. 48, pp. 738–741, 1982. 67. G. C. Sherman and K. E. Oughstun, “Energy velocity description of pulse propagation in absorbing, dispersive dielectrics,” J. Opt. Soc. Am. B, vol. 12, pp. 229–247, 1995. 68. K. E. Oughstun and J. E. Laurens, “Asymptotic description of ultrashort electromagnetic pulse propagation in a linear, causally dispersive medium,” Radio Sci., vol. 26, no. 1, pp. 245–258, 1991. 69. C. M. Balictsis, Gaussian Pulse Propagation in a Causal, Dispersive Dielectric. PhD thesis, University of Vermont, 1993. Reprinted in UVM Research Report CSEE/93/12-06 (December 31, 1993). 70. C. M. Balictsis and K. E. Oughstun, “Uniform asymptotic description of Gaussian pulse propagation of arbitrary initial pulse width in a linear, causally dispersive medium,” in UltraWideband, Short-Pulse Electromagnetics 2 (L. Carin and L. B. Felsen, eds.), pp. 273–283, New York: Plenum, 1994. 71. J. Peatross, S. A. Glasgow, and M. Ware, “Average energy flow of optical pulses in dispersive media,” Phys. Rev. Lett., vol. 84, no. 11, pp. 2370–2373, 2000. 72. M. Ware, S. A. Glasgow, and J. Peatross, “Role of group velocity in tracking field energy in linear dielectrics,” Opt. Exp., vol. 9, no. 10, pp. 506–518, 2001. 73. N. A. Cartwright and K. E. Oughstun, “Pulse centroid velocity of the Poynting vector,” J. Opt. Soc. Am. A, vol. 21, no. 3, pp. 439–450, 2004. 74. P. Pleshko, Transients in Guiding Structures. PhD thesis, New York University, 1969. 75. P. Pleshko and I. Pal´ocz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett., vol. 22, pp. 1201–1204, 1969. 76. J. B. Gunn, “Bouncing ball pulse generator,” Electron. Lett., vol. 2, no. 5, pp. 172–173, 1966. 77. E. D. Torre, Magnetic Hysteresis. New York: IEEE Press, 1999. 78. M. E. Brodwin and D. A. Miller, “Propagation of the quasi-TEM mode in ferrite-filled coaxial line,” IEEE Trans. Microwave Theory Tech., vol. 12, no. 9, pp. 496–503, 1964. 79. H. Suhl and L. R. Walker, “Topics in guided wave propagation through gyromagnetic media,” Bell Syst. Tech. J., vol. 33, no. 9, pp. 1133–1194, 1954. 80. D. D. Stancil, “Magnetostatic wave precursors in thin ferrite films,” J. Appl. Phys., vol. 53, no. 3, p. 2658, 1982. 81. D. D. Stancil, Theory of Magnetostatic Waves. New York: Springer, 1993. 82. O. Avenel, M. Rouff, E. Varoquaux, and G. A. Williams, “Resonant pulse propagation of sound in superfluid 3 He–B,” Phys. Rev. Lett., vol. 50, no. 20, pp. 1591–1594, 1983. 83. E. Varoquaux, G. A. Williams, and O. Avenel, “Pulse propagation in a resonant medium: Application to sound waves in superfluid 3 He  B,” Phys. Rev. B, vol. 34, no. 11, pp. 7617–7640, 1986. ´ Falcon, C. Laroche, and S. Fauve, “Observation of Sommerfeld precursors on a fluid sur84. E. face,” Phys. Rev. Lett., vol. 91, no. 6, pp. 064502–1–064502–4, 2003. 85. J. Aaviksoo, J. Lippmaa, and J. Kuhl, “Observability of optical precursors,” J. Opt. Soc. Am. B, vol. 5, no. 8, pp. 1631–1635, 1988. 86. J. Aaviksoo, J. Kuhl, and K. Ploog, “Observation of optical precursors at pulse propagation in GaAs,” Phys. Rev. A, vol. 44, no. 9, pp. 5353–5356, 1991. 87. J. L. Birman and M. J. Frankel, “Predicted new electromagnetic precursors and altered signal velocity in dispersive media,” Opt. Comm., vol. 13, no. 3, pp. 303–306, 1975. 88. M. J. Frankel and J. L. Birman, “Transient optical response of a spatially dispersive medium,” Phys. Rev. A, vol. 15, no. 5, pp. 2000–2008, 1977. ¨ 89. S.-H. Choi and U. Osterberg, “Observation of optical precursors in water,” Phys. Rev. Lett., vol. 92, no. 19, pp. 1939031–1939033, 2004.

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90. R. R. Alfano, J. L. Birman, X. Ni, M. Alrubaiee, and B. B. Das, “Comment on ‘Observation of optical precursors in water’,” Phys. Rev. Lett., vol. 94, no. 23, p. 239401, 2005. 91. R. Albanese, J. Penn, and R. Medina, “Ultrashort pulse response in nonlinear dispersive media,” in Ultra-Wideband, Short-Pulse Electromagnetics (H. L. Bertoni, L. B. Felsen, and L. Carin, eds.), pp. 259–265, New York: Plenum, 1992. 92. Y. Okawachi, A. D. Slepkov, I. H. Agha, D. F. Geraghty, and A. L. Gaeta, “Absorption of ultrashort optical pulses in water,” J. Opt. Soc. Am. A, vol. 24, no. 10, pp. 3343–3347, 2007. 93. H. Jeong, A. M. C. Dawes, and D. J. Gauthier, “Direct observation of optical precursors in a region of anomalous dispersion,” Phys. Rev. Lett., vol. 96, no. 14, p. 143901, 2006. 94. Y. Park, M. H. Asghari, T. J. Ahn, and J. Aza˜na, “Transform-limited picosecond pulse shaping based on temporal coherence synthesization,” Opt. Express, vol. 15, no. 15, pp. 9584–9599, 2007. 95. R. Landauer, “Light faster than light?,” Nature, vol. 365, pp. 692–693, 1993. 96. G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A, vol. 223, pp. 327–331, 1996. 97. A. Einstein, “Zur elektrodynamik bewegter k¨orper,” Ann. Phys., vol. 17, pp. 891–921, 1905.

Problems 15.1. Derive the approximate expressions given in (15.29)–(15.33) for the main signal, anterior pre-signal, and posterior pre-signal space–time points c .!c /, c1 .!c /, and c2 .!c /, respectively. 15.2. Prove that the energy velocity vE .!/ reduces to the group velocity vg .!/ in the limit as ı ! 0 in a single resonance Lorentz model dielectric. 15.3. Derive the approximate expressions given in (15.67)–(15.69) for the energy velocity in the below resonance, intra-absorption band, and high-frequency domains. 15.4. Prove that the dispersion model for the effective dielectric permittivity of a microstrip transmission line given in (15.94) is causal. 15.5. (a) Derive the approximate expression for .!n ;  / given in (15.137) for n D 0; 1; 2; 3; : : : and compare the real part with that at the distant saddle point SPC d . (b) Derive the approximate expression for .!n ;  / given in (15.142) for n D 0; 1; 2; 3; : : : and compare the real part with that at the near saddle point SPC n for  > 1 . 15.6. Obtain the uniform asymptotic descriptions of the generalized Sommerfeld Ags .z; t / and Brillouin Agb .z; t / precursor fields for the gaussian envelope pulse wavefield Ag .z; t / described by (15.159). 15.7. Prove (15.178), showing that the relative maxima of the real part m .! 0 / of the modified complex phase function defined in (15.166) for the gaussian envelope pulse occur at the real ! 0 -axis crossing points !rcmj , j D 1; 2; 5 that are defined in (15.168) for the saddle points SPmj of ˚m .!;  0 /.

Problems

677

15.8. Use the method of stationary phase to derive the asymptotic expression for the Sommerfeld precursor in an air-filled rectangular waveguide with dispersion relation given by (15.216). Compare this result with the limiting behavior obtained from the uniform asymptotic approximation of the Sommerfeld precursor in a single resonance Lorentz model dielectric given in (13.35) as ı ! 0. 15.9. Derive the approximate expression for the group velocity of surface waves given in (15.223), valid when kh  1.

Chapter 16

Physical Interpretations of Dispersive Pulse Dynamics

The causally interrelated effects of phase dispersion and absorption on the evolution of an electromagnetic pulse as it propagates through a homogeneous linear dielectric, particularly when the pulse is ultrawideband, developed originally by Sommerfeld [1] and Brillouin [2–4] in 1914 in support of Einstein’s 1905 special theory of relativity [5], Brillouin’s signal velocity description partially corrected by Baerwald [6] in 1930, and the theory finally completed in the 1970–1980’s by Oughstun [7] and Sherman [8–12] in a series of papers that forms the basis of the modern asymptotic theory have been described in detail in Chaps. 12–15 of this volume. The results show that after the pulse has propagated sufficiently far in the medium, its spatiotemporal dynamics settle into a relatively simple regime, known as the mature dispersion regime, for the remainder of the propagation. In this regime, the wavefield becomes locally quasimonochromatic with fixed local frequency and wavenumber in small regions of space–time which move with their own characteristic constant velocity. The theory provides accurate but approximate analytic expressions for the local wave properties at any given space–time point in the mature dispersion regime. The expressions are complicated, however, as is their derivation from a well-defined asymptotic theory (presented in Chap. 10), and neither do the results nor their derivations provide complete insight into the physical reasons for the wavefield having the particular local space–time properties it does have in the various subregions of space moving with specific velocities. The mature dispersion regime is well known in the theory of propagation of rather general linear waves in homogeneous dispersive media in which there is no absorption or gain. It is exhibited by all waves whose monochromatic spectral components are described by the Helmholtz equation with real propagation factor; examples include electromagnetic, acoustic, elastic, and gravity waves in lossless, gainless linear systems. Furthermore, a physical explanation is available for the local properties of all of these waves that is based on the concept of the group velocity of time-harmonic waves [13–15]. When either (frequency-dependent) absorption or gain is present in the medium, however, the group velocity description breaks down. As stated by L. B. Felsen in his 1976 review paper [16]: The concept of group velocity vg D .dk=d!/1 for describing the energy propagation characteristics of a wave packet with small frequency spread becomes obscured in a lossy medium since vg is now complex. Furthermore, different propagation speeds may K.E. Oughstun, Electromagnetic and Optical Pulse Propagation 2: Temporal Pulse Dynamics in Dispersive, Attenuative Media, Springer Series in Optical Sciences 144, c Springer Science+Business Media, LLC 2009 DOI 10.1007/978-1-4419-0149-1 16, 

679

680

16 Physical Interpretations of Dispersive Pulse Dynamics

be associated with different features of the pulse (for example, the spatial and temporal maxima). Although attempts have been made to extend the notion of group velocity to wave packets in dissipative media by examining such quantities as Œ 0. The wavefield is taken to be zero for all t < 0 and therefore can expressed in a Fourier–Laplace representation as Z

iaC1

A.z; t / D

Q !/e i!t d!; A.z;

(16.1)

ia1

where a is a positive constant for the Bromwich contour extending along the straight Q !/ satisfies the Helmholtz from ia  1 to ia C 1. The spectral wave function A.z; equation   Q !/ D 0 r 2 C kQ 2 .!/ A.z; (16.2) throughout the half-space z > 0. The angular frequency dispersion of the comQ plex wavenumber k.!/ .!=c/n.!/ is specified by the frequency dispersion of  1=2 the complex index of refraction n.!/ ..!/=0 /.c .!/=0 of the medium, where c .!/ D .!/Ci k4k .!/=!. The analysis presented here considers a simple dielectric, in which case .!/ D 0 ,  .!/ D 0, and c .!/ D .!/, which is taken here to be described by the single resonance Lorentz model, so that  n.!/ D 1 

b2 2 !  !02 C 2i ı!

1=2 ;

(16.3)

where !0 is the undamped angular resonance frequency, b the plasma frequency, and ı  0 the phenomenological damping constant of the medium. As in the classical asymptotic theory, it is assumed here that the pulsed, plane wavefield satisfies the boundary value A.0; t / D f .t /;

(16.4)

where f .t / is a real-valued function that identically vanishes for all negative time [i.e., f .t / D 0 for all t < 0]. Because of its central importance in linear system theory, the analysis focuses on the impulse response of the dispersive medium, in which case f .t / D A0 fı .t / (see Sect. 11.2.1) with fı .t / D ı.t /, where A0 is a constant and ı.t / the Dirac delta function, the impulse response then being given

682

16 Physical Interpretations of Dispersive Pulse Dynamics

by A.t /=A0 . The exact integral solution to this boundary value problem can be expressed in the form [cf. (12.1)] A.z; t / D

1 2

Z

iaC1

fQ.!/e .z=c/.!;/ d!;

(16.5)

ia1

where a is a real constant greater than the abscissa of absolute convergence [see (C.12) of Appendix C in Vol. 1] for the function f .t /, fQ.!/ D

Z

1

f .t /e i!t dt;

(16.6)

  .!; / D i! n.!/   ;

(16.7)

1

and where with

ct (16.8) z being a dimensionless space–time parameter that, for any fixed value of  , travels with the wavefield at the fixed velocity z=t D c=. Sommerfeld’s relativistic causality theorem [1] (see theorem 6, Sect. 13.1) proves that it directly follows from the exact integral solution in (16.5) that the propagated wavefield A.z; t / identically vanishes for all superluminal space–time points  < 1, that is D

A.z; t / D 0;

8 t < z=c;

(16.9)

in keeping with Einstein’s special theory of relativity [5]. The physical description presented here is developed for initial pulse functions f .t /DA.0; t / that possess temporal Fourier spectra fQ.!/ that are entire functions of the complex angular frequency variable !. In that case the propagated pulse wavefield may be expressed in the form A.z; t / D As .z; t / C Ab .z; t /;

8 t  z=c:

(16.10)

The asymptotic behavior as z ! 1 of these two field components is given by [cf. (13.15)] ( ) 1=2 z 2c .! ;/ s Q As .z; t /  2< (16.11) f .!s /e c z 00 .!s / for  > 1, where  0 .!s / D 0 and  00 .!s / ¤ 0, and by [cf. 13.73) and (13.94), respectively] ( ) 1=2 z 2c Ab .z; t /  < fQ.!b /e c .!b ;/ ; 1 <  < 1 ; (16.12) z 00 .!b / ( ) 1=2 z 2c .! ;/ b Q Ab .z; t /  2< (16.13) ;  > 1 ; f .!b /e c z 00 .!b /

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

683

where  0 .!b / D 0 and  00 .!b / ¤ 0. Here !s !SP C . / denotes the first-order d distant saddle point in the right half of the complex !-plane [see (12.201)] and !b !SPC . / denotes the first-order near saddle point along the imaginary axis d for 1 <  < 1 [see (12.228)] and then in the right half of the complex !-plane for  > 1 [see (12.239)]. The asymptotic expressions given in (16.11)–(16.13) are referred to as nonuniform asymptotic results because they break down at certain critical space–time points. In particular, the right-hand sides of (16.12) and (16.13) become infinite at  D 1 and give discontinuous asymptotic behaviors on opposite sides of that space– time point because  00 .!b ; 1 / D 0. Although uniform asymptotic results have been derived in Sects. 13.2.2 and 13.3.2, they are not necessary in the initial analysis as the nonuniform expressions are much simpler to work with. The uniform results are employed in the final analysis, however, in order to provide an energy velocity description that is valid for all   1 .

16.1.1 Approximations Having a Precise Physical Interpretation Approximations of (16.11)–(16.13) are now obtained which have a precise physical interpretation and result in a simple physical model of dispersive pulse dynamics in the mature dispersion regime. These approximations are valid provided that the phenomenological damping constant ı is much smaller than both the angular resonance frequency !0 and plasma frequency b of the Lorentz model dielectric, viz., ı  !0

&

ı  b:

(16.14)

This requires that the medium not be too highly absorbing. This requirement is not overly restrictive as it is satisfied by Brillouin’s choice of the medium parameters [see (12.58)] for which ı=!0 D 0:07 and ı=b ' 0:0626, where this medium is so absorbing that it would be considered to be opaque at nearly all nonzero, finite frequency values. To obtain the desired simplifications, Sherman and Oughstun [21,22,25] replaced the saddle points appearing in the asymptotic expressions by other frequencies which yield approximately the same results but which have clearer physical interpretations. In particular, the saddle points !s !SPC . / and !b !SPC ./ appearing d d in (16.11) and (16.13) are replaced by specific real angular frequencies leading to quasi-time-harmonic (quasimonochromatic) waves with local frequency, phase, and amplitude which are easily understood in physical terms. Similarly, the saddle point !b !SP C . / appearing in (16.12) is replaced with a specific purely imaginary fred quency leading to a nonoscillatory field with local amplitude and growth rate which is also easily understood in physical terms. The analysis presented here is based entirely upon the earlier published analysis of Sherman and Oughstun [21, 22, 25].

684

16.1.1.1

16 Physical Interpretations of Dispersive Pulse Dynamics

The Quasimonochromatic Contribution

To identify the real frequencies of interest, attention is focused on the attenuation of the wavefield with increasing propagation distance z > 0. It is important that the resultant approximation have the correct attenuation because the theory is centered on the properties of an exponentially decaying wavefield after it has propagated a large distance (relative to some characteristic absorption depth) in the dispersive medium. Hence, it is desired to determine those time-harmonic waves (with real frequencies) that are attenuated in the dispersive medium at the same rate as the wavefield components given in (16.11) and (16.13). To that end, first define the notation2 that gr and gi represent, respectively, the real, and imaginary parts of the arbitrary complex quantity g. Then, for any fixed space–time point   1, the attenuation with increasing propagation distance z > 0 of a wave of the form exp f.z=c/.!; /g for complex ! is determined by r .!; /. For any given space–time value   1, define !Ej as the real frequency value nearest the saddle point !j that satisfies r .!Ej .// D r .!j ;  /

(16.15)

with j D s; b. A time-harmonic plane wave with real angular frequency !Ej then has the same attenuation as the pulsed wave described by (16.11) and (16.13) in the mature dispersion regime. The locations of these real angular frequency values !Es D !Es . / and !Eb D !Eb . / in the complex !-plane are indicated in Fig. 16.1 for some fixed value  > 1 as the corresponding intersections with the real ! 0 -axis of the isotimic contours of r .!; / D  r .!j ;  / that pass through the saddle points !j , j D s; b. ''

r

Eb

'

branch cut

r

Es

b

branch cut

'

b

'

s

r

r

s

Fig. 16.1 Location of the real angular frequencies !Eb D !Eb . / and !Es D !Es . / relative to the locations of the near and distant saddle points !b . / and !s . / in the complex !-plane for a fixed space–time value  > 1 . The dashed curves describe the isotimic contours of constant r .!;  / D 1 is bounded away from 1 in (16.13). Hence, the asymptotic expressions given in (16.11) and (16.13) may be, respectively, approximated as ( As .z; t /  2
1, and ( Ab .z; t /  2
1 bounded away from 1 . Greater care must be taken in approximating the exponential function in the above expressions because it is a more rapidly varying function of !. The attenuation has already been expressed in terms of !Ej in (16.15). That result can be expressed in terms of the complex index of refraction n.!/ D nr .!/ C ini .!/ by noting that r .! 0 / D ! 0 ni .! 0 / for real ! 0 . Equation (16.15) can therefore be written as (16.21) r .!j / D !Ej ni .!Ej / D c kQi .!Ej /: The oscillatory portion of the exponential phase is determined by the expression h i z z z i i .!/ D i ! 0 t C ! 0 nr .!/  ! 00 ni .!/ c c c

(16.22)

evaluated at the relevant saddle point ! D !j . If !j is replaced by !Ej in the index of refraction terms and if !j0 is replaced by !Ej elsewhere in this expression, it becomes h i z z (16.23) i i .!j / ' i !Ej t C zkQr .!Ej /  !j00 ni .!Ej / : c c Combination of the expressions given in (16.21) and (16.22), the quantity appearing in the exponential of the integrands of (16.19) and (16.20) can then be approximated by h i z Q Ej /  i z ! 00 ni .!Ej / : (16.24) i .!j / ' i !Ej t C zk.! j c c

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

687

In the case when j D b, (16.24) applies when  > 1 and is accurate only when  is not too close to 1 . The last term appearing in (16.24) includes the location of the saddle point !j00 . That term is not very important, however, because it is negligible except when !Ej is in the absorption band and it contributes only a small phase shift to the field even then. Hence, that term is ignored in this physical model of dispersive pulse dynamics with the understanding that the phase of the wavefield that is obtained using this model may be slightly shifted when !Ej is in the absorption band. In the same spirit, the approximations given in (16.20) and (16.24) with j D b are applied for all  > 1 with the understanding that they become inaccurate as  approaches the critical value 1 from above. The small space–time interval where this problem exists decreases as ı decreases relative to !0 . The numerical calculations of the dynamical wavefield using this physical model that are presented later in this section display the effects of these simplifications in the highy absorbing case. 16.1.1.2

The Nonoscillatory Contribution

Attention is now turned to the contribution to the propagated wavefield that is given in (16.12). This contribution is nonoscillatory and is important only for space–time points .z; t / in the region about the space–time point given by z D vE .0/: t

(16.25)

The group velocity or phase velocity for time-harmonic waves with zero frequency could be applied in (16.25) equally well because all three velocities are equal for zero frequency. Hence, this field is essentially quasistatic. Consider then the nonoscillatory electromagnetic fields of the form h i QQ !/z w.z; t; !/ Q D exp !t Q  k. Q

(16.26)

where the growth rate !Q is a real-valued constant. This wavefield is a solution to QQ !/ Maxwell’s equations in a single resonance Lorentz medium if k. Q is given by !Q QQ !/ Q !/ k. Q D ik.i Q D c

s 1C

!Q 2

b2 ; C !02 C 2ı !Q

(16.27)

which is also real-valued. Because !b is purely imaginary over the initial space–time domain 1    1 , the exponential term appearing in the integrand of (16.12) is a wavefield of the form given in (16.26) and (16.27) except that the growth rate is a function of position and time. Equation (16.16) for the near saddle point !b can be written in the form !b D i !Q b ;

(16.28)

688

16 Physical Interpretations of Dispersive Pulse Dynamics

where !Q b is the real-valued solution to the equation vQ G .!Q b / D

z t

with vQ G .!/ Q defined as Q vQ G .!/

QQ !/ dk. Q d!Q

(16.29) !1 ;

(16.30)

which can be taken as the group velocity of the nonoscillatory waves given in (16.26). This identification does not provide much physical insight, however, because the group velocity of a nonoscillatory wave is more a mathematical object rather than a physical one. To connect it with a more physical quantity, consideration is now given to the velocity of energy transport in a nonoscillatory wave. Take the velocity of energy flow in fields of the form given in (16.26) to be given by (16.17) with the change that the Poynting vector and energy density are not to be time-averaged because the field is nonoscillatory. With this definition, it is shown in [22] that an electromagnetic field of the form given in (16.26) and (16.27) has Q given by energy velocity vQ E .!/ vQ E .!/ Q D

Q cn.i!/Q Q 2 .!/ ; 2 2 n .i!/Q Q .!/ Q  b 2 ı !Q

(16.31)

where Q Q.!/ Q D !Q 2 C !02 C 2ı !; s b2 n.i!/ Q D 1C : Q.!/ Q

(16.32) (16.33)

It is also shown in [22] that the group velocity of the nonoscillatory waves is given by Q cn.i !/Q Q 2 .!/ Q D 2 : (16.34) vQ G .!/ 2 2 n .i!/Q Q .!/ Q  b ı !Q  b 2 !Q 2 Comparison of (16.31) and (16.34) shows that the two velocities differ only by the additional term b 2 !Q 2 appearing in the denominator of (16.34). This term is negligible for all growth rates that satisfy the inequality !Q  !0 I

(16.35)

notice that this is a sufficient condition but that it is far from necessary. Because the attenuation with propagation distance z of the nonoscillatory waves increases with increasing growth rate, it is clear that for z sufficiently large, the nonoscillatory waves which do not satisfy the inequality given in (16.35) will be negligible compared to those that do. In particular, it is shown in [22] that if z satisfies the inequality

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

z>

4Kc !1 .!0 C ı/2 ; r2 ı!0 b 2

689

(16.36)

then w.z; t; !/=w.z; Q t; 0/ < e K for !Q  r!0 , where K and r are arbitrary positive constants which can be chosen to give as good an approximation as desired when neglecting the nonoscillatory waves that do not satisfy the inequality given in Q by vQ E .!/. Q For Brillouin’s choice of the (16.35) well enough to approximate vQ G .!/ medium parameters [see (12.58)] and with K D 2 and r D 0:1, the inequality given in (16.36) gives z > 0:012 cm. It is shown in the next subsection through a numerical example that this approximation is useful even for smaller values of the propagation distance z > 0. Hence, for sufficiently large values of the propagation distance z > 0, a good approximation of the near saddle point location over the initial space–time domain 1    1 can be obtained by taking !Q b as the solution to the equation vQ .!/ Q D

z t

(16.37)

that is closest to the saddle point. One more approximation is useful for the formulation of the physical model. It follows from (12.225) that the critical space–time point 1 can be approximated by the space–time value 0 defined by 0 D

c c D D n.0/; vE .0/ vQ E .0/

(16.38)

which is simply the index of refraction for a static field. The validity of the approximation decreases with increasing ı but is still very good even for the parameter values used by Brillouin [2, 4] (in which case 0 D 1:500 and 1 1:503). This result implies that the second precursor field Ab .z; t / changes from the nonoscillatory form with zero growth rate to the time-harmonic form with zero oscillation frequency at the observation point that is traveling with velocity vE .0/ D vQ E .0/.

16.1.2 Physical Model of Dispersive Pulse Dynamics The physical model of dispersive pulse dynamics is now presented based on the previous results. The simpler nonuniform model is first developed followed by the more complicated, but more accurate, uniform model [22]. 16.1.2.1

The Nonuniform Physical Model

As the propagation distance z tends to infinity with fixed  D ct=z, the wavefield Q t / which, for ı A.z; t / can be expressed as the real part of a complex wavefield A.z; much smaller than both !0 and b, can be approximated as Q t / AQTH .z; t / C AQQS .z; t /; A.z;

(16.39)

690

16 Physical Interpretations of Dispersive Pulse Dynamics

where 1=2 2c Q fQ.!Ej /e i .k.!Ej /z!Ej t / ; z 00 .!Ej / j Ds;b 1=2    QQ 2c !Q E tk. !Q E /z Q Q : AQS .z; t / D f .i !Q E /e z 00 .i !Q E /

AQTH .z; t / D 2

X 

(16.40)

(16.41)

Here, the angular frequency values !Ej are defined to be the nonnegative real-valued solutions of the equation z c (16.42) vE .!Ej / D ;  t and !Q E is defined to be the positive real-valued solution to vQ E .!E /

z c D :  t

(16.43)

The field quantity AQTH .z; t / given in (16.40) is the time-harmonic component discussed in the preceding subsection. It is shown in the following that the sum in (16.40) includes only one term, the Sommerfeld precursor As .z; t /, over the initial space–time domain 1 <  < 0 , and includes two terms, the Sommerfeld precursor As .z; t / and the Brillouin precursor Ab .z; t /, for   0 . The field quantity AQQS .z; t / given in (16.40) is the nonoscillatory component discussed in the preceding subsection. These results constitute a physical model [21,22] of dispersive pulse propagation because they can be used to describe the local dynamics of the pulse in physical terms. They are similar to the mathematical results that lead to the group velocity description that is valid for lossless, gainless systems but are different in three respects: 1. The nonoscillatory contribution is included in addition to the time-harmonic contribution, 2. The energy velocity is used to determine the pulse dynamics in place of the group velocity, 3. The pulse dynamics are strongly affected by the relative attenuation of the various time-harmomic and nonoscillatory contributions. It follows from (16.39)–(16.43) that the principal quantities that determine the local dynamics of the dispersive pulse evolution are the energy velocity vE and attenuation ˛ of time-harmonic waves as functions of frequency and the energy velocity vQ E and attenuation coefficient ˛Q of nonoscillatory waves as a function of the growth rate !. Q Graphs of these functions for a Lorentz model medium with Brillouin’s parameter values [see (12.58)] are given, respectively, in Figs. 16.2–16.5. The energy velocities for the oscillatory and nonoscillatory waves were computed using (16.18) and (16.31), respectively. The attenuation coefficients of the waves are the rates of exponential decay of the wave amplitudes as the propagation distance z > 0 increases with constant  . For time-harmonic waves with real frequencies,

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

691

1.0

0.8

vE /c

0.6

0.4

0.2

0 0

5

10

15

(X1016 r/s)

Fig. 16.2 Angular frequency dispersion of the normalized energy velocity vE .!/=c of a monochromatic electromagnetic wave with angular frequency ! in a single resonance Lorentz model medium

(X108/m)

3.0

2.0

1.0

0 0

5

10

15

(X1016 r/s)

Fig. 16.3 Angular frequency dispersion of the attenuation coefficient ˛.!/ of a monochromatic electromagnetic wave with angular frequency ! in a single resonance Lorentz model medium

692

16 Physical Interpretations of Dispersive Pulse Dynamics 0.70

0.69

v~E /c

0.68

0.67

0.66

0.65 0

0.2

0.4 0.6 ~ (X1016 r/s)

0.8

1.0

Fig. 16.4 Normalized energy velocity vQE .!/=c Q of the nonoscillatory wave components in a single resonance Lorentz model medium

5.0

~(X105/m)

4.0

3.0

2.0

1.0

0 0

0.2

0.4

0.6 ~ (X1016 r/s)

0.8

1.0

Fig. 16.5 Attenuation coefficient ˛. Q !/ Q of the nonoscillatory wave components in a single resonance Lorentz model medium

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

693

the attenuation coefficient ˛.!/ is given by the imaginary part of the complex Q wavenumber k.!/ D ˇ.!/Ci˛.!/ D .!=c/n.!/ with complex index of refraction n.!/ given here by (16.3). For the nonoscillatory waves the attenuation coefficient ˛. Q !/ Q is given by [22] QQ !Q /   !Q ˛. Q !Q E / k. E E c QQ !Q /  !Q E ; D k. E vQ E .!Q E /

(16.44) (16.45)

QQ !/ where k. Q is given by (16.31). Q is given by (16.27) and vQ E .!/ A qualitative description of all of the main features of dispersive pulse dynamics can be obtained through a careful consideration of these four figures (Figs. 16.2– 16.5). Notice first from Fig. 16.2 that there is only one solution to (16.42) over the initial space–time domain 1 <  < 0 , where 0 D 1:5 for Brilllouin’s choice of the medium parameters, so that the summation appearing In (16.40) for the timeharmonic contribution includes only one term. The angular frequency !E of this term is large for  near 1 and decreases monotonically with increasing  . From Fig. 16.3, it is seen that as the frequency decreases from a large value, the attenuation increases. This means that this high-frequency, quasi-time-harmonic term decreases in frequency and amplitude as  increases, in agreement with the asymptotic description of the Sommerfeld precursor (see Sect. 13.2). Continuing on with the physical description, it is noticed in Fig. 16.4 that there is one positive solution !Q E of (16.43) over the initial space–time domain 1 <  < 0 . The growth rate !Q E decreases with increasing , tending toward 0 as  approaches 0 from below. From Fig. 16.5, it is seen that the attenuation with propagation distnce z > 0 of the nonoscillatory contribution is large for large growth rate, but gradually decreases to 0 as !Q E approaches 0. Hence, the nonoscillatory contribuQ t / given in (16.39) is negligible in tion AQQS .z; t / to the complex wavefield A.z; comparison to the Sommerfeld precursor contribution for small  > 1, but gradually increases in amplitude until it dominates the Sommerfeld precursor field as  approaches 0 from below. This marks the arrival of the Brillouin precursor. Finally, notice from Fig. 16.4 that there is no positive solution to (16.43) for   0 . Hence, the nonoscillatory contribution no longer contributes to the complex waveQ t / when   0 . Notice that the nonoscillatory contribution with !Q E D 0 field A.z; at  D 0 has been disallowed by including only positive solutions to (16.43). This has been done so as to avoid the inclusion of the zero frequency solution twice, as it is included in the time-harmonic contribution [because all nonnegative solutions of (16.42) have been included]. Returning to Fig. 16.2, notice that for   0 , there are now two nonnegative solutions of (16.42). The first is a high-frequency solution which is the continuation of the Sommerfeld precursor. The second solution is a low-frequency solution with angular frequency which begins at zero for  D 0 and then increases with increasing  . Consideration of Fig. 16.3 shows that the attenuation of this wave is much less than that for the high-frequency solution. Hence, this low-frequency

694

16 Physical Interpretations of Dispersive Pulse Dynamics

wave contribution dominates the sommerfeld precursor as z increases. Consideration of Fig. 16.3 shows also that the attenuation of this wave increases with increasing !, causing the wave to decrease in amplitude with increasing  . Hence, this contribution has increasing frequency and decreasing amplitude with increasing  , in agreement with the asymptotic description of the Brillouin precursor (see Sect. 13.3). In addition to providing a description of the qualitative pulse behavior in physical terms, the physical model gives approximate analytical expressions which predict the propagated pulse dynamics quantitatively without requiring the evaluation of the saddle point locations in the complex !-plane. Of course, the model does require the solution of (16.42) and (16.43), which are transcendental equations, but these equations are simpler to deal with than the saddle point equations because they involve only real quantities. To investigate the accuracy of the nonuniform physical theory, these expressions have been evaluated numerically for the case of the delta-function pulse A.0; t / D ı.t / in a single resonance Lorentz model medium using Brillouin’s choice of the material parameters with a propagation distance of z D 1 )m. Equations (16.42) and (16.43) were then solved numerically using Mueller’s method [22]. The numerical results of the physical model are described by the solid curve in Fig. 16.6. For comparison, the nonuniform asymptotic result given by (16.10)–(16.13) for the

Ad (z,t)

1.0X1016

0

−1.0X1016 1.0

1.2

1.4

1.6

1.8

2.0

q = ct/z

Fig. 16.6 Nonuniform results for the propagated wavefield due to an input delta function pulse in a single resonance Lorentz model medium. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

695

same parameter values is described by the dotted curve in the figure. These latter numerical results used numerically determined saddle point locations. It is apparent from the graph in Fig. 16.6 that the accuracy of the physical model is very good. The main discrepancy between the physical model and asymptotic description is a minor shift in the phase of the Brillouin precursor that has been discussed in connection with the approximations made leading to (16.24). The scale in Fig. 16.6 was chosen so that the transition between the two precursors was clearly displayed. Because the field amplitudes are off this scale for small space–time values  near unity in that figure, the same numerically determined wavefield values are replotted in Fig. 16.7 for small  near the luminal space–time point  D 1 with an appropriate vertical scale. The results of the physical model are almost indistinguishable from the nonuniform asymptotic results. The large discontinuous peak that occurs in both the physical model and asymptotic results presented in Fig. 16.6 for space–time values  near 0 is a consequence of the nonuniform nature of the results as discussed in Sect. 16.1 following (16.10)– (16.13). This behavior is an artifact of the nonuniform asymptotic analysis which makes the results invalid in that space–time region. To obtain results that are valid there, it is necessary to employ the uniform asymptotic description of the propagated wavefield.

Ad (z,t)

5.0X10

17

0

−5.0X1017 1.0

1.01

1.02

1.03 q = ct/z

1.04

1.05

Fig. 16.7 Expanded view of the nonuniform results for the propagated wavefield due to an input delta function pulse in a single resonance Lorentz model medium for space–time values  near 1. The solid curve is the result of the physical model and the dashed curve is the result of the nonuniform asymptotic theory

696

16.1.2.2

16 Physical Interpretations of Dispersive Pulse Dynamics

The Uniform Physical Model

The asymptotic results (and the resulting physical model derived from them) that have been used so far are nonuniform in the vicinity of two critical space–time points: 1.  D 1, which corresponds to the luminal arrival of the Sommerfeld precursor field, and 2.  D 1 > 1, which occurs during the arrival of the Brillouin precursor. This means that in order for the results to provide useful approximations for large z > 0, the propagation distance must be taken larger and larger as  approaches one of these critical space–time values. Furthermore, the functional form of the results are different for space–time values  on opposite sides of a critical value. These difficulties may be removed from the physical model by using the appropriate uniform asymptotic approximations described in Chap. 10 (see Sects. 10.2 and 10.3) and applied in Sects. 13.2.2 and 13.3.2 of Chap. 13. The propagated wavefield is then expressed in terms of special functions (e.g., the Bessel and Airy functions) which are more complicated than the exponential functions occurring in the nonuniform expressions. The arguments of these functions involve the same saddle points as applied in the nonuniform analysis. As the value of  tends away from either one of the critical values, the uniform results tend asymptotically to the same expressions given in the nonuniform description. To employ the uniform asymptotic results of Chap. 13 in the physical model, the saddle points occurring in the uniform asymptotic expressions must be replaced with the approximations used in the nonuniform physical model, these being given by the real solutions of (16.42) and (16.43). Because the asymptotic expressions involve Bessel and Airy functions instead of quasi-time-harmonic waves and nonoscillatory exponentially growing waves, the physical interpretation of the resulting description is not as apparent as that described by the nonuniform description. Nevertheless, such functions frequently arise in the theory of wave propagation and can certainly be considered as representing physical waves. Moreover, in the uniform physical model, the arguments of these special functions are real-valued, involving real frequencies and growth rates which are clearly connected with the physics of both time-harmonic and nonoscillatory waves in the dispersive medium through (16.42) and (16.43). For the case of the delta function pulse A.0; t / D ı.t /, one really doesn’t have to take the trouble to make the physical model uniform in the vicinity of the luminal space–time point  D 1 because the nonuniform model yields reasonably accurate results for values of  quite close to 1. This is demonstrated in Fig. 16.8 which displays the results of a numerical evaluation3 of the exact integral representation of the propagated wavefield given in (16.5) with fQ.!/ D 1 for the same 3

The numerical algorithm used to evaluate the integral representation of the propagated pulse wavefield for this and subsequent examples in this section is described in [22]. As discussed in that reference, the algorithm begins to produce numerical artifacts in the results as  approaches 1

16.1 Energy Velocity Description of Dispersive Pulse Dynamics

697

17

Ad (z,t)

5.0X10

0

−5.0X1017 1.0

1.01

1.02

1.03

1.04

1.05

q = ct/z

Fig. 16.8 Propagated wavefield evolution due to an input delta function pulse A.0; t / D ı.t / for space–time values near the speed of light point at  D 1. The solid curve describes the result of a numerical integration of the exact integral representation, and the dotted curve describes the result given by the nonuniform physical model

parameter values used in Fig. 16.7. Comparison of Fig. 16.8 with the wavefield plot in Fig. 16.7 shows that the latter result has the same form but with a high-frequency ripple superimposed. It has been verified in [22] that this high-frequency ripple is an artifact of the numerical algorithm by showing that its frequency changed when the numerical value of the initial sum index k was changed in the implementation of this inverse Laplace transform algorithm [27], whereas the rest of the curve remained unchanged. To compare the results of the physical model with the numerical integration results, the results of the nonuniform physical model are presented in Fig. 16.8 by the dotted curve. That curve is barely visible following along the centerline of the rippled curve in Fig. 16.8. This then demonstrates that the nonuniform physical model gives valid results for these small space–time values when   start where start D 1:00055 marks the space–time point where the calculation began. For a smaller starting value, one would need to make the physical model uniform in the vicinity of the point  D 1. The results are not as critical for other initial pulse shapes whose spectra vanish as j!j ! 1. from above for the delta function pulse. This is a consequence of the fact that the integral itself is ill behaved at  D 1 (see Sect. 13.2.5).

698

16 Physical Interpretations of Dispersive Pulse Dynamics

The physical model is now modified in order to make it uniform in the vicinity of the critical space–time point  D 0 . The analysis begins with (16.10)–(16.13) with the exception that the expressions in (16.12) and (16.13) are now replaced by the uniform asymptotic expressions given in Sect. 13.3.2. The asymptotic approximation is made uniform in the vicinity of the critical space–time point  D 1 0 by replacing the nonuniform expressions given in (16.12) and (16.13) for Ab .z; t / with i z AQb .z; t / D e c ˛0 ./ 2

(  i c 1=3 i 2  h Q e 3 f .!C /hC ./ C fQ.! /h . / z   z 2=3  Ai j˛1 ./j c  2=3 h i c 4 C e i 3  fQ.!C /hC ./  fQ.! /h . / z )   z 2=3  1=2 0 ˛1 ./Ai j˛1 . /j ; (16.46) c

where Ab .z; t / D c of the electromagnetic beam wavefield. The angle of incidence u of the ku -component of the angular spectrum for the incident beam wavefield is given by u D 1  arcsin .ku =k/ so that 1 1 u D  arcsin .ku =k/ and ku D k sin . 1 /, where ku D 0 corresponds to u D 1 . One then has that d ku D k cos .1  u /d1 ! kd1 at u D 1 . With this result, (17.89) becomes

17.3 Reflection and Transmission Phenomena

749

Medium 2 n2 n1

>

Medium 1 n D

> 1w

1 w'

1

1

Fig. 17.19 Depiction of the Goos–H¨anchen shift for an electromagnetic beam wavefield incident at a supercritical angle of incidence on the interface S

ˇ ˇ 1 @ ˇˇ  @ ˇˇ DD D ; k @ ˇD1 2 @ ˇD1

(17.92)

where the final expression is appropriate for a monochromatic beam with wavelength  in medium 1. From (17.71), the Goos–H¨anchen shift for s-polarization (TE-mode) is found to be given by DE D

sin 1  ; q  sin2   n2 =n2 1

2

(17.93)

1

and from (17.77), the Goos–H¨anchen shift for p-polarization (TM-mode) is found to be given by DE ; (17.94) DH D 2 2 .1 C n1 =n2 / sin2 1  1 for 1 > c C . Notice that both of these expressions for the Goos–H¨anchen shift are singular at the critical angle 1 D c ; this critical value can be approached from above by making the angular spread of the incident beam wavefield  very small. The angular dependence of the Goos–H¨anchen shift for both s- and p-polarization is illustrated in Fig. 17.20 when n1 D 2:0 and n2 D 1:5. This lateral Goos–H¨anchen shift of the reflected beam upon total internal reflection is equivalent to perfect (i.e., geometrical optics) reflection from a hypothetical interface a distance dj into medium 2. From the simple geometry indicated in Fig. 17.19 it is seen that Dj ; (17.95) dj D 2 sin 1

750

17 Applications 2

q D/ λ 1

DE / λ DH / λ

0 Θc

60

70 Θ1 (degrees)

80

90

Fig. 17.20 Supercritical angular depenedence of the Goos–H¨anchen shift for both s-polarization (solid curve) and p-polarization (dashed curve), where q D DE =DH

for j D E; H . Hence, dH D dE =q and DH D DE =q, where  qD

 n21 n2 C 1 sin2 1  1 D 12 sin2 1  cos2 1 ; 2 n2 n2

(17.96)

for 1 2 .c ; =2/. The supercritical angular dependence of this factor is indicated by the dotted curve in Fig. 17.20. Notice that q D q.1 / has a well-defined limiting value at the critical angle given by q.c / D n22 =n21 . A more detailed analysis of the Goos–H¨anchen shift at and near the critical angle has been given by Chan and Tamir [21]. Generalizations of these results to dispersive absorptive media remains to be fully addressed. The case when the reflecting medium is absorptive has been given by Wild and Giles [29] where it was shown that, under certain conditions, the Goos– H¨anchen shift can become negative. Because this effect is an essential part of the guided mode condition in dielectric optical waveguides [30], particularly in integrated optics [31], its rigorous solution when the frequency dispersion of both the core and cladding materials are properly described by causal models is of fundamental importance, particularly at terabit per second (Tbit=s) transmission rates [32]. For example, a 1 Tbit=s data rate requires that the rectangular envelope pulse bit duration Tb be on the order of Tb  5  1013 s D 500 fs so that a 100 Tbit=s data rate requires that Tb  5 fs where precursor effects become critical.

17.4 Optimal Pulse Penetration through Dispersive Bodies

751

17.3.3 Reflection and Transmission at a Dispersive Layer: The Question of Superluminal Tunneling The problem of the reflection and transmission of an ultrawideband electromagnetic pulse from and through a dispersive absorptive material layer is abundant in physical wave phenomena with a wide variety of practical applications. For example, although the design methodology for antireflection coatings for continuous wave applications is well established, its extension to ultrawideband pulses is not as clearly defined because of the nonzero timing delay between the reflected pulse sequence. A quarter-wave dielectric layer will not extinguish the reflected wavefield when the incident wavefield is a single cycle pulse, and it may even enhance the reflected pulse under certain conditions. At a more fundamental level, the question of superluminal tunneling through a dispersive dielectric layer is of considerable interest [33]. Because the Fresnel transmission coefficients for s- and p-polarization are causal when the material dispersion models are causal, then Sommerfeld’s relativistic causality theorem (Theorem 6 in Sect. 13.1) applies to the transmtted pulse which then identically vanishes for all superluminal transit times when the input pulse vanishes for all t < 0. Of course, for gaussian envelope pulses, pulse reshaping effects can give the appearance of superluminal tunneling, as described in Sects. 15.9 and 15.12.

17.4 Optimal Pulse Penetration through Dispersive Bodies The analysis presented in Sect. 15.8.3 resulted in the identification of a so-called Brillouin pulse that is comprised of a pair of Brillouin precursor structures with the trailing precursor delayed in time and  phase shifted from the leading precursor, given by (15.189) as [9]    .!N .T /; T / .!N ./; /  exp : fBP .t / D exp !c ni .!c / !c ni .!c / 

(17.97)

Here T   cT =zd where T > 0 describes the fixed time delay between the leading and trailing Brillouin precursors. If T is chosen too small then there will be significant destructive interference between the leading and trailing components and the pulse will be rapidly extinguished. For practical reasons, 2T should be chosen near to the inverse of the operating frequency fc of the antenna used to radiate this Brillouin pulse. Because of its dependence on the complex index of refraction along the imaginary axis, the Brillouin pulse shape fBP .t / is dependent on the dispersive properties of the medium, so that each Brillouin pulse is uniquely matched to the dispersive material it is to penetrate.With the normally incident pulse in vacuum,

752

17 Applications

the transmitted plane wave pulse in the dispersive medium with complex index of refraction n.!/ is given by the Fourier integral representation 1 ABP .z; t / D 

Z

1 1

n.!/ Q Q fBP .!/e i .k.!/z!t / d! 1 C n.!/

(17.98)

for all z > 0, where the planar interface is situated at z D 0. Here fQBP .!/ denotes the Fourier transform of the dispersion-matched Brillouin pulse. Notice that the transmitted pulse will be distorted by the frequency-dependent behavior of the transmission coefficient so that the initial pulse just inside the dispersive material is no longer optimal. If necessary, this effect can be corrected by pre-distorting the incident pulse spectrum by the inverse of the transmission coefficient, so that fQBP .!/ ! Œ1 C n.!/=2n.!/ fQBP .!/ in (17.98).

17.4.1 Ground Penetrating Radar With the experimentally measured material dispersion data given by Tinga and Nelson [34], the relative complex dielectric permittivity of loamy soil may be accurately described by a three term Rocard–Powles–Debye model (see Sect. 4.4.3 of Vol. 1) 3 X aj 0 =0 Ci (17.99) c .!/ D 1 C .1  i !j /.1  i !fj / ! j D1 augmented by a static conductivity factor to account for both an ambient material conductivity and that resulting from the moisture content in the soil. The rms best-fit parameters for loamy soil at 25ı C are given in Tables 17.1 and 17.2 for 0% and 2:2% moisture contents, respectively. A comparison of the resultant angular frequency dispersion described by (17.99) with the experimental data for each case is presented in Fig. 17.21. The open circles describe the experimental data for 0% moisture content and the diamonds describe the experimental data for 2:2% moisture content. For comparison, the frequency dependence described by (17.99) for the 0% moisture content case with 0 D 0 is described by the dotted curves in Fig. 17.21. The only discernible difference is in the low-frequency behavior of

Table 17.1 Estimated rms “best-fit” Rocard–Powles–Debye model parameters for loamy soil at 25ı C with 0% moisture content, where 1 D 2:444 and 0 D 6:7  109 mho=m. Here rmsf 0, where 1 D sin # cos '; 2 D sin # sin '; 3 D cos #; with 0  # < =2 and 0  ' < 2. Under these two changes of variables, the Q 2 .r; !/ becomes integral for U Q 2 .r; !/ D U

Z

2

0

Z

A.˛; ˇ/e ikRŒsin # sin ˛ cos .ˇ'/Ccos # cos ˛ d˛dˇ;

C

where A.˛; ˇ/ 2 .p; q/V .p; q; m/e ik.px0 Cqy0 Cmz0 / sin ˛; with V .p; q; m/ defined in (9.204). The contour ofintegration C in the complex p ˛-plane, which extends from ˛1 D arcsin C1  " along the real ˛ 0 -axis to =2 p  C4 C " , depicted in Fig. G.2, and then to the endpoint at ˛4 D =2  i cosh1 then results in a complete, single covering of the original integration domain D2 . Notice that, in contrast with the original form of the integral, the phase ˚ sin # sin ˛ cos .ˇ  '/ C cos # cos ˛

''

'

C

Fig. G.2 Contour of integration C for the ˛-integral in Q 2 .r; !/ U

816

G Proof of Theorem 1

Q 2 .r; !/ is analytic and the ampliappearing in the above transformed integral for U tude function A.˛; ˇ/ is continuous with continuous partial derivatives with respect to ˛ and ˇ up to order N over the entire integration domain D2 . However, the contour of integration C is now complex, as illustrated in Fig. G.2, and this results in the phase ˚ being complex-valued when ˛ D ˛ 0 C i ˛ 00 varies from =2 to ˛4 along C . As ˛ varies over the contour C with the angle ˇ held fixed, the phase function ˚ varies over a simple curve C.ˇ/ of finite length in the complex ˛-plane. Since the partial derivative 

@˚ @˛

 D sin # cos ˛ cos .ˇ  '/  cos # sin ˛ ˇ

is an entire function of complex ˛ for all ˇ, then 

@˛ @˚

"

 D ˇ

@˚ @˛

 #1 ˇ

  is an analytic function of complex ˛ for all ˛; ˇ provided that @˚ ¤ 0. Since @˛ ˇ  @˚  cos # D 3 > ı, then @˛ ˇ ¤ 0 when cos ˛ D 0. It then follows that the zeros of  @˚  occur for those values of ˛ that satisfy the relation @˛ ˇ tan ˛ D tan # cos .ˇ  '/: On the portion of the contour C over which ˛ is real, ˛1  ˛  =2, and since cos # D 3  C1  " (see Fig. G.1), then #  ˛  =2 and one obtains the inequality tan ˛ > tan # cos .ˇ  '/. On the portion of the contour C over which ˛ is complex,