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NONLINEAR FIBER OPTICS NONLINEAR FIBER OPTICS Sixth Edition GOVIND P. AGRAWAL The Institute of Optics University of Rochester Rochester, NY United States of America Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2019 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. 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To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-817042-7 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisition Editor: Tim Pitts Editorial Project Manager: Fernanda Oliveira Production Project Manager: Mohana Natarajan Designer: Greg Harris Typeset by VTeX http://www.elsevier.com/permissions https://www.elsevier.com/books-and-journals In the memory of my parents and for Anne, Sipra, Caroline, and Claire Contents Author biography xvii Preface xix 1. Introduction 1 1.1. Historical perspective 1 1.2. Fiber characteristics 3 1.2.1. Material and fabrication 4 1.2.2. Fiber losses 5 1.2.3. Chromatic dispersion 6 1.2.4. Polarization-mode dispersion 11 1.3. Fiber nonlinearities 15 1.3.1. Nonlinear refraction 15 1.3.2. Stimulated inelastic scattering 16 1.3.3. Importance of nonlinear effects 18 1.4. Overview 19 Problems 22 References 22 2. Pulse propagation in fibers 27 2.1. Maxwell’s equations 27 2.2. Fiber modes 29 2.2.1. Eigenvalue equation 30 2.2.2. Characteristics of the fundamental mode 32 2.3. Pulse-propagation equation 34 2.3.1. Nonlinear wave equation 34 2.3.2. Higher-order nonlinear effects 40 2.3.3. Raman response function and its impact 42 2.4. Numerical methods 46 2.4.1. Split-step Fourier method 46 2.4.2. Finite-difference methods 50 Problems 52 References 53 3. Group-velocity dispersion 57 3.1. Different propagation regimes 57 3.2. Dispersion-induced pulse broadening 59 3.2.1. Gaussian pulses 60 3.2.2. Chirped Gaussian pulses 62 3.2.3. Hyperbolic secant pulses 64 vii viii Contents 3.2.4. Super-Gaussian pulses 65 3.2.5. Experimental results 68 3.3. Third-order dispersion 69 3.3.1. Chirped Gaussian pulses 70 3.3.2. Broadening factor 71 3.3.3. Ultrashort-pulse measurements 74 3.4. Dispersion management 76 3.4.1. Dispersion compensation 76 3.4.2. Compensation of third-order dispersion 78 3.4.3. Dispersion-varying fibers 80 Problems 81 References 82 4. Self-phase modulation 85 4.1. SPM-induced spectral changes 85 4.1.1. Nonlinear phase shift 85 4.1.2. Changes in pulse spectra 88 4.1.3. Effect of pulse shape and initial chirp 90 4.1.4. Effect of partial coherence 93 4.2. Effect of group-velocity dispersion 95 4.2.1. Pulse evolution 96 4.2.2. Broadening factor 98 4.2.3. Optical wave breaking 100 4.2.4. Experimental results 103 4.2.5. Effect of third-order dispersion 104 4.2.6. SPM effects in fiber amplifiers 105 4.3. Semianalytic techniques 108 4.3.1. Moment method 108 4.3.2. Variational method 110 4.3.3. Specific analytic solutions 111 4.4. Higher-order nonlinear effects 113 4.4.1. Self-steepening 114 4.4.2. Effect of GVD on optical shocks 117 4.4.3. Intrapulse Raman scattering 119 Problems 121 References 123 5. Optical solitons 127 5.1. Modulation instability 127 5.1.1. Linear stability analysis 127 5.1.2. Gain spectrum 129 5.1.3. Experimental observation 131 5.1.4. Ultrashort pulse generation 132 5.1.5. Impact of loss and third-order dispersion 134 5.1.6. Spatial modulation of fiber parameters 136 Contents ix 5.2. Fiber solitons 138 5.2.1. Inverse scattering method 139 5.2.2. Fundamental soliton 141 5.2.3. Second and higher-order solitons 143 5.2.4. Experimental confirmation 146 5.2.5. Soliton stability 147 5.3. Other types of solitons 150 5.3.1. Dark solitons 150 5.3.2. Bistable solitons 154 5.3.3. Dispersion-managed solitons 155 5.3.4. Optical similaritons 156 5.4. Perturbation of solitons 159 5.4.1. Perturbation methods 159 5.4.2. Fiber loss 160 5.4.3. Soliton amplification 162 5.4.4. Soliton interaction 165 5.5. Higher-order effects 169 5.5.1. Moment equations for pulse parameters 169 5.5.2. Third-order dispersion 171 5.5.3. Self-steepening 173 5.5.4. Intrapulse Raman scattering 175 5.6. Propagation of femtosecond pulses 180 Problems 182 References 183 6. Polarization effects 189 6.1. Nonlinear birefringence 189 6.1.1. Origin of nonlinear birefringence 190 6.1.2. Coupled-mode equations 192 6.1.3. Elliptically birefringent fibers 193 6.2. Nonlinear phase shift 194 6.2.1. Nondispersive XPM 194 6.2.2. Optical Kerr effect 196 6.2.3. Pulse shaping 200 6.3. Evolution of polarization state 202 6.3.1. Analytic solution 202 6.3.2. Poincaré-sphere representation 204 6.3.3. Polarization instability 207 6.3.4. Polarization chaos 210 6.4. Vector modulation instability 210 6.4.1. Low-birefringence fibers 211 6.4.2. High-birefringence fibers 213 6.4.3. Isotropic fibers 215 6.4.4. Experimental results 217 x Contents 6.5. Birefringence and solitons 220 6.5.1. Low-birefringence fibers 220 6.5.2. High-birefringence fibers 221 6.5.3. Soliton-dragging logic gates 225 6.5.4. Vector solitons 226 6.6. Higher-order effects 228 6.6.1. Extended coupled-mode equations 229 6.6.2. Impact of TOD and Raman nonlinearity 230 6.6.3. Interaction of two vector solitons 233 6.7. Random birefringence 236 6.7.1. Polarization-mode dispersion 236 6.7.2. Vector form of the NLS equation 237 6.7.3. Effects of PMD on solitons 239 Problems 241 References 241 7. Cross-phase modulation 245 7.1. XPM-induced nonlinear coupling 245 7.1.1. Nonlinear refractive index 245 7.1.2. Coupled NLS equations 247 7.2. XPM-induced modulation instability 248 7.2.1. Linear stability analysis 248 7.2.2. Experimental results 250 7.3. XPM-paired solitons 252 7.3.1. Bright–dark soliton pair 252 7.3.2. Bright–gray soliton pair 253 7.3.3. Periodic solutions 254 7.3.4. Multiple coupled NLS equations 256 7.4. Spectral and temporal effects 257 7.4.1. Asymmetric spectral broadening 258 7.4.2. Asymmetric temporal changes 263 7.4.3. Higher-order nonlinear effects 266 7.5. Applications of XPM 267 7.5.1. XPM-induced pulse compression 267 7.5.2. XPM-induced optical switching 270 7.5.3. XPM-induced wavelength conversion 271 7.6. Polarization effects 272 7.6.1. Vector theory of XPM 272 7.6.2. Polarization evolution 273 7.6.3. Polarization-dependent spectral broadening 276 7.6.4. Pulse trapping and compression 278 7.6.5. XPM-induced wave breaking 281 7.7. XPM effects in birefringent fibers 282 7.7.1. Fibers with low birefringence 283 7.7.2. Fibers with high birefringence 287 Contents xi 7.8. Two counterpropagating waves 288 Problems 291 References 292 8. Stimulated Raman scattering 297 8.1. Basic concepts 297 8.1.1. Raman-gain spectrum 298 8.1.2. Raman threshold 299 8.1.3. Coupled amplitude equations 302 8.1.4. Effect of four-wave mixing 305 8.2. Quasi-continuous SRS 307 8.2.1. Single-pass Raman generation 307 8.2.2. Raman fiber lasers 309 8.2.3. Raman fiber amplifiers 312 8.2.4. Raman-induced crosstalk 317 8.3. SRS with short pump pulses 319 8.3.1. Pulse-propagation equations 319 8.3.2. Nondispersive case 320 8.3.3. Effects of GVD 323 8.3.4. Raman-induced index changes 326 8.3.5. Experimental results 327 8.3.6. Synchronously pumped Raman lasers 331 8.3.7. Short-pulse Raman amplification 332 8.4. Soliton effects 334 8.4.1. Raman solitons 334 8.4.2. Raman soliton lasers 338 8.4.3. Soliton-effect pulse compression 341 8.5. Polarization effects 342 8.5.1. Vector theory of Raman amplification 342 8.5.2. PMD effects on Raman amplification 346 Problems 349 References 350 9. Stimulated Brillouin scattering 355 9.1. Basic concepts 355 9.1.1. Physical process 355 9.1.2. Brillouin-gain spectrum 356 9.2. Quasi-CW SBS 360 9.2.1. Brillouin threshold 360 9.2.2. Polarization effects 361 9.2.3. Techniques for controlling the SBS threshold 363 9.2.4. Experimental results 365 9.3. Brillouin fiber amplifiers 368 9.3.1. Gain saturation 368 9.3.2. Amplifier design and applications 370 xii Contents 9.4. SBS dynamics 372 9.4.1. Coupled amplitude equations 372 9.4.2. SBS with Q-switched pulses 374 9.4.3. SBS-induced index changes 378 9.4.4. Relaxation oscillations 383 9.4.5. Modulation instability and chaos 385 9.5. Brillouin fiber lasers 387 9.5.1. CW operation 387 9