Handbook of Nonlinear Optical Crystals (Springer Series in Optical Sciences)

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OPTICAL SCIENCES

V. G. Dmitriev G.G. Gurzadyan D.N.Nikogosyan

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Han 00 of Nonlinear Optical Crystals

Third Revised Edition

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~ Springer

Springer Series in Optical Sciences Editor: A. E. Siegman

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

Volume 64

Springer Series in Optical Sciences Editorial Board: A. L. Schawlow A. E. Siegman T. Tamir Managing Editor: H. K. V. Lotsch

Solid-State Laser Engineering By W. Koechner 5th Edition

21 Laser Spectroscopy IV Editors: H. Walther and K. W. Rothe

2

Table of Laser Lines in Gases and Vapors By R. Beck, W. Englisch, and K. Gurs 3rd Edition

22 Lasers in Photomedicine and Photobiology Editors: R. Pratesi and C A. Sacchi

3

Tunable Lasers and Applications Editors' A. Mooradian, T. Jaeger, and P. Stokseth

4

Nonlinear Laser Spectroscopy By V. S. Letokhov and V. P. Chebotayev Optics and Lasers Including Fibers and Optical Waveguides By M. Young 3rd Edition (available as a textbook)

6

Photoelectron Statistics With Applications to Spectroscopy and Optical Communication By B. Saleh

7

Laser Spectroscopy III Editors: J. L. HaJJ and J. L. Carlsten

8

Frontiers in Visual Science Editors: S. J. Cool and E. J. Smith III

9

High-Power Lasers and Applications Editors: K.~L. Kompa and H Walther

10 Detection of Optical and Infrared Radiation By R. H. Kingston 11 Matrix Theory of Photoclasticity

By P. S . Theocaris and E. E. Gdoutos 12 The Monte Carlo Method in Atmospheric Optics By G I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinian, B A. Kargin, and B. S. Elepov

23 Vertebrate Photoreceptor Optics Editors: J. M. Enoch and F. L. Tobey, Jr. 24 Optical Fiber Systems and Their Components An Introduction By A. B Sharma, S. J. Halme, and M. M Butusov 25 High Peak Power Nd: Glass Laser Systems By D. C Brown 26 Lasers and Applications Editors: W. O. N. Guimaraes, C. T. Lin, and A. Mooradian 27 Color Measurement Theme and Variations By D. L. MacAdam 2nd Edition 28 Modular Opticsl Design By O. N. Stavroudis 29 Inverse Problems of Lidar Sensing of the Atmosphere By V. E. Zuev and I. E. Naats 30 Laser Spectroscopy V Editors: A. R. W McKellar, T. Oka, and B. P. Stoicheff 31 Optics in Biomedical Sciences Editors: G. von Bally and P. Greguss 32 Fiber-Optic Rotation Sensors and Related Technologies Editors: S. Ezekiel and H. J. Arditty

13 Physiological Optics By Y. Le Grand and S. G. El Hage

33 Integrated Optics: Theory and Technology By R. G. Hunsperger 3rd Edition (available as a texthook)

14 Laser Crystals Physics and Properties By A. A. Kaminskii 2nd Edition

34 The High-Power Iodine Laser By G. Brederlow, E. Fill, and K. J Witte

15 X-Ray Spectroscopy

35 Engineering Optics By K. Iizuka 2nd Edition

By B. K. Agarwal

2nd Edition 16 Holographic Interferometry From the Scope of Deformation Analysis of Opaque Bodies By W. Schumann and M. Dubas 17 Nonlinear Optics of Free Atoms and Molecules By D. C. Hanna, M. A. Yuratich, and D. Cotter 18 Holography in Medicine and Biology Editor: G. von Bally 19 Color Theory and Its Application in Art and Design By G. A. Agoston 2nd Edition 20 Interferometry by Holography By Yu. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya

36 Transmission Electron Microscopy Physics of Image Formation and Microanalysis By L. Reimer 4th Edition 37 Opto-Acoustic Molecular Spectroscopy By V. S. Letokhov and V. P. Zharov 38 Photon Correlation Techniques Editor: E. O. Schulz-DuBois 39 Optical and Laser Remote Sensing Editors: D. K Killinger and A. Mooradian 40 Laser Spectroscopy VI Editors· H P. Weber and W. LUthy 41 Advances in Diagnostic Visual Optics Editors: G. M. Breinin and I. M. Siegel

Springer Series in Optical Sciences Editorial Board: A. L. Schawlow A. E. Siegman T. Tamir

42 Principles of Phase Conjugation By B. Ya. Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov

43 X-Ray Microscopy Editors' G. Schmahl and D. Rudolph

44 Introduction to Laser Physics By K. Shimoda

2nd Edition

45 Scanning Electron Microscopy Physics of Image Formation and Microanalysis By L. Reimer 2nd Edition

46 Holography and Deformation Analysis By W. Schumann, J.-P ZUrcher, and D Cuche

47 Tunable Solid State Lasers Editors: P. Hammerling, A. B. Budgor, and A. Pinto

48 Integrated Optics Editors: H. P. Nolting and R. Ulrich

49 Laser Spectroscopy VII Editors' T. W. Hansch and Y. R. Shen

50 Laser..Induced Dynamic Gratings By H. J. Eichler, P. GUnter, and D. W. Pohl

5 J Tunable Solid State Lasers for Remote Sensing Editors: R. L. Byer, E. K. Gustafson, and R. Trebino

52 Tunable Solid-State Lasers II Editors: A. B. Budgor, L Esterowitz, and L. G. DeShazer

53 The CO 2 Laser

By W. J. Witteman

54 Lasers, Spectroscopy and New Ideas A Tribute to Arthur L Schawlow Editors' W. M. Yen and M D. Levenson

55 Laser Spectroscopy VIII Editors W Persson and S. Svanberg

56 X-Ray Microscopy II Editors: D. Sayre, M. Howells, J Kirz, and H. Rarback

57 Single-Mode Fibers

Fundamentals

By E.-G. Neumann

58 Photoacoustic and Photothermal Phenomena Editors: P. Hess and J. Pelzl

59 Photorefractive Crystals in Coherent Optical Systems By M. P. Petrov, S. I. Stepanov, and A V. Khomenko

Volumes 1-41 are listed at the end of the book

60 Holographic Interferometry in Experimental Mechanics By Yu. I Ostrovsky, V P. Shchepinov, and V. V. Yakovlev

61 Millimetre and Sub millimetre Wavelength Lasers A Handbook of cw Measurements By N. G. Douglas

62 Photoacoustic and Photothermal Phenomena II Editors. J. C. Murphy, J. W. Maclachlan Spicer, L. C. Aamodt, and B. S H. Royce

63 Electron Energy Loss Spectrometers The Technology of High Performance By H Ibach

64 Handbook of Nonlinear Optical Crystals By V. G . Dmitriev, G G Gurzadyan, and D. N Nikogosyan

3rd Edition

65 High-Power Dye Lasers Editor: F. J. Duarte

66 Silver Halide Recording Materials for Holography and Their Processing By H. 1. Bjelkhagen

2nd Edition

67 X-Ray Microscopy III Editors. A G. Michette, G. R. Morrison, and C. J. Buckley

68 Holographic Interferometry Principles and Methods Editor: P. K. Rastogi

69 Photoacoustic and Photothermal Phenomena III Editor: D. Bicanic

70 Electron Holography By A. Tonomura 2nd Edition 71 Energy-Filtering Transmission Electron Microscopy Editor: L. Reimer

72 Handbook of Nonlinear Optical Effects and Materials Editor: P. GUnter

73 Evanescent Waves By F. de Fornel

74 International Trends in Optics and Photonics leO IV Editor: T. Asakura

v. G. Drnitriev

G. G. Gurzadyan D. N. Nikogosyan

Handbook of Nonlinear Optical Crystals Third Revised Edition With 39 Figures

Springer

Sci.

Ph.D.

Professor VALENTIN G. DMITRIEV, Ph.D

Dr.

R&D Institute "Polyus", Vvedenskogo St. 3 117342 Moscow, Russia

National Academy of Sciences of Armenia Yerevan, Armenia

GAGIK G. GURZADYAN,

E-mail: [email protected]

Professor DAVID

N. NIKOGOSYAN,

Ph.D.

Institute of Nonlinear Science University College Cork Cork, Ireland E-mail: [email protected]

Editorial Board ARTHUR L. SCHAWLOW, Ph. D.

THEODOR TAMIR,

Department of Physics, Stanford University Stanford, CA 94305-4060, USA

Polytechnic University 333 Jay Street, Brooklyn, NY 11201, USA

Professor ANTHONY E.

SIEGMAN,

Ph. D.

Ph. D.

Electrical Engineering E. L. Ginzton Laboratory, Stanford University Stanford, CA 94305-4060, USA

Managing Editor: Dr.-Ing.

HELMUT

K.V.

LOTSCH

Springer-Verlag, Tiergartenstrasse 17, D-69121 Heidelberg, Germany

ISSN 0342-4111 ISBN 3-540-65394-5 3rd Edition Springer-Verlag Berlin Heidelberg New York ISBN 3-540-61275-0 2nd Edition Springer-Verlag Berlin Heidelberg New York Library of Congress Cataloging-in-Publication Data Gurzadiiin, G. G. (Gagik Grigor'evich), 1957- [Nelineino-opticheskie kristally, English) Handbook of nonlinear optical crystals / V. G. Dmitriev, G. G. Gurzadyan, D. N. Nikogosyan. - 3rd rev. ed. p. cm.(Springer series in optical sciences, ISSN 0342-4111; v. 64) Gurzadfans name appears first on earlier eds. Includes bibliographical references (p. - ) and index. ISBN 3-540-65394-5 (hc.: acid-free paper) 1. Laser materials-Handbooks, manuals, etc. 2. Optical materials-Handbooks. manuals, etc. 3. Crystals-Handbooks, manuals, etc. 4. Nonlinear optics-Handbooks, manuals, etc. I. Dmitriev, V. G. (Valentin Georgievich) II. Nikogosian, D. N., 1946-. III. Title. IV. Series. QC374.G8713 1999 621.36'6-dc21 99-17769 enThis work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991, 1997, 1999 Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Scientific Publishing Services (P) Ltd, Madras Cover concept by eStudio Calamar Steinen using a background picture from The Optics Project. Courtesy of John T. Foley, Professor, Department of Physics and Astronomy, Mississippi State University, USA Cover production: design & production GmbH, Heidelberg SPIN 10706983

56/3144/di-54 32 10- Printed on acid-free paper

To our Parents

Preface to the Second and Third Editions

When we had finished our work on the first edition of our Handbook we never supposed that three years later it would become necessary to greatly revise and update the material into a second edition. It happened because of the following developments. 1. The invention and tremendous development of modem nonlinear optical crystals such as BBO, LBO, KTP, ZnGeP 2 , etc. 2. Rapid progress in laser techniques (femtosecond CPM laser, Ti: sapphire laser, diode-pumped solid-state lasers, etc.). 3. The appearence of numerous organic crystals which can be synthesized with predictable properties. 4. Progress in the theory of nonlinear frequency conversion utilizing biaxial crystals, femtosecond pulses, etc. 5. Accumulation of new data on the properties of nonlinear optical crystals. In accordance with the above, the second edition included many changes in the text. The first chapter was revised by D. N. Nikogosyan, the second one by V.G. Dmitriev and D. N. Nikogosyan, and the fourth one by G. G. Gurzadyan. The third chapter, containing the main reference material on 77 nonlinear optical crystals was completely rewritten and updated by D. N. Nikogosyan. The Appendix, containing the list of most commonly used laser wavelengths, was compiled by D.N. Nikogosyan. This third edition has been further revised in several of the graphical presentations and includes updates in the details of the experimental data. We would like to thank H. K. V. Lotsch for his fruitful and long-standing cooperation. Moscow, Yerevan, Cork Russia, Armenia, Ireland February 1999

v G. Dmitriev G. G. Gurzadyan D. N. Nikogosyan

Preface to the First Edition

Since the invention of the first laser 30 years ago, the frequency conversion of laser radiation in nonlinear optical crystals has become an important technique widely used in quantum electronics and laser physics for solving various scientific and engineering problems. The fundamental physics of three-wave light interactions in nonlinear optical crystals is now well understood. This has enabled the production of various harmonic generators, sum- and differencefrequency generators, and optical parametric oscillators based on nonlinear optical crystals that are now commercially available. At the same time, scientists continue an active search for novel, highly efficient nonlinear optical materials. Therefore, in our opinion, there is a great need for a handbook of nonlinear optical crystals, intended for specialists and practitioners with an engineering background. This book contains a complete description of the properties and applications of all nonliner optical crystals of practical importance reported in the literature up to the beginning of 1990. In addition, it contains the most important equations for calculating the main parameters (such as phase-matching direction, effective nonlinearity, and conversion efficiency) of nonlinear frequency converters. Dolgoprudnyi, Yerevan, Troitzk USSR October 1990

V. G. Dmitriev G. G. Gurzadyan D. N. Nikogosyan

Contents

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Optics of Nonlinear Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14

2.15

Three- and Four-Wave (Three- and Four-Frequency) Interactions in Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . Phase-Matching Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . Optics of Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Phase Matching in Uniaxial Crystals . . . . . . . . . . . .. Calculation of Phase-Matching Angles in Uniaxial Crystals. . .. Reflection and Refraction of Light Waves at the Surfaces of Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Optics of Biaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . .. Types of Phase Matching in Biaxial Crystals. . . . . . . . . . . . . .. Calculation of Phase-Matching Angles in Biaxial Crystals . . . .. Crystal Symmetry and Effective Nonlinearity: Uniaxial Crystals. Crystal Symmetry and Effective Nonlinearity: Biaxial Crystals.. Theory of Nonlinear Frequency-Conversion Efficiency. . . . . . .. Wave Mismatch and Phase-Matching Bandwidth . . . . . . . . . .. Calculation of Nonlinear Frequency-Conversion Efficiency in Some Special Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.14.1 Plane-Wave Fixed-Field Approximation. . . . . . . . . . .. 2.14.2 Fundamental Wave Depletion C'Nonlinear Regime") .. 2.14.3 SHG of a Divergent Fundamental Radiation Beam in the Fixed-Field Approximation . . . . . . . . . . . . . . .. 2.14.4 SHG of a Divergent Fundamental Radiation Beam in the Nonlinear Regime. . . . . . . . . . . . . . . . . . . . . .. 2.14.5 Fixed-Intensity Approximation . . . . . . . . . . . . . . . . .. 2.14.6 Frequency Conversion of Ultrashort Laser Pulses. . . .. 2.14.7 Frequency Conversion of Laser Beams with Limited Aperture in the Stationary Regime. . . . .. 2.14.8 Linear Absorption . . . . . . . . . . . . . . . . . . . . . . . . . .. Additional Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

3 3 5 6 10 13 14 16 18 19 23 25 32 40 48 49 52 54 55 57 59 61 65 65

XII

Contents

3 Properties of Nonlinear Optical Crystals 3.1

Basic Nonlinear Optical Crystals . . . . . . . . . . . . . . . . . . . . . . 3.1.1 LiB 30 s, Lithium Triborate (LBO) . . . . . . . . . . . . . . . 3.1.2 KH 2P04 , Potassium Dihydrogen Phosphate (KDP). . . KD 2P04 , Deuterated Potassium Dihydrogen 3.1.3 Phosphate (DKDP) 3.1.4 NH 4H 2P04 , Ammonium Dihydrogen Phosphate (ADP) J3-BaB 204 , Beta-Barium Borate (BBO) . . . . . . . . . . . . 3.1.5 uro., Lithium Iodate. . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 KTiOP0 4 , Potassium Titanyl Phosphate (KTP) . . . .. 3.1.7 3.1.8 LiNb0 3 , Lithium Niobate . . . . . . . . . . . . . . . . . . . .. 3.1.9 KNb0 3, Potassium Niobate 3.1.10 AgGaS2, Silver Thiogallate . . . . . . . . . . . . . . . . . . .. 3.1.11 ZnGeP 2 , Zinc Germanium Phosphide. . . . . . . . . . . .. 3.2 Frequently Used Nonlinear Optical Crystals. . . . . . . . . . . . .. 3.2.1 KBsO g . 4H 20, Potassium Pentaborate Tetrahydrate (KB5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. CO(NH 2)2, Urea 3.2.2 3.2.3 CsH 2As04 , Cesium Dihydrogen Arsenate (CDA) . . .. CsD1As04, Deuterated Cesium Dihydrogen Arsenate 3.2.4 (DCDA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. KTiOAs0 4 , Potassium Titanyl Arsenate (KTA) .... , 3.2.5 3.2.6 MgO : LiNb0 3 , Magnesium-Oxide-Doped Lithium Niobate. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2.7 Ag 3AsS 3 , Proustite . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2.8 GaSe, Gallium Selenide. . . . . . . . . . . . . . . . . . . . . .. 3.2.9 AgGaSe2, Silver Gallium Selenide . . . . . . . . . . . . . .. 3.2. 10 Cd'Se, Cadmium Selenide 3.2.11 CdGeAs 2 , Cadmium Germanium Arsenide. . . . . . . .. 3.3 Other Inorganic Nonlinear Optical Crystals . . . . . . . . . . . . .. 3.3.1 KB s0 8 . 40 20, Deuterated Potassium Pentaborate Tetrahydrate (DKB5) . . . . . . . . . . . . . . . . . . . . . . .. 3.3.2 CsB 30S , Cesium Triborate (CBG) . . . . . . . . . . . . . .. 3.3.3 BeS04 . 4H 20, Beryllium Sulfate. . . . . . . . . . . . . . .. 3.3.4 MgBaF 4, Magnesium Barium Fluoride. . . . . . . . . . .. 3.3.5 NH 4D2P04 , Deuterated Ammonium Dihydrogen Phosphate (DADP). . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 RbH 2P04 , Rubidium Dihydrogen Phosphate (RDP) . . 3.3.7 RbD 2P04 , Deuterated Rubidium Dihydrogen Phosphate (DRDP). . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.8 KH 2As0 4 , Potassium Dihydrogen Arsenate (KDA). .. K.D 2 As0 4 , Deuterated Potassium Dihydrogen 3.3.9 Arsenate (DKDA) . . . . . . . . . . . . . . . . . . . . . . . . ..

67 68 68 78 85 90 96 103 107 119 126 132 136 142 142 146 149 152 156 159 162 166 169 173 176 179 179 180 182 184 186 188 192 192 195

Contents

3.3.10 NH4H 2As04 , Ammonium Dihydrogen Arsenate (ADA). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.11 NH4D2As04, Deuterated Ammonium Dihydrogen Arsenate (DADA) " 3.3.12 RbH 2As04, Rubidium Dihydrogen Arsenate (RDA) .. 3.3.13 RbD 2As04, Deuterated Rubidium Dihydrogen Arsenate (DRDA) . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.14 LiCOOH· H 20, Lithium Formate Monohydrate (LFM) 3.3.15 NaCOOH, Sodium Formate. . . . . . . . . . . . . . . . . .. 3.3.16 Ba(COOH)2, Barium Formate. . . . . . . . . . . . . . . . .. 3.3.17 Sr(COOH)2, Strontium Formate. . . . . . . . . . . . . . .. 3.3.18 Sr(COOH)2' 2H 20, Strontium Formate Dihydrate. .. 3.3.19 LiGa02, Lithium Gallium Oxide. . . . . . . . . . . . . . .. 3.3.20 (X-HI0 3, ex-Iodic Acid . . . . . . . . . . . . . . . . . . . . . . .. 3.3.21 K 2La(N03)s' 2H 20, Potassium Lanthanum Nitrate Dihydrate (KLN) . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.22 CsTiOAs04 , Cesium Titanyl Arsenate (CTA) . . . . . .. 3.3.23 NaN0 2, Sodium Nitrite . . . . . . . . . . . . . . . . . . . . .. 3.3.24 Ba2NaNbs015' Barium Sodium Niobate ("Banana") .. 3.3.25 K 2Ce(N03)s' 2H 20, Potassium Cerium Nitrate Dihydrate (KCN) . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.26 K3Li2NbsOlS, Potassium Lithium Niobate . . . . . . . .. 3.3.27 HgGa2S4' Mercury Thiogallate . . . . . . . . . . . . . . . .. 3.3.28 HgS, Cinnibar . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.29 Ag 3SbS3, Pyrargyrite. . . . . . . . . . . . . . . . . . . . . . . .. 3.3.30 Se, Selenium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.31 TI3AsS3, Thallium Arsenic Selenide (TAS) . . . . . . . .. 3.3.32 Te, Tellurium. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4 Other Organic Nonlinear Optical Crystals 3.4.1 C12H220lb Sucrose (Saccharose) . . . . . . . . . . . . . . .. 3.4.2 L-Arginine Phosphate Monohydrate (LAP) 3.4.3 Deuterated L-Arginine Phosphate Monohydrate (DLAP) 3.4.4 L-Pyrrolidone-2-carboxylic Acid (L-PCA). . . . . . . . .. 3.4.5 CaC4H40 6 . 4H 20, Calcium Tartrate Tetrahydrate (L-CTT) 3.4.6 (NH4)2C204 . H 20, Ammonium Oxalate (AO) . . . . .. 3.4.7 m-Bis(aminomethyl)benzene (BAMB). . . . . . . . . . . .. 3.4.8 3-Methoxy-4-hydroxy-benzaldehyde (MHBA) . . . . . .. 3.4.9 2-Furyl Methacrylic Anhydride (FMA) 3.4.10 3-Methyl-4-nitropyridine-l-oxide (POM). . . . . . . . . .. 3.4.11 Thienylchalcone (T-17) . . . . . . . . . . . . . . . . . . . . . .. 3.4.12 5-Nitrouracil (5NU) . . . . . . . . . . . . . . . . . . . . . . . .. 3.4.13 2-(N-Prolinol)-5-nitropyridine (PNP) . . . . . . . . . . . ..

XIII

196 198 199 202 204 207 209 210 211 213 214 217 220 221 224 227 229 231 233 235 236 238 240 243 243 245 247 250 251 253 254 256 258 259 261 263 265

XIV

Contents

3.4.14 2-Cyclooctylamino-5-nitropyridine (COANP) . . . . . .. 3.4.15 L-N-(5-Nitro-2-pyridyl)leucinol (NPLO) 3.4.16 C 6H4(N02) 2, rn-Dinitrobenzene (MDNB) . . . . . . . . .. 3.4.17 4-(N,N-Dimethylamino)-3-acetamidonitrobenzene (DAN) 3.4.18 Methyl-(2,4-dinitrophenyl)-aminopropanoate (MAP). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4.19 m.. Nitroaniline (MNA) . . . . . . . . . . . . . . . . . . . . . .. 3.4.20 N-(4-Nitrophenyl)-N -methylaminoacetonitrile (NPAN) 3.4.21 N . . (4-Nitrophenyl)-L-prolinol (NPP). . . . . . . . . . . . .. 3.4.22 3-Methyl-4-methoxy-4'-nitrostilbene (MMONS) .. . .. Properties of Crystalline Quartz (~-Si02) " New Developments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. «

3.5 3.6

4 Applications of Nonlinear Crystals

266 268 270 272 274 276 278 280 281 283 286

"

289

Generation of Neodymium Laser Harmonics . . . . . . . . . . . .. 4.1.1 Second-Harmonic Generation of Neodymium Laser Radiation in Inorganic Crystals . 4.1.2 Second-Harmonic Generation of 1.064 urn Radiation in Organic Crystals. . . . . . . . . . . . . . . . . . . . . . . . .. 4.1.3 Intracavity SHG. . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1.4 Third-Harmonic Generation .. 4.1.5 Fourth-Harmonic Generation " 4.1.6 Fifth-Harmonic Generation. . . . . . . . . . . . . . . . . . .. 4.1.7 Harmonic Generation of 1.318 J.lm Radiation. . . . . .. 4.2 Harmonic Generation of High-Power Large-Aperture Neodymium Glass Laser Radiation " 4.2.1 "Angle-Detuning" Scheme 4.2.2 "Polarization-Mismatch" Scheme 4.2.3 "Polarization-Bypass" Scheme. . . . . . . . . . . . . . . . .. 4.2.4 Comparison of Schemes . . . . . . . . . . . . . . . . . . . . .. 4.2.5 Experimental Results '" . . . . . . . . . . . . . . . . . . . .. 4.2.6 "Quadrature" Scheme. . . . . . . . . . . . . . . . . . . . . . .. 4.3 Harmonic Generation for Other Laser Sources. . . . . . . . . . .. 4.3.1 Ruby Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.2 Ti:sapphire Laser " 4.3.3 Semiconductor Lasers. . . . . . . . . . . . . . . . . . . . . . .. 4.3.4 Dye Lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.5 Gas Lasers " 4.3.6 Iodine Laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.7 CO 2 Laser ... Other Lasers .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.8 4.3.9 Frequency Conversion of Femtosecond Pulses ... . ..

289

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289 294 296 298 301 301 304 306 306 306 308 308 308 310 311 311 312 312 315 320 321 324 324 326

Contents

4.4

Sum-Frequency Generation. . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.1 Up-Conversion to the UV Region . . . . . . . . . . . . . .. 4.4.2 Infrared Up-Conversion . . . . . . . . . . . . . . . . . . . . .. 4.4.3 Up-Conversion of CO 2 Laser Radiation to the Near IR and Visible Regions . . . . . . . . . . . . .. 4.5 Difference-Frequency Generation. . . . . . . . . . . . . . . . . . . . .. 4.5.1 DFG in the Visible Region . . . . . . . . . . . . . . . . . . .. 4.5.2 DFG in the Mid IR Region . . . . . . . . . . .. 4.5.3 DFG in the Far IR Region. . . . . . . . . . . . . . . . . . .. 4.6 Optical Parametric Oscillation " 4.6.1 OPO in the UV, Visible, and Near IR Spectral Regions. . . . . . . . . . . . . . . . .. 4.6.2 OPO in the Mid IR Region. . . . . . . . . . . . . . . . . . .. 4.6.3 Conversion of OPO Radiation to the UV Region . . .. 4.7 Stimulated Raman Scattering and Picosecond Continuum Generation in Crystals "

XV

327 328 333 336 339 339 340 344 345 345 359 360 362

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

367

Appendix: List of Commonly Used Laser Wavelengths. . . . . . . . . . ..

405

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407

List of Abbreviations

a c cont cr cw DF DFG dif DROPO ds, dis e

eff exp f fcg FIHG FOHG ICSFG ICSHG int IR L NCPM NL no pm o

OPO OR P

PM, pm

PL pr qs s SF

Aperture Cut Continuum Critical Continuous wave Difference frequency Difference-frequency generation Diffraction Doubly-resonant optical parametric oscillation Dispersive spreading Extraordinary Effective Experimental Fast Free-carrier generation Fifth-harmonic generation Fourth-harmonic generation Intracavity sum-frequency generation Intracavity second-harmonic generation Internal Infrared Linear Non-critical phase matching Nonlinear No phase matching Ordinary Optical parametric oscillation Optical rectification Pulse Phase matching Parametric luminescence Photorefraction Quasistatic Slow Sum frequency

XVIII

SFG SFM SH SHG SIHG SROPO SRS theor THG thr tsa TWOPO unc UV

List of Abbreviations

Sum-frequency generation Sum-frequency mixing Second harmonic Second-harmonic generation Sixth-harmonic generation Singly-resonant optical parametric oscillation Stimulated Raman scattering Theoretical Third-harmonic generation Threshold Thermal self-action Traveling-wave optical parametric oscillation Unconverted Ultravi0 let

1 Introduction

In 1960, Maiman (USA) created the first source of coherent optical radiation, namely, a ruby laser emitting in the red spectral region (A == 0.6943 urn) [1.1]. Several years later a great family of lasers was already in existence. The following types were known: 1) solid-state lasers, e.g., Nd:CaW0 4 laser emitting at 1.065 urn [1.2], neodymium glass laser (A== 1.06Jlm) [1.3], Nd:YAG laser (A=: 1.064Jlm) [1.4] 2) gas lasers, e.g., He-Ne laser (A == 0.6328, 1.1523, 3.3913 urn) [1.5], argon ion laser (A == 0.4880,0.5145 um) [1.6], C02 laser (A == 9.6, 10.6 um] [1.7]; 3) dye lasers [1.8,9] 4) semiconductor lasers [1.10-12]; and so on. The wavelengths of the above mentioned lasers were either fixed or tunable over a small range. It was a matter of practical importance to widen the range of wavelengths generated by laser sources. The propagation of electromagnetic waves through nonlinear media gives rise to vibrations at harmonics of the fundamental frequency, at sum and difference frequencies, and so on. In the optical frequency range, the same effect is observed when light waves propagate through weakly nonlinear optical dielectrics. When one or two sufficiently powerful beams of laser radiation pass through these dielectrics, the radiation frequency may be transformed to the second, third, and higher harmonics and to combination (sum and difference) frequencies. In this way, the range of wavelengths generated by a certain laser source can be considerably increased. For instance, the second harmonic of the ruby laser radiation lies in the UV region (A == 0.34715IJm), whereas the second harmonic of the neodymium glass laser radiation lies in the green spectral range (A == 0.53 urn). As early as in 1961, Franken et al. [1.13] observed a radiation at the doubled frequency when a ruby laser light was directed into a quartz crystal. However, because of phase mismatch of the waves at the fundamental and doubled frequencies upon propagation in a quartz crystal, the efficiency of conversion to the second harmonic proved to be very low, less than 10- 12 . In 1962, Giordmaine [1.14] and Maker et al. [1.15] simultaneously proposed an ingenious method of matching the phase velocities of the waves at the

2

1 Introduction

fundamental and doubled frequencies. Their technique used the difference between the refractive indices of the waves with different polarizations in an optically anisotropic (uniaxial or biaxial) nonlinear crystal (phase-matching method), and with it the efficiency of conversion of laser radiation to the second harmonic was enhanced to several ten percent. At the beginning of the 1960s, parallel to the research on second-harmonic generation, first experiments were carried out on the generation of optical radiation at combination frequencies, namely: sum-frequency generation of radiation from two lasers [1.16], sum-frequency generation of radiation from a laser and a noncoherent source [1.17], and difference-frequency generation [1.18,19].We should specially mention optical parametric oscillation, which is a nonlinear effect that allows one to obtain continuously tunable coherent optical radiation [1.20]. The ferroelectrics ADP and KDP used in electro-optic and elasto-optic devices were the first crystals applied for nonlinear frequency conversion (nonlinear optical crystals) [1.21]. They were grown by conventional techniques. However, some special nonlinear optical problems called for crystals with improved properties (better transparency, higher nonlinearity, lower hygroscopicity, etc.). The resulting intensive scientific search for new materials has led to the synthesis of a number of nonlinear crystals of high optical quality: LiNb0 3 in 1964 [1.22], BaNaNbs0 1S in 1967 [1.23], proustite in 1967 [1.24], Lil0 3 in 1969 [1.25], KTP in 1976 [1.26], and others. The first reviews comparing the properties of various nonlinear optical crystals have been published [1.27,28]. Very recently two new nonlinear crystals from the borate family, of excellent quality, were invented by Chen et al.: BaB204 (BBO) in 1985 [1.29] and LiB30 s (LBO) in 1989 [1.30].

2 Optics of Nonlinear Crystals

This chapter introduces the main concepts of the physics of nonlinear optical processes: three-wave interactions, phase matching and phase-matching angle, role of phase mismatch for the interaction of quasi-plane waves, group-velocity mismatch and interaction of ultrashort light pulses, optics of uniaxial and biaxial crystals, crystal symmetry and effective nonlinearity, "walk-off" angle, phase-matching bandwidths (angular, temperature, spectral), thermal effects, and so on. It presents the main material required for calculating of phasematching angles and for an assessment (as a rule, in approximation of quasiplane light waves) of frequency conversion efficiency in the case of generation of optical harmonics and combination (sum and difference) frequencies) and optical parametric oscillation in nonlinear optical crystals. For convenience, the so-called "effective lengths" are introduced for the corresponding processes: by comparing the nonlinear crystal's length with the effective length of the corresponding process, we may conclude whether this process must be taken into account for the calculation of the conversion efficiency or not. The chapter contains many tables with the equations for calculating phasematching and "walk-off" angles, bandwidths, effective nonlinearity and conversion efficiency.

2.1 Three- and Four-Wave (Three- and Four-Frequency) Interactions in Nonlinear Media Conversion of a light-wave frequency (multiplication, division, mixing) is possible in nonlinear optical crystals for which the refraction index n is a function of the electric field strength vector E of the light wave

n(E) == no + nlE + n2E2 + ... ,

(2.1)

where no is the refractive index in the absence of the electric field (this quantity is used in conventional "linear" optics), and nl, n2, and so on are the coefficients of the series expansion of n(E). In nonlinear optics a vector of dielectric polarization P (dipole moment of unit volume of the matter) is introduced. It is related to the field E by the matter equation [2.1--4]

4

2 Optics of Nonlinear Crystals

P(E) == K(E)E == KoE+ X(2) E2 + X(3) E 3 + ....,

(2.2)

where K is the linear dielectric susceptibility (denoted as KO in the absence of the electric field), and X(2), X(3), and so on are the nonlinear dielectric susceptibility coefficients (square, cubic, and so on, respectively). The following equations hold true:

(2.3)

where eo is the dielectric constant in the absence of the electric field. In the general case of anisotropic crystals, the quantities eo, n, K, and X are the tensors of the corresponding ranks [2.4]. The square nonlinearity takes place (X(2) =1= 0) only in acentric crystals, i.e., in crystals without symmetry center; in crystals with symmetry center and as well as in isotropic matter X(2) == O. On the contrary the cubic nonlinearity exists in all crystalline and isotropic materials. Propagation of two monochromatic waves with frequencies WI and W2 in crystals with square nonlinearity gives rise to new light waves with combination frequencies W3,4 == W2 ± WI; the sign plus corresponds to sum frequency, the sign minus - to differencefrequency (three-wave or three-frequency interaction). Sum-frequency generation (SFG) is frequently used for conversion of longwave radiation, for instance, infrared (IR) radiation, to short-wave radiation, namely, ultraviolet (UV) or visible light. Difference-frequency generation (DFG) is used for conversion of short-wave radiation to long-wave radiation. At WI == W2 we obtain two special cases of conversion, namely, secondharmonic generation (SHG) as a special case of SFG, W3 == 2WI, and optical rectification (OR) as a special case of (DFG), W4 == o. The effect of parametric luminescence (PL), or optical parametric oscillation (OPO), is the opposite process to SFG and involves the appearance of two light waves with the frequencies WI,2 in the field of the intense light wave with frequency W3 == WI + W2. Generation of more complex combination frequencies is possible with successive SFG and/or SHG processes. For example, the third-harmonic generation (THG) can be realized by using the following SFG process: W3

== 3WI == WI

+ 2WI

;

(2.4)

the fourth-harmonic generation (FOHG, W4 == 4WI) can be realized as SHG process of frequency 2w). In a similar manner, the fifth- and sixth-harmonic generations (FIHG and SIHG) can be realized: (2.5) or (2.6)

2.2 Phase-Matching Conditions W6

== 6WI == WI + SWI

5

(2.7)

or

(2.8) Propagation of two light waves with frequencies WI,2 in substance with cubic nonlinearity gives rise to new light waves with combination frequencies 2WI ± W2 and WI ± 2W2 (four-wave or four-frequency interaction). The special cases with WI == W2 are the direct THG process, W3 == 3Wl, and the process of self-action, W4 == 2Wl - WI == WI, or the generation of the same frequency WI. Because of the relatively seldom usage of frequency conversion in cubic substances (as a rule, X(3) E« X(2) , in this chapter we shall consider only the three-wave interactions occuring in the crystals with square nonlinearity (X(2)

to).

2.2 Phase-Matching Conditions Under usual conditions all optical media are weakly nonlinear, i.e., the inequalities X(3) E 2 « X(2) E4;:.. KO are valid. Noticeable nonlinear effects can be observed only when light propagates through fairly long crystals and the socalled phase-matching conditions are fulfilled: k3 == k 2 + ki

(2.9)

or (2.10)

where k, are the wave vectors corresponding to the waves with frequencies (i == 1,2,3,4):

1k,z 1-- k.z --

Win(Wi) -- ~ -- 2nni -- 2Ten z., Vz C v(Wi) Ai

,

OJi

(2.11)

where the quantities Vi, n, == n(Wi), Ai and Vi are the phase velocity, refractive index, wavelength, and wave number at the frequency Wi, respectively. The relative location of the wave vectors under phase matching can be either collinear (scalar phase matching) or noncollinear (vector phase matching) (Fig. 2.1) Under scalar (collinear) phase matching we have for SFG

== k2 + k1 , or W3n3 == W2 n 2 + wlnl and for SHG ( WI == W2; ill3 == 2WI ) : k3 == 2kl or n3 == n 1 . k3

(2.12)

(2.13)

The physical sense of phase-matching conditions (2.9,10) is the space resonance of the propagating waves, namely, between the wave of nonlinear dielectric

6

2 Optics of Nonlinear Crystals

(a)

polarization at the frequency Q)3 for SFG (or Q)4 for DFG) and produced by her light wave at the same frequency Q)3 (or Q)4, respectively). Note that in the

optical transparency region in isotropic crystals (and also in anisotropic crystals for identically polarized waves), the equality (2.13) for SHG is never fulfilled because of normal dispersion (nl < n3). The use of anomalous dispersion is almost impossible since energy absorption is very high. The phase-matching conditions are fulfilled only in anisotropic crystals under interaction of differently polarized waves. Combination of nonzero square nonlinearity of an optically transparent crystal with phase matching is the necessary and sufficient condition for an effective three-wave interaction.

2.3 Optics of Uniaxial Crystals In uniaxial crystals a special direction exists called the optic axis (Z axis). The plane containing the Z axis and the wave vector k of the light wave is termed the principal plane. The light beam whose polarization (i.e., the direction of the vector E oscillations) is normal to the principal plane is called an ordinary beam or an o-beam (Fig. 2.2). The beam polarized in the principal plane is known as an extraordinary beam or e-beam (Fig. 2.3). The refractive index of the o-beam does not depend on the propagation direction, whereas for the e-beam it does. Thus, the refractive index in anisotropic crystals generally depends both on light polarization and propagating direction. The difference between the refractive indices of the ordinary and extraordinary beams is known as birefringence 1\n. The value of 1\n is equal to zero along the optic axis Z and reaches a maximum in the direction normal to this axis. The refractive indices of the ordinary and extraordinary beams in the plane normal to the Z axis are termed the principal values of the refractive

2.3 Optics of Uniaxial Crystals

7

Fig. 2.2. Principal plane of the crystal (kZ) and ordinary beam

K

E

E Fig. 2.3. Principal plane of the crystal (kZ) and extraordinary beam

y

Fig. 2.4. Polar coordinate system for description of refraction properties of uniaxial crystal (k is the light propagation direction, Z is the optic axis, 0 and ¢ are the coordinate angles)

index and are denoted by no and ne , respectively; the value no should not be confused with the refractive index value no in the absence of electric field in (2.1). The refractive index of the extraordinary wave is, in general, a function of the polar angle ()between the Z axis and the vector k (Fig. 2.4). It is determined by the equation (index e in this case is written as a superscript):

1 + tan- ()

(2.14)

8

2 Optics of Nonlinear Crystals

The following equations are evident: nO(O) == no , ne(O == 0°) == no , ne(O == 90°) == ne , ~n(O

(2.15) (2.16) (2.17)

== 0°) == 0 ,

(2.18)

An(() == 90°) == ne - no , An(0) == nee0) - no .

(2.19) (2.20)

If no > ne , the crystal is negative; if no < ne , it is positive. The quantity ne does not depend on the azimuthal angle 4> (the angle between the projection of k onto the XY plane and the X axis - see Fig. 2.4). The dependence of the refractive index on light propagation direction inside the uniaxial crystal (index surface) is a combination of a sphere with radius no (for an ordinary beam) and

no x(r}

nr > nz eeo

Xy

oeo eoo

tan- ¢ ~ 1 - U

U-s tan? ¢ = 1 - V V-y tarr' ¢ = 1 - T T-Z

-

YZ

D +E Al ' A2 ' A3 ' Al ' Zl Y2 X2 Z3 B)2 ~)2 n n n n - . y- - . A - - ' B - - ' C - - ' E - V= ( C - A ,, - AI' - A2' - A3' - A2 Yl . B = n Z2 . C = n Z3 . D = n Xl 2. Z = {~)2. A = n _ A _ ) T= ( C - B' \lJ' Al ' A2 ' A3 ' Al C'

I

A2

-~---

ooe

tan2

eoe

tan2

oee

U= ( A + B) 2. S = (A + B) 2. A = n rr B = nY2. C = nZ3. D = nXl. E = nX2

e = 1- U W-l

e~ 1- U W-R

tan2

e ~ 1- U

W-Q

B)2 . W= (A B)2 . A=-' nXl B=-' nX2 C=-' n Y3 F=nZ3 -+-+U= ( A C'

U= ( A + B) 2. W = C'

B

F' (A + B) 2. R F'

B)

=

Al ' A2 ' A3 ' A3 (A + B) 2. A = n Y1. B = n X2. C == n Y3. D D + B' Al ' A2 ' A3 '

(A B)

N

\.0

= n zr F = n Z3 A.3

Al '

2 2 2 n n n n n U= ( A + ) . W= (~ . Q= -±- . A =~. B=-B· C==~· E=.-E.· F=-E C' F' A + E' Al ' A2 ' A3 ' A2 ' ;.3

e< Vz

eeo oeo eoo

tarr' e ~ 1 - U

U-S tarr' e = 1 - V V-y tarr' e = 1 - T T-Z

~

E.

s:~ t:S

o

--

XZ

o

n

~

-e

=-

y3 U= ( A + B) 2. S= (A+B)2. A = n Xl . B= n X2 . C= n . D= n Zl . E= n Z2 C' D + E' Al ' A2 ' A3 ' Al ' A2 2 ( 2 n Yl n X2 n Y3 n Z2 - . y- . A - - ' B--' C--' E-V= C- A ' - E ' - AI' - A2' - A3' - A2 A ) 2 (A) 2 nXl nY2 n Y3 n Zl T= ( C-B ; Z== 15 ; A=~; B=;;;; C=~; D==;;

( B)

~

~

~

B)

~

s

=S'

(JQ

> =:s

--

(JQ

XZ

ooe

tarr' e = 1 - U W-l

y2. C = n X3 . F = n Z3 U= ( A +B)2. W = (A +B)2. A = n Yl . B = n

e » Vz

eoe

tan? e ~ 1 - U W-R

U= ( A + B) 2. W == (A

oee

tarr' e ~ 1 - U W-Q

U= ( A + B) 2. W =

C' C' C'

Al ' A2 ' A3 ' A3 R == (A + B) 2. A == nXI. B = nY2. C == nX3. D = n zi F F' D + B' Al ' A2 ' A3 ' Al ' (A + 2. Q = (A + 2. A = n rt = n X2. C = nX3. E = n Z2. F F' A + E' Al ' A2 ' A3 ' A2 '

cr VJ

F'

+ B) 2.

B)

B)

B

= n Z3 =

A3 n Z3 A3

S' t::C ;. >
sin 4> sin b cos 2£5

(2.70)

+(d24 - dIS) cos 0 sin Ocos4> sin ct> sin c5(4 cos2 t5 - 1) -(d31 cos2 ct> + d32 sirr' ct» cos2 (} sin 0 cos c5 sin2 t5 +(d31 sin2 ct> + d 32 cos 2 ct» sin 0 cos b sin2 c5

- 2(dIS cos2 ct> + d 24 sin2 ct» cos2 (} sin 0 cos c5 sin2 c5 -(d IS sin2 ct> + d24 cos 2 ct» sin 0 cos c5 cos 2c5 -d33 sirr' 0 cos c5 sin2 c5 .

(2.71)

28

2 Optics of Nonlinear Crystals

Table 2.5. The effective nonlinearity of mm2 point group biaxial crystal for the different assignments between the dielectric and crystallographic coordinate systems Assignment

d~f/ (Type I)

d~~f (Type II)

X, Y,Z

---+ a,b,c

2d15AH(BDH - CE)(BDE + CH) +2d24AH(BCE - DH)(BCH + DE) +d31 AE(BDH - CE)2 +d32AE(BCH + DE)2 +d 33A3H2E

-d I5(AH(BDE + CH)2 +AE(BDH - CE)(BDE + CH)] -d24[AH(BCE - DH)2 +AE(BCE - DH)(BCH + DE)] -d31AE(BDH - CE)(BDE + CH) -d32AE(BCE - DH)(BCH + DE) -d33A3E2H

X, Y,Z

---+ b.a,c

2d 1SAH(BCE - DH)(BCH + DE) +2d 24AH(BDH - CE)(BDE + CH) +d31 AE(BCH +DE)2 +d 32AE(BDH - CE)2 +d 33A 3H2E

-dIS [AH(BCE - DH)2 +AE(BCE - DH)(BCH + DE)] -d24 [AH(BDE + CH)2 +AE(BDH - CE)(BDE + CH)] -d31AE(BCE - DH)(BCH + DE) -d32AE(BDH - CE)(BDE + CH) -d33A3E2H

X, Y,Z

---+ a,c,b

2d15AH(BDH - CE)(BDE + CH) -2d24A2EH(BCH + DE) -d 31(BCE - DH)(BDH - CE)2 -d 32A2H2(BCE - DH) -d 33 (BeE - DH) (BCH + DE)2

X, Y,Z

---+ b,c,a

d 15[(BCH + DE) (BDE + CH)2 +(BCE - DH)(BDH - CE)(BDE + CH)J +d 24[A 2EH(BCE - DH) +A 2E2(BCH + DE)] +d31 (BCE - DH)(BDH - CE)(BDE + CH) +d32A2EH(BCE - DH) +d33(BCE - DH)2(BCH + DE) d I5[A 2EH(BCE - DH) +A 2E2(BCH +DE)J

x.r.z-:».».«

X, Y,Z

---+ c,a,b

-2d 1sA 2EH(BCH + DE) +2d 24AH(BDH - CE)(BDE + CH) -d 31A2H2(BCE - DH) -d 32(BCE - DH)(BDH - CE)2 -d 33 (BCE -DH)(BCH +DE)2

+d24[(BCH + DE)(BDE + CH)2 +(BCE - DH)(BDH - CE)(BDE + CH)] +d 31A2EH(BCE - DH) +d32 (BCE - DH)(BDH - CE)(BDE + eH) +d33(BCE -DH)2(BCH + DE) dIS[A 2E2(BDH - CE) +A 2EH(BDE + CH)]

-2d 1SA2EH(BDH - CE) -2d 24(BCE - DH) x (BDH - CE)(BCH + DE) -d31A2H2(BDE + CH) -d 32 (BCH + DE)2(BDE + CH) -d 33(BDH - CE)2(BDE + CH)

+d24[(BCE - DH)2(BDH - CE) +(BCE - DH)(BCH + DE) (BDE + CH)J +d31A2Ef/(BDE + CH) +d32(BCE - DH)(BCH + DE)(BDE + CH) +d33(BDH - CE)(BDE + CH)2

-2d 1S(BCE - DH) x (BDH - CE)(BCH + DE) -2d 24A2EH(BDH - CE) -d3l (BCH + DE) 2 (BDE + CH) -d 32A2H2(BDE + CH) -d33 (BDH - CE)2(BDE + CH)

d 15[(BCE - DH)2(BDH - CE) +(BCE - DH)(BCH + DE) (BDE + CH)] +d24[A 2E2(BDH - CE)+A 2EH(BDE + CH) +d3 l (BCE - DH)(BCH + DE) (BDE + CH) d 32A 2EH(BDE + CH) d33(BDH - CE)(BDE + CH)

As it was mentioned above the existence of both the nonzero deff values and of phase-matching direction (Opm, 4>pm), is the necessary and sufficient condition for an effective three-wave interaction. It should be emphasized that when varying 0, 4>, b together with deff some other parameters of three-wave interaction such as angular, thermal (temperature), and spectral bandwidths,

2.11 Crystal Symmetry and Effective Nonlinearity: Biaxial Crystals

29

anisotropy ("walk-ofl") angle, etc., are also changed. Therefore the maximum value of deff in the general case does not correspond to the maximum efficiency of interaction. From the practical point of view the calculation of deff in the particular case of light propagation in the principal planes of a biaxial crystal (XY, YZ, ZX; in the ZX plane two different cases: f) < Vz and f) > Vz should be distinguished) is of significant interest. The corresponding expressions can be deduced from Table 2.5 using values of the angles f), 4>, ~ and coefficients A, B, C, D, E, H for light propagation in principal planes from Table 2.6. It should be noted that when calculating the principal plane values of the angle ~ it is necessary to evaluate correctly the arising indeterminate form of (2.66), each time taking into account the definition range of the angle ~. The "sign" (negative or positive) of the principal plane determinates the assignment between "s.f" and "o.e" indices. For instance, for the case with nx < nr < nz in the ZX plane, an ordinary wave corresponds to a slow wave at f) < Vz and to a fast wave at f) > Vz (Fig. 2.12a); for the case nx > ny > nz the situation is opposite (Fig. 2.12b). Tables 2.7 and 2.8 list the possible types of interactions and Tables 2.9 and 2.10 contain the calculated expressions for deff for the cases of light propagation in principal planes. To use these tables (remember they correspond to the biaxial crystals of the mm2 point group!) it is necessary first to determine the assignment between the coordinate systems (X, Y, Z) and (a, b, c). Then using the data of Table 2.5 for the given assignment, the general expressions for deff and for ss-f or sf-f interactions could

Table 2.6. Meaning of the angles and coefficients for the formulae from Table 2.5 in the case of light propagation in the principal planes of mm2 point group biaxial crystal Angles and coefficients

Principal plane

Xy

YZ

XZ

e < Vz

e > Vz

()

n/2

e

e

e

A B

1 0

sin f

C D

sin cos

sinO cos() 0 0 1

sinO cos() 0 0 1

s

0 0 1

n/2

H

0 0 1

I 0

0 0 1

s

-n/2

nx > ny > nz -n/2

E H

--1 0

-1 0

0 0 1

-1 0

E

cosO n/2 1 0

nx < ny < nz

-n/2

30

2 Optics of Nonlinear Crystals

Table 2.7. The possible types of phase matching in the principal planes of the mm2 point group biaxial crystal for the case nx < ny < nz Principal plane Assignment

XY

YZ

XZ f)
ny > nz Principal plane Assignment

Xy

YZ

e < Vz

(»

X,Y,Z ---+ a,b,c or ---+ b.a,c X,Y,Z ---+ a.c b or ---+ b.c,a X,Y,Z ---+ c,b,a or ---+ c.a.b

1(+)

1(-)

11(+)

1(-)

ee-o

oo-e

oe-o.eo-o

oo-e

11(+)

1(-)

1(+)

11(-)

XZ Vz

oe-o,eo-o

oo-e

ee-o

oe-e,eo-e

11(+)

11(-)

11(+)

1(-)

oe-o.eo-o

oe-e,eo-e

oe-o.eo-o

oo-e

be determined. For the concretization of these expressions it is necessary to substitute the coefficients A, B, etc., using (2.69). Note that the angles f) and 4> determine the direction of three-wave phase-matched collinear interaction of light waves inside the biaxial crystal whereas the angle ~ is deduced from (2.66) using the given values f), 4> angles and taking into account the definition range of ~. In the case of light propagation in the principal planes, Tables 2.7-2.10 should be employed. First using the data of Tables 2.7,8 for the given assignment between the coordinate systems and relation between the principal values of the refraction index, the possible types of phase matching are determined, then from Tables 2.9,10 the formulae for deff can be found. The above-discussed method of calculation of deff values for mm2 point group crystals can be applied to the nonlinear biaxial crystals of other point groups. The calculations performed in the case of the biaxial crystals of the 222 point group show that upon the validity of Kleinman symmetry relations the single nonzero component d xyz exists for all possible assignments between two reference frames (Table 2.11).

2.1 Crystal Symmetry and Effective Nonlinearity: Biaxial Crystals

31

Table 2.9. The deff expressions for the principal planes of the mm2 point group biaxial crystal in the case nx < nr < nz Assignment

Plane

d~fff (Type I)

d~~f (Type II)

X,Y,Z -sab.c

XY YZ XZ, XZ, XY YZ XZ, XZ, XY YZ XZ, XZ, XY

0

d I5 sin lj> + d24cos 2 lj> d I5 sin f

X, Y,Z -sb.a.c

X,Y,Z -va.cb

X,Y,Z -sb.c,a

Vz 0> Vz

f}
Vz

0 < Vz f» Vz

d3I cos

YZ X,Y,Z -sc b.a

X, Y,Z -sc.ab

XZ, XZ, XY YZ XZ, XZ, XY YZ XZ, XZ,

0 d32 sin f} 0 0 0 d 3I sinO 0 d32 cos 0 0 d32 sin 2 f} + d31 cos 28

0 < Vz (J> Vz

0 < Vz 0> Vz

B «; Vz 0> Vz

0 0 d3I sin 2 0 + d32 cos? 0 d 3I sin d3I sin 2 0 + d32 cos? f) d32 cos f)

0 d 32 sin d32 sin 2 0 + d3I cos? (J d3I cos£) 0

2

0 d24 sin f 2 d24 sin t/> + dIS cos? t/> d 24 sin 0 0 d I5 sin f 0 d I 5 cos 0 d24 sin 2 (J + d I5 cos? 0 0 0 d 24 cosO 2 d I5 sin 0 + d24cos 2 0 0 0 0 0 d 24COSO 0 0 0 d I5 cos 0

Concerning the biaxial crystals of 2 point group it should be mentioned that in [2.25] the expressions for effective nonlinearity in the dielectric reference frame (X, Y, Z) using nonlinear coefficients defined in crystallographic reference frame (a, b, c) were deduced for MAP crystal. In all other ensuing works (see, for instance, [2.26-28]) the determination of d-tensor coefficients was made directly in dielectric coordinate system (X, Y, Z). Table 2.12 presents the expressions for deff and possible types of phase matching for biaxial crystals of the 2 point group when Kleinman symmetry relations are valid and nonlinear coefficients are measured in dielectric reference frame. The inclusion of birefringence (anisotropy) in the calculation of dat for light propagation into a biaxial crystal is complicated enough and we haven't done it here. It is possible, however, as a first approximation, (2.65), to substitute instead of 0, the values (0 ± p), depending on the "sign" of the crystal. Usually we have p ~ (}pm; but the inclusion of angle p is necessary for completeness of the physical picture as well as for the increase of calculation accuracy. In conclusion, note that the lack of adherence to uniform nomenclature and conventions in nonlinear crystal optics (first of all, for the biaxial crystals) has resulted in growing confusion in the literature. In [2.29] the standards were

32

2 Optics of Nonlinear Crystals

Table 2.10. The deff expressions for the principal planes of the mm2 point group biaxial crystal in the case nx > ny > nz Assignment

Plane

d:~/(Type I)

d:~f (Type II)

X, Y,Z -sab.c

XY YZ XZ, 8 < Vz XZ, 8> Vz XY YZ XZ, 8 < Vz XZ, 8> Vz XY YZ XZ, 8 < Vz XZ,8> Vz XY YZ XZ, 8 < Vz XZ, 8> Vz XY YZ XZ, 8 < Vz XZ, 8> Vz XY YZ XZ, 8 < Vz XZ, 8> Vz

d 3I sin2 + d32 cos 2 ljJ d 3I sin 8 0 d 32 sin 8 2ljJ d 32 sin + d3I cos 2 ljJ d32 sin 8 0 d 3I sin 8 0 d 3I cos 8 2 d 32 sin 8 + d3I cos? 8 0 0 d 32 cos 8 d3I sin2 8 + d32 cos 2 8 0 0 0 0 d 32 cos 8 0 0 0 d 3I cos 8

0 0

X, Y,Z -sb.a,c

X, Y,Z -sa.cb

X,Y,Z -sb.c.a

X,Y,Z -vcb.a

X, Y,Z -sc.ab

d24 sin 8 0 0 0 d I5 sin 8 0 d 24 cos ljJ 0 0 2 d24 sin f) + d I5 cos? 8 d I5 cos ljJ 0 0 d I5 sirr' 8 + d24 cos? 8 d I5 sinljJ d I5 sin2 f) + d24 cos 2 8 d 24 cos 8 0 d 24 sin ljJ 2 d 24 sin 8 + d I 5 cos? 8 d I5cos 8 0

Table 2.11. Expressions for deffand possible types of phase matching in the principal planes of the 222 point group biaxial crystal when Kleinman symmetry relations are valid Plane

nx < nr < nz

nx > nr > nz

XY YZ XZ, 8 < Vz XZ, 8> Vz

dI4sin2ljJ, type 11(-) d I4 sin 28, type 1(+) -d I4 sin 28, type 11(-) -d I4 sin 28, type 1(+)

-d I4 sin 2ljJ, type 1(+) -d I4 sin 28, type 11(-) d I4 sin 28, type 1(+) dI4 sin 28, type 11(-)

proposed in order to eliminate any ambiguity in the definition of nonlinear tensor components, effective nonlinearity, "walk-off" angle, and so on.

2.12 Theory of Nonlinear Frequency-Conversion Efficiency The initial equation for calculation of the nonlinear frequency-conversion efficiency is the wave equation derived from the Maxwell equations [2.1-4]

2.12 Theory of Nonlinear Frequency-Conversion Efficiency

33

Table 2.12. Expressions for deff and possible types of phase matching in the principal planes of the biaxial crystal of 2 point group when Klienman symmetry relations are valid and nonlinear coefficients are defined in dielectric reference frame Plane

nx < ny < nz

nx > ny > nz

XY

d 25 sin2cP, type 11(-)

sin 2cP, type 1(+) cos cP, type 11(+) d 21 cos 8, type I(-} d25 sin 28, type 11(-) 2 2 d21 cos 8 + d23 sin cP +d25 sin 28, type 1(+) 2 2 d 21 cos 8 + d23 sin cP +d 25 sin 28, type 11(-) d25

cos cP, type 1(-) d21 cos 8, type II(+} d25 sin 28, type 1(+) 2 2 d21 cos 8 + d23 sin cP +d 25 sin 28, type 11(-) d21 cos- 8 + d 23 sirr' cP +d 25 sin 28, type 1(+) d23

YZ XZ, 8< Vz XZ, 8> Vz

d23

eo82E(r,t) __ 4n82PNL(r,t) cur 1 cur 1E(r, t ) + e 2 82 2 82 tc t

(2.72)

in combination with (2.2) for nonlinear polarization (in the approximation of square nonlinearity) (2.73) and with initial and boundary conditions for the electric field E(r,t). In (2.72, 73) r is the radius vector, t is time, and c is light velocity. Let us present the field E as a superposition of three interacting waves

E(r,t)

1

3

==2n~1(PnAn(r,t)exp[j(wnt-kn

·r)] +C.C.),

(2.74)

where A(r,t) are the complex wave amplitudes; W n and k n are frequencies and wave vectors, respectively; and C.C. means "complex conjugate". Substituting of (2.74) into (2.72) with allowance for (2.73) and using the method of slowly varying amplitudes gives the following truncated equations for complex amplitudes [2.4]:

== jO'IA3A~ exp(j~kz) , M2A 2 == j0'2A3A i expU~kz) , M3A3 == j0'3AIA2 exp( -j~kz)

(2.75)

MIA}

(2.76) (2.77)

,

where operator M; has the form A

M;

8

8

j

= f}z + P f}x + 2kn .

(f)2 f}x 2

82

&- )

-1

+ f} y 2 + un

+ Jgn 8t 2 + bn + Qn(A).

8 f}t

(2.78)

The calculation is carried out in the Cartesian coordinates x, y, Z, where z is the propagation direction (not to be confused with the dielectric axes X, Y, Z). In

34

2 Optics of Nonlinear Crystals

(2.75-78) p; are the birefringence (or "walk-off') angles (the "walk-off' of an extraordinary beam being assumed to be in the XZ plane), (In are the nonlinear coupling coefficients, u; are the group velocities, gn are the dispersive spreading coefficients, 11k is the total wave mismatch, bn are the linear absorption coefficients, and Qn takes into account nonlinear (commonly two-photon) absorption. The following relations take place:

== 4nk l,2 n~~ PI,2 d P2,IP3 , (J3 == 2nk3n32 P3 dPI P2 , bn == kn(2n~)-lpn[lm{80(wn)}]Pn , n == 1,2,3 (JI,2

gn == =

Un

1

(a k)

(2.79) (2.80) (2.81)

2

2" aw2

(2.82)

'

OJ=OJ n

(~;) w=w. C[8~:)L=w.

= c

=

[nn + ron( : : ) w=w-l -I

(2.83)

.

In (2.81) 1m{80 ( W n)} is the imaginary part of the linear dielectric susceptibility tensor responsible for linear absorption of radiation. The sequence of writing vectors and tensors in (2.79-81) should not be violated. Finally 11k

== M L + I1ktsa + M pr + I1kfeg ,

(2.84)

where I1kL is the linear wave mismatch: (2.85) I1ktsa is the mismatch due to thermal self-actions (tsa) in nonlinear crystal, Akfeg is the mismatch due to free-carrier generation in the conduction band because of a nonlinear absorption. Thermal mismatch appears in the thermal conductivity equation, which has the following form for a stationary (with respect to heat) process: 3

I1r { I1ktsa (r,z)} = -cp-;/

L s,», (a~(r, t))

.

(2.86)

n=l

Here I1r is the Laplace operator with respect to the transverse coordinates r == (x, y); Per is the critical power of self-focusing equal to (2.87)

where B is the so-called dispersive birefringence, for SHG B == nol - n2; T is the temperature; To is the temperature of the crystal at which the z-axis (the normal to the input crystal surface) coincides with the phase-matching direction; y is the thermal conductivity coefficient of the crystal; and L is the length of the crystal. A mean square of real amplitudes an == I An I is equal to

(a~(r,t)) = f

1:

2.12 Theory of Nonlinear Frequency-Conversion Efficiency

a;(r,z,t)dt,

35

(2.88)

where f is the pulse-repetition frequency of laser radiation. The appearance of heat mismatch is physically related to non-uniform (over the beam cross section) radiative heating of a nonlinear crystal. Thermal conductivity equation given by (2.86) can be solved with the corresponding boundary conditions and with truncated equations (2.75-77). Nonlinear absorption in the crystal (Qn -# 0) must be taken into account not only in the truncated equations but mainly in the value of bn in (2.86). For SHG (at == a2), we have [2.4] QI == PI2a~;

Q2

== 2PI2 aT + P22a~

;

(2.89)

where Pl2 and P22 are the coefficients of mixed (hWI + hW2 > E g ) and twophoton (2hw3 > E g ) nonlinear absorption (E g is the value of the forbidden energy band, i.e., the band gap). Note that for great nonlinear (usually twophoton) absorption at a maximum (sum) frequency W3, total (linear and nonlinear) absorption at frequencies WI,2 and W3 are not equal. This may result in asymmetry and even in hysteresis of the temperature dependence of the resulting radiation power (near the temperature of phase matching). Photorefraction (the photorefractive effect) arises in some nonlinear crystals (of lithium niobate type) and consists in a radiation-induced change of the refractive index. In the case of continuous irradiation of lithium niobate at a frequency W3 with a power density 83 ~ 200 W cm- 2 , M == L\(no - ne)~ 10- 3 . For pulse irradiation of lithium niobate with 83 ~ 108 W cm", M ~ p8:;1/2, where p == 6 x 10-9cm W- I / 2 . At small M, the value of L\kpr can be compensated for at the expense of L\kL, i.e., by phase mismatching (this can always be realized in practice). Remember that the photorefraction may result in coloration of the crystal, increase of absorption, and thermal self-actions. Nonlinear absorption is accompanied with electron transitions from the valency band to the conductivity band, i.e., free-carrier generation (L\kfcg). The fcg-effect leads in turn to two phenomena: an additional absorption at all three frequencies (absorption on free carriers) and an additional wave mismatch. The wave mismatch L\kfcg is proportional to the square of the power density (i.e., to the fourth power of the amplitude) of two-photon absorbed radiation, generally at a maximum frequency L\kfcg == -qaj ,

(2.90)

where q is a coefficient depending on the nonlinear absorption parameters, lifetime of free carriers, and so on. The fcg-effect must be taken into account when crystals of lithium and barium sodium niobates are used; then the absorption on free carriers at all three frequencies may be neglected, but the mismatch M fcg is left in the equations. Now we shall reconsider the operator M; (2.78). Its first term (the derivative with respect to z) describes changes of the amplitudes in the process of

36

2 Optics of Nonlinear Crystals

their propagation and interaction. The second term (the derivative with respect to x) describes the influence of crystal anisotropy (the" walk-off' of an extraordinary beam along the x axis). The third term, containing second derivatives with respect to transverse coordinates x and y, corresponds to the diffraction effect (the diffractive spreading of the beam). The fourth term (the derivative with respect to time) describes the effect of temporary modulation (the pulse mode), including the effect of group-velocity mismatch of the pulses. The term containing the second derivative with respect to time corresponds to the effect of the dispersive spreading ofpulses. The terms bn and Qn(A) describe linear and nonlinear absorption, respectively. The right-hand parts of eqs (2.75-77) describe nonlinear interaction of the waves. An exact calculation of the efficiency of SHG, SFG, and DFG convertors according to (2.75-77) is very complex and generally requires the numerical calculation. Only in some simple cases do analytical solutions allow one to evaluate roughly the conversion efficiency. For proper evaluation of the efficiency, the parameters of the initial (convertible) radiations and of the crystal converter must be known, and an adequate calculation procedure must be chosen on the basis of the recommendations below. Let us introduce the effective lengths of the interaction process: 1) Aperture length La: (2.91)

where do is the characteristic diameter of the beam and p is the anisotropy ("walk-off') angle. 2) Quasi-static interaction length Lqs : L qs == x]»

(2.92)

where 1: is the radiation pulse duration and v is the inverse group-velocity mismatch. For SHG (2.93)

where (2.83).

UI

and

U3

are the group velocities at the corresponding wavelengths

3) Diffraction length

Ldif:

kd~ 4) Dispersive spreading length Lds: Ldif ==

(2.94)

(2.95)

where g is the dispersive spreading coefficient (2.82). A nonlinear interaction length LNL is also introduced:

2.12 Theory of Nonlinear Frequency-Conversion Efficiency

LNL

1 , uao

37

(2.96)

== -

where a is the nonlinear coupling coefficient (2.79,80) and ao is expressed by the equation ao == [ai(O)

+ a~(O) + a~(O)] 1/2

,

(2.97)

where an(O) are the wave amplitudes at the input surface of the crystal (at == 0). Whether or not a given effect must be taken into account in the mathematical description of nonlinear conversion is determined by a comparison of the crystal length L with the corresponding effective length Leff from (2.91-96). If L < Leff, the corresponding effect can be neglected. For instance, when L < La, one may neglect the anisotropy effect and put the second term in operator Un equal to zero; when L < Ldif, the diffractive spreading of the beam can be neglected; and so on. Note the role of the nonlinear interaction length LNL. When the condition L < L N L is fulfilled, the so-called fixed-field approximation is realized; for instance, for SFG it means that the SF field amplitude is

z

(2.98)

and the nonlinear equations given by (2.75-77) are transformed into linear (with respect to the real field amplitudes a == IAI) equations. In particular, for the SF field amplitude we have (2.99)

where 4JI,2 are the wave phases. When L ~ LNL, we must solve exact (nonlinear) equations. When L < L N L and all of Leff == 00, we have plane-wave fixed-field approximation. With L < L qs ~ Ldis (the inequality Lqs ~ Ldis is valid always) the quasi-static approximation takes place, as well as with L < L qs, L < La and L < Ldif we have quasi-plane-wave approximation. The very important difference between plane-wave and quasi-plane-wave approximations lies in the fact that in quasi-plane-wave case it is necessary to take into account the inhomogeneity of spatial (beam) and temporal (pulse) intensity distributions of interacting waves (by simple integrating with respect to the time for pulses or over the area for beams, see below). But in this case it's not necessary to take into account "walk-off' angle, diffraction, group-velocity mismatch and dispersive spreading. Thus, before calculation one should 1) determine all the effective lengths Leff of the process, compare them with the length L of a nonlinear crystal, and find out all of the effects that must be taken into account; 2) calculate the nonlinear interaction length LNL, compare it with the crystal length L, and determine whether the fixed-field approximation is valid or exact nonlinear equations must be solved.

38

2 Optics of Nonlinear Crystals

Here are some practical cases with corresponding recommendations. Under continuous-wave laser irradiation, we may neglect the group-velocity mismatch (Lqs = (0) and the dispersive spreading of pulses (Lds = 00). In the practically used crystals with L ~ 1 em we may neglect the following: the diffraction and anisotropy for the beams of do ~ 1 em in diameter; group velocity mismatch at r ~ 10-9 s; dispersive spreading at 'r ~ 10- 12 s; and nonlinear absorption and the fcg-effect at 2hw3 < E g • The photorefractive effect may be neglected in calculations, because the value dkpr (but not an additional absorption due to photorefractive effect!) is easily compensated for by an additional turn of the crystal (in lithium niobate crystals the photorefractive effect disappears completely at To ~ 170°C). Diffraction must be taken into account only for conversion of focused beams [2.4,30]. If the crystal length L is smaller than each effective length, the operator Un in (2.78) has the form A

M n = bn

d

+ dz

(2.100)

.

When in this case (L < Leff) the radiations being converted are temporally and spatially modulated (pulse duration 'r, beam diameter do) and the modulation shape is nonuniform (for instance, Gaussian beams, Gaussian pulses), the following calculation procedure can be used within fairly good accuracy. The beam (or pulse) envelope of the radiation being converted is approximated by a step-wise function (Fig. 2.13), the field amplitude inside of each step being constant. For each step - i.e., for each field amplitude value,the conversion efficiency is calculated by the equations for plane waves. Then the results are summed (integrated) with respect to transverse coordinates (or time), and the power (or energy) of the beam (or pulse) of the resulting radiation is determined. If the condition Po < Per is fulfilled, where Po is an average (or continuous) power of the radiation being converted at the input surface of the crystal, then the effects of thermal self-actions may be neglected (and the thermal conductivity equation need not be solved). If an opposite inequality is valid, truncated equations must be solved together with the thermal conductivity equation; two variants are possible. In one there is no dispersion of the absorption coefficients (bl = b2 = b3) and therefore the thermal conductivity equation as a first approximation can be solved independently on the truncated equations. If in this case the heat contact of the crystal with the outer medium (thermostat) is ideal - i.e., there is no temperature jump at the crystal-thermostat interface - we have for temperature mismatch dktsa

4ny

= -p [T(r) er

where

- To] ,

(2.101)

2.12 Theory of Nonlinear Frequency-Conversion Efficiency

0123

t,.

39

5678

t,r Fig. 2.13. Pulse (beam) approximation with a step-wise function for calculating the conversion efficiency in quasi-static (for a pulse) and diffraction-free (for a beam) approximations

T(r)

=

T(O) - i>Po

2ny

[In(2 wij,2) - Ei(-2-C) + c] . wij

(2.102)

Here T(O) is the temperature on the beam axis, Wo is the characteristic radius of the convertible radiation beam, C == 0.5772. " is the Euler-Mascheroni constant, and Ei(x) == f~x ex~y) dy is the integral exponential function [2.31]. For instance, for a LiNb0 3 crystal (b ~ 0.01 cm", y = 2.6 X 10- 3 Wcm-IK- I) at Po ~ lOW the temperature gradient between the crystal axis and beam boundary may be about 2K, which exceeds the temperature bandwidth (see below). The temperature mismatch calculated from (2.101) is substituted into the truncated equations. They are solved for each value of the transverse coordinate r, and then the summation over the surface area is carried out to determine the power of the resulting radiation. In the second variant, when bI,2 =1= b3, the thermal conductivity equation cannot be solved independently of the truncated equations, and the solution can be found only by using numerical calculation. The situation is similar for a temperature jump at the crystal-thermostat interface. Figure 2.14 illustrates typical dependences of the SHG conversion efficiency on the average input power of the fundamental laser radiation PI (0) is the widely used CDA and DCDA crystals with a typical absorption. A decrease of losses in nonlinear crystals is a cardinal way of eliminating heat self-action effects.

2 Optics of Nonlinear Crystals

40

Fig. 2.14. SHG conversion efficiency versus the average power of fundamental laser radiation in 3 cm long CDA and DCDA crystals (A. I = 1.06 ,urn)

'l

as

o.J

0.2

0.1

o

5

15

10

~(O)[w]

20

2.13 Wave Mismatch and Phase-Matching Bandwidth In real frequency converters the situation is far from ideal: the convertible radiation is not a plane wave - i.e., it is divergent, pulse and nonmonochromatic - and the temperature of the crystal converter is unstable. Therefore, in practice we must calculate the following parameters of nonlinear frequency converters: angular, spectral, and temperature bandwidths corresponding to maximum permissible divergence, spectral width of the convertible radiation, and instability of temperature. The value ~k is a function of crystal temperature T, frequencies of the interacting waves Vn , and deviation from the phase-matching angle bf} == f} - f}pm. The dependence of ~k of these parameters in the first (linear) approximation is determined by first derivatives:

A.k(T, MJ, v) ~ A.k(O) where ~k(O)

~k(O)

8(dk)

8(~k)

8(dk)

+ ---arA.T + 8(MJ) MJ + --a;-A.v ,

(2.103)

is the mismatch for the exact phase matching (therefore

== 0), and partial derivatives with respect to one argument are taken

under the condition that the other two arguments are constant. Below we will show that the power of the resulting radiation in the fixedfield approximation is halved if wave mismatch is equal to

M == 0.886

tt

L

.

This makes possible the evaluation of the angular spectral (Av) bandwidths:

(2.104) (~f}),

temperature

(~T),

and

2.13 Wave Mismatch and Phase-Matching Bandwidth

41

Table 2.13. Equations for calculating the SHG internal angular bandwidth for the different types of interaction

Type of interaction

Internal angular bandwidth for SHG

ooe

AO

+ (01 = (02)

((01

0.443AI [1 + (no2/ne2)2 tan? OJ L tan Oil - (no2/ne2)2In2(0)

eoe,oee

2J 0.886 n~(O) [1 - (nol/ned n2(0) [1 - (n02/ne2)2J I-I 2 L tan 0 Al [1 + (nodne I) 2 tan 0] - A2 [1 + (nodned tan? 0] 1

M

=

eeo

AO

=

eoo,oeo

AO =

Q.443AI [1 + (nol/n el)2 tan 2 OJ 2

LtanO[I - (nol/ned Jnj(O) 0.886AI [1 + (nOI/ne.)2 tan 2 OJ

----------

L tan 0[1 - (nol/n et}2Jnj (0)

Table 2.14. Equations for calculating the SFG internal angular bandwidth for the different types of interaction

Type of interaction

ooe

eoe

oee

eeo

Internal angular bandwidth for SFG

((01

+ (02 = (03)

0.886A3 [1 + (no3/ne3)2 tan 2 OJ AO = - - - - = - - - - - - - - - ' LtanOII- (no3/ne3)2In3(0) A8

=

0.886

1

nj(8) [1 - (nOI/net}

2J

LtanO Al [1 + (nol/ned2tan28J AO

=

AO

=

_ n3(0) [1 - (n03/ne3)2J I-I ),3[1

+ (n03/ne3)2tan28J

0.886 n2(8) [1 - (no2/ne2)2J _ n3(0) [1 - (n03/ne3)2J I-I L tan 0 A2 [1 + (n02/ ne2)2 tarr' 8J ),3 [1 + (no3/ne3)2 tan? OJ 1

0.886 { nj(8)[1- (nol/neI) L tan (I

Al [1 + (nol /n eI)2

2J

tan 2 OJ

eoo

O.886AI [1 + (nol/n el)2 tarr' 0] A8 = ----=-------~ Ltan8[1 - (nOI/neI)2Jn~(0)

oeo

O.886A2 [1 + (no2/ne2)2 tan 2 OJ A8 = - - - - - - - - - L tan 8[1 - (n02/ne2)2Jn2(8)

+

n 2(8) [1 -

(n02/ne2)2J }-I

A2 [1 + (no2/ne2)2 tan 2 OJ

42

2 Optics of Nonlinear Crystals

Table 2.15. Equations for calculating the SHG internal angular 90° phasematching bandwidth for all types of interaction

Types of Interaction

Internal angular 90° phase-matching bandwidth for SHG (WI + WI = (2)

ooe

eoe oee eeo eoo oeo

Table 2.16. Equations for calculating the SFG internal angular 90° phasematching bandwidth for all types of interaction

Types of interaction

Internal angular 90° phase-matching bandwidth for SFG (WI + (Q2 = (3)

1/2 ooe

~8=2

0.886A3

(Ln e3 [1 - (ne3/n 3 )2] ) 0

eoe

oee

_ (0.886Inel [1 - (n- et)2] - -ne L Al nol A3 e -n-e _ (0.886Ine2 A8-2 - - - [1- (n 2)2]

L

eeo

eoo

oeo

3 [

A8 - 2 - - -

A2

no 2

3 [

1-

3)2]

no3

(ne I-I) 1/2

1- -

A3

_ (0.886Inet [1 - (n- el)2 ] na [ Al nol + -A2 1 -

A8 - 2 - - L

(n- e I-I) 1/2 3)2]

no3

(n- C2) 2]1-)) 1/2 no2

2.13 Wave Mismatch and Phase-Matching Bandwidth

43

Table 2.17. Equations for calculating the SHG temperature bandwith for the different types of interaction

Temperature bandwidth for SHG

Type of interaction

WJ +WJ =W2

lano l _ an;(8)Isr or

ooe

AT = 0.443AJ

eoe,oee

AT == 0.886AJ lan~(8)

eeo

AT

=

0.443Al L

eoo,oeo

AT

=

0.886Al

L L

L

tt

[8(~k)]-1

dO == 1.772 L 8(bO)

_

J

anol _ 2an~(8)I-l

er + or or lan~(8) _ ano21-1 or or lan~(8) anol _ 2ano21-1 et + et nt

(2.105)

'

(J-(Jpm

~T -== 1.772~ [a(~k)] -1 L

~v == 1.772~

L

or

,

[8 Cl1k)] -1 8v

(2.106)

T=T pm

V=V

.

(2.107)

pm

The derivatives used in (2.105-107) depend on the dispersion of the refractive indices and on the type of phase matching. Note that the expressions (2.104107) are valid, in the strict sense, only in the fixed-field approximation, but nevertheless they can be successfully used for quantitative assessments. Table 2.18. Equations for calculating the SFG temperature bandwidth for the different types of interaction

Type of interaction

Temperature bandwidth for SFG WJ

+ (J)2 == (J)3 = 0. 886

11- anol + 1- ano2 _ .Lan3(8)1-

1

ooe

AT

eoe

IlT == 0. 886

oee

IlT = 0.886

eeo

IlT = 0.886

eoo

IlT = 0. 886

11- an~ (8) +..1 ano

-..1 ano31-J

oeo

IlT = 0. 886 11-anoJ + 1-8n;(8) L Al aT A2 aT

_1- ano3!-J

L L

L L L

AJ Bl'

A2 aT

A3 aT

11- an~(8) + -l ano2 _1- an~(8)I-l Al

aT

~

aT

~

aT

11- anoJ + 1-8n;(8) _1- an~(8)I-J Al aT

A2 aT

A3 aT

11- 8nj(O) + 1-8n~(8) _1- ano3[-1 Al Al

aT

st

A2 aT 2

A2

st

A3 aT A3 DT A3 aT

44

2 Optics of Nonlinear Crystals

Table 2.19. Equations for calculating the SHG spectral bandwidth for the different types of interaction Type of interaction

Spectral bandwidth for SHG (WI +Wl = (2)

ooe

AVI

=

0.44318nol _ ani(O)IAlL aAl aA2

eoe,oee

AVI

=

0.8861anol + ani (0) _ 2 ani(O)IAlL aAl aAl aA2

eeo

AVI = 0.443 Iani (0) _ ano21-l AlL aAl aA2

eoo,oeo

AVI

l l

= 0.88618nol + ani (0) _ 2 8no21-l AlL aAl

aAl

aA2

Tables 2.13,14 contain the equations for calculating the internal (inside the crystal) angular bandwidth (~O) for SHG and SFG. The equations used for SFG can also be applied to DFG if polarization designations for the interacting waves are made in the order of increasing frequency. For Opm == 90° (90° phase matching) the first derivative 8(~k)/8(bO) becomes equal to zero and the corresponding second derivative becomes important. Hence, the 90° phase-matching internal angular bandwidth is

~Ole

-900

pm-

(--2) -1] 1/2 .

~ 2 [0.886-Ln

82(~k)

(2.108)

8(bO)

Table 2.20. Equations for calculating the SFG spectral bandwidth when the lower-frequency interacting wave has a wide-band spectrum Type of interaction

Spectral bandwidth for SFG (ml + m2 = (1)3) AI: wide-band spectrum; A2: fixed wavelength

ooe

8861 A _ 0. 1 anol + 1 an 3(0)ILlVI - - nol-ne(o) 11.3-3 -11.1L aAl aA3

eoe

A _ 0. 8861 e(o) 1 8ni(0) + LlVI - - nl -ne(o) -11.1-3 L aAt

oee

AVI

eeo

A _ 0. 8861 e(o) 1 ani (0) + 1 ano31-1 LlVI - - L - n l -no3-1I.1~ 11.3 aA3

eoo

A LlVl

oeo

0.886 anol ano31-l AVI = -L- nol - no3 - Al + A3 8i;

1

l

an 3(0)1-1 aA3

11.3--

0.8861nol -n3(0) e anol an3(0) 1-1 - AI-+A3--

= --

mt

L

a~

0.886 nte(o) -n 1 ani (0) + 1 ano31-1 o3-lI.t-- 11.3L aAl aA3

=--

1

1

a;:;

2.13 Wave Mismatch and Phase-Matching Bandwidth

45

Table 2.21. Equations for calculating the SFG spectral bandwidth when, the higher-frequency interacting wave has a wide-band spectrum Type of interaction

Spectral bandwidth for SFG (WI + W2 = (3) AI: fixed wavelength; A2: wide-band spectrum

ooe

A

- 0. 886 1 L no2

-

e(o) _ A ano2 + A an 3(O)I- 1 n3 2 a A2 3 aA3

eoe

A - 0. 886 V2 L no2

_

e(o) _ A ano2 + A an~(O)I-1 n3 2 aA2 3 aA3

oee

A

eeo

ane(o) 2 + 1 ano3 A _ 0.886 e(o) 1 LlV2 - -L- n2 - n o 3 - A2 A3 a A3

V2 -

1

V2

= 0. 886

1

L

e(o) _ e(o) _ A an~(O) n2 n3 2 aA2

+

1 an 3(O )I- 1 A3 aA3

--a;;;-

I-I

1

!-1

eoo

3 A _ 0. 886 1 1 ano 2 + 1 an 0 LlV2 - -L- nsz - n o 3 - A2 a A2 A3 aA3

oeo

A

- 0. 886 1 e(o) _ _ A an~(O) L n2 no3 2 aA2

V2 -

+ an031-1 1

A3 aA3

For 90° phase matching the angular bandwidth of phase matching for SHG and SFG can be calculated by the equations given in Tables 2.15,16. Temperature and spectral bandwidths of phase matching are calculated by the Table 2.22. Equations for calculating the angular tuning of phase matching in the case of SHG for the different types of interaction Type of interaction

Angular tuning of phase matching for SHG (WI +WI =(2)

ooe

[1 - (nol/ned2] n~ (0)

[1 - (n 2/ne2)2] n~ (0) 0

)-------- - 2 tan 0 1 + (n ol/neI)2 tan? 0 1 + (no2/ ne2)2 tan? ()

eoe, oee A21anol I

eeo

aVI

n~ (0) [1

aAI

+ an~(B) _

2an~(B)1 aA2

aAI

- (nol/ned2] tan B

ao AT [1 + (n ol) 2tan? BJ IanI (B) _ ano21 nel

aAI

aA2

nj (B) [1 - (nol/ned2] tan B eoo,oeo

46

2 Optics of Nonlinear Crystals

Table 2.23. Equations for calculating the angular tuning of phase matching in the case of SFG when the lower..frequency interacting wave has a wide-band spectrum Type of interaction

ooe

eoe

Angular tuning of phase matching for SFG (WI AI: wide-band spectrum; A2: fixed wavelength

a Vl

of)

av] ao

f

+ w2 = (3)

n3(0)ll - (n0 3/ne3)21tan 0

I

I

03 000 1 A3~ 8nHO) A3 [ 1 + (nne3 tan 2] 0 no] - n3(0) - A] M"+

n~(O)[I- (n01/neI)2]

n3(0) [1 - (n03/ne3)2]

A] [1 + (nol/net}2 tarr' 0]

A3 [1 + (n03/ne3)2 tarr' 0]

tan 0

Ine(o) _ne(O) _ A 8nHO) + A 8n3CO) I 1 3 1 a 3 aA3 Al

n~(i1) [I - (n02/ne2)2] oee

2 n3(i1) [I - (n03 / ne3 ) ]

tan 0

2 aVl _ A2 [1 + (n02/ne2)2 tan 0] ao -

A3 [1 + (n03/ne3)2 tan? 0]

I

8n~(o)1

8no l

nO]-n3(0)-AIM"+A3~

{ nHO) eeo

[I - (not/ned]

a Vl ao

Ie n

1(0)

eoo

ovr ao

nHO)

[ COif

A] 1+ n;t

aVl

ao ,1.2

+

[I - (n

o2/ne2)2]

}

~nO

A2 [I + (n02/ne2)2 tan? 0]

8n031

8nHO) A3 8 3 - n03 - Al ~+ A

[I - (not/ned] tan 0 JI

8n031

8nHO) tan 20 n~(0)-n03-Al~+A3aA3

nHO) oeo

n;(O)

Al [1 + (no] /n eI)2 tan 2 0]

[I - (n

0

2!ne2)2]

tan 0

[I + (::~f tan? oJ Inol - n 3 - Al ~oll + ,1.3 ~:31 0

equations presented in Tables 2.17-21. In Tables 2.22-27 the equations are given for calculation of the derivatives 8v/ 8e and 8v/ 8T describing angular and temperature tuning. These derivatives characterize a change of the convertible radiation frequency v with variations in the angle of temperature, respectively. Tables 2.20,21,23,24,26,27 contain the equations for SFG when one of the interacting waves has a wide-band spectrum.

Table 2.24. Equations for calculating the angular tuning of phase matching in the case of SFG

when the higher-frequency interacting wave has a wide-band spectrum Type of interaction

Angular tuning of phase matching for SFG (WI + W2 = (3) Al : fixed wavelength; A2: wide-band spectrum

ooe

8 V2 80

n3(O)!1 - (n0 3/n e3) 2 1 tan 0

n~3 0 A3 ( 1+ 2tan ~

2) In02-n3(O)-A2m-+A3~ ano2 an3(II) I 2

I 8V2 80

eoe

8 V2 80

eeo

8V2 80

Ino2 nHII)(1 -

e ano2 A3-an3(II) I n3(O) - A2-+ 8A2 8A3

n~2/n~2) 2

Adl + (no2/ne2) tan 2 0]

nHII)(1 -

n~l/n~l) 2

Al[I + (no1 /n e1) tan 2 0]

In (0) - no e

3 -

2

8V2 80

8V2 80

oeo

-

I

nj(II)(1 - n~3/n~3) 2 tan 0 A3 [I + (no3/ne3) tan? OJ

Ine(lI) - ne(lI) _;. an;(II) +;. anj(lI) I 2 3 2 8 A2 3 8 A3

I

eoo

-

n;(II)(1 - n~2/n~2) I tan 0 A2 [t + (no2/ne2)2 tan? OJ 12

an; (II) ano31 a;;;+ A3 81 3

I'll (0)( 1 - n~ 1/ n~.) tan 0

I

;.

( n~l 11) no2- no3- 2m-+ ano2 ;.3mano31 A11+2tan ne1 2 3

2

ni(O)(1 - n~2/n~2) tan 0

2) In2«() )-no3-A2~+A3man; (II) ano31

n~2 0 AI ( 1+ 2tan n~

2

Table 2.25. Equations for calculating the temperature tuning of phase matching in the case of SH G for the different types of interaction

Type of interaction

Temperature tuning of phase matching for SHG (WI +W1 = (2)

ooe

8 V1 8T

eoe,oee

eeo

eoo,oeo

8V1 8T

I

nHII)(1 - n~l/n~l) nj(II)(1 - n~3/n~3) tan II Adl + (no1/ne1)2 tan? 0] A3[1 + (no3/ne3)2 tan 2 OJ

I oee

3

18no1/8T - 8ni(O)/8TI Ai18no1/8A1 - 8ni(O)/8A21

18ne1/8T + 8no1/8T - 28ni(O)/8TI AiI8nl(O)/8A1

+ 8no1/8AI -

28ni(O)/8A21

8V1 _ 18ne1/8T- 8no2/8T I 8T - Ai18nl(O)/8A1 - 8no2/8A21 18nl(O)/8T + 8no1/8T - 28no2/8T I A~18nl(O)/8AI

+ 8nol/8AI

- 28no2/8A21

3

48

2 Optics of Nonlinear Crystals

Table 2.26. Equations for calculating the temperature tuning of phase matching in the case of SFG when the lower-frequency interacting wave has a wide-band spectrum. Type of interaction

ooe

Temperature tuning of phase matching for SFG (WI + W2 == (3); AI: wide-band spectrum; A2: fixed wavelength

aVI _ aT -

I-lOnol Al

st

+

~ 07102

et

A2

-

-lonWJ) I A3 or

In l - n3(fJ) - ~ Onol + A3 -----ax;;onH!J) I o

At

II-onj( On- - I 0)I - -0)+I-onH -02

eoe

~

aVt

aT

k aT

aT

~

Ine(o) - n e (0) -,( onj(O) I 3 I aAt

aT

+,( 3

on~(O) aA3

I

I-lOnol -lon~(O) _ -lonHO) I +

oee

At

aVt

aT

In I

aT

A2 aT

ne(o)

-,(1

-

o

3

A3

aT

onHO) +,(3 onHO)! aAI aA3

II-onj( On~-(0) - -0)+I- -I- eeo

aT

et

Al

aVt

Inj(o) -

7103 -

aAt

eoo

aT -

Inj(0) -

071 02 _

+

st

At

A2

7103 -

,(\

st

1

A3 or

,(1 onj(O) +

l-l on~ (0) -l aVt _

07103

et

A2

.L 07103 1 A3 aA3

.L 0710 3 A3

et

1

onj( 0) + .l. 071 0 3 aAI A3 aA3

1

I~ 071 01 + .L on~(O) _ .L 07103 1 oeo

8 VI aT

AI 8T

I

not

A2 aT

-no 3 -

A3 aT

,(1071aAt- +A3I -aA30

1

07103

1

2.14 Calculation of Nonlinear Frequency-Conversion Efficiency in Some Special Cases An accurate calculation of the frequency-converter efficiency in the general case with allowance to the accompanying factors is very complex. The analytical solving can be derived only for some simple special cases, but they can be used for evaluating the limiting efficiency of nonlinear frequency converters. Below, some analytical equations are given for calculating the conversion efficiency.

2.14 Calculation of Nonlinear Frequency-Conversion Efficiency in Some Special Cases

49

Table 2.27. Equations for calculating the temperature tuning of phase matching in the case of SFG when the higher-frequency interacting wave has a wide-band spectrum Type of interaction

Temperature tuning of phase matching for SFG (WI + w2 = (3); AI: fixed wavelength; A2: wide-band spectrum

11- OOo l + 1-on02 _1- onH0) I A1 st

aV2 aT

ooe

A3 ot

0 Ino2 - ne3(f}) - A2 00 onHO) I aA22 + A3 ~

11- OOH 0) + 1-on02 _1- onH0)I ~ aT ~ aT k aT

aV2 aT

eoe

A2 st

Ino2 -

on02 + A3~ onHO)!

e

n3(f}) - A2 aA2

11- onol -+1 -on;(O) ---1 -OOHO) - -1 A1 st

A2

st

A3

st

oee

aV2 aT

eeo

aV2 aT

eoo

onHO) + 1-on02 _1- on031 A2 et A3 ot aV2 _ Al st aT 000 3 0 Ino 2 - no3 - ;. 2 on aA22 + ;. 3lii; 1

oeo

aV2 aT

=:

In;(f}) - n~(f}) -

on~(O) on~(O) A2 ~ + A3 ~

I

11-onHO) on~(O) on031 - - +1- - -1Al st

Ine2(f}) -

A2 st

A3 st

on~(O) on031 no 3 - A2 ~ + A3 aA3

11-

11- onol + 1-on~(O) _ -l on 3 0

A1 st

=:

Ine

A2 st

2(f}) - no 3 -

1

A3 et

00;(0) A2~+

on031 A3aI)

2.14.1 Plane-Wave Fixed-Field Approximation

In this approximation we can neglect such restricting factors as diffraction, anisotropy, group-velocity mismatch, and dispersive spreading. In addition, we neglect heat effects, linear and nonlinear absorption (and hence the fcg-effect). In other words, in this approximation the following conditions must be fulfilled:

< LNL , < Leff , Po < Per, 2hw3 < E g

L L

•

(2.109) (2.110) (2.111) (2.112)

50

2 Optics of Nonlinear Crystals

In particular, (2.109) means for SPG (it is the fixed-field approximation)

«

01,2(0)

(2.113)

01 (z)

«

02,3(0)

(2.114)

02(Z)

«

01,3(0)

(2.115)

03(Z)

for DFG,

or

and for SHG, 03 (z)

«

01 (0) == 02 (0) .

(2.116)

The conditions (2.110-112) signify the plane-wave approximation and together with (2.109) the plane-wave fixed-fie Id approximation. We assume also that the temporal and spatial distributions of beam and pulse are homogenous. Tables 2.28,29 illustrate the equations for calculating the conversion efficiency for SHG, SFG and DFG with the use of (2.94-97) in the SI and CGS systems. Here deff is the effective nonlinearity in the phase-matching direction (see Tables 2.3,14); n, are the refractive indices at wavelengths Ai in the phasematching direction with allowance for wave polarizations; A == is the crosssectional area of the laser beam with a radius WQ (the areas of the beams of all interacting waves are assumed to be equal); and Pi are the powers of the corresponding waves with frequencies Wi' For pulsed (quasi-static) irradiation, by Pi we mean the pulse powers

n%

(2.117) where E, are the pulse energies at frequencies Wi, and Li are the corresponding pulse durations. When the powers of the mixing waves are almost equal, the conversion efficiency for SFG is determined by the equation

Table 2.28. Equations for calculating SHG, SFG, and DFG conversion efficiencyin the plane-wave fixed-field approximation in the SI system Nonlinear process SHG WI +Wl

Conversion efficiency

= W2

ill = 2k 1 -k2 SFG Wt

+W2

= W3

ill = kl +k2 -k3 DFG W3 - W2 ill = k 1 +k2 -k3 Wt =

P3

= 231t2d~L2~2 sin2(IMIL/2)

PI P

--!.

P3

cocnln2n3A3A

=

[deff] Eo =

23 rr;2d2 L2P efT

2

sinc2 (IM IL/ 2)

cocnIn2n3AiA

mjV; [P] = W; [L] = m; [A] = m; [A] 8.854 X 10- 12 AsjVm; c = 3 X 108mjs

=

=

m2 ;

2.14 Calculation of Nonlinear Frequency-Conversion Efficiency in Some Special Cases

51

Table 2.29. Equations for calculating SHG, SFG, and DFG conversion efficiency in the plane-wave fixed-field approximation in the CGS system Nonlinear process

Quantum conversion efficiency

SHG

P 27 n 5d 2 L2PI ---l = eff sinc2(IIllIL/2) PI cnin2A~A

WI

+ WI = W2

III = 2kl 8pm, the inequality n2 < not is valid which corresponds to anomalous dispersion). In region 0 < Opm neither scalar (collinear) nor vector (noncollinear) phase matching are fulfilled, and the SHG efficiency decreases with increase of the focusing parameter. In order to obtain the optimalfocusing and hence the maximum efficiencyit is necessary to optimize the function h (2.164) on the both parameters: v (mismatch) and ~ (focusing). Numerical calculation shows that the maximum h takes place at ~opt :=: 2.84, Vopt == -0.55 (it corresponds to AkoptL/2 == -1.6), and h(Vopt, ~opt) == 1.07 [2.4]. At 8pm < 90 (p i= 0) the calculation is more complicated, see [2.4,36]; note that the crystal anisotropy leads to the decrease of SHG efficiency in the case of non-focused narrow laser beams. 0

2.15 Additional Comments

65

2.14.8 Linear Absorption The absorption parameter (J =1= 0) in the absence of heat effects can be taken into account by multiplying the conversion efficiency calculated for b == 0 by the factor exp( -2bL). Since the nonlinear crystals are generally transparent for the interacting waves, the following expansion can be used: exp( -2bL)

~

1 - 2bL .

(2.170)

Note that the linear absorption coefficient (X (for intensity absorption) widely used in the literature is equal to 2b. More rigorous inclusion of the absorption can be done by substitution of expression L' == ( is the asimuthal angle, Vz is the angle between one of the optical axes and the Z axis

3.1 Basic Nonlinear Optical Crystals

Mohs hardness: 6; Transparency range at "0" transmittance level: 0.155-3.2 urn [3.1,2]; Linear absorption coefficient (X [3.3]:

[em-I]

A [urn]

ex

0.35--0.36 1.0642

0.0031 0.00035

Experimental values of refractive indices:

A [Jlrn]

nx

ny

nz

Ref.

0.2537 0.2894 0.2968 0.3125 0.3341 0.3650

1.6335 1.6209 1.6182 1.6097 1.6043 1.59523 1.5954 1.58995 1.5907 1.5859 1.58449 1.5817 1.58059 1.57906 1.57868 1.5785 1.5782 1.5780 1.57772 1.5765 1.5760 1.57541 1.5742 1.5734

1.6582 1.6467 1.6450 1.6415 1.6346 1.62518 1.6250 1.61918 1.6216 1.6148 1.61301 1.6099 1.60862 1.60686 1.60642 1.6065 1.6212 1.6057 1.60535 1.6039 1.6035 1.60276 1.6014 1.6006 1.59893 1.59615 1.59386 1.59187 1.59072 1.5905 1.6053 1.59005

1.6792 1.6681 1.6674 1.6588 1.6509 1.64025 1.6407

3.1 3.1 3.1 3.1 3.1 3.3 3.1 3.3 3.1 3.1 3.3 3.1 3.3 3.3 3.3 3.1 3.4 3.1 3.3 3.1 3.1 3.3 3.1 3.1 3.3 3.3 3.3 3.3 3.3 3.1 3.4 3.3

0.4000 0.4047 0.4358 0.4500 0.4861 0.5000 0.5250 0.5321 0.5398 0.5461 0.5500 0.5780 0.5893 0.6000 0.6328 0.6563 0.7000 0.8000 0.9000 1.0000 1.0642 1.0796 1.1000

1.56959 1.56764 1.56586 1.56487 1.5656 1.5655 1.56432

1.6353 1.6297 1.62793 1.6248 1.62348 1.62122 1.6212 1.6063 1.6206 1.62014 1.6187 1.6183 1.61753 1.6163 1.6154 1.61363 1.61078 1.60843 1.60637 1.60515 1.6055 1.5902 1.60449

69

70

3 Properties of Nonlinear Optical Crystals

Temperature derivative of refractive indices within interval 20-65 °C for spectral range 0.4 - 1.0 urn [3.3]:

dnx/dT x 106 == -1.8 , dny/dT x 106 == -13.6 , dnz/dT x 106 = -6.3 - 2.1 A , where A in urn and dnx/dT, dny/dT, and dnz/dT are in K- 1.

Experimental values of phase-matching angle (T= 293 K) and comparison between different sets of dispersion relations: XY plane, ()

== 90°

In teracting wavelengths [Jlrn] SHG, 0+0 =} e 1.908 =} 0.954 1.5 =} 0.75 1.0796 =} 0.5398

1.0642 =} 0.5321

0.896 =} 0.448 0.88 =} 0.44 0.84 =} 0.42 0.80 =} 0.40 0.78 =} 0.39 0.75 =} 0.375 0.7094 =} 0.3547

0.63 =} 0.315 0.555 =} 0.2775 0.554 =} 0.277

4>exp

[deg]

23.8 [3.5] 7 [3.5] 10.6 [3.5] 10.7 [3.1] 10.7 [3.4] 11.3 [3.3] 11.4 [3.7] 11.4 [3.8] 11.6 [3.2] 11.6 [3.5] 11.6 [3.9] 11.8 [3.10] 23.3 [3.11] 24.5 [3.11] 27.9 [3.11] 31.7 [3.11] 33.7 [3.11] 37.1 [3.11] 41.8 [3.5] 41.9 [3.12] 42 [3.13] 43.5 [3.14] 55.6 [3.15] 86 [3.5] 90 [3.16]

4>theor [deg]

[3.2]

[3.5]

[3.6]

24.04 7.03 10.64

23.98 6.81 10.39

31.32 10.18 10.42

11.60

11.36

11.39

23.25 24.53 27.94 31.69 33.72 37.02 42.09

23.01 24.29 27.70 31.47 33.51 36.83 41.94

23.14 24.44 27.88 31.66 33.71 37.03 42.12

55.32 85.75 88.97

55.29 85.74 88.87

55.42 85.97 no pm

3.1 Basic Nonlinear Optical Crystals

SFG, 0+0 =} e 1.0642+0.5321 =* =} 0.35473

1.0642 + 0.35473 0.26605

37.1 [3.6] 37.2 [3.2] 37.2 [3.3]

37.21

36.86

37.30

60.7 [3.2] 61 [3.5]

60.63

61.35

61.02

70.2 [3.2]

70.13

78.47

71.32

20 [3.6]

20.02

19.41

20.18

50.3 [3.6]

48.41

68.17

52.48

63.8 (3.6]

60.99

no pm

64.07

88 [3.6]

81.21

no pm

no pm

=}

=:}

1.3188 + 0.26605 => 0.22139 1.3414 + 0.6707 =} =} 0.44713 0.21284 + 2.35524 =} => 0.1952 0.21284 + 1.90007 =} => 0.1914 0.21284 + 1.58910 =} =} 0.1877 =:}

yz plane,

0.954 1.5 =} 0.75 1.0796 => 0.5398 1.0642 => 0.5321

SFG, o+e =} 0 1.0642 + 0.5321 =} :::} 0.35473

XZ plane, e 1.3414 =} 0.6707

Oexp

[deg]

4.2 [3.5] 5.0 [3.9]

Otheor

[deg]

[3.2]

[3.5]

[3.6]

4.67

4.17

5.00

71

72

3 Properties of Nonlinear Optical Crystals

1.3188 :::} 0.6594 1.3 => 0.65 XZ plane, 4>

5.2 [3.2] 5.4 [3.9]

5.10 5.26

4.62 4.78

5.29 5.36

== 0°, () > Vz

In teracting wavelengths [Jlm]

(}exp

[deg]

(}theor

[3.2]

SHG, e+e => 0 1.3414 => 0.6707

86.3 86.6 86.0 86.1

1.3188 => 0.6594 1.3 => 0.65

[3.5] 86.47 [3.9] [3.2] 86.26 [3.9] 86.25

[deg] [3.5]

[3.6]

86.22

88.93

86.03 86.01

87.79 87.41

Note: The other sets of dispersion relations from [3.1, 18,3, 19,20,8,21,22, 23] show worse agreement with the experiment Best set of dispersion relations (A in urn, T == 20°C) [3.2]: 2

0.01125 2 A - 0.01135 0.01277 == 2.5390 + 2 A - 0.01189

nx == 2.4542 + 2

ny

n2

== 2.5865 +

z

A2

2

-

0.01388 A ,

-

0.01848 A ,

2

0.01310 _ 0.01861 A2 • - 0.01223

Calculated values of phase-matching and "walk-off" angles: XY plane, () == 90°

In teracting wavelengths [urn] SHG, 0 + 0 => e 2.098 => 1.049 1.1523 => 0.57615 1.0642 => 0.5321 0.6943 => 0.34715 0.5782 => 0.2891 SFG, 0 + 0 => e 1.0642 + 0.5321 :::} => 0.35473 1.0642 + 0.35473 => => 0.26605 1.3188 + 0.6594 :::} => 0.4396

¢Jpm [deg]

P3 [deg]

31.61 6.06 11.60 44.19 69.91

0.840 0.213 0.403 1.086 0.730

37.21

1.046

60.63

1.006

21.11

0.705

3.1 Basic Nonlinear Optical Crystals

73

yz plane, 4> == 90° In teracting wavelengths [J.!rn] SHG, 0 + e=}o 2.098 ==> 1.049 1.1523 ==> 0.57615 1.0642 ==> 0.5321 SFG, 0 +e => 0 1.0642 + 0.5321 => => 0.35473

XZ plane

(Jpm [deg]

P3 [deg]

72.90 9.28 20.45

0.307 0.169 0.348

42.19

0.533

4J == 0°, (J < Vz

In teracting wavelengths [urn] SHG, e + 0 =} e 1.3188 ==? 0.6594

XZ plane, 4>

(Jpm [deg] PI [deg]

0.248

5.10

P3 [deg]

0.262

== 0°, (J > Vz

In teracting wavelengths [J.!rn] SHG, e +e => 0 1.3188 =} 0.6594

Opm[deg]

PI [deg]

86.26

0.191

Calculated values of inverse group-velocity mismatch for the SHG process in LBO: XY plane, () == 90°

In teracting 4>pm [deg] wavelengths [urn]

SHG,

0

+0

1.2 =} 0.6 1.1 =} 0.55 1.0 =} 0.5 0.9 => 0.45 0.8 => 0.4 0.7 =} 0.35 0.6 =} 0.3

=}

f3

[fs/mm]

e

2.36 9.37 15.74 22.94 31.69 43.38 62.63

18 37 59 86 123 175 257

74

3 Properties of Nonlinear Optical Crystals

YZ plane,

l/J = 90

Interacting wavelengths [urn]

0

f}pm

[deg]

SHG, 0 + e=}o 1.1 =} 0.55 15.98 28.96 1.0 => 0.5 0.9 =} 0.45 45.36 0.8 =} 0.4 76.88

P [fs/mm]

82 106 139 186

Experimental values of NCPM temperature: along X axis Interacting T rC] wavelengths [urn] SHG, type I 1.25 =} 0.625 1.215 => 0.6075 1.211 =} 0.6055 1.2 =} 0.6 1.15 =} 0.575 1.135 =} 0.5675 1.11 =} 0.555 1.0796 => 0.5398 1.0642 =} 0.5321

1.047

=}

0.5235

-2.9 21 20 24.3 61.1 77.4 108.2 112 148 148.5 149 149.5 151 166.5 167 172 175 176.5 180 190.3

1.025 =} 0.5125 SFG, type I 1.908 + 1.0642 =} =} 0.6832 81 1.135 + 1.0642 => 112 => 0.5491

Ref

3.7, 8 3.8 3.2 3.7, 8 3.7, 8 3.10 3.7, 8 3.1 3.7, 8 3.24,25 3.10 3.26 3.17 3.27 3.28 3.29 3.30 3.31 3.32 3.7, 8

3.10 3.10

3.1 Basic Nonlinear Optical Crystals

75

Experimental values of internal angular, temperature and spectral bandwidths: along X axis Interacting wavelengths [urn]

1.047

=}

SFG, type I 1.908 + 1.0642 => =} 0.6832 1.135 + 1.0642 =} =} 0.5491

XY plane, 0

== 90

0

SHG, 0 + 0 =} e 1.0796 =} 0.5398 1.0642 =} 0.5321

0.886 0.870 0.78

=} =}

0.443 0.435

2.3

1.9

2.1

2.1

81

7.4

3.10

112

5.0

3.10

== 20 °C)

0.5675 1.0642 =} 0.5321

A > >

10- 12 [W/m2]

0.4 0.6 0.6 1.0 470000(?) > 1.8

Ref.

Note

3.38 3.39 3.40 3.41 3.42 3.43

10 Hz

3.1 Basic Nonlinear Optical Crystals

A [urn]

Lp

[ns]

0.3547

10 10 8 7 0.03 0.03 0.015 0.018 0.025 0.5145 cw 0.5235 0.055 0.055 cw 0.5321 60 10 0.1 0.035 0.015 0.605 0.0002 0.616 0.0004 0.0004 0.0004 0.02 0.652 0.7--0.9 10 0.71--0.87 25 0.72-0.85 0.001 cw 1.0642 60 18 9 8 1.3 0.1 0.035 0.025 1.0796 0.04

Ithr X

10- 12 [W/rn2 ]

> 0.4 > 2.0 > 1.3 > 1.4 > 94 > 180 > 28 > 50 > 60 > 0.0003 > 11 > 50 > 0.004 > 0.7 > 2.2 > 45 > 31 > 44 > 250 310000 (?) 350000 (?) 380000 (?) > 8.1 > 0.3 11-14 > 80 > 0.01 > 0.6 >6 >9 >5 190 250 > 48 > 33 300

Thermal conductivity coefficient [3.58]: K

=

3.5W/rnK.

Ref. 3.12 3.44 3.19 3.45 3.46 3.47 3.14 3.13 3.48 3.49 3.32 3.50 3.26 3.51 3.9 3.52 3.24 3.20 3.53 3.42 3.54 3.55 3.21 3.11 3.34 3.56 3.26 3.51 3.43 3.57 3.17 3.33 3.1 3.24 3.48 3.42

Note

10 Hz 10 Hz

10 Hz 500 Hz 500 Hz 900 Hz 500 Hz

10 Hz 25 Hz

1333 Hz 10 Hz 10 Hz

10 Hz

77

78

3 Properties of Nonlinear Optical Crystals

3.1.2 KH 2P04 , Potassium Dihydrogen Phosphate (KDP) Negative uniaxial crystal: no > ne ; Point group: 42m; Mass density: 2.3383 g/cm 3 at 293 K [3.59]; Mohs hardness: 2.5; Transparency range at "0" transmittance level: 0.174 - 1.57 JlID [3.60, 59]; Transparency range at 0.5 transmittance level for a 0.8 em long crystal: 0.178 - 1.45 urn [3.60, 59]; Linear absorption coefficient a: A [Jlm]

a [cm"]

Ref.

0.212 0.25725

0.2 0.01-0.2 0.007 < 0.07 0.003 0.00005 0.01 0.01 0.05 0.058 0.02 0.1 0.3 0.1

3.61 3.62 3.63 3.64 3.65 3.62 3.66 3.67 3.66 3.65 3.65 3.68 3.69 3.68

0.3-1.15 0.3513 0.5145 0.5265 0.94 1.053 1.054 1.22 1.3152 1.32

Note e - wave, ..1 c e - wave, ..1 c e - wave, ..1 c wave 0 - wave

0-

wave o - wave e - wave, ..1 c 0 - wave 0-

e - wave, ..1 c

p:

Two-photon absorption coefficient A[urn]

px

0.216 0.2661

60±5 27 ± 8.1 40-80 0.59 ± 0.21

0.3547

1013 [m/W]

Ref. 3.70 3.71 3.72 3.71

Note

() == 410, 4> == 45° e - wave, .L c

Experimental values of refractive indices at T = 298 K [3.73]:

A [urn]

no

ne

A [)lID]

no

ne

0.2138560 0.2288018 0.2446905 0.2464068 0.2536519 0.2800869

1.60177 1.58546 1.57228 1.57105 1.56631 1.55263

1.54615

0.2980628 0.3021499 0.3035781 0.3125663 0.3131545 0.3341478

1.54618 1.54433

1.49824 1.49708 1.49667 1.49434 1.49419 1.48954

1.51586 1.50416

1.54117 1.54098

3.1 Basic Nonlinear Optical Crystals

0.3650146 0.3654833 0.3662878 0.3906410 0.4046561 0.4077811 0.4358350 0.4916036 0.5460740

1.52932 1.52923 1.52909 1.52341 1.52301 1.51990 1.51152

1.48432 1.48423 1.48409 1.48089 1.47927 1.47898 1.47640 1.47254 1.46982

0.5769580 0.5790654 0.6328160 1.0139750 1.1287040 1.1522760 1.3570700 1.5231000 1.5295250

1.50987 1.50977 1.50737 1.49535 1.49205 1.49135 1.48455

79

1.46856 1.46685 1.46041 1.45917 1.45893 1.45521 1.45512

Temperature derivative of refractive indices [3.74]:

A [Jlm]

dno/dT x 105 [K- 1] dne/dT x 105 [K- 1]

0.405 0.436 0.546 0.578 0.633

-3.27 -3.27 -3.28 -3.25 -3.94

-3.15 -2.88 -2.90 -2.87 -2.54

Temperature dependences of refractive indices upon cooling from room temperature to T [K]. for the spectral range 0.365 - 0.690 urn [3.75]:

no(T) == no(298) + 0.402 x 10- 4{[n o(298)]2 - 1.432}(298 - T) ne(T) == n e (298) + 0.221 x 10- 4{[n e(298)]2 - 1.105}(298 - T) for the spectral range 0.436 - 0.589 urn [3.76]:

no(T) == no(300) + 10-4(143.3 - 0.618T + 4.81 x 10-4 T 2 )

,

ne(T) == ne (300) + 10- 4(153.3 - 0.969T + 1.57 x 10- 3 T 2 )

.

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: In teracting wavelengths [urn] SHG, 0 + 0 =} e 0.517 =} 0.2585 0.6576 =} 0.3288 0.6943 =} 0.34715 0.8707 =} 0.43535 1.06 =} 0.53

(Jexp

[deg]

90 [3.74] 53.6 [3.69] 50.4 [3.79] 42.4 [3.80] 41 [3.81] 41 [3.82]

(Jtheor

[deg]

[3.73]

[3.77]

[3.78]K

no pm 53.6 50.6 42.8 41.2

no pm 53.6 50.6 42.7 41.0

73.6 53.2 50.4 42.8 40.9

80

3 Properties of Nonlinear Optical Crystals

44.3 [3.69] 1.3152 ==> 0.6576 SFG, o+o~e 1.415 + 0.22027 ~ ~ 0.1906 88.7 [3.83] 1.3648 + 0.6943 ~ 40.9 [3.80] ~ 0.46019 1.3152 + 0.6576 ~ ~ 0.4384 42.2 [3.69] 1.0642 + 0.2707 ~ 87.6 [3.84] ~ 0.21581 1.0642 + 0.5321 ~ =} 0.35473 47.3 [3.85] 1.06 + 0.53 ~ 47.5 [3.82] =* 0.35333 0.6576 + 0.4384 ~ ~ 0.26304 74 [3.86] SHG, e + 0 =} e 1.3152 =} 0.6576 61.4 [3.69] 1.06::::} 0.53 59 [3.82] SFG, e+o ~ e 1.0642 + 0.5321 ~ ==> 0.35473 58.3 [3.85] 1.06 + 0.53 =} =} 0.35333 59.3 [3.82]

44.6

44.7

44.1

83.7

83.6

54.3

41.7

41.7

41.6

42.1

42.1

42.0

87.5

87.3

62.9

47.3

47.3

47.1

47.4

47.4

47.3

75.2

75.4

68.6

61.8 59.0

61.8 58.8

60.7 58.6

58.2

58.3

57.9

58.5

58.5

58.1

Note: The other sets of dispersion relations from [3.74] and [3, 78]E show worse agreement with the experiment. [3.78]K ==> see [3.78], data of Kirby et al.; [3.78]E ==> see [3.78], data of Eimerl. Experimental values of NCPM temperature: T rOC]

Ref.

-13.7

-11 20 177 177

3.63 3.62 3.74 3.87 3.88

SFG, o+o~e -70 1.06 + 0.265 =} 0.212 1.0642 + 0.26605 =} 0.21284 -40 -35

3.61 3.89 3.90

Interacting wavelengths [)lID] SHG, 0+0 ~ e 0.5145 =} 0.25725 0.517 =} 0.2585 0.5321 ~ 0.26605

3.1 Basic Nonlinear Optical Crystals

Best set of dispersion relations (l in urn, T n2 = 2.259276

n

2

e

+ 13.00522A.2 + l2 _ 400

o

2 132 8 3.2279924l 66 + 2 l - 400

=.

= 20°C)

[3.74] :

0.01008956 l2 - (77.26408)-1 '

2

0.008637494 1 (81.42631)-

+l2 -

.

Temperature-dependent Sellmeier equations (l in urn, Tin K) [3.77] : 2

4

n2 o

=(1.44896 + 3.185 x 1O-5T) + (0.84181 - 1.4114 x 10- T)A. l2 _ (0.0128 - 2.13 x 10-7T) (0.90793 + 5.75 x 10- 7 T)l2 + l2 - 30 '

n2

=(1.42691 _ 1.152 x 10-5 T)

+

e

+

5

(0.22543 - 1.98 x 10- 7 T)l2 2

.

l - 30

Calculated values of phase-matching and "walk-off" angles:

Interacting wavelengths [Jlm] SHG, 0+0 ~ e 0.5321 ~ 0.26605 0.5782 ~ 0.2891 0.6328 ~ 0.3164 0.6594 ~ 0.3297 0.6943 ~ 0.34715 1.0642 ~ 0.5321 1.3188 ~ 0.6594 SFG, o+o~e 0.5782 + 0.5105 ~ 1.0642 + 0.5321 ~ 1.3188 + 0.6594 ~ SHG, e -} o ~ e 1.0642 ~ 0.5321 1.3188 ~ 0.6594 SFG, e+o~e 1.0642 + 0.5321 ~ 1.3188 + 0.6594 ~

0.27112 0.35473 0.4396

0.35473 0.4396

(}pm

2

(0.72722 - 6.139 x 10- T)A. l2 - (0.01213 + 3.104 x 10- 7 T)

[deg]

PI [deg]

P3 [deg]

76.60 64.03 56.15 53.43 50.55 41.21 44.70

0.808 1.391 1.611 1.657 1.687 1.603 1.549

72.46 47.28 42.05

1.025 1.712 1.657

58.98 61.85

1.149 0.922

1.404 1.269

58.23 49.42

1.166 1.104

1.521 1.634

81

82

3 Properties of Nonlinear Optical Crystals

Calculated values of inverse group-velocity mismatch for SHG process in KDP: In teracting wavelengths [J.1m]

SHG, 0 + 0 =} e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5 0.9 => 0.45 0.8 => 0.4 0.7 => 0.35 0.6 => 0.3 SHG, e + 0 => e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5 0.9 => 0.45 0.8 => 0.4

(}pm

[deg]

P[fs/mm]

42.45 41.38 41.22 42.24 44.91 50.14 60.40

42 17 9 40 77 128 208

59.54 58.87 59.75 62.97 70.71

89 67 89 118 158

Experimental values of internal angular and temperature bandwidths: Interacting wavelengths [urn]

SHG, 0 + 0 =} e 1.1523 =} 0.57615 1.0642 =} 0.5321 1.064 => 0.532 1.06 => 0.53 1.054 => 0.527 0.5321 =} 0.26605

0.53 => 0.265

SFG, 0 +0 => e 1.0642 + 0.5321 => => 0.35473 1.054 + 0.527 => => 0.35133 SHG, e + 0 =} e 1.0642 =} 0.5321 1.06 => 0.53

T rOC] 20 20 25 20 20 25 25 177 177 20 20

[deg]

A{fnt [deg]

41 41

0.074 0.070

41 41 41

0.069 0.063 0.060

(}pm

AT rOC]

23

90 90 77

77

1.7 1.9 2 0.059 0.066

5.5

25 25

48

0.046

25 20

59

0.129

Ref.

3.91 3.92 3.93 3.94 3.81 3.95 3.93 3.87 3.88 3.96 3.97

3.93 3.95

18.3

3.93 3.96

3.1 Basic Nonlinear Optical Crystals

1.054 => 0.527 SFG, e+ 0 => e 1.0642 + 0.5321 => :=} 0.35473 1.06 + 0.53 => :=} 0.35333 1.054 + 0.527 =} :=} 0.35133

25

3.95

0.126

59

25

5.2

3.93

20

59

0.062

3.97

25

59

0.059

3.95

Experimental values of spectral bandwidth: In teraeting wavelengths [um] SHG, 0 + 0 =} e 1.06 =} 0.53 0.53 =} 0.265 SHG, e + 0 =} e 1.06 =} 0.53

T

Bpm

L1v

Ref.

[OC] [deg] [cm"] 20 20

41 77

178 1.2

3.81 3.96

20

59

101.5

3.96

Temperature variation of phase-matching angle: Interacting wavelengths [urn] SHG, 0 + 0 =} e 1.0642 =} 0.5321 1.054 =} 0.527 0.5321 =} 0.26605 SFG,o+o=}e 1.0642 + 0.5321 =} 0.35473 1.054 + 0.527 =} 0.35133 SHG, e + 0 =} e 1.0642 =} 0.5321 1.06 =} 0.53 1.054 =} 0.527 SFG, e + 0 ~ e 1.0642 + 0.5321 =} 0.35473 1.054 + 0.527 =} 0.35133

T

rOC] 25 25 25 25 25 25 25 20 25 20 25 25 25 20

lJpm [deg]

dlJpm/dT Ref. [deg/K]

41

0.0028 0.0046 0.0382

3.93 3.95 3.93

0.0073 0.0046

3.93 3.95

0.0069 0.0069 0.0057 0.0086 0.0069

3.98 3.93 3.96 3.95 3.65

0.0106 0.0117 0.0152 0.0075

3.98 3.93 3.95 3.65

59 59 59 59 59 58 59 59

83

84

3 Properties of Nonlinear Optical Crystals

Temperature tuning of noncritical SHG [3.74]: Interacting wavelengths [JlmJ

dAI/dT [nmjKJ

SHG, 0 + 0 => e 0.517 => 0.2585

0.048

Temperature variation of birefringence for noncritical SHG process: Interacting wavelengths [um] 0.5145 => 0.25725 0.5321 => 0.26605

Ref. 3.99 3.87

1.745 1.2

Effective nonlinearity expressions in the phase-matching direction [3.100]:

d ooe == d36 sin 0 sin 2et> d eoe == d oee

,

== d 36 sin 20 cos 2cfJ .

Nonlinear coefficient [3.37]: d 36(1.064

,urn) == 0.39 pmjV ,

Laser-induced bulk-damage threshold: 't p

0.52 0.5265 0.527 0.53 0.5321 0.596 0.6943 1.053

1.054 1.06

1.064

[ns]

330 20 0.6 0.5 0.2 0.005 0.6 0.03 330 20 20 25 1 1 0.14 60 12-25 0.5 0.2 20 1.3

I thr

X

10- 12 [W1m2 ]

2 30 90 > 140 170 10000(?) > 80 300 2.4 30 >4 40 180 200 > 70 2 2.5 > 30 230 3-6 80

Ref. 3.101 3.66 3.66 3.102 3.103 3.104 3.72 3.105 3.101 3.101 3.106 3.66 3.66 3.107 3.108 3.109 3.81 3.110 3.103 3.111 3.33

3.1 Basic Nonlinear Optical Crystals

A [urn]

Lp

1.064

1 1 0.1

[ns]

Ithr X

10- 12 [Wjm 2 ]

30-70 50 70

Ref. 3.111 3.112 3.1

Thermal conductivity coefficient [3.59]: [WjmK], lie" [WjmK], -l c

T[K]

K

302 319

1.21 1.34

3.1.3 KD2P04 , Deuterated Potassium Dihydrogen Phosphate (DKDP) Negative uniaxial crystal: no > ne ; Point group: 42m; Mass density: 2.355 g/cm"; Mohs hardness: 2.5; Transparency range at "0" transmittance level: 0.2 - 2.1 urn [3.113, 114]; Linear absorption coefficient ex: [em-I]

A [um]

it

0.266 0.5321 0.82-1.21 0.94 1.0642 1.315 1.57 1.74

0.035 0.004-0.005 < 0.015 0.005 0.004-0.005 0.025 0.1 0.1

Ref. 3.115 3.116 3.67 3.67 3.116 3.117 3.68 3.68

Note 98-99 % deuteration

98-990/0 deuteration

o - wave, 950/0 deuteration e - wave, 950/0 deuteration

Two-photon absorption coefficient fJ: A [um]

f1

0.2661

2.0 ± 1.0 2.7 ± 0.7 0.54 ± 0.19

0.3547

x 1013 [mjW]

Ref.

Note

3.118 3.115 3.71

e - wave, .-L c

Experimental values of refractive indices at T = 298 K [3.95]:

0.4047 0.4078

1.5189 1.5185

1.4776 1.4772

85

86

3 Properties of Nonlinear Optical Crystals

A [urn] 1.5155 1.5111 1.5079 1.5063 1.5044 1.5022

0.4358 0.4916 0.5461 0.5779 0.6234 0.6907

1.4747 1.4710 1.4683 1.4670 1.4656 1.4639

Temperature derivative of refractive indices [3.74]:

0.405 0.436 0.546 0.578 0.633

-3.00 -3.37 -2.99 -3.00 -3.16

-1.86 -2.13 -1.95 -2.52 -2.03

Temperature dependences of refractive indices upon cooling from room temperature to T [K] for the spectral range 0.365 - 0.690 urn [3.75] :

no(T)

=

no(298) + 0.228 x 10- 4{[n o(298)]2 - 1.047}(298 - T)

ne(T) = ne(298) + 0.955 x 10- 5[n e(298)]2(298 - T) ; for the spectral range 0.436 - 0.589 urn [3.76]:

no(T) = no(300) + 10-4(85.2 - 0.0695 T - 7.25 x 10- 4T2 ) ne(T)

:=

ne(300) + 10-

4(21.8

- 0.445 T - 1.24 x 10-

3T2

,

) •

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: Interacting wavelengths [urn] SHG, 0 + 0 =} e 0.530 =} 0.265 0.6943 => 0.34715 1.062 =} 0.531 SHG, e + 0 =} e 1.3152 =} 0.6576

(Jexp

[deg]

(Jtheor

[deg]

[3.77]

[3.78]K [3.78]£

90 [3.119] 52 [3.79] 37.1 [3.120]

no pm 50.6 38.6

no pm 50.9 36.6

87.4 51.0 36.6

51.3 [3.69]

63.2

51.7

49.4

Note: The set of dispersion relations from [3.74] shows worse agreement with the experiment. [3.78]K ==> see [3.78], data of Kirby et al.;

3.1 Basic Nonlinear Optical Crystals

[3.78]E =} see [3.78], data of Eimerl. Experimental values of NCPM temperature: Interacting wavelengths [urn]

SHG, 0 + 0 =} e 0.528 =} 0.264 0.5321 =} 0.26605

0.536

=}

-30 42 45 46 49.8 60.8 100

0.268

Ref.

Note

3.119 3.89 3.87 3.90 3.121 3.122 3.119

99% deuteration 950/0 deuteration 99% deuteration > 95% deuteration 90% deuteration

Best set of dispersion relations (;, in urn, T n2

= 2.240921

+

o

2 _ 2 12 019 n - . 6 e

+

+

2.246956A? A? _ (11.26591)2 0.784404;,2

A? - (11.10871)

== 20°C) [3.78]K : 0.009676

A? _ (0.124981)2 '

2+

0.008578 2· ;,2 - (0.109505)

Temperature-dependent Sellmeier equations (A in urn, Tin K) [3.77] : n2 =(1.55934 + 3.3935 x 10-4 T)

+

o

4

(0.71098 - 4.1655 x 10- T)A? A2 - (0.01407 + 6.4904 x 10-6 T)

(0.67671 + 4.8281 x 10- 5 T)A? + - - - -2 - - - - - -

A

n2

-

30

=(1.68647 + 3.43 x 10-6 T)

e

+

(0.59614

+ 2.41

+

' 5

2

(0.46629 - 6.26 x 10- T)A. A2 - (0.01663 + 1.3626 x 10-6 T)

x 10- 7 T);,2

2

.

A - 30

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [Jlm]

(Jpm

SHG, 0 + 0 =} e 0.5321 =} 0.26605 0.5782 ::::} 0.2891 0.6328 ::::} 0.3164 0.6594 =} 0.3297 1.6943 ::::} 0.34715 1.0642 =} 0.5321 1.3188 =} 0.6594

86.20 66.87 57.53 54.31 50.86 36.60 36.36

[deg]

PI [deg]

P3 [deg] 0.225 1.197 1.467 1.522 1.558 1.450 1.412

87

88

3 Properties of Nonlinear Optical Crystals

SFG, 0 + 0 =} e 0.5782 + 0.5105 => 0.27112 1.0642 + 0.5321 => 0.35473 1.3188 + 0.6594 => 0.4396 SHG, e + 0 => e 1.0642 => 0.5321 1.3188 => 0.6594 SFG, e + 0 => e 1.0642 + 0.5321 => 0.35473 1.3188 + 0.6594 => 0.4396

77.88 46.82 39.18

0.595 1.580 1.515

53.47 51.70

1.286 1.222

1.427 1.420

59.38 47.70

1.174 1.254

1.378 1.527

Calculated values of inverse group-velocity mismatch for SHG process in

DKDP: Interacting wavelengths [urn]

Opm

SHG, 0 + 0 => e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5 0.9 =* 0.45 0.8 =* 0.4 0.7 =? 0.35 0.6 => 0.3 SHG, e + 0 => e 1.2 => 0.6 1.1 =* 0.55 1.0 =* 0.5 0.9 =? 0.45 0.8 =? 0.4

[deg]

P [fs/mm]

0.5321 0.5321 =? 0.26605 SHG, e+o =* e 1.0642 => 0.5321 1.06 =* 0.53

T rOC]

Opm

[deg]

L\oint [deg]

20 37 60.8 90 45 90

0.081

20 20 20

0.131 0.126 0.143

54 60

L\T rOC]

1.8 1.9

Ref.

3.92 3.122 3.87 3.123 3.124 3.96

3.1 Basic Nonlinear Optical Crystals

Experimental value of spectral bandwidth [3.96]: Interacting wavelengths [urn] SHG, e+o 1.06 ~ 0.53

(Jpm [deg]

~e

20

74.8

60

Temperature variation of phase-matching angle [3.96]: Interacting wavelengths [urn] SHG, e+o ~ e 1.06 ~ 0.53

20

(Jpm [deg]

d(Jpm/dT [deg/K]

60

0.0063

Temperature tuning of noncritical SHG [3.74]: Interacting wavelengths [Jlm] SHG, 0.519

0+0 ~ ~

dAI/dT [nm/K]

e

0.2595

0.068

Effective nonlinearity in the phase-matching direction [3.100]: d ooe

== d 36 sin (J sin 24> ,

d eoe

== d oee == d 36 sin 2(J cos 24> .

Nonlinear coefficient [3.37]: d 36(1.064Jlm)

== 0.37 pm/V.

Laser-induced bulk-damage threshold:

A [urn]

't

0.266 0.532

0.03 30 8 0.6 0.03 330 0.007 40 18 14 1 0.25 1

0.6 1.062 1.064

1.315

p [ns]

Ithr X

10- 12 [W/m2 ]

> 100 > 0.5 170

> 80 > 80 3

> 10 > 2.5 > 1.0 80 60 > 30 15

Ref. 3.115 3.122 3.125 3.72 3.118 3.101 3.120 3.122 3.116 3.125 3.124 3.116 3.69

89

90

3 Properties of Nonlinear Optical Crystals

Thermal conductivity coefficient [3.78]: K

== 1.86 Wm/K (II c) ,

K

== 2.09 Wrn/K (1- c) .

3.1.4 NU,,"2P04, Ammonium Dihydrogen Phosphate (ADP) Negative uniaxial crystal: no > ne ; Point group: 42rn; Mass density: 1.803 g/crn 3 at 293 K [3.59]; Mohs hardness: 2; Transparency range at "0" transmittance level: 0.18 - 1.53 urn [3.60, 126]; Transparency range at 0.5 transmittance level for a 0.8 em long crystal: 0.185 - 1.45 urn [3.60, 59] Linear absorption coefficient

A [urn]

(X

0.25725 0.265 0.266 0.3-1.15 0.5145 1.027 1.083 1.144

[em-I]

0.002 0.07 0.035 < 0.07 0.00005 0.086 0.208 0.150

(X:

Ref.

Note

3.62 3.127 3.115 3.64 3.62 3.67 3.67 3.67

e - wave, 1- c e - wave, 1- c

o - wave, ..L c

Two-photon absorption coefficient

A [urn]

p x 1013 [m/WJ Ref.

0.2661

6±1 11 ± 3 24±7 23 ± 5 0.68 ± 0.24

0.3078 0.3547

p:

Note

3.118 3.115 3.71 3.128 3.71

() == 42°, 4J == 45° e - wave, 1- c

Experimental values of refractive indices at T = 298 K [3.73, 129]:

A [Jlm]

no

ne

A [urn]

no

ne

0.2138560 0.2288018 0.2536519 0.2967278

1.62598 1.60785 1.58688 1.56462

1.56738 1.55138 1.53289 1.51339

0.3021499 0.3125663 0.3131545 0.3341478

1.56270 1.55917 1.55897 1.55300

1.51163 1.50853 1.50832 1.50313

3.1 Basic Nonlinear Optical Crystals

A [urn]

no

ne

A [urn]

0.3650146 0.3654833 0.3662878 0.3906410 0.4046561 0.4077811 0.4358350 0.4916036

1.54615 1.54608 1.54592 1.54174 1.53969 1.53925 1.53578

1.49720 0.5460740 1.52662 1.48079 1.49712 0.5769590 1.52478 1.47939 1.49698 0.5790654 1.52466 1.47930 0.6328160 1.52195 1.47727 1.49159 1.0139750 1.50835 1.46895 1.49123 1.1287040 1.50446 1.46704 1.48831 1.1522760 1.50364 1.46666 1.48390

91

ne

no

Temperature derivative of refractive indices [3.74]:

A [urn] dno/dT x 105 [K- 1] dne/dT x 105 [K- 1] 0.405 0.436 0.546 0.578 0.633

-4.78 -4.94 -5.23 -4.60 -5.08

~O

~O ~O ~O ~O

Temperature dependences of refractive indices upon cooling from room temperature to T [K]. for the spectral range 0.365 - 0.690 urn [3.75]:

no(T) == no(298) + 0.713 x 10- 2 {[no(298)J 2 - 3.0297 n o (298) + 2.3004} (298 - T) , ne(T) == ne(298) + 0.675 x 10-6(298 - T) ; for the spectral range 0.436 - 0.589 urn [3.76]:

no(T) == no(300)

+ 10-4(141.8 -

0.322 T - 5.02

ne(T) == ne(300) + 10-4(2.5 - 0.01763 T + 2.901 Experimental values of phase-matching angle (T between different sets of dispersion relations: In teracting wavelengths [urn]

SHG, 0 + 0 => e 0.524 => 0.262 0.530 => 0.265 0.6943 => 0.34715 0.7035 =>0.35175

(Jexp [deg]

90 [3.74] 81.7 [3.97] 51.9 [3.79] 50.5 [3.130]

10-4 T 2 )

X

=

X

10- 5 T 2 )

, .

293 K) and comparison

(Jtheor [deg] [3.73] [3.129]

[3.77]

[3.78]K

no pm 81.6 51.1 50.4

no pm 82.2 51.1 50.5

83.6 79.6 51.5 50.8

92

3 Properties of Nonlinear Optical Crystals

1.06 =* 0.53 SFG, 0 +0 => e 1.0642 + 0.5321 => => 0.35473 1.0642 + 0.2810 => =* 0.22230 0.81219 + 0.34715 => =* 0.24320 SFG, e + 0 => e 1.0642 + 0.5321 => => 0.35473

41.9 [3.79] 42 [3.81]

41.7

41.7

42.2

46.9 [3.85]

47.8

47.9

48.3

90 [3.84]

89.0

no pm 74.7

90 [3.131]

no pm no pm 81.1

60.2 [3.85]

59.9

60.0

60.4

Note: The other sets of dispersion relations from [3.74] and [3,78]£ show worse agreement with the experiment. [3.78]K =* see [3.78], data of Kirby et a1.: [3.78]E => see [3.78], data of Eimerl. Experimental values of NCPM temperature: In teracting wavelengths [urn] SHG, 0+0 =* e 0.4920 => 0.2460 0.4965 => 0.24825 0.5017 => 0.25085 0.5145 =* 0.25725

0.524 => 0.262 0.52534 =* 0.26267 0.53 => 0.265

0.5321 => 0.26605

0.548 =* 0.274 0.557 =* 0.2785 SFG, 0 + 0 => e 1.0642 + 0.26605 => 0.21284

T rOC]

Ref.

-116 -93.2 -68.4 -11.7 -10.2 -9.2 20 30 43 47 48 49.6 47.1 49.5 50 51.2 44.6 51-52 100 120

3.132 3.133 3.133 3.99 3.133 3.62 3.74 3.134 3.127 3.97 3.135 3.136 3.90 3.137 3.138 3.139 3.139 3.140 3.134 3.119

-55

3.141

Note

0.1-1 Hz 20 Hz

3.1 Basic Nonlinear Optical Crystals

Best set of dispersion relations (l in urn, T n2

= 2.302842 + 15.102464A? + ,12 _ 400

o

n2

0.011125165 ,12 - (75.450861)-1 '

2

= 2.163510 + 5.919896..1. + l2 _ 400

e

== 20°C) [3.73], [3.129] :

0.009616676 . l2 - (76.98751)-1

Temperature-dependent Sellmeier equations (A in 11m, T in K) [3.77] :

n2 = (1.6996 _ 8.7835 x 10-4 T)

+

o

4

(1.10624 - 1.179 x 10-4 T)l2 + ,{2 _ 30 '

n2 = (1.42036 _ 1.089 x 10- 5 T)

+

e

+

6

(0.74453 + 5.14 x 10- T);.2 l2 - (0.013 - 2.471 x 10- 7 T)

(0.42033 - 9.99 x 10- 7 T),1 2 2

.

l - 30

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [11m] SHG, 0 + 0 ~ e 0.5321 :::} 0.26605 0.5782 :::} 0.2891 0.6328 ::::} 0.3164 0.6594 =} 0.3297 0.6943 =} 0.34715 1.0642 =} 0.5321 1.3188 =} 0.6594 SFG, 0 + 0 ~ e 0.5782 + 0.5105 =} 0.27112 1.0642 + 0.5321 =} 0.35473 1.3188 + 0.6594 =} 0.4396 SHG, e + 0 ~ e 1.0642 =} 0.5321 1.3188 ::::} 0.6594 SFG, e + 0 ~ e 1.0642 + 0.5321 =} 0.35473 1.3188 + 0.6594 =} 0.4396

{}pm

[deg]

2

(0.64955 + 7.2007 x 10- T)A l2 - (0.01723 -1.40526 x 10- 5 T)

PI [deg] P3 [deg]

80.15 65.28 56.91 54.07 51.09 41.74 45.55

0.639 1.427 1.703 1.762 1.803 1.746 1.694

74.84 47.82 42.56

0.955 1.836 1.794

61.39 65.63

1.230 0.968

1.449 1.250

59.85 50.86

1.272 1.274

1.582 1.748

93

94

3 Properties of Nonlinear Optical Crystals

Calculated values of inverse group-velocity mismatch for SHG process in ADP: In teracting wavelengths [urn] SHG, 0 + 0 => e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5 0.9 => 0.45 0.8 => 0.4 0.7 => 0.35 0.6 => 0.3 SHG, e+o => e 1.2 => 0.6 1.1=>0.55 1.0 => 0.5 0.9 => 0.45 0.8 => 0.4

(}pm

[deg]

P [fs/mm]

43.10 41.94 41.71 42.68 45.34 50.67 61.39

49 21 8 42 85 142 233

62.50 61.39 62.02 65.24 73.80

105 78 95 127 173

Experimental values of internal angular and temperature bandwidths: In teracting wavelengths [urn] SHG, 0 + 0 => e 1.06 => 0.53 0.5321 => 0.26605 0.53 => 0.265

T

(}pm

~fint

~T

[OC]

[deg]

[deg]

[OC]

20 49.5 51 20 20 20

42 90 90 82 82 82

0.057 1.086 0.118 0.088 0.089

0.60 0.53

0.63

Experimental values of spectral bandwidth: Interacting wavelengths [prn]

T rOC]

(}pm

~v

[deg]

[cm']

SHG, 0 + 0 => e 1.06 => 0.53 0.53 => 0.265

20 20

42 82

178 1.2

Ref.

3.81 3.96

Ref.

3.81 3.137 3.139 3.103 3.96 3.97

3.1 Basic Nonlinear Optical Crystals

95

Temperature variation of phase-matching angle [3.97]: Interacting wavelengths [urn] SHG, 0 + 0 => e 0.53 => 0.265

T

Opm

rOC]

[deg]

dOpm/dT [degjK]

20 47

82 90

0.1418 1.1020

Temperature tuning of noncritical SHG [3.74]: Interacting wavelengths [J.lm]

dAl/dT [nmjK]

SHG, 0 + 0 => e 0.524 => 0.262

0.306

Temperature tuning of noncritical SFG [3.142]: Interacting wavelengths (J.lm]

dA3/dT [nmjK]

SFG, 0 + 0 => e 0.6943 + 0.39961 => 0.25363

0.171

Temperature variation of birefringence for (0.5145 urn => 0.25725 urn, 0 + 0 => e): d(n~ - n~)/dT

== 5.65

noncritical

SHG process

x 10- 5K- 1[3.99].

Effective nonlinearity expressions in the phase-matching direction [3.100]:

d ooe == d 36 sin 0 sin 2

0.23825 0.488 =:} 0.244 0.4965 => 0.24825 0.5145 => 0.25725 0.5321 => 0.26605

0.604 =:} 0.302 0.6156 =? 0.3078 0.70946 =:} 0.35473

1.0642 ==> 0.5321

Oexp

[deg]

90 [3.145] 90 [3.150] 82.8 [3.145] 79.2 [3.145] 57 [3.151] 54.5 [3.151] 52.5 [3.151] 49.5 [3.151] 47.3 [3.148] 47.5 [3.145] 47.5 [3.152] 47.6 [3.153] 47.6 [3.45] 48 [3.154] 40 [3.155] 39 [3.156] 32.9 [3.157] 32.9 [3.158] 33 [3.159] 33 [3.152] 33 [3.160] 33.1 [3.45] 33.3 [3.147] 33.7 [3.161] 22.7 [3.148] 22.8 [3.145] 22.8 [3.152] 22.8 [3.33] 22.8 [3.162] 22.8 [3.45] 22.8 [3.163]

SFG,o+o=>e 0.73865 + 0.25725 => =} 0.1908 81.7 [3.164] 0.72747 + 0.26325 => =:} 0.1933 76 [3.165] 0.5922 + 0.2961 => 88 [3.166] => 0.1974

Otheor

[deg]

[3.149]

[3.148] [3.145]

89.36 87.25 84.11 79.80 57.79 55.53 54.00 51.13 48.67

86.51 85.54 82.99 78.87 56.57 54.29 52.76 49.87 47.42

88.82 86.97 83.77 79.31 56.73 54.46 52.94 50.06 47.62

41.00 40.02 33.65

39.89 38.95 32.94

40.13 39.18 33.15

21.42

22.88

22.78

72.94

75.27

76.11

71.79

73.59

74.22

80.44

82.13

83.22

98

3 Properties of Nonlinear Optical Crystals

0.5964 + 0.2982 => 82.5 [3.167] => 0.1988 0.5991 + 0.29955 => 80 [3.166] => 0.1997 0.60465 + 0.30233 => 76.2 [3.167] => 0.20155 0.5321 + 0.32561 => 83.9 [3.145] => 0.202 0.6099 + 0.30495 => 73.5 [3.166] => 0.2033 0.5321 + 0.34691 => 71.9 [3.145] => 0.21 1.0642 + 0.26605 => 51.1 [3.145] => 0.21284 1.0642 + 0.35473 => 40.2 [3.145] => 0.26605 1.0642 + 0.5321 => 31.1 [3.148] => 0.35473 31.3 [3.145] 31.4 [3.161] 0.5782 + 0.5106 => 46 [3.168] => 0.27115 0.59099 + 0.5321 => 44.7 [3.169] => 0.28 2.68823 + 0.5712 => 21.8 [3.170] => 0.4711 1.41831 + 1.0642 => 21 [3.171] => 0.608

SHG, e+o => e 0.5321 ::::} 0.26605 0.70946 =} 0.35473 1.0642 => 0.5321

SFG, e + 0 => e 1.0642 + 0.35473 => => 0.26605 1.0642 + 0.5321 => => 0.35473

78.02

79.11

79.81

76.71

77.57

78.14

74.41

74.92

75.34

80.88

81.22

81.95

72.51

72.82

73.16

72.11

71.60

71.84

50.69

51.04

51.12

40.75

40.19

40.31

31.52

31.12

31.28

45.23

46.03

46.24

45.23

44.03

44.25

18.37

21.73

21.39

18.40

21.26

20.96

no pm 48.72

82.03 47.61

80.78 47.92

30.00

31.94

32.18

46.6 [3.145]

46.81

46.11

46.31

38.4 [3.148] 38.5 [3.145]

38.39

37.77

38.15

81 [3.145] 48 [3.159] 48.1 [3.152] 31.6 [3.172] 32.4 [3.148] 32.7 [3.152] 32.7 [3.33] 32.9 [3.145]

3.1 Basic Nonlinear Optical Crystals

SFG, 0 +e => e 1.0642 + 0.5321 =} =} 0.35473

59.8 [3.145]

59.46

58.91

99

58.89

Note: The sets of dispersion relations from [3.143, 154, 170] show worse agreement with the experiment. Best set of dispersion relations (2 in urn, T n2

== 2.7359 +

o

n2

== 2.3753 +

e

== 20°C) [3.145]:

A2

0.01878 - 0.013542 2 - 0.01822 '

A2

0.01224 - 0.01516.A? . - 0.01667

Calculated values of phase-matching and "walk-off" angles: In teracting wavelengths [urn]

SHG, 0+0 =? e 0.4880 =} 0.2440 0.5105 =} 0.25525 0.5145 =} 0.25725 0.5321 =} 0.26605 0.5782 =} 0.2891 0.6328 => 0.3164 0.6594 =} 0.3297 0.6943 =} 0.34715 1.0642 => 0.5321 1.3188 =} 0.6594 SFG,o+o=}e 1.3188 + 0.6594 => 0.4396 1.3188 + 0.4396 => 0.3297 1.3188 + 0.3297 => 0.26376 1.3188 + 0.26376 => 0.2198 1.0642 + 0.5321 => 0.35473 1.0642 + 0.35473 => 0.26605 1.0642 + 0.26605 => 0.21284 0.6943 + 0.34715 => 0.23143 0.5782 + 0.5105 => 0.27112 0.5145 + 0.4880 => 0.25045

(Jpm

[deg] P3 [deg]

54.46 50.66 50.06 47.62 42.46 37.87 36.05 33.96 22.78 20.36

4.757 4.861 4.869 4.879 4.782 4.571 4.457 4.306 3.189 2.881

25.39 31.19 37.40 44.52 31.28 40.31 51.12 55.00 46.12 52.17

3.515 4.205 4.897 5.588 4.132 4.941 5.497 4.882 4.872 4.831

100

3 Properties of Nonlinear Optical Crystals

In teracting wavelengths [urn]

(}pm

SHG, e + 0 => e 0.5321 => 0.26605 0.5782 => 0.2891 0.6328 => 0.3164 0.6594 => 0.3297 0.6943 => 0.34715 1.0642 => 0.5321 1.3188 => 0.6594 SPG, e + 0 => e 1.3188 + 0.6594 => 0.4396 1.3188 + 0.4396 => 0.3297 1.3188 + 0.3297 => 0.26376 1.0642 + 0.5321 => 0.35473 1.0642 + 0.35473 => 0.26605 1.0642 + 0.26605 => 0.21284 0.6943 + 0.34715 => 0.23143 0.5782 + 0.5105 => 0.27112 SFG, 0 + e => e 1.3188 + 0.6594 ==> 0.4396 1.3188 + 0.4396 ==> 0.3297 1.0642 + 0.5321 ==> 0.35473 0.5782 + 0.5105 ==> 0.27112

[deg] PI [deg] P2 [deg] P3 [deg]

80.78 65.08 55.98 52.77 49.25 32.18 28.77

1.252 3.068 3.773 3.941 4.070 3.840 3.632

30.88 35.71 41.38 38.15 46.31 56.96 72.50 70.05

3.773 4.013 4.140 4.078 4.108 3.666 2.254 2.555

45.50 78.68 58.89 84.64

1.252 3.068 3.773 3.941 4.070 3.840 3.632

1.446 3.460 4.163 4.310 4.408 3.940 3.663 3.947 4.444 4.973 4.441 4.913 5.048 2.860 2.951

4.164 1.556 3.619 0.737

4.312 1.640 3.831 0.842

Calculated values of inverse group-velocity mismatch for SHG process in BBO: Interacting wavelengths [urn] SHG, 0 + 0 => e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5 0.9 => 0.45 0.8 => 0.4 0.7 => 0.35 0.6 => 0.3 0.5 => 0.25 SHG, e+o => e 1.2 => 0.6 1.1 => 0.55 1.0 => 0.5

(}pm

[deg]

P[fs/rnm]

21.18 22.28 23.85 26.07 29.18 33.65 40.47 52.34

54 76 104 141 194 275 415 695

29.91 31.46 33.73

103 130 164

3.1 Basic Nonlinear Optical Crystals

0.9 0.8 0.7 0.6

=> => => =>

0.45 0.4 0.35 0.3

36.98 41.67 48.74 60.91

101

210 276 373 531

Experimental values of internal angular, temperature and spectral bandwidths at T == 293 K: Interacting wavelengths [J.lm] SHG, 0 + 0 => e 0.5321 => 0.26605 1.0642 => 0.5321

SFG,o+o=>e 1.0642 + 0.5321 => 0.35473 2.44702 + 0.5712 ~ 0.4631 2.68823 + 0.5712 => 0.4711 SHG,e+o=>e 1.0642 => 0.5321 SFG, e + 0 => e 1.0642 + 0.5321 ~ 0.35473 SFG, 0 +e => e 1.0642 + 0.5321 => 0.35473

L\l1llt

L\T rOC]

L\v Ref. [cm"]

4 37

9.7

Opm [deg]

[deg]

47.3 22.8 21.9 22.7

0.010 0.021 0.028 0.030

31.1 22.1 21.8

0.015 0.026 0.028

32.7 32.4

0.034 0.046

37

3.33 3.148

38.4

0.020

13

3.148

58.4

0.050

12

3.148

51 16

3.148 3.170 3.170

8.8

Temperature variation of phase-matching angle at T In teracting wavelengths [J.lm] SHG, 0 + 0 => e 0.5321 => 0.26605 1.0642 => 0.5321 SFG,o+o=>e 1.0642 + 0.5321 => 0.35473 SHG, e + 0 => e 1.0642 => 0.5321 SFG, e + 0 => e 1.0642 + 0.5321 => 0.35473 SFG, o+e => e 1.0642 + 0.5321 => 0.35473

3.148 3.33 3.154 3.148

== 293 K [3.148]:

lJpm [deg]

dOpm/dT [deg/K]

47.3 22.7

0.00250 0.00057

31.1

0.00099

32.4

0.00120

38.4

0.00150

58.4

0.00421

102

3 Properties of Nonlinear Optical Crystals

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe

== d 31sin e- d 22 cos esin 3 0.5

Ref. 3.218 3.199 3.219 3.220 3.221 3.222 3.222 3.220 3.202 3.101 3.206 3.185 3.203 3.220 3.220 3.223 3.202 3.201 3.224

Note

25 Hz 1 Hz 12.5 Hz

10 pulses 500 pulses

100 Hz 1 kHz 50 Hz

Thermal conductivity coefficient [3.182]: K

== 1.47 W/mk

3.1.7 KTiOP0 4 , Potassium Titanyl Phosphate (KTP) Positive biaxial crystal: 2Vz == 37.4° at A == 0.5461 urn [3.225]; Point group: mm2 Assignment of dielectric and crystallographic axes: X, Y, Z =? a, b, C (Fig. 3.2) ; Mass density: 2.945 g/cm 3 [3.226, 227]; 3.023 g/cm 3 [3.228]; 3.03 g/cm 3 [3.229]; Mohs hardness: 5 [3.227]; Vickers hardness: 531 [3.228], 566 [3.230]; Knoop hardness: 702 [3.228];

107

108

3 Properties of Nonlinear Optical Crystals

Z(c)

light

optic axis

Fig. 3.2. Dependence of refractive index on light propagation direction and polarization (index surface) in the first octant of dielectric reference frame (X, Y, Z) of KTP crystal. Designations: ()is the polar angle, ¢ is the asimuthal angle, Vz is the angle between one of the optical axes and the Z axis

Transparency range at "0" transmittance level: 0.35 - 4.5 urn [3.231, 232]; Linear absorption coefficient rx : A [Jlm]

rx [cm"]

Ref.

Note

0.43-0.78 0.5145

< 0.004

2.233 3.186 3.186 3.186 2.233 3.234 3.235 3.186 3.186 3.186 3.229 3.235 3.234 3.186 3.186 3.186

oxygen annealing + cerium doping along a axis along b axis along c axis oxygen annealing along SHG direction

0.53-0.78 0.5321 0.6594

1.06 1.0642

0.013 0.027 0.026 < 0.005 0.04 < 0.02 0.0065 0.0087 0.0065 < 0.01 < 0.006 0.005 0.0002 0.0005 0.0004

along a axis along b axis along c axis

along along along along

SHG direction a axis b axis c axis

3.1 Basic Nonlinear Optical Crystals

A [um]

('J,

1.3188

[em-I]

Ref.

Note

0.0015 0.0004 0.001

3.186 3.186 3.186

along a axis along b axis along c axis

Experimental values of refractive indices: hydrothermally grown KTP [3.229]

A[urn]

nx

ny

nz

0.53 1.06

1.7787 1.7400

1.7924 1.7469

1.8873 1.8304

flux-grown KTP A [urn]

nx

ny

nz

Ref.

0.4047 0.4358 0.4916 0.5343 0.53975 0.5410 0.5461 0.5770 0.5790 0.5853 0.5893 0.6234 0.6328 0.6410 0.6939 0.6943 0.7050 1.0640 1.0642 1.0795 1.3414

1.8249 1.8082 1.7883 1.7780 1.7764

1.8410 1.8222 1.8000 1.7887 1.7869 1.7873 1.7860 1.7803 1.7798 1.7787 1.7780 1.7732 1.7714 1.7709 1.7652 1.7652 1.7642 1.7458 1.7454 1.7450 1.7387

1.9629 1.9359 1.9044 1.8888 1.8863 1.8869 1.8850 1.8769 1.8764 1.8749 1.8740 1.8672 1.8649 1.8641 1.8564 1.8564 1.8550 1.8302 1.8297 1.8291 1.8211

3.225 3.225 3.225 3.225 3.236 3.225 3.225 3.225 3.225 3.225 3.225 3.225 3.236 3.225 3.225 3.225 3.225 3.225 3.236 3.236 3.236

1.7767 1.7756 1.7703 1.7699 1.7689 1.7684 1.7637 1.7622 1.7617 1.7565 1.7564 1.7555 1.7381 1.7379 1.7375 1.7314

Temperature derivative of refractive indices [3.237] : dnx /dT x 105 == 0.1323 A- 3 - 0.4385 A- 2 + 1.2307 A-I + 0.7709 , dny/dT x 105 == 0.5014 A- 3 - 2.0030 A- 2 + 3.3016 A-I + 0.7498 , dnz/dT x 105 == 0.3896 A- 3 -1.3332 A- 2 +2.2762 A-I +2.1151 ,

where A in urn and dnx/dT, dny/dT, and dnz/dT are in K- 1.

109

110

3 Properties of Nonlinear Optical Crystals

Temperature derivative of refractive indices [3.237] :

0.5321 2.41 1.0642 1.65

4.27 3.40

3.21 2.50

Experimental values of phase-matching angle (T between different sets of dispersion relations: hydrothermally grown KTP XY plane, () = 90° In teracting wavelengths [urn]

0.5321

YZ plane,

4J

4Jexp

[deg]

4Jtheor

[deg]

[3.242] [3.232] [3.236] 23.0 23.2 23.3 24.1 25.0 25.2 25.2 25.2 25.3

[3.243] [3.225] [3.244] [3.245] [3.227] [3.231] [3.238] [3.246] [3.230]

21.12

24.59

22.89

= 90°

In teracting wavelengths [urn] SHG, o+e~ 0 1.0642 => 0.5321 1.068 => 0.534 1.182 => 0.591 1.3188 => 0.6594 1.5 => 0.75

Oexp

[deg]

Otheor

[deg]

[3.242] [3.232] [3.236] 69.0 69.2 67.8 57.4 50.0 44.6

[3.247] [3.238] [3.247] [3.247] [3.238] [3.247]

68.03

68.67

68.83

67.52 56.77 49.42 43.80

68.16 57.41 50.25 45.02

68.32 57.64 50.38 44.87

111

3 Properties of Nonlinear Optical Crystals

112

XZ plane, 1> == 0°, fJ > Vz fJexp [deg]

In teracting wavelengths [urn] SHG, o+e* 0 1.0796 * 0.5398

1.3414 * 0.6707 1.54 * 0.77 1.90768 * 0.95384 2.05 * 1.025 2.1284 * 1.0642 SFG, o+e* 0 1.3188 + 0.6594 :::} * 0.4396 1.338 + 0.669 * * 0.446 1.3414 + 0.6707 :::} * 0.44713 1.0642 + 1.90768 * =? 0.68333 1.0796 + 1.3414 :::} * 0.59817 1.54 + 0.78 =? =? 0.51776 1.90768 + 2.40688 =? * 1.0642 1.58053 + 1.54 * *0.78 1.90768 + 1.0642 * * 0.68333

[deg]

[3.242]

[3.232]

[3.236]

85.68

no pm

86.94

59.03

60.38

59.58

58.02 52.02 48.33 48.6 48.63

59.42 53.93 51.32 51.82 52.36

58.58 52.64 49.07 48.82 49.15

87.6 [3.238] 87.1 [3.241]

84.76

86.84

83.14

79.8 [3.241]

79.21

80.23

78.53

78.1 [3.252]

78.52

79.50

77.91

77.2 [3.249]

72.47

75.21

72.73

74.9 [3.236]

75.03

76.49

74.48

61 [3.253]

59.87

60.79

60.17

58.6 [3.249]

52.79

57.08

53.37

52.1 [3.253]

51.21

53.15

51.83

48.7 [3.249]

46.70

48.17

47.22

85.3 [3.248] 86.7 [3.236] 58.3 [3.238] 58.9 [3.249] 58.7 [3.236] 53 [3.250] 51.1 [3.249] 50.8 [3.249] 53.7 [3.251] 54 [3.249]

1.3188 * 0.6594

(}theor

Note: The other sets of dispersion relations from [3.225, 254, 255, 238] show worse agreement with the experiment. Best sets of dispersion relations (A in urn, T == 20°) hydrothermally grown KTP [3.239] : n2 x

= 2.1146 +

2

O.89188A ,1,2 _ (0.20861)2

-

O.01320A2

'

3.1 Basic Nonlinear Optical Crystals

= 2.1518 +

n2 Y

2

nz

- O.01327A?

O.87862A?

,12 _ (0.21801)2

== 2.3136 + 2

1.00012,12

A - (0.23831)

'

2 - 0.01679,1

2

.

flux-grown KTP [3.232] : n 2 == 3.0065

+

0.03901 - 0.01327A? ,12 - 0.04251 '

+

0.04154 - 0.01408,12 ,12 - 0.04547 '

+

0.05694 - 0.01682,12 . ,12 - 0.05658

x

n 2 == 3.0333 Y

n 2 == 3.3134 z

Calculated values of phase-matching and "walk-off" angles for flux-grown KTP: XY plane, () == 90° In teracting wavelengths [urn]

SHG, e+o ~ e 1.0642 =:} 0.5321

cjJpm [deg]

PI [deg] P3 [deg]

24.59

0.202

YZ plane, 4J == 90° In teracting wavelengths [urn]

SHG, o+e ~ 0 1.0642 =} 0.5321 1.1523 =} 0.57615 1.3188 =} 0.6594 2.098 1.049 2.9365 =} 1.46825 SFG, 0 +e 0 1.3188 + 0.6594 => 0.4396

*

*

{}pm

[deg]

P2 [deg]

68.67 59.59 50.25 43.01 57.95

1.829 2.314 2.544 2.481 2.225

65.14

2.210

*

0.268

113

114

3 Properties of Nonlinear Optical Crystals

XZ plane, 4J = 0°, () > V,

Interacting wavelengths [urn] SHG, 0 + e=>o 1.1523 => 0.57615 1.3188 => 0.6594 2.098 =} 1.049 2.9365 => 1.46825 SFG, 0 + e=>o 1.3188 + 0.6594 => => 0.4396

(Jpm

[deg]

P2 [deg]

72.01 60.38 52.13 67.36

1.747 2.487 2.671 1.928

86.84

0.362

Calculated values of inverse group-velocity mismatch for SHG process in flux-grown KTP: XY plane, (J = 90° Interacting wavelengths [urn] SHG, e + 0 => e 1.0=>0.5 1.05 => 0.525

o 1.0=>0.5 1.1 => 0.55 1.2 => 0.6 1.3 => 0.65 1.4 => 0.7 1.5 => 0.75 1.6 => 0.8 1.7 => 0.85 1.8=>0.9 1.9 => 0.95 2.0 => 1.0

[fsjmm]

3.1 Basic Nonlinear Optical Crystals

115

XZ plane, cP = 0°, () > Vz Interacting wavelengths [urn]

SHG, 0 + e=>o 1.1 => 0.55 1.2 => 0.6 1.3 => 0.65 1.4 => 0.7 1.5 => 0.75 1.6 => 0.8 1.7 =} 0.85 1.8 => 0.9 1.9 => 0.95 2.0 => 1.0

()pm

[deg]

80.31 67.47 61.25 57.32 54.70 52.99 51.94 51.42 51.32 51.57

P [fs/mm]

391 307 246 200 164 135 111 90 81 98

Experimental values of NCPM temperature and corresponding temperature bandwidth: hydrothermally grown KTP along X axis T [OC]

~T

1.3188 Y + 0.6594z => 0.4396 Y 1.338 Y + 0.660/ => 0.446Y along Yaxis

47 463

8.5 8.5

Interacting wavelengths [urn]

T (OC]

~T

20

175

3.256

20

122

3.257

T [OC]

~T

153(?) 63

20 30

Interacting wavelengths [urn]

[OC]

Ref.

SFG, type II

SHG, type II 0.9943x + 0.9943 z => 0.49715x SFG, type II 1.0642x + 0.800/ => 0.45961 x

3.241 3.241

(OC]

Ref.

flux-grown KTP along X axis Interacting wavelengths [urn]

[OC]

Ref.

SHG, type II 1.0796Y

+ 1.0796z => 0.5398 Y

SFG, type II 1.090Y + 1.030/ => 0.5321 Y

20 20

3.248 3.258 3.259 3.260

116

3 Properties of Nonlinear Optical Crystals

2.15Y + 1.04z => 0.70094Y 3.09Y + 1.38z => 0.95396Y 3.297Y + 1.571 z => 1.047Y 3.276Y + 1.530/ => 1.0642Y 1.3188 Y + 0.6594z => 0.4396Y 1.338 Y + 0.660/ => 0.446Y

20 20 20 20 60.2 484

3.261 3.261 3.262 3.262 3.241 3.241

8.5 8.5

along Yaxis Interacting wavelengths [Jlm]

T rOC] Ref.

SHG, type II 0.90/ + 0.90/ => 0.495x SFG, type II 1.0642x + O.8068 z => O.458~ 1.0642x + 0.808z => 0.45920/ 1.0642x + 0.9691 z => 0.5072x

20

3.254

20 20 20

3.254 3.238 3.254

Note: Superscripts of interacting wavelengths represent polarization directions Experimental values of internal angular, temperature, and spectral bandwidths: XY plane, () == 900 ( T == 20°C) Interacting wavelengths [urn]

l/>pm

[deg]

SHG, e + 0 => e 1.0582 => 0.5921 1.062 => 0.531 1.0642 => 0.5321

YZ plane, c/J

25 23 23.2 23.3 25 25.2 25.2 25.2

== 90

0(T

Al/>int

L\Oint

[deg]

[deg]

0.43 0.49 0.53 0.58 0.43

2.01 2.23

0.42 0.52

1.82

AT [OC]

Av [cm"]

25 20 24 20

4.9

3.263 3.229 3.243 3.225 3.244 3.227 3.231 3.246 3.230

4.0 6.2

25 17.5 25.7

2.52

Ref.

== 20°C)

In teracting wavelengths [urn]

()pm

A ()int

AljJint

[deg]

[deg]

[deg]

AT [Oe]

[cm"]

SHG, o+e => 0 0.9943 => 0.49715

90

2.96

5.70

175

7.1

Av

Ref.

3.256

3.1 Basic Nonlinear Optical Crystals

1.0642 => 0.5321 2.532 =} 1.266 SFG, type II I.0642x + O.80~ => =} 0.45961 x

XZ plane, 4>

== 0°

1 ()

100

69 69 56

0.11 0.20

90

2.72

6.13

T

()pm

~()int

[Oe]

[deg] [deg]

20 153

85.3 90

47 30.7

3.237 3.264 3.264

17.6(~v2)

3.257

117

> Vz

Interacting wavelengths [urn] SHG, 0 + e=}o 1.0796 =} 0.5398

0.34 1.70

Ref.

3.248 3.248

Note: Superscripts of interacting wavelengths represent polarization directions Effective nonlinearity in the phase-matching direction for three-wave interactions in the principal planes of KTP crystal [3.35, 36]: XYplane d eoe == d oee

== d 31sin2 c/J + d 32 cos 2 c/J ,

YZ plane d oeo == d eoo

== d 31sin () ,

XZ plane, () < Vz d ooe == d 32 sin () , XZ plane, () > Vz

d oeo == d eoo == d32 sin () . Effective nonlinearity for three-wave interactions in the arbitary direction of KTP crystal are given in [3.36] Nonlinear coefficients [3.265] : d31(1.0642,um)

== 1.4 pm/V,

d 32(1.0642,um)

== 2.65 pmjV ,

d 33(1.0642,um)

== 10.7 pmjV .

118

3 Properties of Nonlinear Optical Crystals

Laser-induced damage threshold: hydrothermally grown KTP A [urn]

Lp

[ns]

0.03 0.03 1.0642 125000 30 20 11

0.526

Ithr X

10- 12 [W/m 2 ]

300 300 0.01 1.5 > 1.5 20-30

Ref. 3.239 3.235 3.266 3.267 3.268 3.269

Note 10 Hz

10 Hz

flux-grown KTP

A [urn]

Lp

0.526 0.5291 0.5321

0.03 18 14 8 8 0.06 25 30 25

1.0582 1.0642

10- 12 [W/m 2 ]

Ref.

Note

100 0.8-1.0 0.5 14-22 20-32 > 18 1.8-2.2 > 3.3 >6

3.235 3.263 3.246 3.270 3.270 3.245 3.263 3.249 3.271

10 Hz surface damage 60 pulses 2 Hz, surface damage 2 Hz, bulk damage 5 Hz surface damage

25

>3

3.271

20 11 11 10 9 1.3 1 1

1.5 15-22 24-35 9-10 310 46 150 > 150

3.246 3.270 3.270 3.243 3.272 3.33 3.225 3.112

[ns]

Ithr X

250 000 pulses, bulk darkening 3 500 000 pulses, bulk darkening 60 pulses 2 Hz, surface damage 2 Hz, bulk damage 1 pulse, bulk damage surface damage 1 pulse

Thermal conductivity coefficient [3.235] : K

2

[W/mK], along a

K

3

[W/mK], along b

K

[W/mK], along c

3.3

3.1 Basic Nonlinear Optical Crystals

3.1.8 LiNb0 3 , Lithium Niobate

Negative uniaxial crystal: no > ne ; Point group: 3m; Mass density: 4.628 g/cm 3 [3.273]; Mohs hardness: 5 - 5.5; Transparency range at "0" transmittance level: 0.4 - 5.5 urn [3.274, 275]; Linear absorption coefficient a: A [urn]

0.5145

0.6594 1.0642

1.3188

(J.

[cm']

Ref.

0.025 0.019-0.025 0.035-0.045 0.0021-0.0044 0.0085-0.0096 0.0019-0.0023 0.0014-0.0019 0.0042 0.0028 0.0018-0.0044 0.0017-0.0110

3.276 3.186 3.186 3.186 3.186 3.186 3.186 3.277 3.277 3.186 3.186

Two-photon absorption coefficient A [11m]

0.5288 0.53 0.5321

px

1011 [m/w] Ref.

0.15 (?) 5.0 2.90 1.57

Note

3.278 3.279 3.188 3.188

II c e - wave, 1- c II c e - wave, 1- c II c e - wave, .L c II c ..lc

II c e - wave, 1- c

p:

Note

o-wave e-wave

Experimental values of refractive indices for lithium-rich lithium niobate, T = 293 K [3.280] : A [urn]

no

ne

A [urn]

no

ne

0.3250 0.4545 0.4579 0.4658 0.4727 0.4765

2.6360 2.3751 2.3719 2.3658 2.3604 2.3573

2.4670 2.2608 2.2584 2.2530 2.2489 2.2465

0.4880 0.4965 0.5017 0.5145 0.6328 1.0642

2.3495 2.3437 2.3405 2.3334 2.2878 2.2339

2.2398 2.2352 2.2329 2.2270 2.1890 2.1440

119

3 Properties of Nonlinear Optical Crystals

120

lithium niobate grown from stoichiometric melt (mole ratio Li/Nb = 1.000), T == 293 K [3.274] :

A [urn]

no

ne

A [urn]

no

ne

0.42 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00 1.20 1.40 1.60

2.4089 2.3780 2.3410 2.3132 2.2967 2.2716 2.2571 2.2448 2.2370 2.2269 2.2184 2.2113

2.3025 2.2772 2.2457 2.2237 2.2082 2.1874 2.1745 2.1641 2.1567 2.1478 2.1417 2.1361

1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00

2.2049 2.1974 2.1909 2.1850 2.1778 2.1703 2.1625 2.1543 2.1456 2.1363 2.1263 2.1155

2.1306 2.1250 2.1183 2.1129 2.1071 2.1009 2.0945 2.0871 2.0804 2.0725 2.0642 2.0553

lithium niobate grown from congruent melt (mole ratio Li/Nb = 0.946), T = 293 K [3.281] :

A [urn]

no

ne

0.43584 0.54608 0.63282 1.1523 3.3913

2.39276 2.31657 2.28647 2.2273 2.1451

2.29278 2.22816 2.20240 2.1515 2.0822

T

=

297.5 K [3.282]:

A [urn]

no

ne

A [um]

no

ne

0.40463 0.43584 0.46782 0.47999 0.50858 0.54607 0.57696 0.57897 0.58756 0.64385

2.4317 2.3928 2.3634 2.3541 2.3356 2.3165 2.3040 2.3032 2.3002 2.2835

2.3260 2.2932 2.2683 2.2605 2.2448 2.2285 2.2178 2.2171 2.2147 2.2002

0.66782 0.70652 0.80926 0.87168 0.93564 0.95998 1.01400 1.09214 1.15392 1.15794

2.2778 2.2699 2.2541 2.2471 2.2412 2.2393 2.2351 2.2304 2.2271 2.2269

2.1953 2.1886 2.1749 2.1688 2.1639 2.1622 2.1584 2.1545 2.1517 2.1515

3.1 Basic Nonlinear Optical Crystals

A [11m]

no

ne

A [11m]

no

ne

1.28770 1.43997 1.63821 1.91125 2.18428

2.2211 2.2151 2.2083 2.1994 2.1912

2.1464 2.1413 2.1356 2.1280 2.1211

2.39995 2.61504 2.73035 2.89733 3.05148

2.1840 2.1765 2.1724 2.1657 2.1594

2.1151 2.1087 2.1053 2.0999 2.0946

Temperature derivative of refractive indices for lithium-rich niobate, T == 298 K [3.280] : ;t [11m]

dno/dT x 105 [K- 1]

dne/dT x 105 [K- 1]

0.3250 0.4545 0.6328 1.0642

8.71 1.93 0.522 0.141

12.9 6.22 4.31 3.85

stoichiometric melt (mole ratio Li/Nb = 1.000), ;t ==: 0.45 - 0.70 11m, T == 293 K) [3.283] : dno/dT == 2.0 x 10- 5 K- 1 5

dne/dT == 7.6 x 10- K-

1

,

;

Sellmeier equations (;t in urn, T == 20°C) for lithium-rich niobate [3.280] : n 2 == 4.91296

+

0.116275 - 0.0273;t2 ;t2 _ 0.048398 '

+

0.091649 - 0.0303;t2 . ;t2 - 0.046079 '

o

n 2 == 4.54528 e

stoichiometric melt (mole ratio Li/Nb = 1.000) [3.284] : n 2 == 4.91300

+

o

n 2 == 4.57906

+

e

0.118717 A,2 - 0.045932

- 0.0278;t2 '

0.099318 - 0.0224;t2 . ;t2 - 0.042286 '

congruent melt (mole ratio Li/Nb = 0.946) [3.281] : n 2 == 4.9048

+

0.117680 - 0.027169;t2 ;t2 - 0.047500 '

+

0.099169 - 0.021950;t2 . ;t2 - 0.044432

o

n 2 == 4.5820 e

121

122

3 Properties of Nonlinear Optical Crystals

Temperature-dependent Sellmeier equations (;, in urn, T in K) for lithium-rich lithium niobate [3.280] n~

==

4.913

+

+ 1.6 x

10- 8 (T2 - 88506.25) 8(T2

0.1163 + 0.94 x 10- 88506.25) _ 0.0273A? 2 8(T2 A - [0.2201 + 3.98 x 10- 88506.25)]2 '

n; == 4.546 + 2.72 x 10- (T 7

+

2

-

88506.25) 8

2

0.0917 + 1.93 x 10- (T - 88506.25) _ 0.0303A? . ;,2 _ [0.2148 + 5.3 x 10- 8(T2 - 88506.25)]2

stoichiometric melt (mole ratio Li/Nb = 1.000) [3.284] : n2 = 4.9130

+

o

8

= 4.5567 + 2.605 x

n2

2

0.1173 + 1.65 x 10- T _ 0.0278A? ;,2 _ (0.212 + 2.7 x 10- 8 T2)2 '

1O-7 T 2 +

e

8

2

- 0.0224A? 0.0970 + 2.70 x 10- T _ (0.201 + 5.4 x 10-ST2)2 '

;,2

congruent melt (mole ratio Li/Nb = 0.946) [3.285] : n~

= 4.9048 + 2.1429 x +

10- 8 (T2 - 88506.25) 8

n; = 4.5820 + 2.2971 x 10- (r 7

+

2

0.11775 + 2.2314 x 10- (T - 88506.25) _ 0.027153A,2 8(T2 - 88506.25)]2 ' ;,2 _ [0.21802 - 2.9671 x 102

-

88506.25) 8(T2

0.09921 + 5.2716 x 1O- 88506.25) _ 0.021940A,2 . ;,2 _ [0.21090 - 4.9143 x 10-8(T2 - 88506.25)]2

Experimental and theoretical values of phase-matching angle and calculated values of "walk-off" angle: lithium-rich lithium niobate, T == 295 K Interacting wavelengths [urn] SHG, 1.0642

o+o~e ~

0.5321

(Jexp

[deg]

(Jtheor

[deg]

P3 [deg]

[3.280] 67.5 [3.280] 66.76

1.776

3.1 Basic Nonlinear Optical Crystals

stoichiometric melt (mole ratio Li/Nb = 1.000), T In teracting wavelengths [urn]

(Jexp

[deg]

(Jtheor

[deg]

P3 [deg]

[3.284]

SHG, 0 + 0 => e 1.118=>0.559 1.1523 => 0.57615 SFG, 0 + 0 => e 2.17933 + 0.8529 => 0.613 4.0 + 0.72394 =} 0.613

71.7 [3.284] 67.6 [3.284] 68 [3.274] 69 [3.286]

71.80 67.74

1.312 1.543

55 [3.287] 47.5 [3.287]

54.75 47.48

2.073 2.212

congruent melt (mole ratio Li/Nb = 0.946), T Interacting wavelengths [urn]

(Jexp

[deg]

= 293 K

(Jtheor

[deg]

P3 [deg]

[3.284]

SHG, 0 + 0 => e 1.1523 => 0.57615 2.12 => 1.06 2.1284 => 1.0642 SFG, 0 + 0 => e 1.95160 + 1.0642 => 2.57887 + 1.0642 => 3.22241 + 1.0642 => 4.19039 + 1.0642 =>

= 293 K

0.68867 0.75333 0.80000 0.84867

72 [3.286] 70.39 43.8 [3.288] 45.25 44.6 [3.289] 45.28 47 [3.290]

1.341 1.988 1.987

52.7 [3.291] 48.1 [3.291] 46.5 [3.291] 47 [3.291]

2.000 2.047 2.044 2.026

52.86 48.13 46.50 46.90

Note: The PM angle values are strongly dependent on melt stoichiometry Experimental values of NCPM temperature: lithium-rich lithium niobate In teracting wavelengths [urn] SHG, 0 + 0 => e 0.954 =} 0.477 1.0642 => 0.5321 1.3188 => 0.6594

T

[OC] Ref.

-62.5 233.7 238 520

3.280 3.277 3.280 3.280

123

3 Properties of Nonlinear Optical Crystals

124

stoichiometric melt (mole ratio Li/Nb = 1.000) In teracting wagelengths [urn] SHG, 0 + 0 =} e 1.029 =} 0.5145 1.058 =} 0.529 1.0642 =} 0.5321 1.084 ~ 0.542 1.118 ~ 0.559 1.1523 =} 0.57615

T [OCl

Ref.

15 0 43 72 97 153.5 193 208 211

3.292 3.293 3.294 3.295 3.296 3.284 3.293 3.284 3.295

congruent melt (mole ratio Li/Nb = 0.946) In teracting wavelengths [11m] SHG, 0+0 ~ e 1.029 ~ 0.5145 1.0576 ~ 0.5288 1.0642 ~ 0.5321

1.084

~

1.1523

0.542

~

0.57615

T [OCl

Ref.

-66 -14 -8 6 11.5 38 42 46 172 174

3.292 3.278 3.297 3.298 3.294 3.299 3.297 3.292 3.297 3.282

Note: The NCPM temperature values are strongly dependent on melt stoichiometry Experimental value of internal angular bandwidth [3.81]: Al1nt[deg]

Interacting wagelengths [Jlm] SHG, 0+0 1.06 ~ 0.53

~

e 0.040

Experimental values of temperature and spectral bandwidths: In teracting wavelengths [urn] SHG, 0+0 1.06 ~ 0.53

~

T rOC]

Bpm [deg]

20

68

A T [OC]

LiVI [em-I]

Ref.

3.2

3.81

e

3.1 Basic Nonlinear Optical Crystals

1.0642

=}

*

1.084

0.5321

0.542

1.1523 =} 0.57615 SFG, 0 + 0 ~ e 1.7 + 0.6943 0.493 0.4115 2.65 + 0.488

* *

51 234 38 46 172

90 90 90 90 90

0.72 0.52 0.74 0.74 0.66

70 90

90 90

1.6

125

3.300 3.277 3.292 3.299 3.297 7.9 2.9

3.301 3.302

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe = d 3 1 sin () - d 22 cos ()sin 34> ,

d eoe

= d oee = d 22 cos 2 () cos 31> .

Nonlinear coefficients: stoichiometric melt (mole ratio LijNb

=

1.000)

d22(1.058 urn] = 2.46 ± 0.23 pm/V [3.274,37] , d 31 (1.058 pm) = -4.64 ± 0.66 pmjV [3.274, 37] ,

d33(1.058 pm) = -41.7 ± 7.8 pmjV [3.274,37] . congruent melt (mole ratio LijNb = 0.946)

d 22(1.06 pm) = 2.10 ± 0.21 pmjV [3.303,37] , d31

(1.06 um) = -4.35 ± 0.44 pmjV [3.303,37],

d33(1.06 um) = -27.2 ± 2.7 pmjV [3.303,37] . Laser-induced surface-damage threshold: 'r p

0.53 0.5321 0.59-0.596 0.6943 1.06

1.0642

[ns]

0.007 0.002 ~ 10 25 30 30 10-30 30 0.006 20 30

Ithr X

10- 12 [Wjm 2 ]

> 100

> 700 > 3.5 1.5 1.2 1.7 3.0 12 > 100 >1 150-200

Thermal conductivity coefficient [3.64]: K

= 4.6 W /mK at T = 300 K .

Ref. 3.304 3.305 3.305 3.306 3.307 3.308 3.309 3.307 3.288 3.289 3.310

Note 10 Hz 10 Hz 1 pulse

bulk damage

with coating

126

3 Properties of Nonlinear Optical Crystals

3.1.9 KNb0 3 , Potassium Niobate Negative biaxial crystal: 2Vz = 66.78° at A = 0.5321 urn [3.311]; Point group: mm2; Assignment of dielectric and crystallographic axes: X, Y,Z ~ b,a,c (Fig. 3.3) Transparency range at "0" transmittance level: ~ 0.4- > 4flm [3.312,313]; Linear absorption coefficient ex:

< 0.05 0.015 0.0018-{).0025

0.42-1.06 0.82 1.0642

Ref.

Note

3.314 3.315 3.316

along b axis

Z(c)

optic axis

..... Y(a)

~-~------+-~

X(b) Fig. 3.3. Dependence of refractive index on light propagation direction and polarization (index surface) in the first octant of dielectric reference frame (X, Y, Z) of KNb0 3 crystal. Designations: (J is the polar angle, t/> is the asimuthal angle, Vz is the angle between one of the optical axes and the Z axis

Experimental values of refractive indices at T = 295 K [3.312]:

A [urn]

nx

0.430 0.488 0.514 0.633

2.4974 2.4187 2.3951 2.3296

nz

2.4145 2.3527 2.3337 2.2801

2.2771 2.2274 2.2121 2.1687

3.1 Basic Nonlinear Optical Crystals

A [~m]

nx

ny

nz

0.860 1.064 1.500 2.000 2.500 3.000

2.2784 2.2576 2.2341 2.2159 2.1981 2.1785

2.2372 2.2195 2.1992 2.1832 2.1674 2.1498

2.1338 2.1194 2.1029 2.0899 2.0771 2.0630

127

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: XY plane, () == 90 0

In teracting wavelengths [urn] SHG, e +e => 0.946 =:} 0.473

¢exp

~

[3.311]

[3.312]

[3.317]

30 [3.318]

26.88

29.97

30.43

[deg]

()theor

[3.311]

[3.312]

[3.317]

77.37 69.03 63.36 60.27 46.57

83.13 70.67 63.92 60.43 45.95

87.98 71.92 64.94 61.37 46.52

[3.311]

[3.312]

[3.317]

71.05

71.85

71.16

()exp

SHG, 0 + 0 ==> e 0.43 0.86 0.445 0.89 0.92 0.46 0.94 0.47 0.5321 1.0642

83.5 [3.319] 70.7 [3.319] 64 [3.319] 60.5 [3.319] ~ 47 [3.311]

*

== 0

0

Interacting wavelengths [urn] SHG, 0 1.0642

[deg]

0

In teracting wavelengths [~m]

XZ plane, 1>

¢theor

0

YZ plane, 4> == 90

* * * *

[deg]

, ()

[deg]

> Vz ()exp

[deg]

()theor

[deg]

+ 0 =* e

*

0.5321

70.4 [3.320] 71[3.311] 71 [3.314] 71 [3.317] ~

Note: The dispersion relations given in [3.321] show worse agreement with the experiment

3 Properties of Nonlinear Optical Crystals

128

Experimental values of NCPM temperature: along X axis In teracting wavelengths [urn] SHG, type I 0.972 ~ 0.486 0.982 => 0.491 0.986 =* 0.493 0.988 ~ 0.494 1.047 => 0.5235 1.0642 =* 0.5321

T rOC]

Ref.

-20 20 20 20 162 178 181 182 184 188

3.322 3.323 3.324 3.314 3.325 3.326 3.311 3.320 3.300 3.327

along Yaxis Tnteracting wavelengths [um] SHG, type I 0.8385 =} 0.41925 0.8406 => 0.4203 0.842 ~ 0.421 0.856 ~ 0.428 0.857 =* 0.4285 0.8593 ~ 0.42965 0.86 => 0.43 0.8615 ::::} 0.43075 0.862 ::::} 0.431 0.879 ~ 0.4395 0.9289 ~ 0.46445 0.95 :::} 0.475 SFG, type I 0.6764 + 1.0642 :::} 0.41355 0.6943 + 1.0642 => 0.42017

T [0 C]

Ref.

-34.2 -28.3 -22.8 15 20 20 22 30 34 70 158 180

3.328 3.329 3.330 3.331 3.332 3.328 3.324 3.333 3.334 3.334 3.328 3.324

-4 27.2

3.335 3.335

Best set of dispersion relations (A in urn, T == 22°C) [3.312]: n2 x

= 1+

1.44121874A? _ 0.07439136

A2

+

2.54336918A?

A2 - 0.01877036

- O.02845018A?

3.1 Basic Nonlinear Optical Crystals

n2

= 1 + 1.33660410,1.1 + 2.49710396,1.1 _ 0.02517432,1.2 A2

y

n2

= 1+

z

-

A2 - 0.01666505

0.06664629 2

1.04824955A A2 _ 0.06514225

+

2

2.37108379A A2 - 0.01433172

_

0.01943289A2

•

Temperature-dependent dispersion relations (A in urn, T in K) [3.336]: n2 x

= 1 + (2.5389409 + 3.8636303 x 10- 6 F)A2 A2 _ (0.1371639 + 1.767 x 10- 7 F)2 (1.4451842 - 3.909336 x 10- 6F - 1.2256136 x 10-4 G)A2

+------------------,-A? - (0.2725429 + 2.38 x 10- 7F - 6.78 x 10- 5 G)2 - (2.837

10- 2

X

1.22

-

X

10- 8F)A2

-

3.3 x 10- 10F A4

,

= 1 + (2.6386669 + 1.6708469 x 10- 6 F)A2 Y JL2 - (0.1361248 + 0.796 x 10- 7 F)2

n2

(1.1948477 - 1.3872635 x 10-6 F - 0.90742707

X

10-4 G)A2

+ -2- - - - - - - - - -7 - - - - -5- - A - (0.2621917 + 1.231 x 10- F - 1.82 X 10- G)2 - {2.513 X 10- 2 n2 = z

-

0.558

X

10-8 F)A2

-

4.4 x 10- 10F A4

,

2

1 + (2.370517 + 2.8373545 x 10- 6 F)A A2 _ (0.1194071 + 1.75 x 10- 7 F)2

{1.048952 - 2.1303781 x 10-6 F - 1.8258521

X

10-4 G)A2

+ -2- - - - - - - - - -------A - (0.2553605 + 1.89 x 10-7 F - 2.48 X 10-5 G)2 - (1.939 where F == T 2

-

X

10-2

-

0.27

X

10-8 F);? - 5.7 x 10- 10 F)..,4 ,

295.15 2 , and G == T - 293.15 .

Calculated values of phase-matching and "walk-off" angles: YZ plane, e 1.0642 => 0.5321 1.3188 => 0.6594

Opm

[deg] P3 [deg]

45.95 29.87

3.009 2.507

129

XZ plane, 1> == 0° () > Vz Interacting wavelengths film] SHG, 0 + 0 =} e 1.0642 =} 0.5321 1.3188 =} 0.6594

()pm

[deg] P3 [deg]

71.85 57.47

2.479 3.553

Experimental values of the internal angular bandwidth: XZ plane, 1> == 0° Interacting wavelengths [urn]

SHG, 0+0 =} e 1.0642 => 0.5321

T [OC] ()pm [deg]

20

A()int

[deg]

0.013-0.014

71

Ref.

3.323

along Yaxis Interacting wavelengths [urn]

SHG, type I 0.857 =} 0.4285

T [OC]

()pm

20

90

[deg]

[deg]

A()int

0.659

A1>int

[deg]

1.117

Ref.

3.323

Experimental values of temperature bandwidth: along X axis Interacting wavelengths [Jlm] SHG, type I 1.0642 =} 0.5321

T [OC] ()pm [deg]

181 182 184 188

90 90 90 90

AT [OC]

Ref.

0.27-0.32 0.28 0.28-0.29 0.34

3.311 3.320 3.300 3.327

along Yaxis Interacting wavelengths [urn] SHG, type I 0.8385 =} 0.41925 0.842 =} 0.421 0.855 =} 0.4275 0.92 =} 0.46 SFG, type I 0.6764 + 1.0642 =} 0.41355

-34.2 -22.8 26.4 (?) 163.5 (?) -4

90 90 90 90

0.27 0.30 0.265 0.285

3.328 3.330 3.314 3.314

90

0.35

3.335

3.1 Basic Nonlinear Optical Crystals

131

Temperature of noncritical SHG [3.323] along X axis

Al == 0.97604 + 2.53 x 10- 4 T + 1.146 X 10- 6 T 2 along Yaxis

Al == 0.85040 + 2.94 x 10-4 T + 1.234 X 10-6 T 2 where A.I in urn, and Tin °C. Temperature variation of birefringence for noncritical SHG process [3.314]: along X axis (1.0642 urn => 0.5321Jlm)

d[nz(2w) - ny(w)]/dT == 1.10 x 10- 4 K- I along Yaxis (0.92 urn => 0.46 um)

d[nz(2w) - nx(w)]/dT == 1.43 x 10- 4 K- 1

.

Effective nonlinearity expressions in the phase-matching direction for three.. wave interactions in the principal planes of KNb0 3 crystal [3.35], [3.36]: XYplane d eeo

== d 32 sin2 ljJ + d 3 I cos 2 ljJ ;

YZ plane d ooe == d 32 sin (); XY plane, 0


Vz d ooe == d 3I sin () . d oeo

Effective nonlinearity expressions for three-wave interactions in the arbitrary direction of KNb0 3 crystal are given in [3.36]. Nonlinear coefficients [3.323,37,313]:

== -11.9 pm/V, d 32(1.0642Jlm) == -13.7 prnjV , d 33(1.0642 urn) == -20.6 pmjV .

d31(1.0642Ilm)

Laser-induced surface-damage threshold: A [J.1m]

1:p

0.527

0.5 0.5 10 25 11

0.5321 1.047

[ns]

Ithr X

10- 12 [W/m 2)

88-94 120-150 0.55 1.5-1.8 > 0.3

Ref.

Note

3.337 3.337 3.326 3.300 3.325

along b axis, E II c along b axis, E ..l c

4 kHz, 2000 hours

132

3 Properties of Nonlinear Optical Crystals

A [urn]

[ns] Ithr

1:p

1.054

10- 12 [W/m2 ]

110 180 370 1.5-1.8 > 1000

0.7 0.7 0.7 25 0.1

1.0642

X

Ref.

Note

3.337 3.337 3.337 3.300 3.323

along a axis, E J.. c along b axis, E J.. c along b axis, E J.. c

Thermal conductivity coefficient: K

> 3.5 W/mK [3.316] .

3.1.10 AgGaS2' Silver Thiogallate Negative uniaxial crystal: no > ne (at A < 0.497 urn ne > no); Point group: 42m ; Mass density: 4.58 g/cm 3 [3.338] ; Mohs hardness: 3 - 3.5 ; Transparency range at "0" transmittance level: 0.47 - 13 urn [3.339] ; Linear absorption coefficient ex:

A [urn]

(X

[cm:']

< 0.1

0.5-13 0.6-0.65 0.6--12 0.633 0.9-8.5 1.064 4-8.5

0.04 < 0.09 0.05 < 0.9 0.01 < 0.04

Ref. 3.340 3.341 3.339 3.342 3.343 3.342 3.341

Experimental values of refractive indices [3.344]:

A [~m]

A [um]

no

0.490 0.500 0.525 0.550 0.575 0.600 0.625 0.650 0.675 0.700 0.750 0.800

2.7148 2.7287 0.850 2.6916 2.6867 0.900 2.6503 2.6239 0.950 2.6190 2.5834 1.000 2.5944 2.5537 1.100 2.5748 2.5303 1.200 2.5577 2.5116 1.300 2.5437 2.4961 1.400 2.5310 2.4824 1.500 2.5205 2.4706 1.600 2.5049 2.4540 1.800 2.4909 2.4395 2.000

ne

A [~m]

no

ne

2.4802 2.4716 2.4644 2.4582 2.4486 2.4414 2.4359 2.4315 2.4280 2.4252 2.4206 2.4164

2.4279 2.200 2.4192 2.400 2.4118 2.600 2.4053 2.800 2.3954 3.000 2.3881 3.200 2.3819 3.400 2.3781 3.600 2.3745 3.800 2.3716 4.000 2.3670 4.500 2.3637 5.000

no

ne

2.4142 2.4119 2.4102 2.4094 2.4080 2.4068 2.4062 2.4046 2.4024 2.4024 2.4003 2.3955

2.3684 2.3583 2.3567 2.3559 2.3545 2.3534 2.3522 2.3511 2.3491 2.3488 2.3461 2.3419

3.1 Basic Nonlinear Optical Crystals

133

A [urn]

no

ne

A [um]

no

ne

A [Jlm]

no

ne

5.500 6.000 6.500 7.000 7.500

2.3938 2.3908 2.3874 2.3827 2.3787

2.3401 2.3369 2.3334 2.3291 2.3252

8.000 8.500 9.000 9.500 10.00

2.3757 2.3699 2.3663 2.3606 2.3548

2.3219 2.3163 2.3121 2.3064 2.3012

10.50 11.00 11.50 12.00 12.50

2.3486 2.3417 2.3329 2.3266 2.3177

2.2948 2.2880 2.2789 2.2716

Optical activity [3.339, 345]: p == 522 degjmm at isotropic point (no == ne , A == 0.4973 J,1m) Temperature dependences of refractive indices (A in J,1m)[3.346] :

2 5 10[39.88A -2no x - A.2 _ 0.0676

dno/dT

==

dn dT

== -

e/

4

112.20A

+ -(A-2-_-0-.0-6-7-6)~2

]

'

10-5 [25.50A.2 45.72A,4] x + + Zn; A,2 - 0.107584 (A,2 - 0.107584)2

.

Note: Canarelli et al. [3.347] observed the discrepancy between these dispersion formulas and the experiment Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: Interacting wavelengths [um]

SHG, 0 + 0 =} e 3.3913 =} 1.69565 10.6 =} 5.3

SFG,o + 0 => e 11.538 + 1.17233 => 1.0642 9.9 + 1.19237 =} 1.0642 8.7 + 1.21252 =} 1.0642 6.24 + 1.28301 =} 1.0642 5.89 + 1.29888 =} 1.0642 4.8 + 1.36735 => 1.0642 4.0 + 1.44996 => 1.0642 3.09 + 1.62325 =:} 1.0642 2.53 + 1.83683 =:} 1.0642 6.85 + 1.0642 =} 0.92110 4.43 + 1.0642 =} 0.85807 6.6 + 0.77593 =} 0.6943

()exp

[deg]

8t heor [deg] [3.348]

[3.349]

[3.350]

33 [3.339] 67 [3.351] 67.5 [3.352] 68 [3.339] 70.8 [3.344]

34.1 70.7

33.2 73.3

33.5 71.7

34.7 [3.48] 35.9 [3.353] 37 [3.354] 41.1 [3.355] 42.1 [3.353] 44 [3.354] 47.7 [3.355] 51 [3.350] 53.4 [3.350] 42 [3.356] 55 [3.356] 60 [3.357]

35.9 36.4 37.3 40.9 41.7 44.7 47.7 51.9 54.4 43.9 57.1 60.5

35.3 35.6 36.4 39.8 40.5 43.4 46.1 50.0 52.4 42.7 55.3 60.4

35.7 36.2 37.0 40.4 41.2 44.1 46.9 50.9 53.4 43.6 56.7 61.8

134

3 Properties of Nonlinear Optical Crystals

4.8 + 0.81171 => 0.6943 11.66329 + 0.617 => 0.586 10.12478 + 0.622 => 0.586 SFG, e + 0 => e 10.9 + 1.17934 :::} 1.0642 8.8 + 1.21060 => 1.0642 7.0 + 1.25500 => 1.0642 5.2 + 1.33803 => 1.0642 10.6 + 1.0642 => 0.96711 9.6 + 1.0642 => 0.95800 10.6 + 0.6943 => 0.65162

75.5 [3.357] 64 [3.358] 70 [3.358]

79.5 58.9 64.2

79.0 67.0 75.4

83.9 63.4 70.1

38.3 [3.359] 40.3 [3.359] 43.6 [3.359] 50.6 [3.359] 39.8 [3.360] 41.5 [3.360] 55 [3.361]

38.3 40.2 43.7 50.6 39.7 41.0 54.0

37.5 39.1 42.4 48.7 38.8 40.0 55.3

38.0 39.9 43.2 49.9 39.5 40.8 55.8

Note: The other sets of dispersion relations from [3.348, 362, 48] show worse agreement with the experiment Best of dispersion relations (A in urn, T = 20°C) [3.350]. n2

o

= 3.3970 +

2.3982A? A2 _ 0.09311

+

2.1640A? A2 - 950.0 '

2

2

2 _ 3 5873 1.9533A 2.3391A n-. +2 +2 e A-0.II066 A-I030.7

.

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [urn] SHG, 0 + 0 => e 10.6 => 5.3 9.6 => 4.8 5.3 => 2.65 4.8 => 2.4 2.9365 => 1.46825 2.1284 => 1.0642 SFG,o + 0 => e 10.6 + 3.533 => 2.65 10.6 + 2.65 => 2.12 10.6 + 1.0642 => 0.96711 10.6 + 0.6943 => 0.65162 SFG, e + 0 => e 10.6 + 5.3 => 3.533 10.6 + 1.0642 => 0.96711 10.6 + 0.6943 => 0.65162

(Jpm

[deg]

PI [deg]

P3 [deg]

71.68 58.15 32.00 31.04 37.27 54.23

0.76 1.15 1.17 1.15 1.24 1.18

37.40 34.79 37.31 52.85

1.25 1.21 1.21 1.04

58.15 39.52 55.76

1.18 1.32 1.23

1.15 1.23 1.00

3.1 Basic Nonlinear Optical Crystals

135

Experimental values of internal angular and spectral bandwidths at T = 293 K: In teracting wavelengths [urn] SHG, 0 + 0 => e 10.6 => 5.3 SFG,o + 0 => e 4.6 + 0.8177 => 0.6943 10.53 + 0.589 => 0.56589 6.24 + 1.283 => 1.0642 4.817 + 1.0642 => 0.87163 10.619 + 0.634 => 0.598 10.6 + 0.598 => 0.566 10.6 + 0.5968 => 0.565

Opm [deg]

L\Oint [deg]

67.5

0.41

3.339

82.7 90 41.1 52 90 90 90

0.42 2.34

3.357 3.349 3.355 3.356 3.341 3.363 3.364

i\Vl [cm'] Ref.

9.8 5.9 1.73 1.5 1.44

Temperature variation of phase-matching angle [3.360]: Interacting wavelengths [Jlm]

T [Oe]

Opm [deg]

dOpm /dT [deg/K]

SPG, e + 0 => e 10.6 + 1.0642 => 0.9671

20

39.8

0.03

Temperature tuning of noncritical SPG [3.347]: Interacting wavelengths [urn]

dAl/dT [nm/K]

SHG, 0 + 0 => e 7.8 + 0.65 => 0.6

~4

Experimental value of temperature bandwidth for the noncritical SPG process (10.6 urn + 0.598 urn => 0.566 urn, 0 + 0 => e): ~T

== 2.5 °C [3.346] .

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe = d 36 sin ()sin 24> ,

d eoe

=:

d oee

=:

d 36 sin 20 cos 24> .

Nonlinear coefficient: d 36(10.6 urn) == 0.134 x d 36(GaAs)

± 15%

=:

11.1 ± 1.7 pm/V [3.344], [3.37] , d 36(10.6 urn] == 0.15 x d 36(GaAs) ± 20% 12.5 ± 2.5 pm/V [3.351], [3.37] .

==

136

3 Properties of Nonlinear Optical Crystals

Laser-induced surface-damage threshold: A [um]

'tp

0.59 0.598 0.625 0.6943

500 3 500 30 10 10 35 20 17.5 15 12 0.023 0.025 0.002 0.021 0.020 150 150 220

1.06 1.0642

10.6

[ns]

I thr

X

10- 12 [W/m 2 ]

0.2 0.15 0.25--0.36 0.006 0.1 0.2 0.2--0.25 0.1 > 0.12 0.2 0.35 > 0.75 >7 > 10 > 20 30 0.1 0.2 0.25

Ref.

Note

3.358 3.363 3.358 3.361 3.357 3.348 3.348 3.350 3.365 3.352 3.359 3.366 3.48 3.367 3.355 3.353 3.349 3.368 3.365

10 pulses 10 pulses 1 Hz, 1000 pulses 100 pulses

10 Hz 1000 pulses 10 Hz 10 Hz 10 Hz

1000 pulses

Thermal conductivity coefficient at T == 293 K [3.58]: K

[W/mK],

II

c

1.4

K

[W/mK], 1- c

1.5

3.1.11 ZnGeP2 , Zinc Germanium Phosphide Positive uniaxial crystal: ne > no ; Point group: 42m ; Mass density: 4.12 g/cm 3 [3.338] ; Mohs hardness: 5.5 ; Transparency range at "0" transmittance level: 0.74 - 12 urn [3.369,370] Linear absorption coefficient a: A [urn]

a [cm"]

Ref.

1.9 2.15 2.5-8 2.5-8.3

0.8--0.95 0.6 < 0.1 < 0.2

3.371 3.372 3.373 3.374

Note

3.1 Basic Nonlinear Optical Crystals

A [urn]

~ [cm"]

Ref.

2.5-8.5 2.8-8.3 3-8 3.5-3.9 3.5 3.8 4.5-8 4.65

< 0.1 < 0.1 < 0.1

3.375 3.376 3.377 3.378 3.379 3.371 3.380 3.381 3.382 3.383 3.378 3.374 3.379 3.373 3.381 3.382 3.383 3.384 3.372 3.379 3.378

0.41 0.4 0.1-0.18 0.03 0.4 0.1-0.2 0.16 0.32 < 0.3

4.8 5.3-6.1 8.3-9.5 9 9.28 9.3

~

1

0.4 0.8 0.4-0.5 0.56 0.42 0.6 0.9 0.83

9.6 10.3 10.4 10.6

Note

o - wave, SFG direction

best samples

e - wave, SFG direction

e - wave, SFG direction

Experimental values of refractive indices [3.369]:

A [um]

no

ne

0.64 0.66 0.68 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.60 1.80 2.00 2.20

3.5052 3.4756 3.4477 3.4233 3.3730 3.3357 3.3063 3.2830 3.2638 3.2478 3.2232 3.2054 3.1924 3.1820 3.1666 3.1562 3.1490 3.1433

3.5802 3.5467 3.5160 3.4885 3.4324 3.3915 3.3593 3.3336 3.3124 3.2954 3.2688 3.2493 3.2346 3.2244 3.2077 3.1965 3.1889 3.1829

A [urn] 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.50 4.70 5.00 5.50 6.00 6.50 7.00 7.50

no

ne

3.1388 3.1357 3.1327 3.1304 3.1284 3.1263 3.1257 3.1237 3.1223 3.1209 3.1186 3.1174 3.1149 3.1131 3.1101 3.1057 3.1040 3.0994

3.1780 3.1745 3.1717 3.1693 3.1671 3.1647 3.1632 3.1616 3.1608 3.1595 3.1561 3.1549 3.1533 3.1518 3.1480 3.1445 3.1420 3.1378

137

138

3 Properties of Nonlinear Optical Crystals

A [~m]

no

ne

A [flm]

no

ne

8.00 8.50 9.00 9.50 10.00

3.0961 3.0919 3.0880 3.0836 3.0788

3.1350 3.1311 3.1272 3.1231 3.1183

10.50 11.00 11.50 12.00

3.0738 3.0689 3.0623 3.0552

3.1137 3.1087 3.1008 3.0949

Temperature derivative of refractive indices [3.369]: A [urn] dno/dT x 105 [K- 1]

dne/dT x 105 [K- 1]

A [um] dn.Jd T x 105 [K- 1]

0.64 0.66 0.68 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20

37.58 37.34 32.53 31.82 28.26 26.43 25.39 24.61 24.26 23.01 22.08 20.51 20.12 16.55 16.75 14.40 15.29 15.28 15.49 16.80 16.05 13.96 16.28

3.40 3.60 3.80 4.00 4.20 4.50 4.70 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00

35.94 31.23 29.52 28.63 26.22 24.69 24.12 22.34 21.32 21.18 20.11 18.63 16.84 15.34 15.10 13.20 14.19 14.60 14.14 15.13 15.48 13.26 14.94

14.40 15.58 14.58 14.26 13.57 15.31 15.51 15.05 14.49 14.58 15.60 12.85 18.15 16.10 15.16 15.56 16.27 16.53 15.40 15.25 14.74 14.24

dne/dT x 105 [K- 1]

15.46 16.29 16.53 15.02 15.14 16.60 16.71 16.43 15.42 16.30 16.13 15.01 18.59 17.43 17.37 17.50 17.11 18.41 16.84 16.34 18.32 16.59

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: In teracting wavelengths [urn] SHG, e + e=}o 3.8 ~ 1.9 4.34 ~ 2.17 4.64 ~ 2.32

f}exp

[deg]

57.8 ±0.3 [3.371] 55.8 ±0.2 [3.372] 47.5 [3.386]

Otheor

[deg]

[3.362]

[3.385]

59.8 52.5 50.1

59.7 52.4 49.9

3.1 Basic Nonlinear Optical Crystals

9.2 => 4.6 9.3 => 4.65

9.5 => 4.75

9.6 => 4.8 10.2 => 5.1 10.3 => 5.15 SFG, e + e=}o 10.668 + 4.34 => 3.085 9.74 + 4.2039 => 2.9365 SFG,o + e ::::} 0 6.74 + 5.2036 => 2.9365 6.45 + 5.3908 => 2.9365 6.25 + 5.5389 => 2.9365 6.15 + 5.6199 => 2.9365 6.29 + 5.0173 => 2.791 6.19 + 5.0828 => 2.791 6.06 + 5.1739 => 2.791 6.015 + 5.207 => 2.791 5.95 + 5.2569 => 2.791 5.90 + 5.2965 => 2.791 10.6 + 1.0642 => 0.9671

63.8 [3.387] 61.3 [3.375] 61.3 [3.385] 62.7-64.4 [3.382] 64 [3.381] 62.1 [3.375] 62.1 [3.385] 66.8 [3.387] 64.9 [3.382] 72 [3.375] 74.3 [3.384]

64.4 65.5

64.0 65.1

67.9

67.6

69.3 81.6 86.9

69.0 81.3 86.4

54.3 ± 0.2 [3.372] 51.5 49.6 [3.370] 49.5

51.3 49.3

76 [3.388] 79.2 [3.374] 84.0 [3.374] 85.5 [3.374] 76 [3.376] 77.6 [3.376] 80.5 [3.376] 84 [3.389] 83.4 [3.376] 87 [3.376] 84 [3.379]

74.9 78.8 83.3 89.0 76.5 78.0 80.5 81.6 83.6 85.8 83.4

Best set of dispersion relations (A in urn, T n2

= 4.47330 +

2 _

ne -

5.26576..1.1

+

A? - 0.13381

o

== 20°C) [3.385]:

1.49085..1.1

A? - 662.55 '

2

4 63 18 5.34215A + 2 . 3 A - 0.14255

75.9 80.1 85.9 no pm 77.4 79.1 82.0 83.3 86.1 no pm 83.0

139

2

+

1.45795A 2 A - 662.55

.

dispersion relation for T = 93 K, 173 K, 373 K, 473 K, and 673 K are given in [3.390], for T = 343 K in [3.391]. Calculated values of phase-matching and "walk-off" angles: In teracting wavelengths [urn]

(Jpm

SHG, e + e :::} 9.6 => 4.8 5.3 => 2.65

68.95 47.08

[deg]

PI [deg]

P2 [deg]

0.49 0.70

0.49 0.70

0

3 Properties of Nonlinear Optical Crystals

140

4.8 =* 2.4

SFG, e + e=}o 10.6 + 2.65 =} 2.12 9.6 + 2.4 =} 1.92 10.6 + 1.0642 :::} 0.96711 9.6 + 1.0642 =} 0.958 SFG,o + e=>o 10.6 + 5.3 =} 3.533 9.6 + 4.8 =} 3.2 10.6 + 1.0642 :::} 0.96711

48.97

0.69

0.69

50.11 51.08 72.54 82.66

0.72 0.71 0.42 0.18

0.66 0.69 0.47 0.21 0.20 0.46 0.19

81.66 69.74 83.31

Experimental values of internal angular bandwidth: Interacting wavelengths [urn] SHG, e+e =} 0 3.8 =} 1.9 4.34=>2.17 5.3 =? 2.65 9.3 :::} 4.65 9.6 =} 4.8 10.2 =} 5.1 10.3 =} 5.15

~oint

[deg]

Ref.

1.33 1.05 0.69 0.74-0.80 1.15 0.8 1.35 1.20

3.371 3.372 3.386 3.382 3.381 3.382 3.375 3.384

1.23

3.372

0.55

3.379

SFG, e + e=}o 10.668

+ 4.34

:::} 3.085

SFG,o + e=>o 10.6 + 1.064 =} 0.967

Experimental values of spectral bandwidth: Ref.

Interacting wavelength [urn] SHG, e + e :::}

0

7.9 4.9

4.34 =} 2.17 10.2 => 5.1

3.372 3.375

Experimental value of temperature bandwidth for SHG process =} 5.1 urn, e + e :::} 0);

(10.2 urn ~T

== 50°C

[3.375] .

3.1 Basic Nonlinear Optical Crystals

Temperature variation of phase-matching angle: In teracting wavelengths [um] SHG, e+e => 0 9.2 => 4.6 10.3 => 5.15 10.6 => 5.3 SFG,o + e=>o 10.6 + 1.0642 => 0.9671

dOpm/dT [deg/K]

Ref.

0.014 0.072 0.107

3.387 3.375 3.375

0.007

3.379

Effective nonlinearity expressions in the phase-matching direction [3.100]:

deeo == d36 sin 20 cos 2ifJ, doeo == deoo

:::::

d36 sin ()sin 2ifJ.

Nonlinear coefficient: d36(10.6Ilm) == 0.83 x d 36(GaAs) ± 15% == 68.9 ± 10.3 pm/V [3.369], [3.37] , d36(9.6Ilm) :::: 75 ± 8 pm/V [3.383] .

Laser-induced surface-damage threshold: A [urn]

Lp

[ns]

1.064

30 10 2.79 0.15 0.1 0.11 2.94 0.11 cw 5.3-6.1 cw 9.28 2 9.3-10.6 125 125 9.3 100r 9.6 129 10.2-10.8 105 - 107 cw cw 10.6 cw

Ithr X

10- 12 [W/m 2 ]

> 0.03 0.03 300 350 300 300 > 0.0001 0.0025 12.5 0.3-0.4 0.25 0.12 0.78 0.6 > 0.00001 > 0.0000001 0.002

Ref.

Note

3.392 3.369 3.376 3.389 3.388 3.370 3.386 3.378 3.373 3.384 3.384 3.381 3.383 3.375 3.375 3.392 3.378

12.5 Hz

Thermal conductivity coefficient at T = 293 K [3.58]: K

[W/mK],

36

II

c

K

[W/rnK] , 1- c

35

2 Hz 20 Hz 100 Hz

1500 Hz

141

142

3 Properties of Nonlinear Optical Crystals

3.2 Frequently Used Nonlinear Optical Crystals 3.2.1 KBsOs . 4H 20, Potassium Pentaborate Tetrahydrate (KB5) Positive biaxial crystal: 2Vz == 126.3° at A = 0.5461 urn [3.393]; Point group: mm2; Assignment of dielectric and crystallographic axes: X, Y,Z => a,b,c (Fig. 3.4); Molecular mass: 1.74 g/cm 3 [3.394]; Mohs hardness: 2.5 [3.394]; Transparency range at "0" transmittance level: 0.162 - 1.5 urn [3.395]; z(c)

X(a) Fig. 3.4. Dependence of refractive index on light propagation direction and polarization (index surface) in the first octant of dielectric reference frame (X, Y, Z) of KB5 crystal. Designations: (J is the polar angle, 0.434 => 0.217

cPexp

[deg]

cPtheor

293 K) and comparison

[3.401]

[3.402]

81.6

no pm

0

90 [3.400]

144

3 Properties of Nonlinear Optical Crystals

0.4342 =} 0.2171 0.4384 =} 0.2192 0.50 => 0.25 0.630 =} 0.315 SFG, e + e=}o 0.5435 + 0.3511 =} 0.2133 0.6943 + 0.3472 =} 0.2314 0.5737 + 0.3345 =} 0.2113 0.6522 + 0.3261 =} 0.2174 0.6219 + 0.3110 =} 0.2073 0.6943 + 0.30519 => 0.2120 0.6943 + 0.28409 => 0.2016 1.06415 + 0.26604 => 0.2128 0.78971 + 0.26604 => 0.1990 0.75322 + 0.26604 => 0.1966 0.79737 + 0.25725 => 0.1945 0.79235 + 0.25725 => 0.1942 0.9 + 0.23287 => 0.185

YZ plane,

¢

90 [3.403] 80.5 [3.86] 52.8 [3.400] 31 [3.403]

81.3 77.3 53.8 33.0

no pm 80.4 54.1 32.8

90 [3.404] 57 [3.405] 90 [3.404] 68 [3.398] 90 [3.398] 70 [3.406] 90 [3.406] 53 [3.397] 75 [3.407] 90 [3.407] 84 [3.408] 90 [3.408] 90 [3.409]

78.9 56.3 77.9 65.7 76.9 66.2 74.2 48.5 67.5 72.5 70.0 70.7 68.4

87.3 57.9 87.2 68.8 no pm 70.5 no pm 52.1 76.1 no pm 83.3 85.6 no pm

== 90°

In teracting wavelengths [urn] SHG, 0 + 0 => e 0.4346 => 0.2173 0.4690 => 0.2345 0.4796 => 0.2398 SFG,o + 0 => e 0.5634 + 0.3511 ==> 0.2163 0.5948 + 0.3345 ==> 0.2141 0.6264 + 0.3132 ==> 0.2088 0.7621 + 0.26604 => 0.1972

¢exp

[deg]

¢theor

[deg]

[3.401]

[3.402]

90 [3.405] 17 [3.405] o [3.403]

69.1 no pm no pm

83.4 12.8 no pm

63 63 68 68

49.2 47.0 52.2 38.5

59.9 59.8 72.0 75.3

[3.404] [3.404] [3.398] [3.407]

Best set of dispersion relations (revised data of [3.401], given in [3.402], A in urn, T == 293 K): ,12 2 n == 1 + x 0.848117,12 - 0.0074477 ' ,12 2 n == 1 + y 0.972682,12 - 0.0087757 ' ,12 n~ == 1 + 2 . 1.008157,1 - 0.0094050

3.2 Frequently Used Nonlinear Optical Crystals

145

Calculated values of phase-matching and "walk-off" angles: XY plane, f) = 90°

Interacting wagelengths [urn]

cjJpm [deg]

SHG, e+e => 0 0.5105 => 0.25525 51.62 0.532075 => 0.26604 47.19 39.57 0.5782 => 0.2891 25.83 0.6973 => 0.34715 SFG, e + e=>o 1.06415 + 0.532075 => 0.35473 20.65 36.35 1.06415 + 0.35473 => 0.26604 52.12 1.06415 + 0.26604 => 0.21283 57.93 0.6943 + 0.34715 => 0.23143 0.5782 + 0.5105 :::} 0.27112 45.17

PI [deg]

P2 [deg]

2.037 2.073 2.020 1.585

2.037 2.073 2.020 1.585

1.324 1.946 2.015 1.889 2.017

1.332 1.979 2.078 1.918 2.075

Experimental values of NCPM temperature: along b axis Interacting wavelengths [urn]

T roC]

Ref.

SFG, type I 0.6943 + 0.28334 => 0.20122 0.6943 + 0.28361 => 0.20136 0.6943 + 0.28405 => 0.20158 0.6943 + 0.28449 => 0.20180 0.79202 + 0.25725 => 0.19418 0.79344 + 0.25725 => 0.19427

-15 0 20 35 25 40

3.406 3.406 3.406 3.406 3.408 3.408

Effective nonlinearity expressions in the phase-matching direction for threewave interactions in the principal planes of KB5 crystal; [3.35, 36]: XYplane d eeo

= d 3I sin 2 4J + d 32 cos 2 4J

YZ plane

= d 31 sin f) ; XZ plane, () < V» d oeo = d eoo = d 32 sin f) XZ plane, f) > Vz d ooe

d ooe = d32 sin f)

;

.

Effective nonlinearity expressions for three-wave interactions in the arbitrary direction of KB5 crystal are given in [3.36].

Nonlinear coefficients [3.37]: d31 (0.5321 um] = 0.04 pm/V ,

d 32 (0.5321 urn) = 0.003 pmjV , d33(0.5321 um) = 0.05 pm/V .

Laser-induced surface-damage threshold: A [Ilm]

't p

0.2661

8 0.03 10 8 7 10 10 30 12

0.311 0.3472 0.45 0.622 0.6943 0.74-0.91 1.0642

[ns]

Ithr X

10- 12 [Wjm 2 ]

> > > >

0.43 4.8 0.13 0.9 10 > 0.4 > 0.8 > 0.5 > 0.85

Ref.

Note

3.397 3.410 3.398 3.393 3.405 3.398 3.393 3.409 3.397

10 Hz 1 Hz 10 Hz 15 Hz 10 Hz

10 Hz

3.2.2 CO(NH2)2, Urea Positive uniaxial crystal: ne > no; Point group: 42m; Mass density: 1.318 g/cm'; Mohs hardness: < 2.5 ; Transparency range at 0.5 transmittance level for a 0.5 em long crystal cut at f) == 74° : 0.2 - 1.43 urn [3.411]; Linear absorption coefficient ex [3.411]:

A [urn]

rx [cm"]

Note

0.213 0.266 1.064

0.10 0.04 0.02

o - wave, FIHG direction e - wave, FIHG direction e - wave, FIHG direction

The graph of no and n« dependences versus wavelength is given in [3.412, 413]. Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: In teracting wavelengths [urn] SHG, e+e => 0.476 => 0.238 0.500 => 0.250 0.550 => 0.275 0.600 => 0.300

f)exp

[deg]

f)theor

[deg]

[3.414]

[3.415]*

[3.416]

82.2 67.5 54.2 46.5

no pm 76.7 55.9 46.5

no pm 72.2 55.2 46.5

0

90 [3.414] 67.6 [3.414] 54 [3.414] 46.6 [3.414]

3.2 Frequently Used Nonlinear Optical Crystals

SFG, e + e ~ 0 0.6943 + 0.34715 => => 0.23143 1.0642 + 0.26605 => => 0.21284 SHG, 0 + e=>o 0.597 => 0.2985 0.650 => 0.325 0.700 => 0.350 SFG,o + e=>o 1.0642 + 0.29146 => 0.2288 1.0642 + 0.29668 => 0.2320 1.0642 + 0.30656 => 0.2380 1.0642 + 0.42792 => 0.3052 1.0642 + 0.63501 => 0.3977 0.720 + 0.53764 => 0.3078 0.646 + 0.58793 => 0.3078 0.62875 + 0.5321 => 0.2882 0.63980 + 0.5321 ::::} 0.2905 0.66406 + 0.5321 => 0.2954 SFG, e + 0 => 0 1.0642 + 0.50787 => 0.3438 1.0642 + 0.53 => 0.3538 1.0642 + 0.575 => 0.3733 1.0642 + 0.63195 => 0.3965

77 [3.411]

81.5

no pm

no pm

72 [3.411]

86.7

no pm

no pm

90 [3.414] 63.6 [3.414] 55.6 [3.414]

no pm 65.4 56.6

no pm 63.5 54.6

no pm 64.6 55.6

90 [3.414] 80 [3.414] 70.4 [3.414] 47.5 [3.414] 37.7 [3.414] 63 [3.417] 69 [3.418] 90 [3.414] 80.5 [3.414] 73.4 [3.414]

no pm 83.6 75.0 49.9 39.1 64.7 71.6 no pm 85.1 75.3

no pm 80.9 70.0 48.8 37.1 63.1 70.0 no pm 84.3 74.3

76.6 72.8 67.8 47.0 37.6 62.7 70.3 no pm 81.5 73.1

90 [3.414] 72.2 [3.414] 62.5 [3.414] 53.5 [3.414]

79.2 70.9 61.4 53.9

84.5 72.3 61.5 53.4

no pm 74.8 63.0 54.5

*with correction given in [3.419]. Best set of dispersion relations (A. in urn, T == 293 K) [3.415, 419]: n 2 == 2.1548

+

o

2-25527 n-. e

0.01310 0.0318'

A? -

0.01784 A. - 0.0294

+2

+

147

O.0288(A-l.5) 2 (A - 1.5) + 0.03371

.

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [urn]

(Jpm

SHG, 0 +e => 0 0.6118 => 0.3059 0.6328 => 0.3164 0.6594 => 0.3297 0.6943 => 0.34715

75.95 68.01 61.49 55.40

[deg]

PI [deg]

P2 [deg]

2.31 3.35 3.98 4.34

148

3 Properties of Nonlinear Optical Crystals

SFG,o + e=>o 1.0642 + 0.5321 =? 1.3188 + 0.6594 =? SFG, e + 0 => 0 1.0642 + 0.5321 => 1.3188 + 0.6594 =>

0.35473 0.4396

41.10 30.46

0.35473 0.4396

71.63 48.98

4.52 3.82 2.69 3.97

Experimental value of internal angular bandwidth [3.411]: In teracting wavelengths (flm] FIHG, e + e=>o 1.064 + 0.266 => 0.213

~f)int

[deg]

0.017

Temperature tuning for noncritical SHG [3.414]: In teracting wavelengths [urn] SHG, e + 0 => e 0.597 :::} 0.2985

dAIIT [nm/K]

-0.013

Effective nonlinearity expressions in the phase-matching direction [3.100]: deeo

== d36 sin 20 cos 2fjJ,

doeo == deoo == d36 sin 0 sin 2fjJ. Nonlinear coefficient: d36(0.6 um) ~ 3 x d 36(KDP) = 1.17 pm/V [3.412,37], d 36(O.597 urn) == 2.4 x d36(ADP) ± 80/0 == 1.13 ± 0.09 pm/V [3.419,37] .

Laser-induced bulk-damage threshold:

A [Jlm]

Lp

0.266 0.355

10 10 10 10 10

0.532 1.064

[ns]

Ithr X

5 14 1.5 30 50

10- 12 [W/m2 ]

Ref.

Note

3.420 3.420 3.421 3.420 3.420

single pulse single pulse 3000 pulses single pulse single pulse

3.2 Frequently Used Nonlinear Optical Crystals

149

3.2.3 CsH 2As04 , Cesium Dihydrogen Arsenate (CDA) Negative uniaxial crystal: no > ne ; Point group: 42m ; Mass density: 3.53 gjcm 3 ; Transparency range at 0.5 transmittance level for a 17.5 mm long crystal cut at f) = 90°, 1J = 45° : 0.26 - 1.43 urn [3.422] ; UV edge of transmission spectrum at "0" transmittance level: 0.2161lm [3.113] ; IR edge of transmission spectrum at "0" transmittance level: 1.871lm for 0 - wave, 1.671lm for e - wave [3.78] ; Linear absorption coefficient lJ. : Ref. 0.35-1.4 1.062 1.064

0.6 0.041 0.041

3.113 3.120 3.422

Two-photon absortion coefficient fJ( f) = 90°, 1J = 45°) [3.71]:

fJ 0.355

X

1013 [mjW]

Note e - wave

2.81

Experimental values of refractive indices [3.422]: A [um]

no

ne

0.3472 0.5321 0.6943 1.0642

1.6027 1.5733 1.5632 1.5516

1.5722 1.5514 1.5429 1.5330

Temperature derivative of refractive indices [3.74]:

0.405 0.436 0.546 0.578 0.633

-3.15 -3.05 -2.59 -2.76 -2.80

-1.89 -2.09 -2.12 -2.39 -2.56

150

3 Properties of Nonlinear Optical Crystals

Experimental values of phase-matching angle (T between different sets of dispersion relations: Interacting wavelengths [urn]

[deg]

Oexp

=

Otheor

293 K) and comparison

[deg]

[3.74]

[3.78]K

[3.78] E

59.8 59.7 59.0

no pm

no pm

no pm no pm

no pm no pm

58.6

no pm

88.7

58.3

88.2

86.5

SHG, o+o:::::} e 1.05 ~ 0.525 1.052 => 0.526 1.06 =* 0.53

90 [3.119] 90 [3.74] 87 [3.423] 87 [3.96] 83.5 [3.424] 83.5 [3.425] 84.2 [3.422] 84.4 [3.426]

1.0642 => 0.5321

1.068 => 0.534

Note: [3.78] K => see [3.78], set of Kirby et al. ; [3.78] E =* see [3.78], set of Eimerl. Experimental values of NCPM temperature: Interacting wavelengths [urn]

T roC]

Ref.

20 20 31 40.3 41 42 43 44.5 45 46 48 39.6 49.2 61 100

3.119 3.74 3.423 3.427 3.425 3.428 3.426 3.90 3.120 3.424 3.422 3.422 3.429 3.428 3.119

Note

SHG, 0+0 =* e 1.05 => 0.525 1.052 ==> 0.526 1.06 => 0.53 1.0642 => 0.5321

1.073 ==> 0.5365 1.078 => 0.539

10 Hz

12.5 Hz 0.1-1 Hz 20 Hz 10 Hz

Best set of dispersion relations (A. in urn, T = 293 K) [3.78]E:

n~ = 1.8776328 -

O.03602222A?

+ O.00523412U4 +

2

O.550395U?

A. - (0.1625700)

2'

3.2 Frequently Used Nonlinear Optical Crystals

n; = 1.6862889 - O.01372244A? + O.003948463A.

4

+

2

O.669457U 2 A - (0.1464712) 2

Calculated values of phase-matching and "walk-off" angles: In teracting wavelengths [urn]

Opm

SHG, 0 +0 =} e 1.0642 => 0.5321 1.3188 => 0.6594

[deg]

P3 [deg]

0.035 0.384

88.72 74.52

Experimental values of internal angular and temperature bandwidths: In teracting wavelengths [urn] SHG, 0 + 0 => e 1.06 => 0.53

1.062 => 0.531 1.0642 => 0.5321

T[OC]

22 31 20 63 (?) 45 40.3 24 46 20 48 20 43

fJpm [deg]

87 90 87 90 90 90 83.5 90 84.2 90 84.4 90

Arfnt

AT[OC] Ref.

[deg] ~O.4 ~

3.8 0.43 3.03 2.85 0.86 3.2 0.70 2.91 0.70

~3

6.5 6.8 rv8

6

~3

3.423 3.423 3.96 3.96 3.120 3.427 3.424 3.424 3.422 3.422 3.426 3.426

Temperature variation of phase-matching angle: In teracting wavelengths [urn] SHG, 0 + 0 => e 1.06 => 0.53 1.0642 => 0.5321

151

T [OC]

fJpm [deg]

dfJpm/dT [deg/K]

Ref.

20 63(?) 24 20 35 39 41

87 90 83.5 84.4 86.5 87.6 88.3

0.085 0.481 0.129 0.131 0.194 0.251 0.537

3.96 3.96 3.424 3.426 3.426 3.426 3.426

.

152

3 Properties of Nonlinear Optical Crystals

Temperature tuning of noncritical SHG [3.74]: In teracting wavelengths [urn]

dAI/dT [nm/K]

SHG, 0 + 0 => e 1.052 :=} 0.526

0.308

Temperature variation of birefringence for (1.0642 urn ~ 0.5321 urn, 0 + 0 => e): d(n~ - nf)/dT

noncritical SHG process

== 7.2 x 10- 6 K- 1 [3.427] ,

d(ni - n1)/dT == (8.0 ± 0.2) x 10-6 K- 1 [3.422] . Effective nonlinearity expressions in the phase-matching direction [3.100]:

== d36 sin 8 sin 24>, d eoe == d oee = d36 sin 28 cos 24> . dooe

Nonlinear coefficient: d36(1.0642 urn) == 0.40

± 0.05 pm/V

[3.422] .

Laser-induced bulk-damage threshold:

A [urn] 0.532 1.062 1.064

'tp

[ns]

10 0.007 12 10 18

Ithr

x 10- 12 [W/m 2]

>3 > 40 > 2.6 3.5 4

Ref.

Note

3.429 3.120 3.422 3.424 3.427

10-20 Hz 12.5 Hz 2-50 Hz

3.2.4 CsD2As04 , Deuterated Cesium Dihydrogen Arsenate (DCDA) Negative uniaxial crystal: no > n.: Point group: 42m ; Transparency range at 0.5 transmittance level for a 13.5 mm long crystal cut at 8 == 90°, t/J == 45° : 0.27 - 1.66 urn [3.422] IR edge of transmission spectrum at "0" transmittance level: 2.03 urn for 0 - wave, 1.78 urn for e - wave [3.78] ;

3.2 Frequently Used Nonlinear Optical Crystals

Linear absorption coefficient

A [Ilm]

(J.

1.062 1.064

0.01 0.02

[em-I]

153

lJ. :

Ref. 3.120 3.422

Two-photon absorption coefficient fJ(8

A [urn] fJ x 1013 [m/W]

Note

0.355

0 - wave e - wave

8.0 5.1

= 90°,4> = 45°) [3.71]:

Experimental values of refractive indices [3.422]: A [urn]

no

ne

0.3472 0.5321 0.6943 1.0642

1.5895 1.5681 1.5596 1.5503

1.5685 1.5495 1.5418 1.5326

Temperature derivative of refractive indices [3.74]:

0.405 0.436 0.546 0.578 0.633

-2.26 -2.26 -2.47 -2.31

-1.77 -1.51 -1.64 -1.71 -1.70

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: Interacting wavelengths [urn] SHG, 0+0 =* e 1.034 =* 0.517 1.037 =* 0.5185 1.046 =* 0.523

8exp [deg]

90 [3.119] 90 [3.74]

8theor

[deg]

[3.74]

[3.78]K [3.78]E

65.2 64.8 63.7

no pm no pm no pm no pm 88.4 88.1

154

3 Properties of Nonlinear Optical Crystals

1.0642 ==> 0.5321

79.35 [3.422] 80.8 [3.426]

61.8

82.4

82.3

[3.78]K =:} see [3.78], data of Kirby et al. ; [3.78]E =:} see [3.78], data of Eimerl.

Note:

Experimental values of NCPM temperature: In teracting wavelengths [flm] SHG, 0 + 0 =:} e 1.034 =:} 0.517 1.037 => 0.5185 1.0642 => 0.5321

T [OC]

Ref.

20 20 102 102 112.3 109.8 96.4 108

3.119 3.74 3.428 3.425 3.422 3.422 3.426 3.119

Note

90% deuteration, < 1 Hz 90% deuteration, 20 Hz 70 % deuteration

Best set of dispersion relations (A in urn, T n~

== 1.6278496 - 0.018220310A2 +0.000281333U4 +

n~

== 293 K) [3.78]E :

0.7808170;,2 A2 - (0.1407699)2 '

== 1.6236063 - 0.009338692A? + 0.001965413014 +

0.7249589A?

.

A? - (0.1414850)2

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [Ilm]

SHG,

0

fJpm [deg] P3 [deg]

+ 0 => e

1.0642 =:} 0.5321 1.3188 =:} 0.6594

82.32 69.54

0.188 0.449

3.2 Frequently Used Nonlinear Optical Crystals

Experimental values of internal angular and temperature bandwidths: In teracting wavelengths [urn] SHG,o+o=:}e 1.0642 =:} 0.5321

T [OC]

Opm [deg]

L\Oint [deg]

L\T [OC]

20 112.3 20 96.4

79.35 90 80.8 90

0.41 2.90 0.50

6.1

~3.5

Ref.

3.422 3.422 3.426 3.426

Temperature variation of the phase-matching angle [3.426]: Interacting wavelengths [urn]

T

[OC]

SHG, 0 + 0 =:} e 1.0642 =:} 0.5321 20 66.3 80 87.7

Opm [deg]

dOpm/dT [deg /K]

80.8 84.3 86.4 88.1

0.042 0.081 0.270 0.533

Temperature tuning of noncritical SHG [3.74]: Interacting wavelengths [Ilm] SHG, 0 + 0 =:} e . 1.037 =:} 0.5185

dA,l/dT [nm/K] 0.317

Temperature variation of birefringence for noncritical SHG process (1.0642Ilm =:} 0.5321Ilm, 0 + 0 =:} e) :

d(n~; n1) = (7.8 ± 0.2)

x 10-6 K- 1 [3.422] .

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe = d36 sin 0 sin 2¢ , d eoe

= d oee = d 36 sin 20 cos 2¢ .

Nonlinear coefficient: d36(1.0642Ilm) = 0.40

± 0.05 pm/V

[3.422] .

Laser-induced bulk-damage threshold:

A [urn] !p [ns]

Ithr X

1.064

>2.6 >2.5

12 12

10- 12 [W/m 2 ] Ref. 3.422 3.139

Note 10-20 Hz 0.1-20 Hz

155

156

3 Properties of Nonlinear Optical Crystals

3.2.5 KTiOAs0 4 , Potassium Titanyl Arsenate (KTA) Positive biaxial crystal: 2Vz == 34.5° at A == 0.5321 urn; Point group: mm2; Assignment of dielectric and crystallographic axes: X, Y,Z::::} a,b,c;

Transparency range at "0" transmittance level: 0.35 - 5.3 urn [3.430, 431]; Linear absorption coefficient a [3.432] :

A [Jlm]

rJ,

4.0 5.0

0.2 1.0

[em-I]

Experimental values of refractive indices [3.433]:

A [urn]

nx

nr

nz

0.6328

1.8083

1.8142

1.9048

Experimental values of phase-matching angle (T == 293 K) and comparison between different sets of dispersion relations: XY plane, f}

== 90°

Interacting wavelengths [um] SHG, e + 0 => e 1.053 => 0.5265 1.0642 :::} 0.5321 SFG, e + 0 =:} e 1.3188 + 0.6594 =:} =? 0.4396 1.0642 + 1.5791 =:} =? 0.6358 YZ plane,

¢exp

[deg]

[deg]

¢theor

[3.433]

[3.434]

65 [3.434] no pm 57.8 [3.434] no pm

64.97 57.58

47.8 [3.434] 68.84

47.79

19.8 [3.434] 16.64

19.63

4J == 90°

Interacting wavelengths [urn] SHG, 0 + e=>o 1.0642 =* 0.5321 1.1523 =* 0.57615

f}exp

[deg]

f}theor

[deg]

[3.433] 76.3 [3.434] no pm 64 [3.434] 69.30

[3.434] 76.28 63.94

3.2 Frequently Used Nonlinear Optical Crystals

1.3188 =} 0.6594 SFG, 0 + e=}o 1.3188 + 0.6594 =} =} 0.4396 1.0642 + 1.5791 =} =} 0.6358 4.15 + 1.0642 =} =} 0.847

XZ plane,

55.9 [3.433] 56.22

53.09

71.2 [3.434] 82.37

71.15

67.3 [3.434] 73.04

67.29

30.3 [3.431] 31.19

31.87

4J == 0°, 0 >

Interacting wavelengths [Ilm]

Vz

(}exp

[deg]

(}theor

[deg]

[3.433]

SHG, 0 + e=}o 1.1523 =} 0.57615 1.3188 =} 0.6594 SFG, 0 +e =} 0 1.5791 + 0.6358 =} =} 0.4533

[3.434]

82.9 [3.434] 80.61 64.2 [3.434] 63.28

83.00 64.25

73.7 [3.434] 72.82

73.74

Best set of dispersion relations (A in urn) [3.434]: n2 x

n2

== 3.1533 + == 3.1775 +

Y

n2 z

== 3.4487 +

0.04029

- 0.01320,12

0.04353

- 0.01444,12

.,t2 _ 0.04932

A? - 0.05640

'

'

0.06334 _ 0.01646 A2 A2 - 0.05887

.

Calculated values of phase-matching and "walk-off" angles: XY plane, 0 == 90°

Interacting wavelengths [urn]

SHG, e + 0 =} e 1.0642 =} 0.5321 SFG, e + 0 => e 1.3188 + 0.6594 => =} 0.4396

(jJpm [deg]

PI [deg]

P3 [deg]

57.58

0.211

0.337

47.79

0.217

0.511

157

158

3 Properties of Nonlinear Optical Crystals

YZ plane, ¢

= 90

0

In teracting wavelengths [urn]

SHG, 0 + e =* 0 1.0642 =* 0.5321 1.1523 =} 0.57615 1.3188 =} 0.6594 2.098 =} 1.049 2.9365 =} 1.46825 SFG, 0 + e =* 0 1.3188 + 0.6594 =} =} 0.4396

XZ plane, ¢ = 0

0,0

Interacting wavelengths [urn] SHG, 0 + e=}o 1.1523 =} 0.57615 1.3188 =} 0.6594 2.098 =} 1.049 2.9365 =} 1.46825

Opm [deg]

P2 [deg]

76.28 63.94 53.09 44.71 59.80

1.179 1.978 2.344 2.345 2.042

71.15

1.708

> Vz Opm [deg]

P2 [deg]

83.00 64.25 53.50 69.37

0.676 2.119 2.445 1.657

Experimental values of internal angular and temperature bandwidths: XY plane, 0 = 90 0

Interacting wavelengths [urn]

SHG, e+o =} e 1.053 =} 0.57615 1.0642 =} 0.5321

yz plane,

4> = 90

Interacting wavelengths [11m] SHG, 0 +e =} 0 1.3188 =} 0.6594

4Jpm [deg]

AcjJint [deg]

AT rOC]

Ref.

65 57.8

0.4 0.37

10.4

3.430 3.434

Opm [deg]

AOint [deg]

Ref.

55.9

0.093

3.433

0

3.2 Frequently Used Nonlinear Optical Crystals

159

Effective nonlinearity expressions in the phase-matching direction for threewave interactions in the principal planes of KTA crystal [3.35, 36]: XY plane

d eoe == d oee == d31 sirr' ¢

+ d 32 cos 2 ¢

;

yz plane d oeo == d eoo == d 31sin () ; XZ plane, ()

< Vz

d ooe == d 32 sin () ;

XZ plane, ()

> Vz

d oeo == d eoo == d32 sin () . Effective nonlinearity expressions for three-wave interactions in the arbitrary direction of KTA crystal are given in [3.36] Nonlinear coefficients: d 31(1.0642Ilm) == 2.5 ± 0.3 pm/V [3.434] , 2.8

d32(1.0642Ilm)

± 0.3

pm/V [3.433] ;

== 4.2 ± 0.4 pm/V [3.433] , 4.5 ± 0.5 pm/V [3.434] ;

d33(1.06421lffi) == 16.2 ± 1.0 pru/V [3.433] . Laser-induced surface-damage threshold:

A [urn]

'r p

0.85 1.0642

2 8

[ns]

Ithr X

>10 >12

10- 12 [W/m 2 ]

Ref.

Note

3.431 3.432

20 Hz, 1000 pulses

3.2.6 MgO : LiNb03, Magnesium-Oxide-Doped Lithium Niobate (5 mole % MgO) Negative uniaxial crystal: no > ne ; Point group: 3m ; Transparency range at "0" transmittance level: Linear absorption coefficient ex:

0.5321 1.0642

0.02 0.96711 10.6 + 0.6943 => 0.65162 SFG,o+e*e 10.6 + 5.3 * 3.533 9.6 + 4.8 =} 3.2

*

* *

PI [deg]

P2

[deg]

P3 [deg]

22.13 19.90 14.71 15.01 21.02 29.46

3.42 3.17 2.50 2.56 3.45 4.44

19.57 25.28

3.39 4.52

30.28 27.27 29.45 42.46

3.84 3.70 4.21 4.77

4.19 3.97 4.32 4.96

19.78 18.70 20.29 25.83

2.87 2.81 2.93 3.49

3.19 3.06 3.49 4.59

28.65 26.89

4.06 3.93

4.13 3.99

Experimental values of internal angular bandwidth: Interacting wavelengths [urn] AOint [deg] Ref.

*

SHG, 0+0 e 10.6 =} 5.3 9.2 =} 4.6 SFG, e + 0 =} e 10.6 + 0.6943 =} 0.6516

0.098 0.082

3.450 3.452

0.031

3.467

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe == d eoe

d 31 sin

0-

d22

cos 0 sin 34> , 2

== doee == d 22 cos 0 cos 3lfJ .

Nonlinear coefficients [3.455, 37]: jd22(10.6 Jlm)j == (0.2 ± 0.03) x jd36(GaAs)j = 16.6 ± 2.5pmjV ,

165

166

3 Properties of Nonlinear Optical Crystals

Id31(10.6 Jlm)1

± 0.1)-1 X Id22(Ag3AsS3)I 10.4 ± 2.2 pmjV .

== (1.6 ==

Laser-induced surface-damage threshold: A [Jlm]

't p

[ns]

106 14 1.0642 cw 18 2.098 200 10.6 190 150 0.6943

Ithr X

10- 12 [Wjm 2 ]

0.00006 0.03 0.000001 >0.12 >0.1 >0.46 0.53

Ref. 3.469 3.365 3.469 3.365 3.365 3.365 3.450

3.2.8 GaSe, Gallium Selenide Negative uniaxial crystal: no > n e ; Point group: 62m; Mass density: 5.03 gjcm 3 [3.338]; Mohs hardness: ~ 0; Transparency range at "0" transmittance level: 0.62 - 20 urn [3.388]; Linear absorption coefficient ex: ex [em-I]

A [urn]

0.65-18 0.7 1.06

n e (at A < 0.804 urn ne > no) ; Point group: 42m; Molecular mass: 5.71 g/cm 3[3.338] ; Mohs hardness: 3 - 3.5 ; Transparency range at "0" transmittance level: 0.71 - 19 urn [3.477, 478]; Linear absorption coefficient (:J. :

A [11m] 1 1.3 2.0

2.05 2.1 2.2 5-11 10.6

a

[em-I]

5.15 6:::}3 5.2 => 2.6 4.1 => 2.05

55.3 53.7 53.1 42.2 40.3 49.7

SFG, 0 + 0 => e 12.15 + 10.63 => 5.67 10.63 + 5.33 => 3.55 5.515 + 3.3913 => 2.1 4.84 + 3.55 =} 2.0479 5.13+2.685=> 1.763 6.00 + 2.586 => 1.807 7.43 + 2.484 => 1.862 9.93 + 2.384 => 1.923 6.95 + 1.66 =} 1.34 7.4 + 1.604 =} 1.318 8.8 + 1.550 =} 1.318 12.3 + 1.476 => 1.318

[3.488] 54.7 [3.488] 53.1 [3.488] 52.5 [3.488] 39.5 [3.488] 41.5 [3.483] 50.6

61 [3.488] 42.7 [3.488] ~48 [3.478] 49.2 [3.483] 61.3 [3.474] 56 [3.474] 49.5 [3.474] 45.8 [3.474] ~78 [3.483] 80 [3.477] 70 [3.477] 60 [3.477]

56.7 54.9 54.3 39.4 40.8 48.3

57.4 55.7 55.1 40.1 41.3 48.6

63.5 42.7 46.2 48.0 53.3 51.7 46.6 42.9 68.6 no pm 69.8 69.0 61.2 58.2 53.1

63.6 43.3 46.5 48.2 53.5 51.9 46.9 43.1 69.2 70.4 61.7 53.4

60.7 42.1 48.1 50.1 57.1 54.9 49.0 44.6 83.1

Best set of Sellmeier equations (2 in urn, T == 293 K) [3.488]: n2

= 3.9362 +

o

n2

= 3.3132 +

e

+ 1.7954-1.2

2.9113-1.2 22 _ (0.38821)2

22 - 1600 '

+

3.3616 -1.2 ,12 _ (0.38201)2

1.7677 -1.2 . A2 - 1600

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [urn] SHG, 0 + 0 =} e 10.6 =} 5.3 9.6 =} 4.8 5.3 => 2.65 4.8 =} 2.4 SFG,o+o=>e 10.6 + 2.65 =} 2.12 9.6 + 2.4 => 1.92 SHG, e + 0 =} e 5.3 =} 2.65

Opm

[deg]

PI [deg]

P3 [deg]

55.02 49.00 41.10 43.63

0.68 0.71 0.69 0.68

43.71 46.36

0.67 0.66

72.03

0.42

0.40

171

172

3 Properties of Nonlinear Optical Crystals

SFG, e + 0 => e 10.6 + 5.3 => 3.533 9.6 + 4.8 => 3.2

55.60 55.36

0.68 0.70

0.66 0.67

Experimental values of internal angular bandwidth Interacting wavelengths [Jlm]

SHG, 0 + 0 => e 10.25 =::} 5.125 SFG, 0 + 0 => e 5.515 + 3.3913 ==> 2.1

~oint [deg]

Ref.

0.84

3.486

0.54

3.478

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe == d36 sin () sin 24J , d eoe == d oee == d 36 sin 20 cos 24J .

Nonlinear coefficient: d 36(10.6 Jlm ) == 33pmjV [3.37] , d 36(9.5 urn) == 32 ± 4pmjV [3.489] .

Laser-induced surface-damage threshold: A [JlID] 1.064

2.0

2.05 2.1

9.5 10.25 10.6

'r p

[ns]

23 35 35 30 30 20-30 50 50 180 180 30 75 150

Ithr X

10- 12 [Wjm 2 ]

Ref. 3.483 3.488 3.488 3.481 3.481 3.482 3.483 3.478 3.485 3.485 3.489 3.486 3.368

0.13--0.4 0.3 0.11 0.083 no ; Point group: 6mm; Mass density: 5.81 g/cm 3 [3.338]; Mohs hardness: 3.25 [3.59]; Transparency range at "0" transmittance level: 0.75 - 25 urn [3.490, 59]; Linear absorption coefficient fY. :

A [urn]

fY.

[em-I]

0.75-20 2.87 15.96 + 2.28 =} 1.995 14.1 + 3.604 =} 2.87 13.7 + 2.8492 =} 2.3587 10.6 + 2.72 => 2.1646 10.361 +2.227 =} 1.833 9.871 + 2.251 =} 1.833 9.776 + 2.256 =} 1.833 8.278 + 4.3 ==> 2.83 8.253 + 4.4 ==> 2.87 8.236 + 4.5 ==> 2.91 7.88 + 3.36 ==> 2.3587 7.86 + 3.37 =} 2.3587

[deg]

f}theor

73.7 [3.493] 62.2 [3.498] 70.9 [3.493] 65 [3.499] 70.5 [3.491] 78 [3.500] 84 [3.500] 90 [3.500] 84 [3.492] 84 [3.492] 84 [3.492] 90 [3.499] 90 [3.490]

[3.362]

71.3 64.2 68.9 65.2 70.4 78.7 83.9 85.8 81.4 82.4 83.6 no pm no pm

72.4 64.6 69.7 65.5 70.5 78.5 83.5 85.1 81.9 83.0 84.5 no pm no pm

== 20°C) [3.362]:

= 4.2243 + 1.7680A? + 3.1200A?

o 2 _ 4 2009 n-.

e

[deg]

[3.468]

Best set of dispersion relations (A in urn, T n2

== 293 K) and comparison

,1.2 _ 0.2270 1.8875,1.2

+2

A - 0.2171

,1.2 - 3380 '

3.6461 ,1.2

+2

A - 3629

.

dispersion relations for the temperatures 73 K, 173 K, 373 K, 573 K are given in [3.390]. Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [urn]

f}pm

SFG, 0 + e==>o 22 + 2.9365 =} 2.5907 20 + 2.9365 =} 2.5605 15 + 2.9365 => 2.4557 10 + 2.9365 => 2.2699 9 + 2.9365 =} 2.2141

83.2 74.0 66.0 71.6 77.6

[deg]

P2 [deg] 0.11 0.24 0.34 0.27 0.19

3.2 Frequently Used Nonlinear Optical Crystals

15 + 3.6513 =} 10 + 4.1573 =} 20 + 3.2437 =} 15 + 3.4290 =} 10 + 3.8715 =}

71.5 73.3 80.9 69.3 71.8

2.9365 2.9365 2.791 2.791 2.791

0.27 0.25 0.14 0.30 0.27

Experimental value of internal angular bandwidth [3.491]: Interacting wavelengths [Jlm]

~()int [deg]

SFG,o+e=}o 10.6 + 2.72 =} 2.1646

1.24

Experimental value of spectral bandwidth [3.491]: Interacting wavelengths [Jlm]

~v

SFG, 0 +e =} 0 10.6 + 2.72 =} 2.1646

15

[em-I]

Effective nonlinearity expression in the phase-matching direction [3.100]: d oeo

== d eoo == d31 sin () .

Nonlinear coefficients [3.37]: d31(10.6Jlm)

== -18pmjV ,

d 33( 10.6 urn)

== 36pmjV .

Laser-induced surface-damage threshold: A [urn]

1"p

[ns]

Ithr X

10- 12 [W/m2 ]

0.3 >0.5 0.5 0.6

1.833 200 1.995 20 2.36 35 200 10.6

Ref. 3.494 3.498 3.490 3.365

Thermal conductivity coefficient at T == 293 K [3.58]: K

[WjmK],

6.9

II

c

K

[WjmK], -.L c

6.2

175

176

3 Properties of Nonlinear Optical Crystals

3.2.11 CdGeAs2 , Cadmium Germanium Arsenide Positive uniaxial crystal: ne > no ; Point group: 42m; Mass density: 5.60 g/cm 3 [3.338]; Mohs hardness: 3.5-4 ; Transparency range at "0" transmittance level: 2.4 - 18 urn [3.501]; Linear absorption coefficient (X :

A [urn]

T [K]

(X

[cm'] Ref.

3.39 4-18 5.3

300 5.7 300 + d32 cos2 4> ;

YZ plane

d ooe = d31 sin fJ ;

< Vz d oeo = d eoo == d32 sin fJ ; plane, fJ > Vz

XZ plane, fJ

XZ

d ooe = d32 sin fJ .

Effective nonlinearity expressions for three-wave interactions in the arbitrary direction of DKB5 crystal are given in [3.36]. Nonlinear coefficients [3.395, 37]: d31 2:: d 31(KB5)

::=

0.04 pm/V ,

180

3 Properties of Nonlinear Optical Crystals

d 32

2 d 32(KB5)

= 0.003 pm/V.

Laser-induced bulk-damage threshold [3.395, 405]:

A [J.1m]

Lp

0.43

7

[ns]

I thr

X

10- 12 [W/rn2 ]

10

3.3.2 CsB30S ' Cesium Triborate (CBO) Negative biaxial crystal: 2Vz = 97.3° at A = 0.5321 urn [3.507]; Point group: 222 ; Mass density: 3.357 g/cm 3 ; Transparency range at "0" transmittance level: 0.167 - 3.0 J.1m [3.507] ; Experimental values of refractive indices [3.507]:

A [J.1m]

nx

ny

nz

0.3547 0.4765 0.4880 0.4965 0.5145 0.5321 0.6328 1.0642

1.5499 1.5370 1.5367 1.5362 1.5349 1.5328 1.5294 1.5194

1.5849 1.5758 1.5736 1.5716 1.5690 1.5662 1.5588 1.5505

1.6145 1.6031 1.6009 1.5996 1.5974 1.5936 1.5864 1.5781

Dispersion relations (A in J.1m, T = 20 DC) [3.507]:

n2

= 2.2916 +

x n 2 = 2.3731

+

y

n2 = 2.4607 Z

+

0.02105

A2 + 0.06525 0.03437

A2 + O.11600 0.03202

A2 + 0.08961

- 3.1848 x 10- 5 A?

. '

_ 7.2632 x 10- 5 A2

.

'

- 5.6332 x 10- 5 A2

.

Note: The dispersion relations in [3.507] are given with a mistake. The numerator of the second term in the equation for n} should be 0.03437 instead of 0.3437

3.3 Other Inorganic Nonlinear Optical Crystals

181

Experimental and theoretical values of phase-matching angle and calculated values of "walk-off" angles: XZ plane, 4> == 0° , 8 > Vz In teracting wavelengths [urn]

SHG, e + e :::} 0 1.0642 =} 0.5321 SFG, e + e :::} 0 1.0642 + 0.5321 =} 0.35473

[deg]

8theor [deg] [3.507]

PI [deg]

P2 [deg]

62 [3.507]

67.53

1.54

1.54

76 [3.507]

76.31

1.01

1.08

8 exp

Experimental value of internal angular bandwidth [3.507]: XZ plane, 4> == 0° Interacting wavelengths [um]

SHG, e + e :::} 0 1.0642 =} 0.5321

Opm

[deg]

A8int [deg]

0.064

62

Effective nonlinearity expressions in the phase-matching direction for threewave interactions in the principal planes of CsB 30S' crystal [3.35]: XY plane d eoe

== d oee == dl 4 sin 24J ;

YZ plane

== d 14 sin 28 ; XZ plane, 8 < Vz d eoe == d oee == d 14 sin 28 ; XZ plane, (j > Vz d eeo == d 14 sin 28 . d eeo

Nonlinear coefficient: d I4(1.064 urn) == 0.648xd22 (BBO) == 1.49 pm/V [3.507,37] .

Laser-induced damage threshold [3.507]:

A [urn] 1.053

Lp

[ns]

I thr

260

X

10- 12 [W/m 2 ]

182

3 Properties of Nonlinear Optical Crystals

3.3.3 BeS04 . 4820, Beryllium Sulfate Negative uniaxial crystal: no > ne ; Point group: 42m; Mass density: 1.713 g/cm 3[3.508] ; Mohs hardness: > 2.5 [3.509] ; Transparency range at "0" transmittance level: 0.17 - 1.58 J.1m [3.508, 510] Linear absorption coefficient (X : (X

0.3164 0.6328 0.187-1.3

[cm"] Ref.

0.6 0.17 e 1.1523 => 0.5762 0.6328 => 0.3164 0.5400 => 0.2700 0.5340 => 0.2670 0.5321 => 0.2661 0.5266 => 0.2633 SHG, e+o ~ e 1.1523 => 0.5762 0.7606 => 0.3803

f}exp

[deg]

== 293 K) and comparison

f}theor

[deg]

[3.510]

[3.511]

30.4 59.9

42.9 56.2

79.0 81.9 83.1

76.7 80.1 81.5

42 [3.508] 55 [3.508] 60 [3.509] 77 [3.511] 80 [3.511] 81.5 [3.510] 81.6 [3.511] 90 [3.511]

no pm* no pm#

64 [3.508] 90 [3.511]

43.7 78.3

65.3 89.3

3.3 Other Inorganic Nonlinear Optical Crystals

SFG, 0 +0 => e 1.0642 + 0.5321 => 1.0642 + 0.3547 => 0.9070 + 0.3547 => 0.8468 + 0.3547 => 0.8209 + 0.3547 =>

47.4 [3.511] 62.4 [3.511] 72.3 [3.511] 80 [3.511] 90 [3.511]

0.3547 0.2661 0.2550 0.2500 0.2477

183

47.4 62.5 72.4 80.0 89.5

47.8 59.4 67.3 71.8 74.2

* NCPM corresponds to the SHG with Al = 0.5271 J.1m; # NCPM corresponds to the SHG with Al = 0.52681Jll1.

Best set of dispersion relations (A in J.1m, T = 20 DC) [3.511]: n2 == 2.1545

+

o

n2

== 2.0335 +

e

0.00835

A2 - 0.01606 A2

- 0.03573 A2 '

0.00806 - 0.01970 A2 - 0.01354

.

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [J.1mJ SHG, 0 + 0 => e 1.0642 ==> 0.5321 0.6943 ==> 0.34715 0.5782 ==> 0.2891 0.5321 ==> 0.26605 SFG, 0 + 0 => e 1.0642 + 0.3547 => 0.26605 0.5782 + 0.5105 => 0.2711 SHG, e + 0 => e 1.0642 ::::} 0.5321 SFG, e + 0 => e 1.0642 + 0.5321 => 0.3547

(}pm

[degJ PI [deg]

P3 [deg]

41.88 50.32 64.99 81.46

1.59 1.60 1.25 0.48

62.50 75.34

1.36 0.80

64.07

1.11

1.23

60.87

1.20

1.37

Experimental values of internal angular, temperature, and spectral bandwidths at T = 293 K: Interacting wavelengths [J.1m] SHG, 0 + 0 => e 0.5321 ::::} 0.2661

(}pm

81.5 81.6

[deg]

Atfnt [deg]

AT [DC]

Av [cm'] Ref.

0.09 0.11

1.45

4.9

3.510 3.511

184

3 Properties of Nonlinear Optical Crystals

Temperature variation of phase-matching angle [3.511]: Interacting wavelengths [J.1m]

TrCJ

Opm [degJ dOpmj dT [deg jKJ

20

81.6

*

SHG, 0+0 e 0.5321 0.2661

*

0.077

Effective nonlinearity expressions in the phase-matching direction [3.100]: d ooe

:::::

d 36 sin 0 sin 21> ,

d eoe

:::::

d oee == d 36 sin 20 cos 24> .

Nonlinear coefficient: d36(0.5321 J.1m)

== 0.62 x d 36(DKDP) ± 10% == 0.23 ± 0.02 pmjV [3.510, 37] .

Laser-induced surface-damage threshold:

0.2661 0.5321

8 8

1 >2.2

Ref.

Note

3.510 3.511

10 Hz 3 Hz

3.3.4 MgBaF4, Magnesium Barium Fluoride Negative biaxial crystal: 2Vz == 117.5° at ,1== 0.5321 J.1m [3.512]; Point group: mm2; Assignment of dielectric and crystallographic axes: X, Y,Z b,c,a; Transparency range: 0.17 - 8 J.1m [3.513]; Experimental values of refractive indices [3.512]:

*

A [J.1m] nx

nz

ny

0.5321 1.4508 1.4678 1.4742 1.0642 1.4436 1.4604 1.4674

== 20°C) [3.512]:

Sellmeier equations (A in J.1m, T n2

x

== 2.0770 + 0.00760

2 _ 2 1238 n y -.

,12 - 0.0079 '

+

0.00860 ,12

'

n~ == 2.1462 + 0.00736 Z

,12 - 0.0090

.

3.3 Other Inorganic Nonlinear Optical Crystals

185

Experimental and theoretical values of phase-matching angle and calculated values of "walk-off" angle: XY plane, ()

== 90°

In teracting wavelengths [urn] SHG, 0+0

*

¢

1.0642

*

¢theor

[deg]

P3 [deg]

[3.512]

9.2 [3.512]

9.65

0.223

== 0° , () < Vz

Interacting wavelengths [urn] SHG, e+o

[deg]

e

1.0642 => 0.5321

XZ plane,

¢exp

*

()exp

[deg]

()theor

[deg]

PI [deg]

P3 [deg]

0.525

0.516

[3.512]

e 18.9 [3.512]

0.5321

17.39

Effective nonlinearity expressions in the phase-matching direction for threewave interactions in the principal planes of MgBaF4 crystal [3.35], [3.36]: XY plane

== d31 cos ¢ ;

dooe

yz plane

== deoo == d32 cos () ; XZ plane, () < Vz d oeo

== d eoe == d 31 sirr' () + d 32 cos 2 () XZ plane, () > Vz d oee

deeo

== d 31 sirr' () + d 32 COS 2 ()

;

.

Effective nonlinearity expressions for three-wave interactions in the arbitrary direction of MgBaF 4 crystal are given in [3.36]. Nonlinear coefficient: d31

(1.0642 urn) == ±0.057 x d 36 (KDP) ± 23%

== ±0.022 ± 0.005 pmjV [3.512, 37] , d 32 ( 1.0642 um)

== ±0.085 x

d 36

(KDP) ± 12%

== ±0.033 ± 0.012 pmjV [3.512, 37] ,

186

3 Properties of Nonlinear Optical Crystals

d33(1.0642 um) = ± 0.023 X d36 (KDP) ± 14% = ± 0.009 ± 0.001 pm/V [3.512, 37] . Laser-induced surface-damage threshold [3.513]:

[nsJ

A [llmJ

'rp

1.0642

~20

Ithr X

10- 12 [W/m 2J

>10

3.3.5 NH 4D2P04 , Deuterated Ammonium Dihydrogen Phosphate (DADP) Negative uniaxial crystal: no > ne ; Point group: 42m; IR edge of transmission spectrum (at "0" transmittance level): 1.9 urn [3.78]; Linear absorption coefficient: rx < 0.013cm- 1 in the range 0.78 - 1.03 urn [3.67]; Experimental values of refractive indices:

A [urn] no

ne

Ref.

0.3472 0.4358 0.53 0.5461 0.6943 1.06

1.4923 1.4831 1.4784 1.4759 1.4737 1.4712

3.126 3.126 3.79 3.126 3.126 3.79

1.5414 1.5278 1.5198 1.5194 1.5142 1.5088

Experimental values of phase-matching angle (T = 293 K) and comparison between different sets of dispersion relations: Interacting wavelengths [urn]

(}exp

[deg]

(}theor

[deg]

[3.78]K

[3.78]E

*

SHG, 0+0 e 90 [3.119] 82.2 0.264 0.528 0.34715 47 [3.514] 50.3 0.6943

* *

Note: [3.78]K [3.78]E

no pm (?) no pm (?)

* see [3.78], data of Kirby et al.; * see [3.78], data of Eimerl

Experimental values of NCPM temperature [3.119]: Interacting wavelengths [Jlm] SHG, 0.516

0+0

*

*

0.258

T rOC]

e -20

3.3 Other Inorganic Nonlinear Optical Crystals

o

0.524 => 0.262 0.528 => 0.264 0.554 => 0.277

20 100

Best set of dispersion relations (A in urn, T = 20°C) [3.78]K: n2 = 2.279481

+

n2

+

1.215879 A? ,12 _ (7.614168)2

o

= 2.151161 +

+

1.199009 A? ,12 _ (11.25169)2

e

0.010761 ,12 - (0.115165)2 '

0.009652

.

,12 - (0.098550)2

Calculated values of phase-matching and "walk-off" angles: Interacting wavelengths [urn] SHG, 0+0 * e 0.5321 ==} 0.26605 0.5782 ==} 0.2891 0.6328 ==} 0.3164 0.6594 ==} 0.3297 0.6943 ==} 0.34715 1.0642 ==} 0.5321 1.3188 ==} 0.6594 SFG, 0 + 0 => e 0.5782 + 0.5105 * 1.0642 + 0.5321 * 1.3188 + 0.6594 * SHG, e+o * e 1.0642 ==} 0.5321 1.3188 ==} 0.6594 SFG, e+o *e 1.0642 + 0.5321 * 1.3188 + 0.6594 *

0.27112 0.35473 0.4396

0.35473 0.4396

(}pm

[deg] PI [deg]

P3 [deg]

79.53 65.24 56.61 53.58 50.31 36.93 37.18

0.652 1.357 1.611 1.664 1.700 1.599 1.569

74.57 46.44 39.29

0.930 1.728 1.659

54.47 53.55

1.411 1.339

1.547 1.533

59.17 48.09

1.308 1.399

1.504 1.668

Effective nonlinearity expressions in the phase-matching direction [3.100]:

d ooe = d36 sin (} sin 2et> , d eoe = d oee = d 36 sin 2(} cos 2et> . Nonlinear coefficient:

d36(0.6943 um) = 1.10 x d 36(K DP) ± 15% = 0.43

± 0.06pmjV [3.514, 37] .

187

188

3 Properties of Nonlinear Optical Crystals

3.3.6 RbH2P04, Rubidium Dihydrogen Phosphate (RDP)

Negative uniaxial crystal: no > n.; Point group: 42m; Mass density: 2.805 g/crrr': Transparency range at 0.5 transmittance level for a 15.3 mm long crystal cut at () == 50°,