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Laser Beam Propagation in Nonlinear Optical Media PDF

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Laser Beam Propagation in Nonlinear Optical Media Laser Beam Propagation in Nonlinear Optical Media Shekhar Guha and Leonel P. Gonzalez Air Force Research Laboratory Wright-Patterson Air Force Base, OH, USA MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MAT- LAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20131009 International Standard Book Number-13: 978-1-4398-6639-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To Sudeshna, Rahul and Rakesh – from Shekhar and Trina, Noah, Ian, Connor and Joshua – from Leonel Contents List of Figures xiii List of Tables xxi Preface xxv Author Biographies xxvii Acknowledgements xxix 1 Light Propagation in Anisotropic Crystals 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vectors Associated with Light Propagation . . . . . . . . . . 2 1.2.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Non-plane waves . . . . . . . . . . . . . . . . . . . . . 4 1.3 Anisotropic Media . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3.1 The principal coordinate axes . . . . . . . . . . . . . . 6 1.3.2 Three crystal classes . . . . . . . . . . . . . . . . . . . 7 1.3.3 The principal refractive indices . . . . . . . . . . . . . 7 1.4 Light Propagationin an Anisotropic Crystal . . . . . . . . . 8 1.4.1 Allowed directions of D and E in an anisotropic medium 9 1.4.2 Values of n for a given propagationdirection . . . . . 11 1.4.3 Directions of D and Eefor theeslow and fast waves . . 12 1.5 Characteristics of the Slow and the Fast Waves in a Biaxial Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 n and n . . . . . . . . . . . . . . . . . . . . . . . . . 15 s f 1.5.2 ρ and ρ . . . . . . . . . . . . . . . . . . . . . . . . . 17 s f 1.5.3 The components of dˆ and dˆ . . . . . . . . . . . . . . 17 s f 1.5.4 The components of eˆ and eˆ . . . . . . . . . . . . . . 19 s f 1.6 Double Refraction and Optic Axes . . . . . . . . . . . . . . . 21 1.6.1 Expressions for components of dˆin terms of the angles θ, φ and Ω . . . . . . . . . . . . . . . . . . . . . . . . 24 1.6.2 Relating the angle δ to Ω, θ and φ . . . . . . . . . . . 27 1.6.3 Directions of E and S . . . . . . . . . . . . . . . . . . 29 1.6.4 The walk-off angles ρ and ρ . . . . . . . . . . . . . . 31 s f 1.6.5 An interim summary . . . . . . . . . . . . . . . . . . . 32 vii viii Contents 1.7 Propagationalong the PrincipalAxes and along the Principal Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 33 1.7.2 Propagationalong the principal axes X, Y and Z . . 34 1.7.3 Propagationalong the principal plane YZ . . . . . . . 35 1.7.4 k along YZ plane, Case 1: n < n < n . . . . . 35 X Y Z 1.7.5 k along YZ plane, Case 2: n > n > n . . . . . 37 X Y Z 1.7.6 Propagationalong the principal plane ZX . . . . . . . 38 1.7.7 k along ZX plane, Case 1a: n < n < n , θ <Ω 38 X Y Z 1.7.8 k along ZX plane, Case 1b: n < n < n , θ >Ω 40 X Y Z 1.7.9 k along ZX plane, Case 2a: n > n > n , θ <Ω 40 X Y Z 1.7.10 k along ZX plane, Case 2b: n > n > n , θ >Ω 41 X Y Z 1.7.11 Propagationalong the principal plane XY . . . . . . . 41 1.7.12 k along XY plane, Case 1: n < n < n . . . . . 42 X Y Z 1.7.13 k along XY plane, Case 2: n > n > n . . . . . 43 X Y Z 1.7.14 Summary of the cases of propagation along principal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.8 Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . 45 1.8.1 FielddirectionsoftheDandEvectorsforextraordinary and ordinary waves . . . . . . . . . . . . . . . . . . . . 47 1.8.2 ρ=0 Case (extraordinary wave) . . . . . . . . . . . . 48 6 1.8.3 Another expression relating ρ and θ . . . . . . . . . . 50 1.8.4 ρ = 0 Case (ordinary wave) . . . . . . . . . . . . . . . 52 1.8.5 Two special cases: θ =0 and θ =90◦ . . . . . . . . . . 53 1.9 PropagationEquation in the Presence of Walk-off . . . . . . 54 1.9.1 Transformation between laboratory and crystal coordinate systems . . . . . . . . . . . . . . . . . . . . 55 1.9.2 The propagationequation in the presence of walk-off . 55 2 Nonlinear Optical Processes 61 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Second Order Susceptibility . . . . . . . . . . . . . . . . . . . 62 2.3 Properties of χ(2) . . . . . . . . . . . . . . . . . . . . . . . . 65 2.3.1 Properties of χ(2) away from resonance. . . . . . . . . 67 2.3.2 Kleinman’s symmetry . . . . . . . . . . . . . . . . . . 67 2.4 d Coefficients and the Contracted Notation . . . . . . . . . . 67 2.4.1 d Coefficients under Kleinman symmetry . . . . . . . 68 2.5 The Non-Zero d Coefficients of Biaxial Crystals . . . . . . . 69 2.6 The Non-Zero d Coefficients of Uniaxial Crystals . . . . . . . 70 2.7 Nonlinear Polarizations . . . . . . . . . . . . . . . . . . . . . 72 2.7.1 Nondegenerate sum frequency generation . . . . . . . 72 2.7.2 Difference frequency generation . . . . . . . . . . . . . 73 2.7.3 Second harmonic generation (SHG). . . . . . . . . . . 74 2.7.4 Optical rectification . . . . . . . . . . . . . . . . . . . 75 Contents ix 2.7.5 Convention used for numbering the three interacting beams of light. . . . . . . . . . . . . . . . . . . . . . . 75 2.7.6 Summaryofpolarizationcomponentsfornon-degenerate three wave mixing . . . . . . . . . . . . . . . . . . . . 76 2.7.7 Summary of polarization components for degenerate three wave mixing (SHG and degenerate parametric mixing) . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.8 Frequency Conversion and Phase Matching . . . . . . . . . . 78 2.8.1 Phase matching in birefringent crystals . . . . . . . . 80 2.8.2 Calculation of phase matching angles. . . . . . . . . . 83 2.9 Walk-Off Angles . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.9.1 Calculationofwalk-offanglesinthephasematchedcase in KTP . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3 Effective d Coefficient for Three-Wave Mixing Processes 89 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.1.1 Definition of d . . . . . . . . . . . . . . . . . . . . . 90 eff 3.1.2 Effective nonlinearity for nondegenerate three wave mixing processes . . . . . . . . . . . . . . . . . . . . . 91 3.1.3 Effective nonlinearity for the degenerate three wave mixing process . . . . . . . . . . . . . . . . . . . . . . 92 3.1.4 Type I degenerate three wave mixing process . . . . . 93 3.1.5 Type II degenerate three wave mixing process . . . . . 93 3.2 Expressions for d . . . . . . . . . . . . . . . . . . . . . . . 94 eff 3.2.1 d of biaxial crystals under Kleinman Symmetry eff Condition . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2.2 Reduction of d to expressions in the literature . . . 100 eff 3.3 d Values for Some Biaxial and Uniaxial Crystals of Different eff Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.3.1 d for KTP for propagationin a general direction . . 102 eff 3.3.1.1 d for KTP for a Type I (ssf) mixing process 103 eff 3.3.1.2 d for KTP for a Type II(sff) mixing process 104 eff 3.3.1.3 d for KTP for a Type II(fsf) mixing process 105 eff 3.3.2 d for KTP for propagationalong principal planes . . 106 eff 3.4 d for Uniaxial Crystals . . . . . . . . . . . . . . . . . . . . 109 eff 3.5 d for Isotropic Crystals . . . . . . . . . . . . . . . . . . . . 114 eff 3.5.1 The direction of the nonlinear polarization . . . . . . 119 3.5.2 Propagationalong principal planes . . . . . . . . . . 121 3.5.3 Propagationthrough orientation patterned material . 124 4 Nonlinear Propagation Equations and Solutions 137 4.1 Nonlinear PropagationEquations . . . . . . . . . . . . . . . 137 4.1.1 Normalized form of the three wave mixing equations . 140 4.2 Solutions to the Three Wave Mixing Equations in the Absence of Diffraction, Beam Walk-off and Absorption . . . . . . . . 141

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