Nonlinear Fiber Optics [6 ed.] 0128170425, 9780128170427

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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. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. 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

In the memory of my parents and for Anne, Sipra, Caroline, and Claire

Contents

Author biography Preface

1. Introduction 1.1. Historical perspective 1.2. Fiber characteristics 1.2.1. Material and fabrication 1.2.2. Fiber losses 1.2.3. Chromatic dispersion 1.2.4. Polarization-mode dispersion 1.3. Fiber nonlinearities 1.3.1. Nonlinear refraction 1.3.2. Stimulated inelastic scattering 1.3.3. Importance of nonlinear effects 1.4. Overview Problems References

2. Pulse propagation in fibers 2.1. Maxwell’s equations 2.2. Fiber modes 2.2.1. Eigenvalue equation 2.2.2. Characteristics of the fundamental mode 2.3. Pulse-propagation equation 2.3.1. Nonlinear wave equation 2.3.2. Higher-order nonlinear effects 2.3.3. Raman response function and its impact 2.4. Numerical methods 2.4.1. Split-step Fourier method 2.4.2. Finite-difference methods Problems References

3. Group-velocity dispersion 3.1. Different propagation regimes 3.2. Dispersion-induced pulse broadening 3.2.1. Gaussian pulses 3.2.2. Chirped Gaussian pulses 3.2.3. Hyperbolic secant pulses

xvii xix

1 1 3 4 5 6 11 15 15 16 18 19 22 22

27 27 29 30 32 34 34 40 42 46 46 50 52 53

57 57 59 60 62 64

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3.2.4. Super-Gaussian pulses 3.2.5. Experimental results 3.3. Third-order dispersion 3.3.1. Chirped Gaussian pulses 3.3.2. Broadening factor 3.3.3. Ultrashort-pulse measurements 3.4. Dispersion management 3.4.1. Dispersion compensation 3.4.2. Compensation of third-order dispersion 3.4.3. Dispersion-varying fibers Problems References

4. Self-phase modulation 4.1. SPM-induced spectral changes 4.1.1. Nonlinear phase shift 4.1.2. Changes in pulse spectra 4.1.3. Effect of pulse shape and initial chirp 4.1.4. Effect of partial coherence 4.2. Effect of group-velocity dispersion 4.2.1. Pulse evolution 4.2.2. Broadening factor 4.2.3. Optical wave breaking 4.2.4. Experimental results 4.2.5. Effect of third-order dispersion 4.2.6. SPM effects in fiber amplifiers 4.3. Semianalytic techniques 4.3.1. Moment method 4.3.2. Variational method 4.3.3. Specific analytic solutions 4.4. Higher-order nonlinear effects 4.4.1. Self-steepening 4.4.2. Effect of GVD on optical shocks 4.4.3. Intrapulse Raman scattering Problems References

5. Optical solitons 5.1. Modulation instability 5.1.1. Linear stability analysis 5.1.2. Gain spectrum 5.1.3. Experimental observation 5.1.4. Ultrashort pulse generation 5.1.5. Impact of loss and third-order dispersion 5.1.6. Spatial modulation of fiber parameters

65 68 69 70 71 74 76 76 78 80 81 82

85 85 85 88 90 93 95 96 98 100 103 104 105 108 108 110 111 113 114 117 119 121 123

127 127 127 129 131 132 134 136

Contents

5.2. Fiber solitons 5.2.1. Inverse scattering method 5.2.2. Fundamental soliton 5.2.3. Second and higher-order solitons 5.2.4. Experimental confirmation 5.2.5. Soliton stability 5.3. Other types of solitons 5.3.1. Dark solitons 5.3.2. Bistable solitons 5.3.3. Dispersion-managed solitons 5.3.4. Optical similaritons 5.4. Perturbation of solitons 5.4.1. Perturbation methods 5.4.2. Fiber loss 5.4.3. Soliton amplification 5.4.4. Soliton interaction 5.5. Higher-order effects 5.5.1. Moment equations for pulse parameters 5.5.2. Third-order dispersion 5.5.3. Self-steepening 5.5.4. Intrapulse Raman scattering 5.6. Propagation of femtosecond pulses Problems References

6. Polarization effects 6.1. Nonlinear birefringence 6.1.1. Origin of nonlinear birefringence 6.1.2. Coupled-mode equations 6.1.3. Elliptically birefringent fibers 6.2. Nonlinear phase shift 6.2.1. Nondispersive XPM 6.2.2. Optical Kerr effect 6.2.3. Pulse shaping 6.3. Evolution of polarization state 6.3.1. Analytic solution 6.3.2. Poincaré-sphere representation 6.3.3. Polarization instability 6.3.4. Polarization chaos 6.4. Vector modulation instability 6.4.1. Low-birefringence fibers 6.4.2. High-birefringence fibers 6.4.3. Isotropic fibers 6.4.4. Experimental results

138 139 141 143 146 147 150 150 154 155 156 159 159 160 162 165 169 169 171 173 175 180 182 183

189 189 190 192 193 194 194 196 200 202 202 204 207 210 210 211 213 215 217

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6.5. Birefringence and solitons 6.5.1. Low-birefringence fibers 6.5.2. High-birefringence fibers 6.5.3. Soliton-dragging logic gates 6.5.4. Vector solitons 6.6. Higher-order effects 6.6.1. Extended coupled-mode equations 6.6.2. Impact of TOD and Raman nonlinearity 6.6.3. Interaction of two vector solitons 6.7. Random birefringence 6.7.1. Polarization-mode dispersion 6.7.2. Vector form of the NLS equation 6.7.3. Effects of PMD on solitons Problems References

7. Cross-phase modulation 7.1. XPM-induced nonlinear coupling 7.1.1. Nonlinear refractive index 7.1.2. Coupled NLS equations 7.2. XPM-induced modulation instability 7.2.1. Linear stability analysis 7.2.2. Experimental results 7.3. XPM-paired solitons 7.3.1. Bright–dark soliton pair 7.3.2. Bright–gray soliton pair 7.3.3. Periodic solutions 7.3.4. Multiple coupled NLS equations 7.4. Spectral and temporal effects 7.4.1. Asymmetric spectral broadening 7.4.2. Asymmetric temporal changes 7.4.3. Higher-order nonlinear effects 7.5. Applications of XPM 7.5.1. XPM-induced pulse compression 7.5.2. XPM-induced optical switching 7.5.3. XPM-induced wavelength conversion 7.6. Polarization effects 7.6.1. Vector theory of XPM 7.6.2. Polarization evolution 7.6.3. Polarization-dependent spectral broadening 7.6.4. Pulse trapping and compression 7.6.5. XPM-induced wave breaking 7.7. XPM effects in birefringent fibers 7.7.1. Fibers with low birefringence 7.7.2. Fibers with high birefringence

220 220 221 225 226 228 229 230 233 236 236 237 239 241 241

245 245 245 247 248 248 250 252 252 253 254 256 257 258 263 266 267 267 270 271 272 272 273 276 278 281 282 283 287

Contents

7.8. Two counterpropagating waves Problems References

8. Stimulated Raman scattering 8.1. Basic concepts 8.1.1. Raman-gain spectrum 8.1.2. Raman threshold 8.1.3. Coupled amplitude equations 8.1.4. Effect of four-wave mixing 8.2. Quasi-continuous SRS 8.2.1. Single-pass Raman generation 8.2.2. Raman fiber lasers 8.2.3. Raman fiber amplifiers 8.2.4. Raman-induced crosstalk 8.3. SRS with short pump pulses 8.3.1. Pulse-propagation equations 8.3.2. Nondispersive case 8.3.3. Effects of GVD 8.3.4. Raman-induced index changes 8.3.5. Experimental results 8.3.6. Synchronously pumped Raman lasers 8.3.7. Short-pulse Raman amplification 8.4. Soliton effects 8.4.1. Raman solitons 8.4.2. Raman soliton lasers 8.4.3. Soliton-effect pulse compression 8.5. Polarization effects 8.5.1. Vector theory of Raman amplification 8.5.2. PMD effects on Raman amplification Problems References

9. Stimulated Brillouin scattering 9.1. Basic concepts 9.1.1. Physical process 9.1.2. Brillouin-gain spectrum 9.2. Quasi-CW SBS 9.2.1. Brillouin threshold 9.2.2. Polarization effects 9.2.3. Techniques for controlling the SBS threshold 9.2.4. Experimental results 9.3. Brillouin fiber amplifiers 9.3.1. Gain saturation 9.3.2. Amplifier design and applications

288 291 292

297 297 298 299 302 305 307 307 309 312 317 319 319 320 323 326 327 331 332 334 334 338 341 342 342 346 349 350

355 355 355 356 360 360 361 363 365 368 368 370

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9.4. SBS dynamics 9.4.1. Coupled amplitude equations 9.4.2. SBS with Q-switched pulses 9.4.3. SBS-induced index changes 9.4.4. Relaxation oscillations 9.4.5. Modulation instability and chaos 9.5. Brillouin fiber lasers 9.5.1. CW operation 9.5.2. Pulsed operation Problems References

10. Four-wave mixing 10.1. Origin of four-wave mixing 10.2. Theory of four-wave mixing 10.2.1. Coupled amplitude equations 10.2.2. Approximate solution 10.2.3. Effect of phase matching 10.2.4. Ultrafast four-wave mixing 10.3. Phase-matching techniques 10.3.1. Physical mechanisms 10.3.2. Nearly phase-matched four-wave mixing 10.3.3. Phase matching near the zero-dispersion wavelength 10.3.4. Phase matching through self-phase modulation 10.3.5. Phase matching in birefringent fibers 10.4. Parametric amplification 10.4.1. Review of early work 10.4.2. Gain spectrum and its bandwidth 10.4.3. Single-pump configuration 10.4.4. Dual-pump configuration 10.4.5. Effects of pump depletion 10.5. Polarization effects 10.5.1. Vector theory of four-wave mixing 10.5.2. Polarization dependence of parametric gain 10.5.3. Linearly and circularly polarized pumps 10.5.4. Effect of residual fiber birefringence 10.6. Applications of four-wave mixing 10.6.1. Parametric amplifiers and wavelength converters 10.6.2. Tunable fiber-optic parametric oscillators 10.6.3. Ultrafast signal processing 10.6.4. Quantum correlation and noise squeezing 10.6.5. Phase-sensitive amplification Problems References

372 372 374 378 383 385 387 387 391 394 395

401 401 403 404 405 406 408 409 410 411 412 413 416 419 419 421 423 427 432 434 435 437 440 443 446 446 448 451 453 456 457 458

Contents

11. Highly nonlinear fibers 11.1. Nonlinear parameter 11.1.1. Units and values of n2 11.1.2. SPM-based techniques 11.1.3. XPM-based technique 11.1.4. FWM-based technique 11.1.5. Variations in n2 values 11.2. Fibers with silica cladding 11.3. Tapered fibers with air cladding 11.4. Microstructured fibers 11.4.1. Design and fabrication 11.4.2. Modal and dispersive properties 11.4.3. Hollow-core photonic crystal fibers 11.4.4. Bragg fibers 11.5. Non-silica fibers 11.5.1. Lead-silicate fibers 11.5.2. Chalcogenide fibers 11.5.3. Bismuth-oxide fibers 11.6. Theory of narrow-core fibers Problems References

12. Novel nonlinear phenomena 12.1. Soliton fission and dispersive waves 12.1.1. Fission of second- and higher-order solitons 12.1.2. Generation of dispersive waves 12.2. Intrapulse Raman scattering 12.2.1. Enhanced RIFS through soliton fission 12.2.2. Cross-correlation technique 12.2.3. Wavelength tuning through RIFS 12.2.4. Effects of birefringence 12.2.5. Suppression of Raman-induced frequency shifts 12.2.6. Soliton dynamics near a zero-dispersion wavelength 12.2.7. Multipeak Raman solitons 12.3. Frequency combs and cavity solitons 12.3.1. CW-pumped ring cavities 12.3.2. Nonlinear dynamics of ring cavities 12.3.3. Frequency combs without a cavity 12.4. Second-harmonic generation 12.4.1. Physical mechanisms 12.4.2. Thermal poling and quasi-phase matching 12.4.3. SHG theory 12.5. Third-harmonic generation 12.5.1. THG in highly nonlinear fibers 12.5.2. Effects of group-velocity mismatch

463 463 463 465 468 469 471 474 476 480 480 482 485 486 487 488 491 492 493 498 499

503 503 503 507 512 512 516 518 521 523 527 530 532 533 534 537 538 539 541 544 546 546 547

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12.5.3. Effects of fiber birefringence Problems References

13. Supercontinuum generation 13.1. Pumping with picosecond pulses 13.1.1. Nonlinear mechanisms 13.1.2. Experimental progress after 2000 13.2. Pumping with femtosecond pulses 13.3. Temporal and spectral evolution of pulses 13.3.1. Numerical modeling of supercontinuum 13.3.2. Role of cross-phase modulation 13.3.3. XPM-induced trapping 13.3.4. Role of four-wave mixing 13.4. CW or quasi-CW pumping 13.4.1. Nonlinear mechanisms 13.4.2. Experimental results 13.5. Polarization effects 13.6. Coherence properties 13.6.1. Effect of pump coherence 13.6.2. Spectral incoherent solitons 13.6.3. Techniques for improving spectral coherence 13.7. Ultraviolet and mid-infrared supercontinua 13.7.1. Extension into ultraviolet region 13.7.2. Extension into mid-infrared region 13.8. Optical rogue waves 13.8.1. L-shaped statistics of pulse-to-pulse fluctuations 13.8.2. Techniques for controlling rogue-wave statistics 13.8.3. Modulation instability revisited Problems References

14. Multimode fibers 14.1. Modes of optical fibers 14.1.1. Step-index fibers 14.1.2. Graded-index fibers 14.1.3. Multicore fibers 14.1.4. Excitation of fiber modes 14.2. Nonlinear pulse propagation 14.2.1. Multimode propagation equations 14.2.2. Few-mode fibers 14.2.3. Random linear mode coupling 14.2.4. Graded-index fibers 14.3. Modulation instability and solitons 14.3.1. Modulation instability

549 551 552

557 557 558 560 563 568 569 572 575 580 581 582 585 588 593 593 596 599 601 602 604 607 607 608 611 614 615

621 621 621 625 627 630 631 631 633 635 638 640 641

Contents

14.3.2. Multimode solitons 14.3.3. Solitons in specific fiber modes 14.4. Intermodal nonlinear phenomena 14.4.1. Intermodal FWM 14.4.2. Intermodal SRS 14.4.3. Intermodal SBS 14.5. Spatio-temporal dynamics 14.5.1. Spatial beam cleanup 14.5.2. Supercontinuum generation 14.6. Multicore fibers Problems References

644 649 651 651 657 662 665 666 668 674 678 679

A. System of units

685

B. Nonlinear response of fibers

687

References

688

C. Derivation of the generalized NLS equation

689

D. Numerical code for the NLS equation

693

E. List of acronyms

695

Index

697

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Author biography

Govind P. Agrawal is the James C. Wyant Professor of Optics at the Institute of Optics of University of Rochester, USA. His previous appointments were at Ecole Polytechnique, France, City University of New York, and AT&T Bell Laboratories. He is an author or coauthor of more than 450 research papers and eight books. Professor Agrawal is a Fellow of the Optical society (OSA) and a Life Fellow of IEEE. He has served on the Editorial Board of many optics journals and was the Editor-in-Chief of the OSA journal Advances in Optics and Photonics from 2014 to 2019. In 2012, IEEE Photonics Society honored Dr. Agrawal with its Quantum Electronics Award. He received in 2013 the Riker University Award for Excellence in Graduate Teaching. Agrawal was awarded in 2015 the Esther Hoffman Beller Medal of the Optical Society. In 2019, Agrawal received the 2019 Max Born Award of the Optical Society and the Quantum Electronics and Optics Prize of the European Physical Society.

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Preface Since the publication of the first edition of this book in 1989, the field of nonlinear fiber optics has remained an active area of research and has continued to grow at a rapid pace. During the 1990s, a major factor behind such a sustained growth was the advent of fiber amplifiers and lasers, made by doping silica fibers with rare-earth materials such as erbium and ytterbium. Optical amplifiers permitted propagation of light over thousands of kilometers as they could compensate for all losses encountered in the optical domain. At the same time, fiber amplifiers enabled the use of massive wavelength-division multiplexing. Nonlinear fiber optics played an important role in the design process of such high-capacity telecommunication systems. Starting around 2000, a new development occurred in the field of nonlinear fiber optics that changed the focus of research and led to a number of advances and novel applications. Several kinds of new fibers, classified as highly nonlinear fibers, were developed. They are referred to with names such as microstructured fibers, holey fibers, or photonic crystal fibers, and share the common property that a relatively narrow core is surrounded by a cladding containing a large number of air holes. The nonlinear effects are enhanced dramatically in such fibers to the extent that they can be observed even when the fiber is only a few centimeters long. Their dispersive properties are also quite different from those of conventional fibers. Because of these changes, microstructured fibers exhibit a variety of novel nonlinear effects that are finding applications in fields as diverse as optical coherence tomography and high-precision frequency metrology. Starting around 2010, multimode fibers were considered for enhancing the capacity of optical communication systems through a technique known as spatial-division multiplexing. This led to a renewed interest in studying the nonlinear phenomena inside multimode and multicore fibers. A new feature of such fibers is the spatio-temporal coupling that becomes relevant when a pulsed optical beam propagates inside them. In addition, different modes become coupled through the fiber’s nonlinearity and several intermodal nonlinear effects lead to a multitude of practical applications. The sixth edition is intended to bring the book up-to-date so that it remains a unique source of comprehensive coverage on the subject of nonlinear fiber optics. It retains most of the material that appeared in the fifth edition. However, an attempt was made to include recent research results on most topics relevant to the field of nonlinear fiber optics, resulting in an increase in the size of the book. Major changes occur in Chapters 12 and 13. Specifically, Chapter 12 has a new section on frequency combs and cavity solitons. Chapter 13, devoted to supercontinuum generation, has been revised considerably and a new section has been added to discuss extension of a supercontinuum into the ultraviolet and mid-infrared regions (Section 13.7).

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The most significant change is the addition of Chapter 14 devoted to nonlinear phenomena in multimode and multicore fibers. All other chapters have also been updated, as found appropriate for improving the book. For example, a new section is added to Chapter 6 to discuss the higher-order nonlinear effects (Section 6.6). Also, Appendices B and C have been added to discuss the nonlinear response of fibers and to present a mathematically rigorous derivation of the pulse-propagation equation that is of singular importance for this book. It is my hope that the these additions will prove useful to the readers of this book. The potential readers of this book are likely to consist of senior undergraduate students, graduate students enrolled in the M.S. and Ph.D. degree programs, engineers and technicians involved with the fiber-optics industry, and scientists working in the fields of fiber optics and optical communications. This revised edition should continue to be a useful text for graduate and senior-level courses dealing with nonlinear optics, fiber optics, or optical communications. Some universities may even opt to offer a high-level graduate course devoted to solely nonlinear fiber optics. The problems provided at the end of each chapter should be useful to instructors of such a course. Many individuals have contributed, either directly or indirectly, to the completion of the sixth edition. I am thankful to all of them, especially to my graduate students whose curiosity and involvement led to several improvements. Several of my colleagues have helped me in preparing this edition, and I thank them for reading drafts of selected chapters and for making helpful suggestions. I am grateful to many readers for their occasional feedback. Last, but not least, I thank my wife, Anne, and my daughters, Sipra, Caroline, and Claire, for their support of this project. Govind P. Agrawal Rochester, New York May 2019

CHAPTER 1

Introduction This introductory chapter is intended to provide an overview of the fiber characteristics that are important for understanding the nonlinear effects discussed in later chapters. Section 1.1 provides a historical perspective on the progress in the field of fiber optics. Section 1.2 discusses various fiber properties such as optical loss, chromatic dispersion, and birefringence. Particular attention is paid to chromatic dispersion because of its importance in the study of nonlinear effects probed by using ultrashort optical pulses. Section 1.3 introduces various nonlinear effects resulting from the intensity dependence of the refractive index and stimulated inelastic scattering. Among the nonlinear effects that have been studied extensively using optical fibers as a nonlinear medium are self-phase modulation, cross-phase modulation, four-wave mixing, stimulated Raman scattering, and stimulated Brillouin scattering. Each of these effects is considered in detail in separate chapters. Section 1.4 gives an overview of how this book is organized for discussing such a wide variety of nonlinear effects in optical fibers.

1.1. Historical perspective Total internal reflection—the basic phenomenon responsible for guiding of light in optical fibers—is known from the nineteenth century. The reader is referred to a 1999 book for the interesting history behind the development of optical fibers [1]. Although unclad glass fibers were fabricated during the decade of 1920s [2–4], the field of fiber optics was not born until the 1950s when the use of a cladding layer led to considerable improvement in the fiber characteristics [5–8]. The idea that optical fibers would benefit from a dielectric cladding was not obvious and has a remarkable history [1]. The field of fiber optics developed rapidly during the 1960s, mainly for the purpose of image transmission through a bundle of glass fibers [9]. These early fibers were extremely lossy (loss >1000 dB/km) from the modern standard. However, the situation changed drastically in 1970 when, following an earlier suggestion [10], losses of silica fibers were reduced to below 20 dB/km [11]. Further progress in fabrication technology [12] resulted by 1979 in a loss of only 0.2 dB/km in the 1.55-µm wavelength region [13], a loss level limited mainly by the fundamental process of Rayleigh scattering. The availability of low-loss silica fibers led not only to a revolution in the field of optical fiber communications [14–16] but also to the advent of the new field of nonlinear fiber optics; see Refs. [17] and [18] for a historical account. Raman and Brillouin scattering processes were studied as early as 1972 using optical fibers [19–21]. This work stimulated the study of other nonlinear phenomena such as optically induced birefringence, parametric four-wave mixing, and self-phase modulation [22–26]. An important Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00008-7

Copyright © 2019 Elsevier Inc. All rights reserved.

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contribution was made in 1973 when it was suggested that optical fibers can support soliton-like pulses as a result of an interplay between the dispersive and nonlinear effects [27]. Optical solitons were observed in a 1980 experiment [28] and led to a number of advances during the 1980s in the generation and control of ultrashort optical pulses [29–33]. The decade of the 1980s also saw the development of pulse-compression and optical-switching techniques that exploited the nonlinear effects in fibers [34–41]. Pulses as short as 6 fs were generated by 1987 [42]. Several reviews have covered the progress made during the 1980s [43–47]. The field of nonlinear fiber optics continued to grow during the decade of the 1990s. A new dimension was added when optical fibers were doped with rare-earth elements and used to make amplifiers and lasers. Erbium-doped fiber amplifiers attracted the most attention because they operate in the wavelength region near 1.55 µm and are thus useful for fiber-optic telecommunication systems [48]. Their use led to a virtual revolution in the design of multichannel lightwave systems [14–16]. After 2000, two nonlinear effects occurring inside optical fibers, namely stimulated Raman scattering and four-wave mixing, were employed to develop new types of fiber-optic amplifiers. Such amplifiers do not require doped fibers and can operate in any spectral region. Indeed, the use of Raman amplification has become quite common in modern telecommunication systems [49]. Fiber-optic parametric amplifiers based on four-wave mixing are also attractive because of their potential for ultrafast signal processing [50]. The advent of fiber amplifiers also fueled research on optical solitons and led eventually to new types of solitons such as dispersion-managed solitons and dissipative solitons [51–54]. In another development, fiber gratings, first made in 1978 [55], were developed during the 1990s to the point that they became an integral part of lightwave technology [56]. Starting in 1996, new types of fibers, known under names such as photonic crystal fibers, holey fibers, and microstructure fibers were developed [57–61]; Chapter 11 is devoted to these new types of fibers. Structural changes in such fibers affect their dispersive as well as nonlinear properties. In particular, the wavelength at which the group-velocity dispersion vanishes shifts toward the visible region, and some fibers exhibit two such wavelengths. At the same time, the nonlinear effects are enhanced considerably inside them because of a relatively small core size. This combination leads to a variety of novel nonlinear phenomena covered in Chapter 12. Supercontinuum generation [62–64], a phenomenon in which optical spectrum of incident light broadens by a factor of more than 100 over a relatively short length of fiber, is covered in Chapter 13. Starting in 2010, multimode fibers attracted renewed attention for enhancing the capacity of modern lightwave systems through the technique of space-division multiplexing. This interest led to a thorough study of nonlinear phenomena in such fibers, especially in the contest of graded-index fibers [65–68]. A new feature of such fibers is the spatio-temporal coupling that may occur when an optical beam is coupled into multiple modes of a single fiber. Chapter 14 is devoted to the nonlinear effects occurring

Introduction

Figure 1.1 Schematic illustration of the cross section and the refractive-index profile of a step-index fiber.

inside multimode fibers. It should be evident from these developments that the field of nonlinear fiber optics has grown considerably over the last 30 years and is expected to remain vibrant in the near future.

1.2. Fiber characteristics In its simplest form, an optical fiber consists of a central glass core surrounded by a cladding layer whose refractive index nc is slightly lower than the core index n1 . Such fibers are generally referred to as step-index fibers to distinguish them from graded-index fibers in which the refractive index of the core decreases gradually from center to core boundary [69–71]. Fig. 1.1 shows schematically the cross section and refractive-index profile of a step-index fiber. Two parameters that characterize an optical fiber are the relative core–cladding index difference =

n1 − nc , n1

(1.2.1)

and the so-called V parameter defined as V = k0 a(n21 − n2c )1/2 ,

(1.2.2)

where k0 = 2π/λ, a is the core radius, and λ is the wavelength of light. The V parameter determines the number of modes supported by the fiber. Fiber modes are discussed in Section 2.2, where it is shown that a step-index fiber supports a single mode if V < 2.405. Optical fibers designed to satisfy this condition are called single-mode fibers. The main difference between the single-mode and multimode fibers

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Nonlinear Fiber Optics

Figure 1.2 Schematic diagram of the MCVD process commonly used for fiber fabrication. (After Ref. [72]; ©1985 Elsevier.)

is the core size. The core radius a is typically 25 µm for multimode fibers. However, single-mode fibers with  ≈ 0.003 require a to be 0 (see Fig. 1.5). In the normal-GVD region, high-frequency (blue-shifted) components of an optical pulse travel slower than lowfrequency (red-shifted) components of the same pulse. By contrast, the opposite occurs

Introduction

in the anomalous (negative) GVD region in which β2 is negative. As seen in Fig. 1.5, silica fibers exhibit anomalous GVD when the incident wavelength exceeds the zerodispersion wavelength (λ > λD ). The anomalous-GVD region is of considerable interest for the study of nonlinear effects because optical fibers can support solitons in this region through a balance between the dispersive and nonlinear effects. It is important to differentiate between the anomalous GVD and anomalous dispersion; the latter occurs when the derivative dn/dω is negative. An important feature of chromatic dispersion is that pulses at different wavelengths propagate at different speeds inside a fiber because of a mismatch in their group velocities. This feature leads to a walk-off effect that plays an important role in the description of the nonlinear phenomena involving two or more closely spaced optical pulses. More specifically, the nonlinear interaction between two optical pulses ceases to occur when the faster moving pulse completely walks through the slower moving pulse. This feature is governed by the walk-off parameter d12 defined as d12 = β1 (λ1 ) − β1 (λ2 ) = vg−1 (λ1 ) − vg−1 (λ2 ),

(1.2.12)

where λ1 and λ2 are the center wavelengths of two pulses and β1 at these wavelengths is evaluated using Eq. (1.2.9). For pulses of width T0 , one can define the walk-off length LW by the relation LW = T0 /|d12 |.

(1.2.13)

Fig. 1.5 shows variation of d12 with λ1 for fused silica using Eq. (1.2.12) with λ2 = 0.8 µm. In the normal-GVD region (β2 > 0), a longer-wavelength pulse travels faster, while the opposite occurs in the anomalous-dispersion region. For example, if a pulse at λ1 = 1.3 µm copropagates with the pulse at λ2 = 0.8 µm, it will separate from the shorter-wavelength pulse at a rate of about 20 ps/m. This corresponds to a walkoff length LW of only 50 cm for T0 = 10 ps. The group-velocity mismatch plays an important role for nonlinear effects involving cross-phase modulation [47].

1.2.4 Polarization-mode dispersion As discussed in Chapter 2, even a single-mode fiber is not truly single mode because it can support two degenerate modes that are polarized in two orthogonal directions. Under ideal conditions (perfect cylindrical symmetry and a stress-free fiber), a mode excited with its polarization in the x direction would not couple to the mode with the orthogonal y-polarization state. In real fibers, small departures from cylindrical symmetry, occurring because of random variations in the core shape along the fiber length, result in a mixing of the two polarization states by breaking the mode degeneracy. The stress-induced anisotropy can also break this degeneracy. Mathematically, the modepropagation constant β becomes slightly different for the modes polarized in the x and

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Nonlinear Fiber Optics

y directions. This property is referred to as modal birefringence. The strength of modal birefringence is defined by a dimensionless parameter [83] Bm =

|βx − βy |

k0

= |nx − ny |,

(1.2.14)

where nx and ny are the modal refractive indices for the two orthogonally polarized states. For a given value of Bm , the two modes exchange their powers in a periodic fashion as they propagate inside the fiber with the period [83] LB =

2π |βx − βy |

=

λ

Bm

.

(1.2.15)

The length LB is called the beat length. The axis along which the mode index is smaller is called the fast axis because the group velocity is larger for light propagating in that direction. For the same reason, the axis with the larger mode index is called the slow axis. In standard optical fibers, Bm is not constant along the fiber but changes randomly because of fluctuations in the core shape and anisotropic stress. As a result, light launched into the fiber with a fixed state of polarization changes its polarization in a random fashion. This change in polarization is typically harmless for continuous-wave (CW) light because most photodetectors do not respond to polarization changes of the incident light. It becomes an issue for optical communication systems when short pulses are transmitted over long lengths [14]. If an input pulse excites both polarization components, the two components travel along the fiber at different speeds because of their different group velocities. The pulse becomes broader at the output end because group velocities change randomly in response to random changes in fiber birefringence (analogous to a random-walk problem). This phenomenon, referred to as polarization-mode dispersion (PMD), has been studied extensively because of its importance for long-haul lightwave systems [84–86]. The extent of pulse broadening can be estimated from the time delay T occurring between the two polarization components during propagation of an optical pulse. For a fiber of length L and constant birefringence Bm , T is given by   L L  = L |β1x − β1y | = L (β1 ), T =  − vgx vgy 

(1.2.16)

where β1 is related to group-velocity mismatch. Eq. (1.2.16) cannot be used directly to estimate PMD for standard telecommunication fibers because of random changes in birefringence occurring along the fiber. These changes tend to equalize the propagation times for the two polarization components. In fact, PMD is characterized by the rootmean-square (RMS) value of T obtained after averaging over random perturbations.

Introduction

The variance of T is found to be [85] σT2 = (T )2  = 2(β1 lc )2 [exp(−L /lc ) + L /lc − 1],

(1.2.17)

where β1 ≡ τ/L, τ represents the differential group delay along the principal states of polarization [84], and the correlation length lc is defined as the length over which two polarization components remain correlated; typical values of lc are of the order of 10 m. For L > 0.1 km, we can use lc  L to find that  √ σT ≈ β1 2lc L ≡ Dp L ,

(1.2.18)

For most fibers, values of Dp are in the range of 0.1 to where√Dp is the PMD parameter. √ 1 ps/ km. Because of its L dependence, PMD-induced pulse broadening is relatively small compared with the GVD effects. However, PMD becomes a limiting factor for high-speed communication systems designed to operate over long distances near the zero-dispersion wavelength of the fiber [14]. For some applications it is desirable that fibers transmit light without changing its state of polarization. Such fibers are called polarization-maintaining fibers [87–92]. A large amount of birefringence is introduced intentionally in these fibers through design modifications so that relatively small birefringence fluctuations are masked by it and do not affect the state of polarization significantly. One scheme breaks the cylindrical symmetry by making the fiber core elliptical in shape [92]. The degree of birefringence achieved by this technique is typically ∼10−6 . An alternative scheme makes use of stress-induced birefringence and permits Bm ∼ 10−4 . In a widely adopted design, two rods of borosilicate glass are inserted on the opposite sides of the fiber core at the preform stage. The resulting birefringence depends on the location and the thickness of the stressinducing elements. Fig. 1.8 shows how Bm varies with thickness d for four shapes of stress-inducing elements located at a distance of five times the core radius [89]. Values of Bm ≈ 2 × 10−4 can be realized for d in the range of 50–60 µm. Such fibers are often named after the shape of the stress-inducing element, resulting in whimsical names such as “panda” and “bow-tie” fibers. The use of polarization-maintaining fibers requires identification of the slow and fast axes before an optical signal can be launched into the fiber. Structural changes are often made to the fiber for this purpose. In one scheme, cladding is flattened in such a way that the flat surface is parallel to the slow axis of the fiber. Such a fiber is called the “D fiber” after the shape of the cladding [92] and makes axes identification relatively easy. When the polarization direction of the linearly polarized light coincides with the slow or the fast axis, the state of polarization remains unchanged during propagation. In contrast, if the polarization direction makes an angle with these axes, polarization changes continuously along the fiber in a periodic manner with a period equal to the

13

14

Nonlinear Fiber Optics

Figure 1.8 Variation of birefringence parameter Bm with thickness d of the stress-inducing element for four different polarization-maintaining fibers. Different shapes of the stress-applying elements (shaded region) are shown in the inset. (After Ref. [89]; ©1986 IEEE.)

Figure 1.9 Evolution of state of polarization along a polarization-maintaining fiber when input signal is linearly polarized at 45◦ from the slow axis.

beat length [see Eq. (1.2.15)]. Fig. 1.9 shows schematically the evolution of polarization over one beat length of a birefringent fiber. The state of polarization changes over one-half of the beat length from linear to elliptic, elliptic to circular, circular to elliptic, and then back to linear but is rotated by 90◦ from the incident linear polarization. The process is repeated over the remaining half of the beat length such that the initial state is recovered at z = LB and its multiples. The beat length is typically ∼1 m but can be as small as 1 cm for a strongly birefringent fiber with Bm ∼ 10−4 .

Introduction

1.3. Fiber nonlinearities The response of any dielectric to light becomes nonlinear for intense electromagnetic fields, and optical fibers are no exception. On a fundamental level, the origin of nonlinear response is related to anharmonic motion of bound electrons under the influence of an applied field. As a result, the total polarization P induced by electric dipoles is not linear in the electric field E, but satisfies the more general relation [93–95] 

.



P = 0 χ (1) · E + χ (2) : EE + χ (3) .. EEE + · · · ,

(1.3.1)

where 0 is the vacuum permittivity and χ (j) is jth order susceptibility. In general, χ (j) is a tensor of rank j + 1. The linear susceptibility χ (1) represents the dominant contribution to P. Its effects are included through the refractive index n and the attenuation coefficient α discussed in Section 1.2. The second-order susceptibility χ (2) is responsible for such nonlinear effects as second-harmonic generation and sum-frequency generation [94]. However, it is nonzero only for media that lack an inversion symmetry at the molecular level. As SiO2 is a symmetric molecule, χ (2) vanishes for silica glasses. As a result, optical fibers do not normally exhibit second-order nonlinear effects. Nonetheless, the electric-quadrupole and magnetic-dipole moments can generate weak second-order nonlinear effects. Defects or color centers inside the fiber core can also contribute to second-harmonic generation under certain conditions.

1.3.1 Nonlinear refraction The lowest-order nonlinear effects in optical fibers originate from the third-order susceptibility χ (3) , which is responsible for phenomena such as third-harmonic generation, four-wave mixing, and nonlinear refraction. Unless special efforts are made to achieve phase matching, the nonlinear processes that involve generation of new frequencies (e.g., third-harmonic generation and four-wave mixing) are not efficient in optical fibers. Most of the nonlinear effects in optical fibers therefore originate from nonlinear refraction, a phenomenon referring to the intensity dependence of the refractive index. In its simplest form, the refractive index can be written as [94] n˜ (ω, I ) = n(ω) + n2 I = n(ω) + n¯ 2 |E|2 ,

(1.3.2)

where n(ω) is the linear part given by Eq. (1.2.6), I is the optical intensity associated with the electromagnetic field E, and n¯ 2 is the nonlinear-index coefficient related to χ (3) by the relation (see Section 2.3.1 for its derivation) n¯ 2 =

3 (3) ), Re(χxxxx 8n

(1.3.3)

where Re stands for the real part and the optical field E is assumed to be linearly (3) of the fourth-rank tensor contributes to polarized so that only one component χxxxx

15

16

Nonlinear Fiber Optics

the refractive index. The tensorial nature of χ (3) can affect the polarization properties of optical beams through nonlinear birefringence. Such nonlinear effects are covered in Chapter 6. The intensity dependence of the refractive index leads to a large number of interesting nonlinear effects; the two most widely studied are known as self-phase modulation (SPM) and cross-phase modulation (XPM). SPM refers to the self-induced phase shift experienced by an optical field during its propagation in optical fibers. Its magnitude can be obtained by noting that the phase of an optical field changes by φ = n˜ k0 L = (n + n¯ 2 |E|2 )k0 L ,

(1.3.4)

where k0 = 2π/λ and L is the fiber length. The intensity-dependent nonlinear phase shift, φNL = n¯ 2 k0 L |E|2 , is due to SPM. Among other things, SPM is responsible for spectral broadening of ultrashort pulses [26] and formation of optical solitons in the anomalous-GVD region of fibers [27]. XPM refers to the nonlinear phase shift of an optical field induced by another field having a different wavelength, direction, or state of polarization. Its origin can be understood by noting that the total electric field E in Eq. (1.3.1) is given by



E = 12 xˆ E1 exp(−iω1 t) + E2 exp(−iω2 t) + c.c. ,

(1.3.5)

when two optical fields at frequencies ω1 and ω2 , polarized along the x axis, propagate simultaneously inside the fiber. (The abbreviation c.c. stands for complex conjugate.) The nonlinear phase shift for the field at ω1 is then given by φNL = n¯ 2 k0 L (|E1 |2 + 2|E2 |2 ),

(1.3.6)

where we have neglected all terms that generate polarization at frequencies other than ω1 and ω2 because of their non-phase-matched character. The two terms on the right-

hand side of Eq. (1.3.6) are due to SPM and XPM, respectively. An important feature of XPM is that, for equally intense optical fields of different wavelengths, the contribution of XPM to the nonlinear phase shift is twice that of SPM. Among other things, XPM is responsible for asymmetric spectral broadening of copropagating optical pulses. Chapters 6 and 7 discuss the XPM-related nonlinear effects.

1.3.2 Stimulated inelastic scattering The nonlinear effects governed by the third-order susceptibility χ (3) are elastic in the sense that no energy is exchanged between the electromagnetic field and the dielectric medium. A second class of nonlinear effects results from stimulated inelastic scattering in which the optical field transfers part of its energy to the nonlinear medium. Two important nonlinear effects in optical fibers fall in this category; both of them are related

Introduction

to vibrational excitation modes of silica. These phenomena, known as stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS), were among the first nonlinear effects studied in optical fibers [19–21]. The main difference between the two is that optical phonons participate in SRS while acoustic phonons participate in SBS. In a simple quantum-mechanical picture applicable to both SRS and SBS, a photon of the incident field (called the pump) is annihilated to create a photon at a lower frequency (belonging to the Stokes wave) and a phonon with the right energy and momentum to conserve the energy and the momentum. Of course, a higher-energy photon at the so-called anti-Stokes frequency can also be created if a phonon of right energy and momentum is available. Even though SRS and SBS are very similar in their origin, different dispersion relations for acoustic and optical phonons lead to some basic differences between the two. A fundamental difference is that SBS in single-mode fibers occurs only in the backward direction whereas SRS can occur in both directions. Although a complete description of SRS and SBS in optical fibers is quite involved, the initial growth of the Stokes wave can be described by a simple relation. For SRS, this relation is given by dIs = gR Ip Is , dz

(1.3.7)

where Is is the Stokes intensity, Ip is the pump intensity, and gR is the Raman-gain coefficient. A similar relation holds for SBS with gR replaced by the Brillouin-gain coefficient gB . Both gR and gB have been measured experimentally for silica fibers. The Raman-gain spectrum is found to be very broad, extending up to 40 THz [19]. The peak gain gR ≈ 6 × 10−14 m/W at pump wavelengths near 1.5 µm and occurs for a spectral shift of about 13.1 THz. In contrast, the Brillouin-gain spectrum is extremely narrow and has a bandwidth of 1 THz), the Raman gain can amplify the low-frequency components of a pulse by transferring energy from the high-frequency components of the same pulse. This phenomenon is called intrapulse Raman scattering. As a result of it, the pulse spectrum shifts toward the low-frequency (red) side as the pulse propagates inside the fiber, a feature referred to as the soliton self-frequency shift [12]. The physical origin of this

Pulse propagation in fibers

Raman-induced effect is related to the delayed nature of the Raman response [13]. Mathematically, Eq. (2.3.6) cannot be used in the derivation of Eq. (2.3.28); one must use the general form of the nonlinear polarization given in Eq. (2.1.10). The delayed nonlinear response of molecules to an optical field has been studied extensively in the Born–Oppenheimer approximation [14–16]. The general treatment is somewhat complex and is discussed in Appendix B. Here, we ignore the polarization effects and use Eq. (B.11) in the form 30 (3) PNL (r, t) = χ E(r, t) 4 xxxx



∞

R(t )|E(r, t − t )|2 dt ,

(2.3.32)

0

where the lower limit is set to 0 to satisfy the causality requirement. The nonlinear response function R(t) includes both the Kerr and Raman contributions and is given by [17–22] R(t) = (1 − fR )δ(t) + fR hR (t),

(2.3.33)

where fR represents the fractional contribution of the delayed Raman response to nonlinear polarization PNL . The form of the Raman response function hR (t) is set by vibrations of silica molecules and is discussed later in this section. The analysis of Section 2.3.1 can still be used by working in the frequency domain, but the perturbation of the refractive index is now given by 

∞

n(ω) = n2 (ω)

R(t )|E(r, t − t )|2 dt +

0

iα(ω) . 2k0

(2.3.34)

This form of n(ω) leads to a different expression for β(ω) appearing in Eq. (2.3.22). When converting back this equation from frequency to time domain, we expand β(ω) in a Taylor series as indicated in Eq. (2.3.25) and truncate this series after two terms and retain both β0 and β1 . The frequency dependence of α and n2 can also be included by treating γ and α as frequency-dependent parameters and writing them in the form γ (ω) = γ (ω0 ) + γ1 (ω − ω0 ),

α(ω) = α(ω0 ) + α1 (ω − ω0 ),

(2.3.35)

where γ1 = dγ /dω and α1 = dα/dω, both evaluated at ω = ω0 . For short pulses, it is also useful to retain multiple terms in the expansion (2.3.23). With these additions, we obtain the following generalized pulse-propagation equation [20]:   ∞ n ∂A 1 ∂ i βn ∂ n A + A−i α(ω0 ) + iα1 ∂z 2 ∂t n! ∂ tn n=1     ∞ ∂   2  = i γ (ω0 ) + iγ1 A(z, t) R(t )|A(z, t − t )| dt . ∂t 0

(2.3.36)

41

42

Nonlinear Fiber Optics

The integral in this equation accounts for the phenomenon of intrapulse Raman scattering. Eq. (2.3.36) can be used for pulses as short as a few optical cycles if enough higher-order dispersive terms are included [23–27]. Multiple dispersive terms are indeed included when dealing with supercontinuum generation in optical fibers, a phenomenon discussed in Chapter 13. It is important to note that the use of γ1 in Eq. (2.3.36) includes automatically the frequency dependence of both n2 and Aeff . Noting that γ1 = (dγ /dω)ω=ω0 , the ratio γ1 /γ consists of the following three terms: γ1 (ω0 ) 1 1 + = γ (ω0 ) ω0 n2



dn2 dω



1 − Aeff ω=ω0



dAeff dω

 .

(2.3.37)

ω=ω0

The first term provides the dominant contribution, but the second and third terms become important in the case of a supercontinuum that may extend over 100 THz or more [27]. If spectral broadening is not too large, we can use γ1 ≈ γ /ω0 , and Eq. (2.3.36) reduces to Eq. (C.13) of Appendix C (assuming negligible losses). If we combine the terms containing the derivative ∂ A/∂ t, we find that the γ1 term makes the group velocity to depend on the optical intensity and leads to the phenomenon of self-steepening [28–32].

2.3.3 Raman response function and its impact It is not easy to calculate hR (t) because of the amorphous nature of silica fibers. An indirect experimental approach is used in practice by noting that the Raman gain spectrum is related to the imaginary part of the Fourier transform of hR (t) as [17] gR ( ω) = fR (ω0 /c )n2 Im[h˜ R ( ω)],

(2.3.38)

where ω = ω − ω0 and Im stands for the imaginary part. The real part of h˜ R can be obtained from the imaginary part through the Kramers–Kronig relation [3]. The inverse Fourier transform of h˜ R ( ω) then provides the Raman response function hR (t). Fig. 2.2 shows (A) the experimentally measured Raman-gain spectrum and (B) the temporal form of hR (t) deduced from it [17]. Attempts have been made to find an approximate analytic form of the Raman response function. If we assume that a single vibrational frequency R of silica molecules is involved in the Raman process, hR (t) can be approximated as [18] hR (t) = (τ1−2 + τ2−2 )τ1 exp(−t/τ2 ) sin(t/τ1 ),

(2.3.39)

where τ1 = 1/ R and τ2 is the damping time of vibrations. In practice, τ1 and τ2 are two adjustable parameters with the values τ1 = 12.2 fs and τ2 = 32 fs for silica fibers [18]. The fraction fR can also be estimated from Eq. (2.3.38). Using the known numerical value of peak Raman gain, fR is found to be about 0.18 for silica fibers [17–19].

Pulse propagation in fibers

Figure 2.2 (A) Measured Raman gain spectrum of silica fibers; (B) temporal form of the Raman response function deduced from the gain data. (Based on the Raman-gain data provided by R.H. Stolen.)

One should be careful in using Eq. (2.3.39) for hR (t) because it approximates the actual Raman gain spectrum with a single Lorentzian profile, and thus fails to reproduce the hump seen in Fig. 2.2(A) at frequencies below 5 THz. This hump, known as the boson peak, is a general feature of amorphous materials and results from a vibrational instability of glasses [33]. A modified form of hR (t) was proposed in 2006 to include this hump [34]: 2 hnew R (t ) = (1 − fb )hR (t ) + fb [(2τb − t )/τb ] exp(−t /τb ),

(2.3.40)

where fb = 0.21 represents the relative contribution of the boson peak with τb = 96 fs. Using it, the value fR = 0.245 was found to fit well the Raman gain spectrum in Fig. 2.2. This new form of the Raman response function reproduces better the experimental data on the Raman-induced frequency shifts of femtosecond pulses inside silica fibers [35]. Eq. (2.3.36) has proven extremely useful for studying nonlinear phenomena in optical fibers. Its accuracy has been verified by showing that it preserves the number of photons during pulse evolution if fiber loss is ignored by setting α = 0 [19]. Energy of a pulse is not conserved in the presence of intrapulse Raman scattering because a part of this energy is taken by silica molecules. Eq. (2.3.36) includes this source of nonlinear loss. It is easy to see that it reduces to the simpler equation (2.3.28) for pulses much longer than the time scale of the Raman response function hR (t). Noting that hR (t)

43

44

Nonlinear Fiber Optics

becomes nearly zero for t > 1 ps (see Fig. 2.2), R(t) for such pulses can be replaced by δ(t). As the higher-order dispersion terms (with n > 2), the loss term involving α1 , and the nonlinear term involving γ1 are also negligible for such pulses, Eq. (2.3.36) reduces to Eq. (2.3.28). For pulses shorter than 1 ps but wide enough to contain many optical cycles, we can simplify Eq. (2.3.36) considerably by setting α1 = 0 and γ1 = γ /ω0 and using the following Taylor-series expansion for the intensity: |A(z, t − t )|2 ≈ |A(z, t)|2 − t

∂ |A(z, t)|2 . ∂t

(2.3.41)

This approximation is reasonable if the pulse envelope does not vary rapidly over short time scales. Defining the first moment of the Raman response function as 

TR = and noting that

∞ 0

∞

−∞



tR(t) dt ≈ fR

∞

hR (t) t dt,

(2.3.42)

0

hR (t) dt = 1, Eq. (2.3.36) can be approximated by

  ∂A α iβ2 ∂ 2 A β3 ∂ 3 A i ∂ ∂|A|2 2 2 + A+ − = i γ | A| A + (|A| A) − TR A , (2.3.43) ∂z 2 2 ∂T 2 6 ∂T 3 ω0 ∂ T ∂T

where a frame of reference moving with the pulse at the group velocity vg (the so-called retarded frame) was introduced by making the transformation T = t − z/vg ≡ t − β1 z.

(2.3.44)

A second-order term involving the ratio TR /ω0 was neglected in arriving at Eq. (2.3.43) because of its smallness. It is easy to identify the origin of the three higher-order terms in Eq. (2.3.43). The term proportional to β3 results from the cubic term in the expansion in Eq. (2.3.23). It governs the effects of third-order dispersion and becomes important for ultrashort pulses because of their wide bandwidth. The term proportional to ω0−1 results from the frequency dependence of β in Eq. (2.3.20). It is responsible for the phenomenon of self-steepening [28–32]. The last term, proportional to TR in Eq. (2.3.43), has its origin in the delayed Raman response and is responsible for the frequency shift induced by intrapulse Raman scattering. Using the analytic form of hR (t) given in Eq. (2.3.39), TR is found to be 

TR = fR 0

∞

t hR (t) dt = fR τ1 τ22 /(τ12 + τ22 )

(2.3.45)

and has a value of about 2 fs for silica fibers. However, this model underestimates the true value. The use of hnew shows that TR is close to 3 fs, a value that agrees with the R

Pulse propagation in fibers

experimental value [36] deduced from the slope of the Raman gain spectrum [13]. For pulses shorter than 0.1 ps, the Raman gain may not vary linearly over the entire pulse bandwidth, and the use of Eq. (2.3.43) becomes questionable for such short pulses. For picosecond pulses (T0 > 1 ps), the parameters (ω0 T0 )−1 and TR /T0 become so small that the last two terms in Eq. (2.3.43) can be neglected. As the contribution of the third-order dispersion term is also quite small for such pulses (as long as the carrier wavelength is not too close to the zero-dispersion wavelength), one can employ the simplified equation i

∂ A iα β2 ∂ 2 A + A− + γ |A|2 A = 0. ∂z 2 2 ∂T 2

(2.3.46)

This equation can also be obtained from Eq. (2.3.28) by using the transformation given in Eq. (2.3.44). In the special case of α = 0, Eq. (2.3.46) is referred to as the NLS equation because it resembles the Schrödinger equation with a nonlinear potential term (variable z playing the role of time). To extend the analogy further, Eq. (2.3.43) is called the generalized (or extended) NLS equation. The NLS equation is a fundamental equation of nonlinear science and has been studied extensively in the context of solitons [37–43]. Eq. (2.3.46) is the simplest nonlinear equation for studying third-order nonlinear effects in optical fibers. If the peak power associated with an optical pulse becomes so large that one needs to include the fifth and higher-order terms in Eq. (1.3.1), the NLS equation needs to be modified. A simple approach replaces the nonlinear parameter γ in Eq. (2.3.46) with γ = γ0 (1 − bs |A|2 ), where bs is a saturation parameter governing the power level at which the nonlinearity begins to saturate. For silica fibers, bs |A|2 1 in most practical situations, and one can use Eq. (2.3.46). However, this term may become relevant when the peak intensity approaches 1 GW/cm2 . The resulting equation is often called the cubic-quintic NLS equation [42] because it contains terms involving both the third and fifth powers of the amplitude A. For the same reason, Eq. (2.3.46) is referred to as the cubic NLS equation. The cubic-quintic NLS equation is often relevant for fibers whose core is doped with high-nonlinearity materials such as organic dyes and semiconductors. Eq. (2.3.46) appears in optics in several different contexts [41]. For example, the same equation holds for propagation of CW beams in planar waveguides when the variable T is interpreted as a spatial coordinate. The dispersion term then governs beam diffraction in the plane of the waveguide. This analogy between “diffraction in space” and “dispersion in time” is often exploited to advantage and leads to novel concepts such as time lens and temporal imaging.

45

46

Nonlinear Fiber Optics

2.4. Numerical methods The NLS equation is a nonlinear partial differential equation that does not generally lend itself to analytic solutions, except for some specific cases in which the inverse scattering method [37] can be employed. A numerical approach is often necessary for an understanding of the nonlinear effects in optical fibers. Several numerical methods were developed during the 1970s for solving the NLS equation [44–48]. They can be classified into two broad categories known as the finite-difference and pseudospectral methods. Generally speaking, pseudospectral methods are faster by up to an order of magnitude for the same accuracy [49]. One method that has been used extensively for solving pulse-propagation problems in nonlinear dispersive media is the split-step Fourier method [44,45]. The relative speed of this method compared with most finite-difference schemes can be attributed in part to the use of the fast Fourier transform (FFT) algorithm [50].

2.4.1 Split-step Fourier method To understand the philosophy behind the split-step Fourier method (SSFM), it is useful to write Eq. (2.3.36) or Eq. (2.3.43) in the form ∂A ˆ +N ˆ )A, = (D ∂z

(2.4.1)

ˆ is a differential operator that accounts for dispersion and losses within a linear where D medium and Nˆ is a nonlinear operator that governs the effect of fiber nonlinearities on pulse propagation. In the case of Eq. (2.3.43) these operators are given by

iβ2 ∂ 2 β3 ∂ 3 α + − , 2 ∂T 2 6 ∂T 3 2   i 1 ∂ ∂|A|2 Nˆ = iγ |A|2 + (|A|2 A) − TR . ω0 A ∂ T ∂T ˆ =− D

(2.4.2) (2.4.3)

In general, dispersion and nonlinearity act together along the length of a fiber. The SSFM obtains an approximate solution by assuming that the dispersive and nonlinear effects can be assumed to act independently for propagating the optical field over a small distance h. More specifically, propagation from z to z + h is carried out in two ˆ = 0 in Eq. (2.4.1). In the steps. In the first step, the nonlinearity acts alone, and D second step, dispersion acts alone, and Nˆ = 0 in Eq. (2.4.1). Mathematically, ˆ ) exp(hN ˆ )A(z, T ). A(z + h, T ) ≈ exp(hD

(2.4.4)

ˆ ) can be evaluated in the Fourier domain using the The exponential operator exp(hD prescription ˆ )A(z, T ) = FT−1 exp[hD(ω)]FT A(z, T ), exp(hD

(2.4.5)

Pulse propagation in fibers

where FT denotes the Fourier-transform operation, D(ω) is obtained from Eq. (2.4.2) by replacing the operator ∂/∂ T by −iω, and ω is the frequency in the Fourier domain. As D(ω) is just a complex number in the frequency domain, the use of the FFT algorithm [50] makes numerical evaluation of Eq. (2.4.5) relatively fast. To estimate the accuracy of the SSFM, we note that a formally exact solution of Eq. (2.4.1) is given by ˆ +N ˆ )]A(z, T ), A(z + h, T ) = exp[h(D

(2.4.6)

if Nˆ is assumed to be z independent. At this point, it is useful to recall the Baker– ˆ Hausdorff formula [51] for two noncommuting operators aˆ and b,   1 ˆ 1 ˆ ˆ ˆ ˆ exp(ˆa) exp(b) = exp aˆ + b + [ˆa, b] + [ˆa − b, [ˆa, b]] + · · · ,

2

12

(2.4.7)

where [ˆa, bˆ ] = aˆ bˆ − bˆ aˆ . A comparison of Eqs. (2.4.4) and (2.4.6) shows that the SSFM ˆ and N. ˆ By using Eq. (2.4.7) with ignores the non-commuting nature of the operators D ˆ ˆ ˆ aˆ = hD and b = hN, the dominant error term is found to result from the commutator 1 2 ˆ ˆ 2 h [D, N ]. Thus, the SSFM is accurate to second order in the step size h. The accuracy of the SSFM can be improved by adopting a different procedure to propagate the optical pulse over one segment from z to z + h. In this procedure Eq. (2.4.4) is replaced by 



hˆ exp A(z + h, T ) ≈ exp D 2



z+h



Nˆ (z ) dz exp



z



hˆ D A(z, T ). 2

(2.4.8)

The main difference is that the effect of nonlinearity is included in the middle of the segment rather than at the segment boundary. Because of the symmetric form of the exponential operators in Eq. (2.4.8), this scheme is known as the symmetrized SSFM method [47]. The integral in the middle exponential is useful to include the z deˆ If the step size h is small enough, it can be pendence of the nonlinear operator N. ˆ approximated by exp(h N ), similar to Eq. (2.4.4). The most important advantage of using the symmetrized form in Eq. (2.4.8) is that the leading error term results from the double commutator in Eq. (2.4.7) and is of third order in the step size h. This can be verified by applying Eq. (2.4.7) twice in Eq. (2.4.8). The accuracy of the SSFM can be further improved by evaluating the integral in Eq. (2.4.8) more accurately than approximating it by hNˆ (z). A simple approach is to employ the trapezoidal rule and approximate the integral by [48]  z

z+h

h Nˆ (z ) dz ≈ [Nˆ (z) + Nˆ (z + h)]. 2

(2.4.9)

47

48

Nonlinear Fiber Optics

Figure 2.3 Schematic illustration of the symmetrized SSFM used for numerical simulations. Fiber length is divided into a large number of segments of width h. Within a segment, the effect of nonlinearity is included at the midplane shown by a dashed line.

However, the implementation of Eq. (2.4.9) is not simple because Nˆ (z + h) is unknown at the mid-segment located at z + h/2. It is necessary to follow an iterative procedure that is initiated by replacing Nˆ (z + h) by Nˆ (z). Eq. (2.4.8) is then used to estimate A(z + h, T ) which in turn is used to calculate the new value of Nˆ (z + h). Although the iteration procedure is time-consuming, it can still reduce the overall computing time if the step size h can be increased because of the improved accuracy of the numerical algorithm. Two iterations are generally enough in practice. The implementation of the SSFM is relatively straightforward. As shown in Fig. 2.3, the fiber length is divided into a large number of segments that need not be spaced equally. The optical pulse is propagated from segment to segment using the prescription of Eq. (2.4.8). More specifically, the optical field A(z, T ) is first propagated for a distance h/2 with dispersion only using the FFT algorithm and Eq. (2.4.5). At the midplane z + h/2, the field is multiplied by a nonlinear term that represents the effect of nonlinearity over the whole segment length h. Finally, the field is propagated for the remaining distance h/2 with dispersion only to obtain A(z + h, T ). In effect, the nonlinearity is assumed to be lumped at the midplane of each segment (dashed lines in Fig. 2.3). In practice, the SSFM can be made faster by noting that the application of Eq. (2.4.8) over M successive steps results in the following expression:

A(L , T ) ≈ e

ˆ − 12 hD

M 

 ˆ hN ˆ hD

e e

1

ˆ

e 2 hD A(0, T ),

(2.4.10)

m=1

where L = Mh is the total fiber length and the integral in Eq. (2.4.9) was approximated ˆ Thus, except for the first and last dispersive steps, all intermediate steps can with hN. be carried over the whole segment length h. This feature reduces the required number of FFTs roughly by a factor of 2 and speeds up the numerical code by the same factor.

Pulse propagation in fibers

Note also that a different algorithm is obtained if we use Eq. (2.4.7) with aˆ = hNˆ and ˆ In that case, Eq. (2.4.10) is replaced with bˆ = hD.

A(L , T ) ≈ e

ˆ − 12 hN

M 

 ˆ hNˆ hD

e e

1

ˆ

e 2 hN A(0, T ).

(2.4.11)

m=1

Both of these algorithms provide the same accuracy and are easy to implement in practice (see Appendix D). One must pay attention to several issues before writing a SSFM code. The FFT algorithm replaces the Fourier integral, ˜ (ω) = A



∞

−∞

A(t) eiωt dt ≈ δ t

N −1

A(tn ) eiωtn ,

(2.4.12)

n=0

with a finite Fourier series with N terms by assuming that A(t) is a periodic function over the range −Tm ≤ t ≤ Tm , where the boundaries ±Tm must be chosen suitably. Clearly, the temporal window used for simulations must be much wider than the pulse width. The temporal resolution δ t = 2Tm /N is set by the number of FFT points. It is important to realize that the FFT algorithm sets the frequency grid automatically once Tm and N are specified. More specifically, the frequency range in hertz is just the inverse of δ t, or the frequency resolution is δω =

2π 1 π = , δt N Tm

(2.4.13)

where we used the relation 2Tm = N δ t. Thus, the angular frequency ω lies in the range −(π/δ t) ≤ ω ≤ (π/δ t). For an accurate calculation, N and Tm must be large enough, or ˜ (ω) are sampled properly over δ t and δω must be small enough, that both A(t) and A their entire ranges. Typically, Tm is chosen to be 20 to 25 times the width of input pulses with N = 210 (because FFT runs faster when N is a power of 2). However, the window size also depends on the length of the fiber over which simulations are carried out. If a pulse breaks up into several parts propagating at different speeds, its energy may spread so rapidly that it may hit the window boundary before a simulation is complete. This can lead to numerical instabilities because energy reaching one edge of the window automatically re-enters at the other edge (in view of the periodic nature of the function A). An “absorbing window” is sometimes deployed so that the radiation reaching window edges is artificially absorbed, even though such an implementation does not preserve pulse energy. Some problems, such as supercontinuum generation (see Chapter 13), require much larger values of Tm (> 100 times pulse width) and N (> 216 ) before one obtains accurate results.

49

50

Nonlinear Fiber Optics

Two more points are noteworthy. First, most FFT algorithms require just the array A(tn ) as the input and return its Fourier transform in the same array, ignoring the multiplication by δ t in Eq. (2.4.12). The inverse FFT algorithm also ignores multiplication by δω but divides the result by N to ensure that A(tn ) is recovered. The division by N can be understood from Eq. (2.4.13) by noting that δ t(δω/2π) = 1/N. However, one must remember that the Fourier transform of A(t) calculated through FFT does not ˜ (ω) = (δ t)FFT(A). This is important to know have correct units and is related to it as A if one wants to match the numerical values to an analytical formula for a known function A(t). Second, any FFT algorithm must make a choice of sign for the exponential term in Eq. (2.4.12). Most of them assume a negative sign and thus do not calculate the sum as indicated in Eq. (2.4.12). This problem can be solved by using the inverse FFT algorithm first and then using FFT to take the inverse Fourier transform. With these clarifications, the reader should be able to follow the SSFM code provided in Appendix D and to use it effectively to solve the problems discussed in later chapters. Another practical issue is related to the step size h in the z direction. The SSFM code provided in Appendix D assumes a constant h over the entire fiber length. This works reasonably well in practice but becomes time-consuming in the case of long fibers with multiple segments in which pulses do not evolve in a uniform fashion. The use of an adaptive step size along z can help in reducing the computational time in such problems [52]. Several generalizations of the SSFM method have been developed that retain the basic idea behind the SSFM but employ an expansion other than the Fourier series; examples include splines and wavelets [53].

2.4.2 Finite-difference methods Although the SSFM is commonly used for analyzing the nonlinear effects in optical fibers, its use becomes quite time-consuming when the NLS equation is solved for simulating modern telecommunication systems or the process of supercontinuum generation. In both cases, the optical field A(t) typically has a wide spectral bandwidth (> 1 THz) but also occupies a wide temporal region (> 0.1 ns). Clearly, temporal resolution should be a fraction of 1/B0 , where B0 is the entire bandwidth of A(t). If total bandwidth approaches 10 THz, temporal resolution should be ∼10 fs. If the required temporal window is 1-ns wide, we need more than 100,000 FFT points in the time domain. Even though each FFT operation is relatively fast, a large number of FFT operations on a large-size array may lead to an overall computation time measured in days even on a state-of-the art computer. As an alternative, a finite-difference scheme can be employed in the time or the frequency domain for solving the NLS equation [49,54]. Some of the common ones are: the Crank–Nicholson scheme and its variants, the hopscotch scheme and its variants, and the leap-frog method. A fourth-order Runge–Kutta scheme has also been used with considerable success. However, it is difficult to recommend a specific finite-difference

Pulse propagation in fibers

scheme because the speed and accuracy depend to some extent on the number and the form of the nonlinear terms included in the generalized NLS equation. A situation in which a spectral-domain finite-difference scheme is useful corresponds to supercontinuum generation (see Chapter 13) requiring the solution of the generalized NLS equation (2.3.36). In this case, the optical spectrum becomes so broad that it may extend over a bandwidth exceeding 100 THz. As spectral evolution is of the real interest in this situation, it is useful to solve Eq. (2.3.36) in the frequency domain [55–57]. Consider again the time-domain equation (2.4.1) with arbitrary dispersion and nonlinear operators. By taking its Fourier transform, we obtain ˜ ∂A ˜ + FT (NA ˆ ). = D(ω)A ∂z

(2.4.14)

ˆ becomes a known function D(ω) in the frequency As the time-domain operator D domain, the partial differential equation (2.4.1) is reduced to a set of ordinary differential equations on a frequency grid chosen to cover the entire spectral bandwidth. In the absence of the nonlinear effects (Nˆ = 0), the analytic solution of Eq. (2.4.14) is ˜ (z, ω) = BeD(ω)z with a constant B. The nonlinear effects can be included by making A B frequency-dependent. It is easy to deduce from Eq. (2.4.14) that B(ω) satisfies ∂B ˆ )e−D(ω)z . = FT (NA ∂z

(2.4.15)

In one finite-difference method, this set of equations is solved with a fourth-order Runge–Kutta (RK4) scheme that is accurate to the order h4 in the step size h. This scheme was called RK4IP in Ref. [55], where IP stands for the interaction picture, a term borrowed from quantum mechanics. It is justified by noting that the linear dispersive part in Eq. (2.4.14) is handled analytically before solving the problem numerically. It should be noted that the implementation of RK4IP scheme requires knowledge of A(z, t) at every step in z by using A(z, t) = FT−1 [B(z, ω)eD(ω)z ]. Thus, the RK4IP algorithm still uses FFT and its inverse operation at each step. However, it is relatively easy to implement a variable step size in z with error control [56]. As a result, accuracy is improved considerably compared to the SSFM, without sacrificing the speed too much. Note also that one can easily include the known frequency variations of fiber’s loss α(ω), propagation constant β(ω), and nonlinear parameter γ (ω) in this frequencydomain method. The reader is referred to Ref. [57] for more details on the numerical implementation of the RK4IP algorithm (and the website where the source code is available). There are several limitations inherent in the use of the NLS equation for pulse propagation in optical fibers. The slowly varying envelope approximation may not hold if pulse width becomes comparable to a single optical cycle. A unidirectional pulse propagation equation may be more suitable in this situation [58]. Another approximation

51

52

Nonlinear Fiber Optics

is related to the total neglect of backward propagating waves. If both the forward and backward propagating waves are present simultaneously (because of injection from both fiber ends or the presence internal index discontinuities), the NLS equation cannot be used. All backward waves are included automatically when Maxwell’s equations are solved with a finite-difference time-domain (FDTD) method [59–61]. Such algorithms have also been extended to the case of nonlinear media [62–67]. The FDTD method is certainly more accurate because it solves Maxwell’s equations directly with a minimum number of approximations. However, improvement in accuracy is achieved only at the expense of a vast increase in the computational effort. This can be understood by noting that the time step needed to resolve the optical carrier is by necessity a fraction of the optical period and should be 100 ps and P0 < 1 mW. However, LD and LNL become smaller as pulses become shorter and more intense. For example, LD and LNL are ∼0.1 km for T0 ∼ 1 ps and P0 ∼ 1 W. For such optical pulses, both the dispersive and nonlinear effects need to be included if fiber length exceeds 10 meters. When the fiber length is such that L  LNL but L ∼ LD , the last term in Eq. (3.1.4) is negligible compared to the other two. The pulse evolution is then governed by GVD, and the nonlinear effects play a relatively minor role. The effect of GVD on propagation of optical pulses is discussed in this chapter. The dispersion-dominant regime is applicable whenever the fiber and pulse parameters are such that γ P0 T02 LD =  1. LNL |β2 |

(3.1.6)

Group-velocity dispersion

As a rough estimate, P0 should be 1 W for 1-ps pulses, if we use typical values for the fiber parameters γ and |β2 | at λ = 1.55 µm. When the fiber length L is such that L  LD but L ∼ LNL , the dispersion term in Eq. (3.1.4) is negligible compared to the nonlinear term. In that case, pulse evolution in the fiber is governed by the nonlinear phenomenon of SPM, which produces changes in the pulse spectrum without affecting its shape. This nonlinear effect is discussed in Chapter 4. The nonlinearity-dominant regime is applicable for relatively wide pulses (T0 > 100 ps) with high peak powers (P0 > 1 W) such that γ P0 T02 LD =  1. LNL |β2 |

(3.1.7)

We note that SPM can lead to pulse shaping in the presence of weak GVD effects. If an optical pulse develops a sharp leading or trailing edge, the dispersion term may become important even when Eq. (3.1.7) is initially satisfied. When the fiber length L is longer or comparable to both LD and LNL , dispersion and nonlinearity act together as the pulse propagates along the fiber. The interplay of the GVD and SPM effects then leads to qualitatively different behavior compared with that expected from GVD or SPM alone. In the anomalous-GVD region (β2 < 0), the fiber can support solitons (see Chapter 5). In the normal-GVD region (β2 > 0), the GVD and SPM effects can be used for pulse compression. Eq. (3.1.4) is extremely helpful in understanding pulse evolution in optical fibers when both dispersive and nonlinear effects should be taken into account. However, this chapter is devoted to the linear regime, and the following discussion is applicable to pulses whose parameters satisfy Eq. (3.1.6).

3.2. Dispersion-induced pulse broadening The effects of GVD on optical pulses propagating inside an optical fiber are well known in the absence of any nonlinearity [4–10]. We can study them by setting γ = 0 in Eq. (3.1.1). If we define the normalized amplitude U (z, T ) according to Eq. (3.1.3), U (z, T ) satisfies the following linear partial differential equation: ∂U iβ2 ∂ 2 U =− . ∂z 2 ∂T 2

(3.2.1)

This equation is similar to the paraxial wave equation that governs diffraction of CW light and becomes identical to it when diffraction occurs in only one transverse direction and β2 is replaced by −λ/(2π), where λ is the wavelength of light. For this reason, the dispersion-induced temporal effects have a close analogy with the diffraction-induced spatial effects [2].

59

60

Nonlinear Fiber Optics

Eq. (3.2.1) is readily solved with the Fourier-transform method. Let U˜ (z, ω) be the Fourier transform of U (z, T ) such that 1 U (z, T ) = 2π



∞

−∞

U˜ (z, ω) exp(−iωT ) dω.

(3.2.2)

It is easy to show that U˜ satisfies an ordinary differential equation ˜ ∂U ˜, = i 12 β2 ω2 U ∂z

(3.2.3)

whose solution is given by U˜ (z, ω) = U˜ (0, ω) exp





i β2 ω2 z . 2

(3.2.4)

Eq. (3.2.4) shows that GVD modifies the phase of each spectral component of the pulse by an amount that depends on its frequency and the propagated distance. Even though such phase changes do not affect the optical spectrum, they can modify the pulse shape. By substituting Eq. (3.2.4) in Eq. (3.2.2), the general solution of Eq. (3.2.1) is found to be U (z, T ) =

1 2π



∞ −∞

U˜ (0, ω) exp





i β2 ω2 z − iωT dω, 2

(3.2.5)

where the Fourier transform of the incident field at z = 0 is obtained using U˜ (0, ω) =



∞ −∞

U (0, T ) exp(iωT ) dT .

(3.2.6)

Eqs. (3.2.5) and (3.2.6) can be used for input pulses of arbitrary shapes.

3.2.1 Gaussian pulses As a simple example, consider the case of a Gaussian pulse for which the incident field is of the form [5]   T2 U (0, T ) = exp − 2 ,

2T0

(3.2.7)

where T0 is the half-width (at 1/e-intensity point) introduced in Section 3.1. In practice, it is customary to use the full width at half maximum (FWHM) of |U |2 in place of T0 . For a Gaussian pulse, the two are related as TFWHM = 2(ln 2)1/2 T0 ≈ 1.665T0 .

(3.2.8)

Group-velocity dispersion

Figure 3.1 Normalized (A) intensity |U|2 and (B) frequency chirp δωT0 as functions of T /T0 for a Gaussian pulse at z = 2LD and 4LD . Dashed lines show the input profiles at z = 0.

If we use Eqs. (3.2.5) through (3.2.7) and carry out the integration over ω using the well-known identity [11] 

∞

−∞

 exp(−ax ± bx) dx = 2





b2 exp , a 4a

π

(3.2.9)

the amplitude U (z, T ) at any point z along the fiber is found to be 

U (z, T ) =



T0 T2 exp − . 2 2 1 / 2 (T0 − iβ2 z) 2(T0 − iβ2 z)

(3.2.10)

Thus, a Gaussian pulse maintains its shape on propagation but its width T1 increases with z as T1 (z) = T0 [1 + (z/LD )2 ]1/2 ,

(3.2.11)

where the dispersion length LD = T02 /|β2 |. Eq. (3.2.11) shows that the extent of broadening is governed by the dispersion length LD . For a given fiber length, short pulses broaden more because of a smaller dispersion length. At z = LD , a Gaussian pulse broad√ ens by a factor of 2. Fig. 3.1(A) shows the extent of GVD-induced broadening for a Gaussian pulse by plotting |U (z, T )|2 at z/LD = 2 and 4. Notice the decrease in the peak power as pulse broadens. A comparison of Eqs. (3.2.7) and (3.2.10) shows that although the incident pulse is unchirped (with no phase modulation), the transmitted pulse becomes chirped. This can be seen clearly by writing U (z, T ) in the form U (z, T ) = |U (z, T )| exp[iφ(z, T )],

(3.2.12)

61

62

Nonlinear Fiber Optics

where the time dependence of the phase φ is given by 

φ(z, T ) = −



sgn(β2 )(z/LD ) T 2 1 z + tan−1 sgn(β2 ) . 2 2 1 + (z/LD ) 2T0 2 LD

(3.2.13)

The phase varies quadratically across the pulse at any distance z. It also depends on whether the pulse experiences normal or anomalous GVD inside the fiber. The time dependence of φ(z, T ) implies that the instantaneous frequency differs across the pulse from the central frequency ω0 . The difference δω is just the time derivative −∂φ/∂ T [the minus sign is due to the choice e−iω0 t in Eq. (2.3.2)] and is given by δω(T ) = −

sgn(β2 )(z/LD ) T ∂φ = . ∂T 1 + (z/LD )2 T02

(3.2.14)

Eq. (3.2.14) shows that the frequency changes linearly with T, i.e., a fiber imposes linear frequency chirp across the pulse. The chirp δω also depends on the sign of β2 . In the normal-GVD region (β2 > 0), δω is negative at the leading edge (T < 0) and increases linearly across the pulse. The opposite occurs in the anomalous-GVD region (β2 < 0); this case is shown in Fig. 3.1(B). As seen there, chirp imposed by the GVD is perfectly linear for Gaussian pulses. Dispersion-induced pulse broadening can be understood by recalling from Section 1.3 that different frequency components of a pulse travel at slightly different speeds along the fiber because of GVD. More specifically, red components travel faster than blue components in the normal-GVD region (β2 > 0), while the opposite occurs in the anomalous-GVD region (β2 < 0). The pulse can maintain its width only if all spectral components arrive together. Any time delay in the arrival of different spectral components leads to pulse broadening.

3.2.2 Chirped Gaussian pulses For an initially unchirped Gaussian pulse, Eq. (3.2.11) shows that dispersion-induced broadening of the pulse does not depend on the sign of the GVD parameter β2 . Thus, for a given value of the dispersion length LD , the pulse broadens by the same amount in the normal- and anomalous-GVD regions of the fiber. This behavior changes if the Gaussian pulse has an initial frequency chirp [6]. In the case of linearly chirped Gaussian pulses, the incident field can be written as [compare with Eq. (3.2.7)] 

U (0, T ) = exp −

(1 + iC ) T 2

2

T02

 ,

(3.2.15)

where C is a chirp parameter. By using Eq. (3.2.12) one finds that the instantaneous frequency increases linearly from the leading to the trailing edge (up-chirp) for C > 0,

Group-velocity dispersion

while the opposite occurs (down-chirp) for C < 0. It is common to refer to the chirp as being positive or negative, depending on whether C is positive or negative. The numerical value of C can be estimated from the spectral width of the Gaussian pulse. By substituting Eq. (3.2.15) in Eq. (3.2.6) and using Eq. (3.2.9), U˜ (0, ω) found to be U˜ (0, ω) =



2π T02 1 + iC

1/2

  ω2 T02 exp − . 2(1 + iC )

(3.2.16)

The spectral half-width (at 1/e-intensity point) from Eq. (3.2.16) is given by ω = (1 + C 2 )1/2 /T0 .

(3.2.17)

In the absence of frequency chirp (C = 0), the spectral width is the smallest and satisfies the relation ωT0 = 1. Clearly, spectral width of a pulse is enhanced by a factor of (1 + C 2 )1/2 if the pulse is linearly chirped. Eq. (3.2.17) can be used to estimate |C | from measurements of ω and T0 . To obtain the U (z, T ) at a distance z, we substitute U˜ (0, ω) from Eq. (3.2.16) in Eq. (3.2.5). The integration can again be performed analytically using Eq. (3.2.9) to obtain the result 

U (z, T ) =



T0 (1 + iC )T 2 exp − . 2 2 1 / 2 [T0 − iβ2 z(1 + iC )] 2[T0 − iβ2 z(1 + iC )]

(3.2.18)

Since |U (z, T )|2 remains a Gaussian function, we conclude that a chirped pulse maintains its initial Gaussian shape on propagation even though its width changes. The width T1 after propagating a distance z is related to the initial width T0 by the relation [6] T1 = T0



C β2 z 1+ T02



2 +

β2 z

2 1/2

T02

.

(3.2.19)

The chirp parameter of the pulse also changes from C to C1 such that C1 (z) = C + (1 + C 2 )(β2 z/T02 ).

(3.2.20)

It is useful to define a normalized distance ξ as ξ = z/LD , where LD ≡ T02 /|β2 | is the dispersion length introduced earlier. Fig. 3.2 shows (A) the broadening factor T1 /T0 and (B) the chirp parameter C1 as a function of ξ in the case of anomalous GVD (β2 < 0). An unchirped pulse (C = 0) broadens monotonically by a factor of (1 + ξ 2 )1/2 and develops a negative chirp such that C1 = −ξ (the dotted curves). Chirped pulses, on the other hand, may broaden or compress depending on whether β2 and C have the same or opposite signs. When β2 C > 0, a chirped Gaussian pulse broadens monotonically at

63

64

Nonlinear Fiber Optics

Figure 3.2 Broadening factor (A) and the chirp parameter (B) as functions of distance for a chirped Gaussian pulse propagating in the anomalous-GVD region of a fiber. Dashed curves correspond to the case of an unchirped Gaussian pulse. The same curves are obtained for normal dispersion (β2 > 0) if the sign of C is reversed.

a rate faster than that of the unchirped pulse (the dashed curves). The reason is related to the fact that the dispersion-induced chirp adds to the input chirp because the two contributions have the same sign. The situation changes dramatically for β2 C < 0. In this case, the contribution of the dispersion-induced chirp is of a kind opposite to that of the input chirp. As seen from Fig. 3.2(B) and Eq. (3.2.20), C1 becomes zero at a distance ξ = |C |/(1 + C 2 ), and the pulse becomes unchirped. This is the reason why the pulse width initially decreases in Fig. 3.2(A) and becomes minimum at that distance. The minimum value of the pulse width depends on the input chirp parameter as T1min =

T0 . (1 + C 2 )1/2

(3.2.21)

Since C1 = 0 when the pulse attains its minimum width, it satisfies the Fourier relation ω0 T1min = 1, where ω0 is the input spectral width of the pulse.

3.2.3 Hyperbolic secant pulses Although pulses emitted from many lasers can be approximated by a Gaussian shape, it is necessary to consider other pulse shapes. Of particular interest is the hyperbolic secant pulse shape that occurs naturally in the context of optical solitons (see Chapter 5) and for pulses emitted from some mode-locked lasers. The optical field associated with such pulses takes the form 







T iCT 2 U (0, T ) = sech exp − , T0 2T02

(3.2.22)

Group-velocity dispersion

Figure 3.3 Normalized (A) intensity |U|2 and (B) frequency chirp δωT0 as a function of T /T0 for a “sech” pulse at z = 2LD and 4LD . Dashed lines show the input profiles at z = 0. Compare with Fig. 3.1 where the case of a Gaussian pulse is shown.

where the chirp parameter C controls the initial chirp as in Eq. (3.2.15). The transmitted field U (z, T ) is obtained by using Eqs. (3.2.5), (3.2.6), and (3.2.22). Unfortunately, it is not easy to evaluate the integral in Eq. (3.2.5) in a closed form for non-Gaussian pulse shapes. Fig. 3.3 shows the intensity and chirp profiles calculated numerically at z = 2LD and z = 4LD for initially unchirped pulses (C = 0). A comparison of Figs. 3.1 and 3.3 shows that the qualitative features of dispersion-induced broadening are nearly identical for the Gaussian and “sech” pulses. The main difference is that the dispersion-induced chirp is no longer purely linear across the pulse. Note that T0 appearing in Eq. (3.2.22) is not the FWHM but is related to it by √

TFWHM = 2 ln(1 + 2)T0 ≈ 1.763 T0 .

(3.2.23)

This relation should be used if the comparison is made on the basis of FWHM. The same relation for a Gaussian pulse is given in Eq. (3.2.8).

3.2.4 Super-Gaussian pulses So far we have considered pulse shapes with relatively smooth leading and trailing edges. As one may expect, dispersion-induced broadening is sensitive to the steepness of pulse edges. In general, a pulse with steeper leading and trailing edges broadens more rapidly with propagation simply because such a pulse has a wider spectrum to start with. A super-Gaussian shape can be used to model the effects of steep leading and trailing edges on GVD-induced pulse broadening. For a super-Gaussian pulse, Eq. (3.2.15) is generalized to take the form [9]

1 + iC U (0, T ) = exp − 2



T T0

2m

,

(3.2.24)

65

66

Nonlinear Fiber Optics

Figure 3.4 Normalized (A) intensity |U|2 and (B) frequency chirp δωT0 as a function of T /T0 for a super-Gaussian pulse at z = 2LD and 4LD . Dashed lines show the input profiles at z = 0. Compare with Fig. 3.1 where the case of a Gaussian pulse is shown.

where the parameter m controls the degree of edge sharpness. For m = 1 we recover the case of a chirped Gaussian pulse. For large value of m, the pulse becomes nearly rectangular with sharp leading and trailing edges. If the rise time Tr is defined as the duration during which the intensity increases from 10 to 90% of its peak value, it is related to the parameter m as Tr = (ln 9)

T0 T0 ≈ . 2m m

(3.2.25)

Thus the parameter m can be determined from the measurements of Tr and T0 . Fig. 3.4 shows the intensity and chirp profiles at z = 2LD , and 4LD in the case of an initially unchirped super-Gaussian pulse (C = 0) by using m = 3. It should be compared with Fig. 3.1 where the case of a Gaussian pulse (m = 1) is shown. The differences between the two can be attributed to the steeper leading and trailing edges associated with a super-Gaussian pulse. Whereas the Gaussian pulse maintains its shape during propagation, the super-Gaussian pulse not only broadens at a faster rate but is also distorted in shape. The chirp profile is also far from being linear and exhibits high-frequency oscillations. Enhanced broadening of a super-Gaussian pulse can be understood by noting that its spectrum is wider than that of a Gaussian pulse because of steeper leading and trailing edges. As the GVD-induced delay of each frequency component is directly related to its separation from the central frequency ω0 , a wider spectrum results in a faster rate of pulse broadening. For complicated pulse shapes such as those seen in Fig. 3.4, the FWHM is not a true measure of the pulse width. The width of such pulses is more accurately described by the root-mean-square (RMS) width σ defined as [5] σ = [ T 2  − T 2 ]1/2 ,

(3.2.26)

Group-velocity dispersion

Figure 3.5 Broadening factor σ/σ0 as a function of z/LD in the anomalous-GVD regime for chirped super-Gaussian pulses with C = 5. The case m = 1 corresponds to a Gaussian pulse. Pulse edges become steeper with increasing values of m.

where the angle brackets denote averaging over the intensity profile as ∞

T n |U (z, T )|2 dT . 2 −∞ |U (z, T )| dT

∞

T  = −∞ n

(3.2.27)

The moments T  and T 2  can be calculated analytically for some specific cases. In particular, it is possible to evaluate the broadening factor σ/σ0 analytically for superGaussian pulses using Eqs. (3.2.24) through (3.2.27) with the result [10] (1/2m) C β2 z (2 − 1/2m) σ2 =1+ + m2 (1 + C 2 ) (3/2m) T02 (3/2m) σ02



β2 z

T02

2 ,

(3.2.28)

where (x) is the gamma function. For a Gaussian pulse (m = 1) the broadening factor reduces to that given in Eq. (3.2.19). To see how pulse broadening depends on the steepness of pulse edges, Fig. 3.5 shows the broadening factor σ/σ0 of super-Gaussian pulses as a function of the propagation distance for values of m ranging from 1 to 4. The case m = 1 corresponds to a Gaussian pulse; the pulse edges become increasingly steeper for larger values of m. Noting from Eq. (3.2.25) that the rise time is inversely proportional to m, it is evident that a pulse with a shorter rise time broadens faster. The curves in Fig. 3.5 are drawn for the case of initially chirped pulses with C = 5. When the pulses are initially chirped, the magnitude of pulse broadening depends on the sign of the product β2 C. The qualitative behavior

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is similar to that shown in Fig. 3.2 for the case of a Gaussian pulse (m = 1), although pulse compression is reduced considerably for super-Gaussian pulses.

3.2.5 Experimental results Compression of chirped pulses was observed experimentally during the 1980s using pulses emitted from semiconductor lasers. In a 1982 experiment [12], relatively wide (1.7 ns) pulses emitted from such a laser at a wavelength of 1.54 µm were chirped positively (C > 0). They were compressed by a factor of 5 after propagating 104 km in the anomalous-GVD region of a fiber with β2 ≈ −20 ps2 /km. In another experiment, the semiconductor laser emitted negatively chirped pulses (C < 0) at a wavelength of 1.21 µm [13]. After propagating them in the normal-GVD region (β2 = 15 ps2 /km) of a 1.5-km-long fiber, their width decreased from 190 to 150 ps. When the fiber length was increased to 6 km, the pulse width increased to 300 ps, in agreement with the qualitative behavior seen in Fig. 3.2. In a different experiment, much shorter optical pulses (initial FWHM ≈ 26 ps) at 1.3 µm were obtained from a distributed-feedback (DFB) semiconductor laser by using the gain-switching technique [14]. As the pulses were negatively chirped (C < 0), a dispersion-shifted fiber was employed with a positive GVD at 1.3 µm (β2 ≈ 12 ps2 /km). The pulses compressed by a factor of three after propagating inside a 4.8-km-long fiber and then started to broaden with a further increase in the fiber length. Compression of chirped picosecond pulses inside optical fibers through GVD was used to advantage in several experiments in which a gain-switched DFB semiconductor laser was used as a source of solitons [15–18]. Even though relatively broad optical pulses (duration 20–40 ps) emitted from such lasers were far from being transform limited, their passage through a fiber of optimized length with positive GVD produced compressed optical pulses that were nearly chirp-free. In a 1989 demonstration of this technique [17], 14-ps pulses were obtained at the 3-GHz repetition rate by passing the gain-switched pulse through a polarization-preserving, dispersion-shifted, 3.7-km-long optical fiber with β2 = 23 ps2 /km at the 1.55-µm operating wavelength. In another experiment, a narrowband optical filter was used to control the spectral width of the gain-switched pulse before its compression [18]. An erbium-doped fiber amplifier then amplified and compressed the pulse simultaneously. It was possible to produce nearly unchirped, 17-ps-wide optical pulses at repetition rates of 6–24 GHz. Pulses as short as 3 ps were obtained by 1990 with this technique [19]. In a related scheme, amplification of picosecond pulses in a semiconductor optical amplifier produced optical pulses chirped such that they could be compressed by using fibers with anomalous GVD [20–22]. The method is useful in the wavelength region near 1.5 µm because silica fibers commonly exhibit anomalous GVD in that spectral region. The technique was demonstrated in 1989 by using 40-ps input pulses from

Group-velocity dispersion

a 1.52-µm mode-locked semiconductor laser [20]. The pulse was first amplified in a semiconductor optical amplifier and then compressed by a factor of two by propagating it through an 18-km-long fiber with β2 = −18 ps2 /km. Such compression pulses were used for transmitting a 16-Gb/s signal over 70 km of standard telecommunication fiber [21].

3.3. Third-order dispersion Dispersion-induced pulse broadening discussed in Section 3.2 is due to the lowest-order GVD term proportional to β2 in Eq. (2.3.23). Although the contribution of this term dominates in most cases of practical interest, it is sometimes necessary to include the third-order dispersion (TOD) term governed by β3 . For example, if the pulse wavelength nearly coincides with the zero-dispersion wavelength of the fiber and β2 ≈ 0, the β3 term provides the dominant contribution to the GVD effects [4]. For ultrashort pulses (with width T0 < 1 ps), it is necessary to include the β3 term even when β2 = 0 because the expansion parameter ω/ω0 is no longer small enough to justify the truncation of the expansion in Eq. (2.3.23) after the β2 term. In this section we consider pulse broadening by including both the β2 and β3 terms, while still neglecting the nonlinear effects. The appropriate propagation equation for the amplitude A(z, T ) is obtained from Eq. (2.3.43) after setting γ = 0. Using Eq. (3.1.3), U (z, T ) satisfies the following equation: ∂U iβ2 ∂ 2 U β3 ∂ 3 U =− + . ∂z 2 ∂T 2 6 ∂T 3

(3.3.1)

This equation can also be solved with the Fourier-transform technique of Section 3.2. In place of Eq. (3.2.5) the transmitted field is obtained from U (z, T ) =

1 2π



∞ −∞

U˜ (0, ω) exp





i i β2 ω2 z + β3 ω3 z − iωT dω, 2 6

(3.3.2)

where the Fourier transform U˜ (0, ω) of the incident field is given by Eq. (3.2.6). Eq. (3.3.2) can be used to study the effects of higher-order dispersion if the incident field U (0, T ) is specified. In particular, one can consider Gaussian, super-Gaussian, or hyperbolic-secant pulses in a manner analogous to Section 3.2. As an analytic solution in terms of the Airy function can be obtained for Gaussian pulses [4], we consider this case first.

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3.3.1 Chirped Gaussian pulses In the case of a chirped Gaussian pulse, we use A˜ (0, ω) from Eq. (3.2.16) in Eq. (3.3.2) and introduce x = ωp as a new integration variable, where p2 =

T02 2



1 1 + iC



−

iβ2 z . T02

(3.3.3)

We then obtain the following expression: A0 U (z, T ) = √



π

∞

−∞

  ib iT exp −x2 + x3 − x dx,

3

p

(3.3.4)

where b = β3 z/(2p3 ). The x2 term can be eliminated with the transformation x = b−1/3 u − i/b. The result can be written in terms of the Airy function Ai(x) as √









2A0 π 2p − 3bT p − bT U (z, T ) = exp Ai , |b|1/3 3pb2 p|b|4/3

(3.3.5)

where p depends on the fiber and pulse parameters as indicated in Eq. (3.3.3). For an unchirped pulse whose spectrum √ is centered exactly at the zero-dispersion wavelength of the fiber (β2 = 0), p = T0 / 2. As one may expect, pulse evolution along the fiber depends on the relative magnitudes of β2 and β3 . To compare the relative importance of the β2 and β3 terms in Eq. (3.3.1), it is useful to introduce a dispersion length associated with the TOD as LD = T03 /|β3 |.

(3.3.6)

The TOD effects play a significant role only if LD ≤ LD or T0 |β2 /β3 | ≤ 1. For a 100-ps pulse, this condition implies that β2 < 10−3 ps2 /km when β3 = 0.1 ps3 /km. Such low values of β2 are realized only if λ0 and λD differ by 0. Third-order dispersion is responsible for the oscillatory structure near the trailing edge of the pulse.

I˜ (z, ω) of the pulse intensity |U (z, T )|2 , I˜ (z, ω) =



∞ −∞

|U (z, T )|2 exp(iωT ) dT ,

(3.3.7)

and differentiating it n times, we obtain lim

ω→0

∂n I˜ (z, ω) = (i)n ∂ωn



∞

−∞

T n |U (z, T )|2 dT .

(3.3.8)

Using Eq. (3.3.8) in Eq. (3.2.27) we find that (−i)n

T n  =

Nc

∂n I˜ (z, ω), ω→0 ∂ωn lim

(3.3.9)

where the normalization constant is given by 

Nc =

∞

−∞

 |U (z, T )| dT ≡ 2

∞

−∞

|U (0, T )|2 dT .

(3.3.10)

It follows from the convolution theorem that I˜ (z, ω) =

1 2π



∞

−∞

U˜ (z, ω − ω )U˜ ∗ (z, ω ) dω .

(3.3.11)

Performing the differentiation and limit operations indicated in Eq. (3.3.9), we obtain

T n  =

(i)n 2π Nc



∞

−∞

U˜ ∗ (z, ω)

∂n U˜ (z, −ω) dω. ∂ωn

(3.3.12)

Group-velocity dispersion

In the case of a chirped Gaussian pulse, U˜ (z, ω) can be obtained from Eqs. (3.2.16) and (3.3.2) in the form U˜ (z, ω) =



2π T02 1 + iC

1/2









iT02 iω 2 i exp β2 z + + β3 ω3 z . 2 1 + iC 6

(3.3.13)

If we differentiate Eq. (3.3.13) two times and substitute the result in Eq. (3.3.12), we find that the integration over ω can be performed analytically. Both T  and T 2  can be obtained by this procedure. Using the resulting expressions in Eq. (3.2.26), we find the result [6] σ = σ0



C β2 z 1+ 2σ02

2



β2 z + 2σ02

2

2 21

+ (1 + C )

2



β3 z 4σ03

2 1/2 ,

(3.3.14) √

where σ0 is the initial RMS width of the chirped Gaussian pulse (σ0 = T0 / 2). As expected, Eq. (3.3.14) reduces to Eq. (3.2.19) for β3 = 0. Eq. (3.3.14) can be used to draw several interesting conclusions. In general, both β2 and β3 contribute to pulse broadening. However, the dependence of their contributions on the chirp parameter C is qualitatively different. Whereas the contribution of β2 depends on the sign of β2 C, the contribution of β3 is independent of the sign of both β3 and C. Thus, in contrast to the behavior shown in Fig. 3.2, a chirped pulse propagating exactly at the zero-dispersion wavelength never experiences width contraction. However, even small departures from the exact zero-dispersion wavelength can lead to initial pulse contraction. This behavior is illustrated in Fig. 3.8 where the broadening factor  σ/σ0 is plotted as a function of z/LD for C = 1 and LD = LD . The dashed curve shows, for comparison, the broadening expected when β2 = 0. In the anomalous-GVD region, the contribution of β2 can counteract the β3 contribution in such a way that dispersive broadening is less than that expected when β2 = 0 for z ∼ LD . For large values of z such that z  LD /|C |, Eq. (3.3.14) can be approximated by  2 1/2 σ/σ0 = (1 + C 2 )1/2 [1 + (1 + C 2 )(LD /2LD ) ] (z/LD ),

(3.3.15)

where we have used Eqs. (3.1.5) and (3.3.6). The linear dependence of the RMS pulse width on the propagation distance for large values of z is a general feature that holds for arbitrary pulse shapes, as discussed in the next section. Eq. (3.3.14) can be generalized to include the effects of a finite source bandwidth [6]. Spontaneous emission in any light source produces amplitude and phase fluctuations that manifest as a finite bandwidth δω of the source spectrum centered at ω0 [24]. If the source bandwidth δω is much smaller than the pulse bandwidth ω, its effect on the pulse broadening can be neglected. However, for many light sources such as light-emitting diodes (LEDs) this condition is not satisfied, and it becomes necessary to

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Figure 3.8 Broadening factor as a function of z/LD for a chirped Gaussian pulse in the vicinity of λD such that LD = LD . Dashed curve corresponds to the case of λ0 = λD (β2 = 0).

include the effects of a finite source bandwidth. In the case of a Gaussian pulse and a Gaussian source spectrum, the generalized form of Eq. (3.3.14) is given by [6]      2 β2 z 2 σ2 C β2 z 2 2 2 2 2 1 β3 z = 1+ + (1 + Vω ) + (1 + C + Vω ) , 2 4σ03 σ02 2σ02 2σ02

(3.3.16)

where Vω = 2σω σ0 and σω is the RMS width of the Gaussian source spectrum. This equation describes broadening of chirped Gaussian pulses in a linear dispersive medium under quite general conditions. It can be used to discuss the effect of GVD on the performance of lightwave systems [25].

3.3.3 Ultrashort-pulse measurements As the GVD and TOD effects can change the shape and width of ultrashort pulses considerably, one should consider how such pulses can be measured experimentally. For pulses broader than 100 ps, pulse characteristics can be measured directly by using a high-speed photodetector. Streak cameras can be used for pulses as short as 0.5 ps. However, most of them operate in the visible spectral region, although infrared operation (up to 1.6 µm) has become possible in recent years. A common technique for characterizing ultrashort optical pulses is based on the nonlinear phenomenon of second-harmonic generation. In this method, known as the autocorrelation technique, the pulse is sent through a nonlinear crystal together with a delayed replica of its own [26]. A second-harmonic signal is generated inside the crystal only when two pulses overlap in time. Measuring the second-harmonic power as a

Group-velocity dispersion

function of the delay time produces an autocorrelation trace. The width of this trace is related to the width of the original pulse. The exact relationship between the two widths depends on the pulse shape. If pulse shape is known a priori, or it can be inferred indirectly, the autocorrelation trace provides an accurate measurement of the pulse width. This technique can measure widths down to a few femtoseconds but provides little information on details of the pulse shape. In fact, an autocorrelation trace is symmetric even when the pulse shape is known to be asymmetric. The use of cross correlation, a technique in which an ultrashort pulse of known shape and width is combined with the original pulse inside a second-harmonic crystal, solves this problem to some extent. The auto- and cross-correlation techniques can also make use of other nonlinear effects such as third-harmonic generation [27] and two-photon absorption [28]. All such methods, however, record intensity correlation and cannot provide any information on the phase or chirp variations across the pulse. An interesting technique, called frequency-resolved optical gating (FROG) and developed during the 1990s, solves this problem quite nicely [29–31]. It not only can measure the pulse shape but can also provide information on how the optical phase and the frequency chirp vary across the pulse. The technique works by recording a series of spectrally resolved autocorrelation traces and uses them to deduce the intensity and phase profiles associated with the pulse. Mathematically, the FROG output is described by 

S(τ, ω) =

∞ −∞

2

A(L , t)A(L , t − τ )eiωt dt ,

(3.3.17)

where τ is an adjustable delay and L is the fiber length. Experimentally, the output pulse is split into two parts that are combined inside a nonlinear crystal after introducing the delay τ in one path. A series of the second-harmonic spectra is recorded as τ is varied from negative to positive values. The FROG technique has been used to characterize pulse propagation in optical fibers with considerable success [32–37]. As an example, Fig. 3.9 shows the measured FROG traces and the retrieved intensity and phase profiles at the output of a 700-mlong fiber when 2.2-ps pulses with a peak power of 22 W are launched into it [32]. The parts (C) and (D) show the results of numerical simulations using the NLS equation with the nonlinear terms included. Such complicated pulse shapes cannot be deduced from the autocorrelation and spectral measurements alone. A related technique, known as the cross-correlation FROG, can also be used for measuring the intensity and phase profiles of ultrashort pulses [35]. It makes use of a reference pulse and is discussed in Section 12.2.2. Another technique, known as timeresolved optical gating, can also be employed [38]. In this method, the pulse is passed through a dispersive medium (e.g., an optical fiber) whose GVD is varied over a certain

75

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Figure 3.9 (A) Measured FROG spectrogram and (B) retrieved intensity (solid curve) and phase profiles (dashed curve) at the output of a 700-m-long fiber; Parts (C) and (D) show corresponding numerical results. (After Ref. [32]; ©1997 OSA.)

range, and a number of autocorrelation traces are recorded for different GVD values. Both the intensity and phase profiles can be deduced from such autocorrelation traces.

3.4. Dispersion management In a fiber-optic communication system [39], information is transmitted over an optical fiber by using a coded sequence of optical pulses whose width is set by the bit rate B of the system. Dispersion-induced broadening of pulses is undesirable as it interferes with the detection process and leads to errors, if the pulse spreads outside its allocated bit slot (TB = 1/B). Clearly, the fiber’s GVD can limit the bit rate of a telecommunication system. The idea of dispersion management was introduced during the 1990s to solve this problem. In its simplest form, two or more pieces of fibers of suitable lengths are combined to reduce the average dispersion of the entire fiber link and to limit the extent of GVD-induced pulse broadening.

3.4.1 Dispersion compensation The technique of dispersion compensation combines two fibers with different characteristics such that the average GVD of the entire fiber link is reduced to zero. Even though an optical pulse would broaden if propagated in each fiber individually, it must compress in the second fiber to recover its original width if the average GVD of both fibers is zero, provided the nonlinear effects are negligible.

Group-velocity dispersion

Figure 3.10 Changes in the width (top) and chirp (bottom) of a Gaussian pulse over 100 km when the GVD of a 80-km-long fiber with is compensated by using a 20-km-long fiber such that the average GVD is zero. The dotted line shows the location where the second fiber begins.

In the absence of the nonlinear effects, dispersion compensation can be understood from the linear nature of Eq. (3.2.1). Eq. (3.2.5) represents the general solution of Eq. (3.2.1). If two fibers of lengths L1 and L2 are used, this equation leads to U (L , T ) =

1 2π



∞

−∞

U˜ (0, ω) exp





i 2 ω (β21 L1 + β22 L2 ) − iωt dω, 2

(3.4.1)

where L = L1 + L2 is the total length and β2j is the GVD parameter of the fiber with length Lj (j = 1 and 2). It follows from this equation that U (L , t) = U (0, t) when the GVD parameters of two fibers satisfy the condition β21 L1 + β22 L2 = 0.

(3.4.2)

Clearly, the input pulse is reproduced at L = L1 + L2 when this condition is satisfied. Fig. 3.10 shows how the width and chirp of a Gaussian pulse (T0 = 10 ps and C = 0 initially) change when the GVD of a 80-km-long fiber with β21 = −20 ps2 /km is compensated by using a 20-km-long fiber with β22 = +80 ps2 /km. It is easy to verify that the condition (3.4.2) is satisfied for the combination of these two fibers, and both pulse parameters indeed recover their original values after 100 km, indicating that the pulse has returned to its initial shape. It is important to note that the pulse broadens considerably in the first fiber (by a factor of more than 15) as it is chirped by its GVD. However, the pulse undergoes a rapid compression in the second fiber such that it recovers its original width while also becoming fully unchirped in the process. We can understand the compression of the optical pulse in the second fiber from Eq. (3.2.20), which shows how the chirp parameter changes inside each fiber. Let C = 0

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Nonlinear Fiber Optics

for the input pulse. At the end of first fiber, a Gaussian pulse becomes broader and is chirped such that C1 = β21 L1 /T02 . The chirp at the end of second fiber is obtained by using C = C1 in Eq. (3.2.20) and is given by C2 = C1 + (1 + C12 )(β22 L2 /T12 ).

(3.4.3)

Using T1 = T0 1 + C12 for the width of the pulse at the input of the second fiber, it is easy to conclude that C2 = 0 when the dispersion-compensation condition (3.4.2) is satisfied. Thus, the second fiber precisely removes the chirp imposed by the first fiber. The pulse compresses because the product β22 C1 is negative in the second fiber. Eq. (3.4.2) can be satisfied in only one way. The GVD parameter of the second fiber must have opposite sign and its length must satisfy L2 = −(β21 /β22 )L1 . Fibers with opposite values of GVD at the same wavelength can be made by shifting the zero-dispersion wavelength appropriately during the manufacturing stage. For example, if standard fibers with β21 ≈ 20 ps2 /km near 1550 nm are employed, we need a dispersion-compensating fiber (DCF) designed to have “normal” GVD at 1550 nm. Such fiber s with values of β22 > 100 ps2 /km were developed during the 1990s for telecom applications. Several other devices (such as fiber gratings) can also be used for dispersion management [25]. In practice, dispersion management can be used to reduce the impact of fiber nonlinearities when short intense pulses must be transmitted through a fiber. As the pulse spreads in the first fiber, its peak power is reduced. As a result, all nonlinear effects (including SPM) become weaker. The second fiber then compresses the pulse and restores it to its original shape. However, the nonlinear effects remain negligible because the peak power remained considerably small over most of the fiber (except near the two ends). When pulses shorter than 1 ps are used, they may still broaden even when the condition (3.4.2) is satisfied. The reason is that the average TOD of the two fibers does not generally vanish. For example, the TOD parameter β3 may have the same sign for the two fibers with opposite GVD signs. This feature can be used to study the impact of the TOD effects in isolation. For example, the results shown in Fig. 3.7 assume β2 = 0 at the input wavelength. If the fiber has a finite β2 at the pulse wavelength, we can combine it with a piece of DCF to ensure that β2 = 0 on average.

3.4.2 Compensation of third-order dispersion One may ask whether it is possible to compensate for both the GVD and TOD simultaneously. To answer this equation, we consider again two fibers of lengths L1 and L2 and ask under what conditions the input pulse can be recovered at the end of the second fiber. Extending Eq. (3.4.1) so that both GVD and TOD are included, it is easy to conclude that the input pulse will be recovered when the following two conditions are simultaneously satisfied: β21 L1 + β22 L2 = 0

and

β31 L1 + β32 L2 = 0,

(3.4.4)

Group-velocity dispersion

where β2j and β3j are the GVD and TOD parameters for fiber of length Lj (j = 1, 2). It is generally difficult to satisfy both conditions simultaneously over a wide wavelength range. However, for a 1-ps pulse, it is sufficient to satisfy Eq. (3.4.4) over a 4–5 nm bandwidth. This requirement is easily met in especially designed DCFs [40] and other devices such as fiber gratings and liquid-crystal modulators. During the 1990s several experiments demonstrated transmission of short pulses at relatively high repetition rates (>100 GHz) over distances > 100 km with simultaneous compensation of both GVD and TOD [41–47]. In a 1996 experiment, a 100-Gb/s signal was transmitted over 560 km with 80-km amplifier spacing [41]. In a later experiment, bit rate was extended to 400 Gb/s by using 1-ps optical pulses within the 2.5-ps time slot [42]. Without compensation of the TOD, the pulse broadened to 2.3 ps after 40 km and exhibited a long oscillatory tail extending over 5–6 ps similar to that seen in Fig. 3.6. With partial compensation of TOD, the oscillatory tail disappeared and the pulse width was reduced to 1.6 ps. In another experiment [43], a spacial device known as a planar lightwave circuit was fabricated with a dispersion slope of −15.8 ps/nm2 over a 170-GHz bandwidth. It was used to compensate the TOD over 300 km of a dispersion-shifted fiber for which β3 ≈ 0.05 ps/(km-nm2 ) at the operating wavelength. The dispersion compensator eliminated the long oscillatory tail and reduced the width of the main peak from 4.6 to 3.8 ps. The dispersion-compensation technique has also been used for femtosecond optical pulses. For a pulse with T0 = 0.1 ps, the TOD length LD is only 10 m for a typical value β3 = 0.1 ps3 /km. Such pulses cannot propagate more than a few meters before becoming severely distorted even when β2 is compensated fully so that its average value is zero. Nonetheless, a 0.5-ps pulse (T0 ≈ 0.3 ps) was transmitted over 2.5 km of fiber by combining it with a 445-m-long DCF with β2 ≈ 98 ps2 /km and β3 ≈ −0.5 ps3 /km [44]. The output pulse remained slightly distorted because β3 was not fully compensated. In a later experiment, a liquid-crystal modulator was used to compensate for the residual β3 [45]. In a 1999 experiment [46], the use of the same technique with a different DCF (length 1.5 km) permitted transmission of a 0.4-ps pulse (T0 ≈ 0.25 ps) over 10.6 km of fiber with little distortion in the pulse shape. The main advantage of a liquid-crystal modulator is that it acts as a programmable pulse shaper. It can even be used to enhance the TOD effects artificially. As an example, Fig. 3.11 shows the pulse shapes at the output of a 2.5-km GVD-compensated fiber link when the effective value of β3 was changed from 0.124 in part (A) to −0.076 ps3 /km in part (B) [45]. It clearly shows negative values of β3 produce distortion on the leading edge of the pulse. The observed pulse shapes are in agreement with those predicted by Eq. (3.3.2), as long as the nonlinear effects remain negligible. In a 2000 experiment, femtosecond pulses at a bit rate of 640 Gb/s were transmitted over 92 km by compensating β2 and β3 over the entire link [47]. When both β2 and β3 are nearly compensated, propagation of femtosecond optical pulses is limited by the fourth-order dispersive effects governed by the parameter β4 . In

79

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Nonlinear Fiber Optics

Figure 3.11 Experimentally observed shapes of a 0.5-ps input pulse at the output of a 2.5-km GVDcompensated fiber link. The value of β3 was changed from 0.124 (left) to −0.076 ps3 /km (right) using a liquid-crystal modulator. (After Ref. [45]; ©1998 OSA.)

a 1999 experiment, the combination of a DCF and a programmable dispersion compensator allowed the management of β2 , β3 , and β4 simultaneously over a 30-nm-wide wavelength range [48]. This scheme led to the transmission of a 200-fs pulse train over a distance of 85 km. In a later experiment, 250-fs pulses could be transmitted over 139 km when dispersion up to fourth order was compensated using a DCF with a negative slope [49]. In a 2001 experiment, the bit rate was extended to 1.28 Tb/s by transmitting 380-fs pulses over 70 km of fiber, while compensating its dispersion up to fourth order [50]. A 2013 experiment carried out data transmission at the same bit rate in the 1.1-µm spectral band [51]. By 2018, a bit rate of 7.68 Tb/s was realized over 150 km using a coherent modulation format with 1.4-ps pulses at a wavelength of 1550 nm [52].

3.4.3 Dispersion-varying fibers A fiber can also be designed such that its dispersive properties vary along its length in a continuous fashion. This is realized in practice by tapering the fiber such that its core diameter decreases with length. As discussed in Section 1.2.3, the waveguide contribution to the GVD parameter β2 depends on the core size. As a result, we expect β2 to change with distance z in a gradual fashion. Typically, tapering reduces the core diameter, which in turn reduces the magnitude of |β2 | in the spectral region near 1550 nm. For this reason, they are referred to as dispersion-decreasing fibers (DDF). The use of DDFs was proposed in 1987 in the context of optical solitons [53]. Since then, DDFs have been used for a variety of applications [54–57]. The use of the phrase “dispersionvarying fiber” is more accurate as GVD of any such fiber will increase or decrease along its length, depending on which end of the fiber the input signal is launched. The NLS equation (3.1.1) can be easily extended for DDFs by making both β2 and γ a function of z. The nonlinear parameter γ also becomes a function of z for any DDF because changes in the core’s diameter along its length also modify the effective mode area. Ignoring fiber losses for the time being, the evolution of optical pulses inside a

Group-velocity dispersion

DDF is governed by ∂ A iβ2 (z) ∂ 2 A = + iγ (z)|A|2 A. ∂z 2 ∂T 2

(3.4.5)

This equation will be useful in Section 5.4.2. As the focus in this chapter is on the pure GVD effects, we neglect the nonlinear term. The resulting linear equation can be solved readily with the Fourier-transform method used in Section 3.2. The amplitude at the end of a fiber of length L is given by 1 A(L , T ) = 2π



∞ −∞

˜ (0, ω) exp A





i 2¯ ω β2 L − iωt dω, 2

(3.4.6)

where β¯2 is the average GVD of the entire fiber defined as β¯2 =

1 L



L

β2 (z)dz.

(3.4.7)

0

It follows from this solution that an input pulse will exit undistorted from any DDF designed such that β¯2 = 0. Physically, Eq. (3.4.6) shows that it is the total GVD accumulated over the entire fiber length β2 L that determines how much an optical pulse is affected by fiber’s dispersion. The solution in Eq. (3.4.6) applies to all dispersion-varying fibers, including those making use of several fiber sections with different but constant values of β2 . As an example, it is easy to see from Eq. (3.4.7) that β¯2 = 0 when two fiber sections of lengths L1 and L2 with constant GVD parameters are used, if the condition in (3.4.2) is satisfied. An important design criterion for any DDF is how fast should β2 change along its length. In other words, what is the appropriation form of the function β2 (z) for such a fiber? It will be seen in Section 5.4.2, that DDFs with an exponentially decreasing dispersion play an important role for solitons. Such fibers can be made by tailoring the cored diameter of a DDF appropriately. One can also realize exponentially decreasing dispersion by approximating it with a stair-case profile through a combination of several fibers cascaded in a serial fashion. Another class of dispersion-varying fibers that have attracted attention in recent years have their GVD vary in a periodic fashion along the fiber’s length [58].

Problems 3.1 A dispersion-shifted fiber is measured to have D = 2 ps/(km-nm) at 1.55 µm. It has an effective core area of 40 μm2 . Calculate the dispersion and nonlinear lengths when (i) 10-ps pulses with 100-mW peak power and (ii) 1-ps pulses with 1-W peak power are launched into the fiber. Are nonlinear effects important in both cases?

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3.2 A chirped Gaussian pulse is well described by Eq. (3.2.15) with C = 5 and T0 = 50 ps. Determine the temporal and spectral widths (FWHM) of this pulse. 3.3 Prove that for an unchirped Gaussian pulse of arbitrary width, the product ν t approximately equals 0.44, where t and ν are the temporal and spectral widths (both measured as FWHM), respectively. 3.4 Repeat Problem 3.3 for an unchirped “sech” pulse and prove that ν t approximately equals 0.315. 3.5 Starting with Eq. (3.2.24), derive an expression for the RMS width of a superGaussian pulse. 3.6 Show that a chirped Gaussian pulse is compressed initially inside a single-mode fiber when β2 C < 0. Derive expressions for the minimum width and the fiber length at which the minimum occurs. 3.7 Evaluate the integral in Eq. (3.3.2) numerically for an unchirped Gaussian pulse with 1-ps width (FWHM) assuming β2 = 0 and β3 = 0.1 ps3 /km. Plot the pulse shapes for L = 2 and 4 km. What happens to the pulse shape if the sign of β3 is reversed? 3.8 Repeat Problem 3.7 for an unchirped “sech” pulse and compare the pulse shapes for L = 2 and 4 km with the Gaussian case. What happens to the pulse shape if the sign of β3 is reversed? 3.9 Evaluate the integral in Eq. (3.3.17) analytically for an unchirped Gaussian pulse and plot S(τ, ω) as a surface plot in the case of 1-ps width (FWHM) of the Gaussian pulse. 3.10 Unchirped Gaussian pulses with 10 ps FWHM are sent through a 100-km-long fiber with β2 = −20 ps2 /km. Find the width of output pulses. A second fiber with β2 = 100 ps2 /km is used for dispersion compensation. How long this fiber has to be? 3.11 A pulse is sent through a fiber whose dispersion parameter D decreases exponentially as e−(z/z0 ) along its 10 km length starting from D = 20 ps/km/nm. Assuming z0 = 5 km, what would be the width of an unchirped Gaussian pulse at the fiber’s output end if its FWHM is 1 ps initially.

References [1] I.N. Sisakyan, A.B. Shvartsburg, Sov. J. Quantum Electron. 14 (1984) 1146. [2] S.A. Akhmanov, V.A. Vysloukh, A.S. Chirkin, Optics of Femtosecond Laser Pulses, American Institute of Physics, 1992, Chap. 1. [3] G.P. Agrawal, in: R.R. Alfano (Ed.), Supercontinuum Laser Source, 3rd ed., Springer, 2016, Chap. 3. [4] M. Miyagi, S. Nishida, Appl. Opt. 18 (1979) 678. [5] D. Marcuse, Appl. Opt. 19 (1980) 1653. [6] D. Marcuse, Appl. Opt. 20 (1981) 3573. [7] F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. 21 (1985) 292. [8] K. Tajima, K. Washio, Opt. Lett. 10 (1985) 460.

Group-velocity dispersion

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

G.P. Agrawal, M.J. Potasek, Opt. Lett. 11 (1986) 318. D. Anderson, M. Lisak, Opt. Lett. 11 (1986) 569. A. Jeffrey, D. Zwillinger (Eds.), Table of Integrals, Series, and Products, 6th ed., Academic Press, 2003. K. Iwashita, K. Nakagawa, Y. Nakano, Y. Suzuki, Electron. Lett. 18 (1982) 873. C. Lin, A. Tomita, Electron. Lett. 19 (1983) 837. A. Takada, T. Sugie, M. Saruwatari, Electron. Lett. 21 (1985) 969. A. Takada, T. Sugie, M. Saruwatari, J. Lightwave Technol. 5 (1987) 1525. K. Iwatsuki, A. Takada, M. Saruwatari, Electron. Lett. 24 (1988) 1572. K. Iwatsuki, A. Takada, S. Nishi, M. Saruwatari, Electron. Lett. 25 (1989) 1003. M. Nakazawa, K. Suzuki, Y. Kimura, Opt. Lett. 15 (1990) 588. R.T. Hawkins, Electron. Lett. 26 (1990) 292. G.P. Agrawal, N.A. Olsson, Opt. Lett. 14 (1989) 500. N.A. Olsson, G.P. Agrawal, K.W. Wecht, Electron. Lett. 25 (1989) 603. G.P. Agrawal, N.A. Olsson, IEEE J. Quantum Electron. 25 (1989) 2297. D.M. Bloom, L.F. Mollenauer, C. Lin, D.W. Taylor, A.M. DelGaudio, Opt. Lett. 4 (1979) 297. G.P. Agrawal, N.K. Dutta, Semiconductor Lasers, 2nd ed., Van Nostrand Reinhold, 1993, Chap. 6. G.P. Agrawal, Lightwave Technology: Telecommunication Systems, Wiley, 2005. J.C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, 3rd ed., Academic Press, 2017. D. Meshulach, Y. Barad, Y. Silberberg, J. Opt. Soc. Am. B 14 (1997) 2122. D.T. Reid, W. Sibbett, J.M. Dudley, L.P. Barry, B.C. Thomsen, J.D. Harvey, Appl. Opt. 37 (1998) 8142. K.W. DeLong, D.N. Fittinghoff, R. Trebino, IEEE J. Quantum Electron. 32 (1996) 1253. D.J. Kane, IEEE J. Quantum Electron. 35 (1999) 421. R. Trebino, Frequency-Resolved Optical Gating, Springer, 2000. J.M. Dudley, L.P. Barry, P.G. Bollond, J.D. Harvey, R. Leonhardt, P.D. Drummond, Opt. Lett. 22 (1997) 457. J.M. Dudley, L.P. Barry, P.G. Bollond, J.D. Harvey, R. Leonhardt, Opt. Fiber Technol. 4 (1998) 237. F.G. Omenetto, B.P. Luce, D. Yarotski, A.J. Taylor, Opt. Lett. 24 (1999) 1392. N. Nishizawa, T. Goto, Opt. Express 8 (2001) 328. J.M. Dudley, F. Gutty, S. Pitois, G. Millot, IEEE J. Quantum Electron. 37 (2001) 587. C. Finot, G. Millot, S. Pitois, C. Bille, J.M. Dudley, IEEE J. Sel. Top. Quantum Electron. 10 (2004) 1211. R.G.M.P. Koumans, A. Yariv, IEEE J. Quantum Electron. 36 (2000) 137, IEEE Photonics Technol. Lett. 12 (2000) 666. G.P. Agrawal, Fiber-Optic Communication Systems, 4th ed., Wiley, 2010. C.C. Chang, A.M. Weiner, A.M. Vengsarakar, D.W. Peckham, Opt. Lett. 21 (1996) 1141. S. Kawanishi, H. Takara, O. Kamatani, T. Morioka, M. Saruwatari, Electron. Lett. 32 (1996) 470. S. Kawanishi, H. Takara, T. Morioka, O. Kamatani, K. Takiguchi, T. Kitoh, M. Saruwatari, Electron. Lett. 32 (1996) 916. K. Takiguchi, S. Kawanishi, H. Takara, K. Okamoto, Y. Ohmori, Electron. Lett. 32 (1996) 755. C.C. Chang, A.M. Weiner, IEEE J. Quantum Electron. 33 (1997) 1455. C.C. Chang, H.P. Sardesai, A.M. Weiner, Opt. Lett. 23 (1998) 283. S. Shen, A.M. Weiner, IEEE Photonics Technol. Lett. 11 (1999) 827. T. Yamamoto, E. Yoshida, K.R. Tamura, K. Yonenaga, M. Nakazawa, IEEE Photonics Technol. Lett. 12 (2000) 353. F. Futami, K. Taira, K. Kikuchi, A. Suzuki, Electron. Lett. 35 (1999) 2221. M.D. Pelusi, F. Futami, K. Kikuchi, A. Suzuki, IEEE Photonics Technol. Lett. 12 (2000) 795. T. Yamamoto, M. Nakazawa, Opt. Lett. 26 (2001) 647. K. Koizumi, M. Yoshida, T. Hirooka, M. Nakazawa, Opt. Express 21 (2013) 29055.

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[52] [53] [54] [55] [56]

K. Kimura, J. Nitta, M. Yoshida, K. Kasai, T. Hirooka, M. Nakazawa, Opt. Express 26 (2018) 17418. K. Tajima, Opt. Lett. 12 (1987) 54. V.A. Bogatyrev, M.M. Bubnov, E.M. Dianov, et al., J. Lightwave Technol. 9 (1991) 561. S.V. Chernikov, E.M. Dianov, D.J. Richardson, D.N. Payne, Opt. Lett. 18 (1993) 476. J.C. Travers, J.M. Stone, A.B. Rulkov, B.A. Cumberland, A.K. George, S.V. Popov, J.C. Knight, J.R. Taylor, Opt. Express 15 (2007) 13203. [57] D.A. Korobko, O.G. Okhotnikov, D.A. Stoliarov, A.A. Sysolyatin, I.O. Zolotovskii, J. Opt. Soc. Am. B 32 (2015) 692. [58] A. Mussot, M. Conforti, S. Trillo, F. Copie, A. Kudlinski, Adv. Opt. Photonics 10 (2018) 1.

CHAPTER 4

Self-phase modulation The first nonlinear effect that we focus on is the self-phase modulation (SPM), a phenomenon that leads to spectral broadening of optical pulses [1–9]. SPM is the temporal analog of self-focusing of CW beams occurring inside any nonlinear medium with n2 > 0. It was first observed in 1967 in the context of transient self-focusing of optical pulses propagating through a CS2 -filled cell [1]. By 1970, SPM had been observed in solids and glasses by using picosecond pulses. The earliest observation of SPM in optical fibers was made with a fiber whose core was filled with CS2 liquid [7]. This work led by 1978 to a systematic study of SPM in a silica-core fiber [9]. This chapter considers SPM as a simple example of the nonlinear effects that can occur inside optical fibers. Section 4.1 is devoted to the case of pure SPM as it neglects the GVD effects and focuses on spectral changes induced by SPM. The combined effects of GVD and SPM are discussed in Section 4.2 with emphasis on the SPM-induced frequency chirp. Section 4.3 presents two analytic techniques and uses them to solve the NLS equation approximately. Section 4.4 extends the analysis to include the higher-order nonlinear effects such as self-steepening.

4.1. SPM-induced spectral changes A general description of SPM in optical fibers requires numerical solutions of the pulsepropagation equation (2.3.43) obtained in Section 2.3. The simpler equation (2.3.46) can be used for picosecond pulses (T0 > 1 ps). A further simplification occurs if the effect of GVD on SPM is negligible so that the β2 term in Eq. (2.3.46) can be set to zero. The conditions under which GVD can be ignored were discussed in Section 3.1 by introducing the length scales LD and LNL [see Eq. (3.1.5)]. In general, the GVD term can be ignored LD  L > LNL for a fiber of length L. Eq. (3.1.7) shows that the GVD effects are negligible for relatively wide pulses (T0 > 50 ps) launched into an optical fiber with a large peak power (P0 > 1 W).

4.1.1 Nonlinear phase shift In terms of the normalized amplitude U (z, T ) defined in Eq. (3.1.3), the pulsepropagation equation (3.1.4) in the limit L  LD becomes ∂U ie−αz 2 = |U | U , ∂z LNL Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00011-7

(4.1.1) Copyright © 2019 Elsevier Inc. All rights reserved.

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where α accounts for fiber losses. The nonlinear length is defined as LNL = (γ P0 )−1 , where P0 is the peak power of input pulses. Eq. (4.1.1) can be solved by using U = V exp(iφNL ) and equating the real and imaginary parts so that ∂V = 0; ∂z

∂φNL e−αz 2 = V . ∂z LNL

(4.1.2)

As the amplitude V does not change along the fiber length L, the phase equation can be integrated analytically to obtain the general solution U (L , T ) = U (0, T ) exp[iφNL (L , T )],

(4.1.3)

where U (0, T ) is the field amplitude at z = 0 and φNL (L , T ) = |U (0, T )|2 (Leff /LNL ).

(4.1.4)

The effective length Leff for a fiber of length L is defined as Leff = [1 − exp(−α L )]/α.

(4.1.5)

Eq. (4.1.3) shows that SPM gives rise to an intensity-dependent phase shift but the pulse shape remains unaffected. The nonlinear phase shift φNL in Eq. (4.1.4) increases with fiber length L. The quantity Leff plays the role of an effective length and is smaller than L because of fiber losses. In the absence of fiber losses, α = 0, and Leff = L. The maximum phase shift φmax occurs at the peak of the pulse, assumed to be located at T = 0. With U normalized such that |U (0, 0)| = 1, it is given by φmax = Leff /LNL = γ P0 Leff .

(4.1.6)

The physical meaning of the nonlinear length LNL is clear from Eq. (4.1.6). It is the effective fiber length over which φmax = 1. If we use a typical value γ = 2 W−1 /km, the nonlinear length is 50 km for P0 = 10 mW but is reduced to 100 m for P0 = 5 W. Spectral changes induced by SPM are a direct consequence of the time dependence of φNL . This can be understood by recalling from Section 3.2 that a temporally varying phase implies that the instantaneous optical frequency differs across the pulse from its central value ω0 . The difference δω is given by δω(T ) = −

  ∂φNL ∂ Leff =− |U (0, T )|2 , ∂T LNL ∂ T

(4.1.7)

where the minus sign is due to the choice of e−iω0 t in Eq. (2.3.2). The time dependence of δω is referred to as frequency chirping. The chirp induced by SPM increases in magnitude with the propagated distance. In other words, new frequency components are

Self-phase modulation

Figure 4.1 Temporal variation of SPM-induced (A) phase shift φNL and (B) frequency chirp δω for Gaussian (dashed curve) and super-Gaussian (solid curve) pulses.

generated continuously as the pulse propagates down the fiber. These SPM-generated frequency components broaden the spectrum over its initial width at z = 0 for initially unchirped pulses. The qualitative features of frequency chirp depend on the pulse shape. Consider, for example, the case of a super-Gaussian pulse with U (0, T ) given in Eq. (3.2.24). The SPM-induced chirp δω(T ) for such a pulse is 2m Leff δω(T ) = T0 LNL



T T0

2m−1

    T 2m , exp −

T0

(4.1.8)

where m = 1 for a Gaussian pulse. For larger values of m, the incident pulse becomes nearly rectangular with increasingly steeper leading and trailing edges. Fig. 4.1 shows variation of (A) the nonlinear phase shift φNL and (B) the induced frequency chirp δω at Leff = LNL for the Gaussian (m = 1) and super-Gaussian (m = 3) pulses. As φNL is proportional to |U (0, T )|2 in Eq. (4.1.4), the nonlinear phase shift mimics the shape of the pulse. The temporal profile of the SPM-induced frequency chirp shown in Fig. 4.1(B) has several interesting features. First, δω is negative near the leading edge (a red shift) and becomes positive near the trailing edge (a blue shift) of the pulse. Second, the chirp is linear and positive (up-chirp) over the central region of a Gaussian pulse. Third, the chirp is considerably larger for pulses with steeper leading and trailing edges. Fourth, super-Gaussian pulses behave differently than Gaussian pulses because the chirp occurs only near pulse edges and does not vary in a linear fashion. The main point is that chirp variations depend considerably on the exact shape of optical pulses launched into the fiber.

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4.1.2 Changes in pulse spectra The SPM-induced chirp can produce spectral broadening or narrowing depending on how the input pulse is chirped. In the case of unchirped input pulses, SPM always leads to spectral broadening, and we focus on this case first. The magnitude of SPM-induced spectral broadening can be estimated from the peak value of δω in Fig. 4.1. We can calculate the peak value by maximizing δω(T ) in Eq. (4.1.8). By setting its time derivative to zero, the maximum value of δω is found to be δωmax =

mf (m) φmax , T0

(4.1.9)

where φmax is given in Eq. (4.1.6) and f (m) is defined as 

f (m) = 2 1 −

1 2m

1−1/2m

   1 exp − 1 − .

2m

(4.1.10)

The numerical value of f depends on m only slightly; f = 0.86 for m = 1 and tends toward 0.74 for large values of m. To obtain the spectral broadening factor, the width parameter T0 should be related to the initial spectral width ω0 of the pulse. For an unchirped Gaussian pulse, ω0 = T0−1 from Eq. (3.2.17), where ω0 is the 1/e halfwidth. Eq. (4.1.9) then becomes (for m = 1) δωmax = 0.86 ω0 φmax ,

(4.1.11)

showing that the spectral broadening factor is approximately given by the numerical value of the maximum phase shift φmax . In the case of a super-Gaussian pulse, it is difficult to estimate ω0 because its spectrum is not Gaussian. However, if we use Eq. (3.2.24) to obtain the rise time, Tr = T0 /m, and assume that ω0 approximately equals Tr−1 , Eq. (4.1.9) shows that the broadening factor of a super-Gaussian pulse is also approximately given by φmax . As φmax ∼ 100 is possible for intense pulses or long fibers, SPM can broaden the spectrum considerably. In the case of intense ultrashort pulses, the output spectrum can extend over 100 THz or more, especially when SPM is accompanied by other nonlinear processes. Such an extreme spectral broadening is referred to as supercontinuum generation [4]; see Chapter 13 for further details. The actual shape of the pulse spectrum S(ω) is found by taking the Fourier transform of Eq. (4.1.3). Using S(ω) = |U˜ (L , ω)|2 , we obtain   S(ω) = 

∞ −∞

2  U (0, T ) exp[iφNL (L , T ) + i(ω − ω0 )T ] dT  .

(4.1.12)

Fig. 4.2 shows the predicted spectra of an unchirped Gaussian pulse for several values of the maximum phase shift φmax . For a given fiber length, φmax increases linearly

Self-phase modulation

Figure 4.2 Predicted optical spectra for several values of φmax when unchirped Gaussian pulses are launched into a fiber. (After Ref. [9]; ©1978 APS.)

with peak power P0 according to Eq. (4.1.6). Thus, spectral evolution seen in Fig. 4.2 can be observed experimentally by increasing the peak power of input pulses. Fig. 4.3 shows the experimentally recorded spectra [9] of nearly Gaussian pulses (T0 ≈ 90 ps) at the output of a 99-m-long fiber with 3.35-µm core diameter (parameter V = 2.53). The experimental spectra are also labeled with φmax and should be compared with the calculated spectra in Fig. 4.2. The asymmetry seen in the experimental traces can be attributed to small departures of input pulses from a perfect Gaussian shape [9]. The overall agreement between theory and the experiment is remarkably good. The most notable feature of Figs. 4.2 and 4.3 is that SPM-induced spectral broadening is accompanied by an oscillatory structure covering the entire frequency range. In general, the spectrum consists of many peaks, and the outermost peaks are the most intense. The number of peaks depends on φmax and increases linearly with it. The origin of the oscillatory structure can be understood by referring to Fig. 4.1 where the time dependence of the SPM-induced frequency chirp is shown. In general, the same chirp occurs at two values of T, showing that the pulse has the same instantaneous frequency at two distinct points. Qualitatively speaking, these two points represent two waves of the same frequency that can interfere constructively or destructively depending on their relative phase difference. The multipeak structure in the pulse spectrum is a result of such interference [1]. Mathematically, the Fourier integral in Eq. (4.1.12) gets dominant contributions from two values of T at which the chirp is the same. These contributions, being complex quantities, may add up in phase or out of phase. Indeed, one can use the method of stationary phase to obtain an analytic expression of S(ω) that is valid for large values of φmax . This expression shows that the number of peaks M in the SPM-broadened spectrum is given approximately by the relation [3] φmax ≈ (M − 12 )π.

(4.1.13)

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Figure 4.3 Experimentally observed spectra at the output of a 99-m-long fiber labeled by φmax . Input pulses are nearly Gaussian. (After Ref. [9]; ©1978 APS.)

Eq. (4.1.11) together with Eq. (4.1.13) can be used to estimate the extent of SPM-induced spectral broadening [6]. To obtain a more accurate measure of spectral broadening, one should use the RMS spectral width ωrms defined as 2 ωrms = (ω − ω0 )2  − (ω − ω0 )2 ,

(4.1.14)

where the angle brackets denote an average over the SPM-broadened spectrum given in Eq. (4.1.12). More specifically,

∞ (ω − ω0 )  = n

−∞ (ω − ω0 ) S(ω) d ω

∞ . −∞ S(ω) d ω

n

(4.1.15)

Using a procedure similar to that of Section 3.3, the spectral broadening factor for a Gaussian pulse is found to be [10]   ωrms 4 2 1/2 = 1 + √ φmax , ω0 3 3

(4.1.16)

where ω0 is the initial RMS spectral width of the pulse.

4.1.3 Effect of pulse shape and initial chirp As mentioned earlier, the shape of the SPM-broadened spectrum depends on the pulse shape, and on the initial chirp if input pulse is chirped [11–13]. Fig. 4.4 compares the

Self-phase modulation

Figure 4.4 Evolution of SPM-broadened spectra for fiber lengths in the range 0 to 50LNL for unchirped (C = 0) Gaussian (m = 1) and super-Gaussian (m = 2) pulses. A gray scale with a 20-dB range is used for the spectral density.

evolution of pulse spectra for Gaussian (m = 1) and super-Gaussian (m = 2) pulses over 50LNL using Eq. (3.2.24) in Eq. (4.1.12) and performing the integration numerically. In both cases, input pulses are assumed to be unchirped (C = 0) and fiber losses are ignored (α = 0). The qualitative differences between the two spectra can be understood by referring to Fig. 4.1, where the SPM-induced chirp is shown for the Gaussian and super-Gaussian pulses. The spectral range is larger for the super-Gaussian pulse because the maximum chirp from Eq. (4.1.9) is larger in that case. Even though both spectra in Fig. 4.4 exhibit multiple peaks, most of the energy remains in the central peak for the super-Gaussian pulse. This is so because the chirp is nearly zero over the central region in Fig. 4.1 for such a pulse, a consequence of the nearly uniform intensity of superGaussian pulses for |T | < T0 . A triangular shape of the spectral evolution in Fig. 4.4 indicates that the SPM-induced spectral broadening increases linearly with distance. An initial frequency chirp can also lead to drastic changes in the SPM-broadened pulse spectrum. This is illustrated in Fig. 4.5, which shows the spectra of a Gaussian pulse for both positive and negative chirps using φmax = 4.5π . It is evident that the sign of the chirp parameter C plays a critical role. For C > 0, spectral broadening increases and the oscillatory structure becomes less pronounced, as seen in Fig. 4.5(B). However, a negatively chirped pulse undergoes spectral narrowing through SPM, as seen clearly

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Figure 4.5 Comparison of output spectra for Gaussian pulses for four values of chirp parameter C when fiber length and peak powers are chosen such that φmax = 4.5π . Spectrum broadens for C > 0 but becomes narrower for C < 0 when compared with that of the input pulse.

in parts (C) and (D) of Fig. 4.5. The spectrum contains a central dominant peak for C = −20 and narrow further as C decreases. This behavior can be understood from Eq. (4.1.9) by noting that the SPM-induced chirp is linear and positive (frequency increases with increasing T) over the central portion of a Gaussian pulse (see Fig. 4.1). Thus, it adds to the initial chirp for C > 0, resulting in a broader spectrum. In the case of C < 0, the two chirp contributions are of opposite signs (except near the pulse edges), and the pulse becomes less chirped. If we employ the approximation that φNL (t) ≈ φmax (1 − t2 /T02 ) near the pulse center for Gaussian pulses, the SPM-induced chirp is nearly canceled for C = −2φmax . This relation provides a rough estimate of the value of C for which narrowest spectrum occurs for a given value of φmax . The SPM-induced spectral narrowing has been observed in several experiments [11–13]. In a 1993 experiment, 100-fs pulses, obtained from a Ti:sapphire laser operating near 800 nm, were first chirped with a prism pair before launching them into a 48-cm-long fiber [11]. The 10.6-nm spectral width of input pulses was nearly unchanged at low peak powers but became progressively smaller as the peak power was increased: It was reduced to 3.1 nm at a 1.6-kW peak power. The output spectral width

Self-phase modulation

Figure 4.6 Observed output spectra at several average power levels when negatively chirped 110-fs pulse was transmitted through a 50-cm-long fiber. The inset shows the spectral FWHM as a function of average power. (After Ref. [12]; ©2000 OSA.)

also changed with the fiber length at a given peak power and exhibited a minimum value of 2.7 nm for a fiber length of 28 cm at the 1-kW peak power. The spectrum rebroadened for longer fibers. In a 2000 experiment, spectral compression from 8.4 to 2.4 nm was observed when a negatively chirped 110-fs pulse was transmitted through a 50-cm-long fiber. Fig. 4.6 shows the measured pulse spectra for several values of the average power [12]. The FROG technique was used to verify that the phase of the spectrally compressed pulse was indeed constant over the pulse envelope. This feature resulted in a transform-limited pulse when peak power was suitably adjusted. For a quantitative modeling of the experimental data it was necessary to include the effects of GVD on the optical pulse. This issue is covered in Section 4.2.

4.1.4 Effect of partial coherence In the preceding discussion, SPM-induced spectral changes occur only for optical pulses because, as seen in Eq. (4.1.4), the nonlinear phase shift mimics temporal variations of the pulse shape. Indeed, the SPM-induced chirp in Eq. (4.1.7) vanishes for continuouswave (CW) radiation, implying that a CW beam would not experience any spectral broadening in optical fibers. This conclusion, however, is a consequence of an implicit assumption that the input optical field is perfectly coherent. In practice, all optical beams are only partially coherent. The degree of coherence for laser beams is large enough that the effects of partial coherence are negligible in most cases of practical interest. For example, SPM-induced spectral broadening of optical pulses is relatively unaffected by the partial temporal coherence of a laser source as long as the coherence time Tc of the laser beam is much larger than the pulse width T0 .

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When the coherence time becomes shorter than the pulse width, effects of partial coherence must be included [14–21]. In the case of a CW beam, SPM can lead to spectral broadening during its propagation inside an optical fiber. The physical reason behind such broadening can be understood by noting that partially coherent light exhibits both intensity and phase fluctuations. The SPM converts intensity fluctuations into additional phase fluctuations [see Eq. (4.1.4)] and broadens the optical spectrum. Alternatively, SPM reduces the coherence time Tc as the CW beam propagates inside the fiber, making it less and less coherent. The optical spectrum of partially coherent light at the fiber output is obtained using the Wiener–Khintchine theorem [25]:

S(ω) =

∞ −∞

(z, τ )eiωτ dτ,

(4.1.17)

where the coherence function (z, τ ) is defined as (z, τ ) = U ∗ (z, T )U (z, T + τ ).

(4.1.18)

The optical field U (z, T ) inside the fiber at a distance z is known from Eq. (4.1.3). The angle brackets denote an ensemble average over fluctuations in the input field U (0, T ). The statistical properties of U (0, T ) depend on the optical source and are generally quite different for laser and nonlaser sources. The average in Eq. (4.1.18) can be performed analytically for thermal sources because both the real and imaginary parts of U (0, T ) follow a Gaussian distribution for such a source. Even though the laser light used commonly in nonlinear-optics experiments is far from being thermal, it is instructive to consider the case of a thermal field. The coherence function in Eq. (4.1.18) in that specific case is found to evolve as [15] (Z , τ ) = (0, τ )[1 + Z 2 (1 − |(0, τ )|2 )]−2 ,

(4.1.19)

where Z = Leff /LNL is the normalized propagation distance. For a perfectly coherent field, (0, τ ) = 1. Eq. (4.1.19) shows that such a field remains perfectly coherent on propagation. In contrast, partially coherent light becomes progressively less coherent as it travels inside the fiber. The spectrum is obtained by substituting Eq. (4.1.19) in Eq. (4.1.17). The integral can be performed analytically in some specific cases [14], but in general requires numerical evaluation (through the FFT algorithm, for example). As an example, Fig. 4.7 shows the optical spectra at several propagation distances assuming a Gaussian form for the input coherence function, (0, τ ) = exp[−τ 2 /(2Tc2 )],

(4.1.20)

Self-phase modulation

Figure 4.7 SPM-induced spectral broadening of a partially coherent CW beam for several values of Z. The curve marked Z = 0 shows the input Gaussian spectrum.

where Tc is the coherence time of the input field. As expected, shortening of the coherence time is accompanied with SPM-induced spectral broadening. Little broadening occurs until light has propagated a distance equal to the nonlinear length LNL , but the spectrum broadens by a large factor at a distance of 10LNL . The spectral shape is quite different qualitatively compared with those seen in Fig. 4.2 for the case of a completely coherent pulse. In particular, note the absence of a multipeak structure. One may ask how the SPM-broadened spectrum of an optical pulse is affected by the partial coherence of the optical source. Numerical simulations show that each peak of the multipeak structure seen in Fig. 4.2 begins to broaden when the coherence time becomes comparable to or shorter than the pulse width. As a result, individual peaks begin to merge together. In the limit of very short coherence time, the multipeak structure disappears altogether, and spectral broadening has features similar to those seen in Fig. 4.7. The SPM-induced coherence degradation and the associated spectral broadening were first observed in a 1991 experiment by using stimulated Raman scattering (see Chapter 8) as a source of partially coherent light [16]. The coherence time of output pulses was reduced by a factor of >2.5 in this experiment. The dispersive effects have been included in several studies by solving the NLS equation numerically and employing a statistical kinetic description of fluctuations [22–24].

4.2. Effect of group-velocity dispersion The SPM effects discussed in Section 4.1 describe the situation only for relatively long pulses (T0 > 10 ps) for which the dispersion length LD is much larger compared with both the fiber length L and the nonlinear length LNL . As pulses become shorter and the dispersion length becomes comparable to the fiber length, it becomes necessary

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to consider the combined effects of GVD and SPM [8]. New qualitative features arise from an interplay between GVD and SPM. In the anomalous-GVD region of an optical fiber, the two phenomena can cooperate in such a way that the pulse propagates as an optical soliton (see Chapter 5). In the normal-GVD region, the combined effects of GVD and SPM can be used for pulse compression. This section considers the temporal and spectral changes that occur when the effects of GVD are included in the description of SPM [26–35].

4.2.1 Pulse evolution The starting point is the NLS equation (2.3.46) or its form given in Eq. (3.1.4). The later equation can be written in a normalized form as i

∂U 1 ∂ 2U − N 2 e−αz |U |2 U , = sgn(β2 ) ∂ξ 2 ∂τ 2

(4.2.1)

where the normalized variables ξ and τ are defined as ξ = z/LD ,

τ = T /T0 ,

(4.2.2)

and the parameter N is introduced by using N2 =

γ P0 T02 LD ≡ . LNL |β2 |

(4.2.3)

The physical significance of N will become clear in Chapter 5. The practical significance of this dimensionless parameter is that solutions of Eq. (4.2.1) obtained for a specific N value are applicable to multiple practical situations through the scaling relation in Eq. (4.2.3). For example, if N = 1 for T0 = 1 ps and P0 = 1 W, the calculated results apply equally well for T0 = 10 ps and P0 = 10 mW, or for T0 = 0.1 ps and P0 = 100 W. As evident from Eq. (4.2.3), N governs the relative importance of the SPM and GVD effects on pulse evolution along the fiber. Dispersion dominates for N  1, while SPM dominates for N  1. For values of N ∼ 1, both SPM and GVD play an equally important role during pulse evolution. In Eq. (4.2.1), sgn(β2 ) = −1 in the case of anomalous GVD (β2 < 0) and 1 otherwise. The split-step Fourier method of Section 2.4.1 can be used to solve Eq. (4.2.1) numerically. Fig. 4.8 shows (A) temporal and (B) spectral evolutions of an initially unchirped Gaussian pulse in the normal-GVD region of a fiber using N = 1 and α = 0. The qualitative behavior is quite different from that expected when either GVD or SPM dominates. In particular, the pulse broadens much more rapidly compared with the N = 0 case (no SPM). This can be understood by noting that SPM generates new frequency components that are red-shifted near the leading edge and blue-shifted near the trailing edge of the pulse. As the red components travel faster than the blue components

Self-phase modulation

Figure 4.8 Evolution of (A) pulse shapes and (B) optical spectra over a distance of 5LD for an initially unchirped Gaussian pulse propagating in the normal-GVD region of the fiber (β2 > 0) with parameters such that N = 1.

in the normal-GVD region, SPM enhances rate of pulse broadening compared with that expected from GVD alone. This in turn affects spectral broadening as the SPMinduced phase shift φNL becomes less than that occurring if the pulse shape were to remain unchanged. Indeed, φmax = 5 at z = 5LD , and a two-peak spectrum is expected in the absence of GVD. The single-peak spectrum at z/LD = 5 in Fig. 4.8 implies that the effective φmax is below π because of pulse broadening. The situation is different for pulses experiencing anomalous GVD inside the fiber. Fig. 4.9 shows the pulse shapes and spectra under conditions identical to those of Fig. 4.8 except that the sign of the GVD parameter has been reversed (β2 < 0). The pulse broadens initially but much less than that expected in the absence of SPM, and its broadening ceases to occur after 2LD or so. Moreover, the pulse shape appears to stop changing its shape for z > 4LD . At the same time, the spectrum narrows rather than exhibiting broadening expected by SPM in the absence of GVD and its shape also stops changing for z > 4LD . This behavior can be understood by noting that the SPM-induced chirp given in Eq. (4.1.8) is positive, while the dispersion-induced chirp in Eq. (3.2.14) is negative for β2 < 0. The two chirp contributions nearly cancel each other along the center portion of the Gaussian pulse when LD = LNL (N = 1). The pulse adjusts itself during propagation to make such cancellation as complete as possible, i.e., GVD and SPM cooperate to maintain a chirp-free pulse. The preceding scenario corresponds to the formation of an optical soliton. Initial broadening of a Gaussian pulse occurs because the Gaussian profile is not the character-

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Figure 4.9 Evolution of (A) pulse shapes and (B) optical spectra under conditions identical to those of Fig. 4.8 except that the Gaussian pulse propagates in the anomalous-GVD region (β2 < 0) of the fiber.

istic shape associated with a fundamental soliton. Indeed, if the input pulse is chosen to be a “sech” pulse [Eq. (3.2.22) with C = 0] keeping all other parameters the same, both its shape and spectrum do not change during its propagation inside the fiber. When the input pulse deviates from this ideal shape, the combination of GVD and SPM affects the pulse in such a way that it evolves to become a ‘sech’ pulse, as seen in Fig. 4.9. Solitons are discussed in detail in Chapter 5.

4.2.2 Broadening factor Figs. 4.8 and 4.9 show that the main effect of SPM is to alter the broadening rate imposed on the pulse by the GVD alone. Fig. 4.10 shows the broadening factor σp /σ0 as a function of z/LD for N = 1 when unchirped Gaussian pulses are launched into the fiber. Here σ is the RMS width defined as in Eq. (3.2.26) and σ0 is its initial value. The dashed line shows for comparison the broadening factor in the absence of SPM (N = 0). The SPM enhances the broadening rate in the normal-GVD region and decreases it in the anomalous-GVD region. The slower broadening rate for β2 < 0 is useful for 1.55-µm optical communication systems designed using standard fibers for which β2 ≈ −20 ps2 /km at wavelengths near 1.55-µm. When the performance of such systems is limited by dispersion, it can be improved by increasing the peak power of input pulses [32]. To study the combined effects of GVD and SPM, it is often necessary to solve Eq. (4.2.1) numerically. However, an approximate analytic expression for the pulse

Self-phase modulation

Figure 4.10 Broadening factor of Gaussian pulses in the cases of normal (β2 > 0) and anomalous (β2 < 0) GVD. The parameter N = 1 in both cases. Dashed curve shows for comparison the broadening expected in the absence of SPM (N = 0).

width is useful to see the functional dependence of the broadening rate on various parameters. Several approaches have been used to solve Eq. (4.2.1) approximately [36–43]. A variational approach was used as early as 1983 [36]. The moment method has also been used with success [41–43]. Both of these techniques are discussed in Section 4.3; they assume that the pulse maintains a certain shape during propagation inside the fiber, even though its amplitude, phase, width, and chirp change with z. A variant of the moment method has been used to include the effects of fiber losses and to predict not only the pulse width but also the spectral width and the frequency chirp [42]. In a different approach [39], the NLS equation is first solved by neglecting the GVD effects. The result is used as the initial condition, and Eq. (4.2.1) is solved again by neglecting the SPM effects. This approach is similar to the split-step Fourier method of Section 2.4, except that the step size is equal to the fiber length. The RMS pulse width can be calculated analytically by following the method discussed in Section 3.3. When an unchirped Gaussian pulse is incident at the input end of a fiber of length L, the broadening factor is given by [39]    2 1/2 √ σ L 4 2 L = 1 + 2φmax + 1 + √ φmax , σ0 LD LD2 3 3

(4.2.4)

where φmax is the SPM-induced maximum phase shift given in Eq. (4.1.6). This expression is fairly accurate for φmax < 1. In another approach, Eq. (4.2.1) is solved in the frequency domain [40]. Such a spectral approach shows that SPM can be viewed as a four-wave mixing process [28] in which two photons at pump frequencies are annihilated to create two photons at

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frequencies shifted toward the blue and red sides. These newly created spectral components result in SPM-induced spectral broadening of the pulse. The oscillatory structure of the SPM spectra is due to the phase-matching requirement of four-wave mixing (see Chapter 10). Although in general the equation describing evolution of the spectral components should be solved numerically, it can be solved analytically in some cases if the pulse shape is assumed not to change significantly.

4.2.3 Optical wave breaking Eq. (4.2.1) suggests that the effects of SPM should dominate over those of GVD for values of N  1, at least during the initial stages of pulse evolution. In fact, by introducing a new distance variable as Z = N 2 ξ = z/LNL , Eq. (4.2.1) can be written as i

∂U d ∂ 2U − + |U |2 U = 0, ∂Z 2 ∂τ 2

(4.2.5)

where fiber losses are neglected and d = β2 /(γ P0 T02 ) is a small parameter. If we use the transformation 



U (z, T ) = ρ(z, T ) exp i



T

v(z, T ) dT

(4.2.6)

0

in Eq. (4.2.5), we find that the pulse-propagation problem reduces approximately to a fluid-dynamics problem in which the variables ρ and v play, respectively, the role of density and velocity of a fluid [44]. In the optical case, these variables represent the power and chirp profiles of the pulse. For a square-shape pulse, the pulse-propagation problem becomes identical to the one related to “breaking of a dam” and can be solved analytically. Such a solution provides considerable physical insight [45–47]. The approximate solution, although useful, does not account for a phenomenon termed optical wave breaking [48–54]. It turns out that GVD cannot be treated as a small perturbation even when N is large. The reason is that, because of a large amount of the SPM-induced frequency chirp imposed on the pulse, even weak dispersive effects lead to significant pulse shaping. In the case of normal dispersion (β2 > 0), the pulse becomes nearly rectangular, with relatively sharp leading and trailing edges, and is chirped linearly across its entire width [26]. It is this linear chirp that can be used to compress the pulse by passing it through a dispersive delay line. The GVD-induced pulse shaping has another effect on pulse evolution. It increases the importance of GVD because the second derivative in Eq. (4.2.1) becomes large near the pulse edges. As a consequence, the pulse develops a fine structure near its edges. Fig. 4.11 shows the temporal and spectral evolutions of an initially unchirped Gaussian pulse using N = 30. An oscillatory structure near pulse edges develops at z/LD = 0.06. Further increase in z leads to broadening of the pulse tails. The oscillatory structure

Self-phase modulation

Figure 4.11 (A) Temporal and (B) spectral evolution of an initially unchirped Gaussian pulse with N = 30 at a distance z = 0.1LD in the normal-GVD region of an optical fiber.

depends considerably on the pulse shape. Fig. 4.12 shows the pulse shape and the spectrum at z/LD = 0.08 for an unchirped “sech” pulse. A noteworthy feature is that rapid oscillations near pulse edges are always accompanied by the sidelobes in the spectrum. The central multipeak part of the spectrum is also considerably modified by the GVD. In particular, the minima are not as deep as expected from SPM alone. The physical origin of temporal oscillations near the pulse edges is related to optical wave breaking [48]. Both GVD and SPM impose frequency chirp on the pulse as it travels down the fiber. However, as seen from Eqs. (3.2.14) and (4.1.8), although the GVD-induced chirp is linear in time, the SPM-induced chirp is far from being linear across the entire pulse. Because of the nonlinear nature of the composite chirp, different parts of the pulse propagate at different speeds [53]. In particular, in the case of normal GVD (β2 > 0), the red-shifted light near the leading edge travels faster and overtakes the unshifted light in the forward tail of the pulse. The opposite occurs for the blue-shifted light near the trailing edge. In both cases, the leading and trailing regions of the pulse contain light at two different frequencies that interfere. Oscillations near the pulse edges in Fig. 4.11 are a result of such interference. The phenomenon of optical wave breaking can also be understood as a four-wave mixing process (see Chapter 10). Nonlinear mixing of two different frequencies ω1 and ω2 in the pulse tails creates new frequencies at 2ω1 − ω2 and 2ω2 − ω1 . The spectral sidelobes in Fig. 4.12 represent these new frequency components. Temporal oscillations near pulse edges and the spectral sidelobes are manifestations of the same phenomenon.

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Figure 4.12 (A) Shape and (B) spectrum of an initially unchirped secant hyperbolic pulse with N = 30 at a distance z = 0.08LD . Spectral side lobes and temporal oscillations near pulse edges are due to optical wave breaking.

Figure 4.13 (A) Shape and (B) spectrum at a distance z = 0.08LD for an input Gaussian pulse chirped such that C = −20. The input peak power corresponds to N = 30.

It is interesting to note that optical wave breaking does not occur in the case of anomalous GVD. The reason is that the red-shifted part of the pulse cannot take over the fast-moving forward tail. Instead, the energy in the pulse tail spreads out, and the pulse acquires a pedestal [53]. The results shown in Figs. 4.11 and 4.12 are obtained for an unchirped pulse (C = 0). Pulses emitted from practical laser sources are often chirped and may follow quite a different evolution pattern depending on the sign and magnitude of the chirp parameter C [50]. Fig. 4.13 shows the pulse shape and the spectrum for a chirped Gaussian pulse using C = −20 and N = 30. The dramatic changes in the pulse shape and spectrum compared with Fig. 4.12 illustrate how much an initial chirp can modify the propagation behavior. For an initially chirped pulse, the shape becomes nearly triangular

Self-phase modulation

rather than rectangular. At the same time, the spectrum exhibits an oscillatory structure in the wings while the central SPM-like structure (seen in Fig. 4.12 in the case of an unchirped pulse) has almost disappeared. These changes in the pulse shape and spectrum can be understood qualitatively by recalling that a positive initial chirp adds to the SPM-induced chirp. As a result, optical wave breaking sets in earlier for chirped pulses. Pulse evolution is also sensitive to fiber losses. For an actual comparison between theory and experiment, it is necessary to include both the chirp and losses in numerical simulations.

4.2.4 Experimental results The combined effects of GVD and SPM in optical fibers were first observed in an experiment in which 5.5-ps (FWHM) pulses from a mode-locked dye laser (at 587 nm) were propagated through a 70-m fiber [26]. For an input peak power of 10 W (N ≈ 7), output pulses were nearly rectangular and had a positive linear chirp. The pulse shape was deduced from autocorrelation measurements as pulses were too short to be measured directly (see Section 3.3.3). In a later experiment, much wider pulses (FWHM ≈ 150 ps) from a Nd:YAG laser operating at 1.06 µm were transmitted through a 20-km-long fiber [29]. As the peak power of the input pulses was increased from 1 to 40 W (corresponding to N in the range 20–150), the output pulses broadened, became nearly rectangular and then developed substructure near its edges, resulting in an evolution pattern similar to that shown in Fig. 4.11. For such long fibers, it is necessary to include fiber losses. The experimental results were indeed in good agreement with the predictions of Eq. (4.2.1). The evidence of optical wave breaking was seen in a 1985 experiment in which 35-ps (FWHM) pulses at 532 nm (obtained from a frequency-doubled Nd:YAG laser) with peak powers of 235 W were propagated through a 93.5-m-long polarization-maintaining fiber [48]. Fig. 4.14 shows the experimentally observed spectrum of output pulses. Even though N ≈ 173 in this experiment, the formal similarity with the spectrum shown in Fig. 4.12 is evident. In fact, the phenomenon of optical wave breaking was discovered in an attempt to explain the presence of the side lobes in Fig. 4.14. In a 1988 experiment [51], the frequency chirp across the pulse was directly measured by using the combination of a streak camera and a spectrograph. The spectral sidebands associated with the optical wave breaking were indeed found to be correlated with the generation of new frequencies near the pulse edges. In a later experiment [52], rapid oscillations across the leading and trailing edges of the optical pulse were observed directly by using a cross-correlation technique that permitted subpicosecond temporal resolution. The experimental results were in excellent agreement with the predictions of Eq. (4.2.1).

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Figure 4.14 Output spectrum of 35-ps input pulses showing SPM-induced spectral broadening. Initial pulse spectrum is also shown for comparison. (After Ref. [48]; ©1985 OSA.)

4.2.5 Effect of third-order dispersion If the input wavelength λ0 nearly coincides with the zero-dispersion wavelength λD of the fiber (β2 ≈ 0), it becomes necessary to include the effects of third-order dispersion (TOD) on SPM-induced spectral broadening [55–63]. The pulse-propagation equation is obtained from Eq. (2.3.43) by setting β2 = 0 and neglecting the higher-order nonlinear terms. If we introduce the dispersion length LD from Eq. (3.3.6) and define ξ = z/LD as the normalized distance, we obtain ∂U 1 ∂ 3U 2 = sgn(β3 ) + iN e−αz |U |2 U , ∂ξ 6 ∂τ 3

(4.2.7)

where we have introduced N as 2

N =

γ P0 T03 LD = . LNL |β3 |

(4.2.8)

Similar to Eq. (4.2.1), the parameter N governs the relative importance of the TOD and SPM effects during pulse evolution; TOD dominates for N  1 while SPM dominates for N  1. Eq. (4.2.7) can be solved numerically with the split-step Fourier method of Section 2.4.1. In the following discussion, we assume β3 > 0 and neglect fiber losses by setting α = 0. Fig. 4.15 shows the shape and the spectrum of an initially unchirped Gaussian pulse at ξ = 5 using N = 1. The pulse shape should be compared with that shown in Fig. 3.6 where SPM effects were absent (N = 0). The effect of SPM is to increase the number of oscillations seen near the trailing edge of the pulse. At the same time, the intensity does not become zero at the oscillation minima. The effect of TOD on the spectrum is also evident in Fig. 4.15. In the absence of TOD, a symmetric two-peak spectrum

Self-phase modulation

Figure 4.15 (A) Shape and (B) spectrum of an unchirped Gaussian pulse propagating exactly at the zero-dispersion wavelength with N = 1 and z = 5L D .

is expected (similar to the one shown in Fig. 4.2 for the case φmax = 1.5π ) as φmax = 5 for the parameter values used in Fig. 4.15. The effect of TOD is to introduce spectral asymmetry without affecting the two-peak structure. This behavior is in sharp contrast with the one shown in Fig. 4.8 where GVD hindered splitting of the spectrum. Pulse evolution exhibits qualitatively different features for large values of N. As an example, Fig. 4.16 shows the evolution of the shape and spectrum of an initially unchirped Gaussian pulse in the range of ξ = 0 to 0.2 for N = 10. The pulse develops an oscillatory structure with deep modulations. Because of rapid temporal variations, the third derivative in Eq. (4.2.7) becomes large locally, and the TOD effects become more important as the pulse propagates inside the fiber. The most noteworthy feature of the spectrum is that the pulse energy is distributed in multiple spectral bands, a feature common to all values of N ≥ 1. As one of the spectral bands lies in the anomalous-GVD region, pulse energy in that band can form a soliton [62]. The energy in other spectral bands, lying in the normal-GVD region of the fiber, disperses with propagation. The soliton-related features are discussed later in Chapter 5. The important point to note is that, because of SPM-induced spectral broadening, the pulse does not really propagate at the zero-dispersion wavelength even if β2 ≈ 0 initially. In effect, the pulse creates its own β2 through SPM. Roughly speaking, the effective value of β2 is given by |β2 | ≈ β3 |δωmax |/2π,

(4.2.9)

where δωmax is the maximum chirp given in Eq. (4.1.9). Physically, β2 is set by the dominant outermost spectral peak in the SPM-broadened spectrum.

4.2.6 SPM effects in fiber amplifiers In a fiber amplifier, input field is amplified as it propagates down the fiber. In the case of a CW beam, the input power increases exponentially by a factor G = exp(gL ) for

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Figure 4.16 Evolution of the (A) shape and (B) spectrum of an initially Gaussian pulse over 0.2L D for N = 10. The gray intensity scale is over a 20-dB range.

an amplifier of length L, where g is the gain coefficient, provided the gain remains unsaturated. In the case of optical pulses, as the pulse energy grows exponentially, the impact of SPM becomes stronger. This is evident from Eq. (4.2.1) if we replace the loss parameter α with −g. In the absence of dispersion, the results of Section 4.1 still hold, but the effective length appearing in Eq. (4.1.6) is defined as Leff = [exp(gL ) − 1]/g, and it may become much larger than the actual amplifier length L, depending on the value of gL. As a result, spectral broadening depends on the amplifier gain and is enhanced considerably. When dispersive effects are included, the impact of amplification depends on the nature of GVD (normal versus anomalous). In the case of anomalous GVD and for values of N close to 1, the pulse begins to compress as it is amplified [64]. The reason is related to the soliton nature of pulse propagation (see Chapter 5). As the pulse is amplified, N can be maintained close to 1 only if its width T0 is reduced simultaneously. In the case of normal GVD, pulse broadens rapidly when g = 0. However, it turns out that when g > 0, the pulse evolves asymptotically to become nearly parabolic, while maintaining a linear chirp across its temporal profile [65–71]. In fact, the asymptotic solution of the NLS equation (3.1.1) with α = −g can be obtained analytically in the form [65] A(z, T ) = Ap (z)[1 − T 2 /Tp2 (z)] exp[iφp (z, T )]

(4.2.10)

Self-phase modulation

for |T | ≤ Tp (z) with A(z, T ) = 0 for |T | > Tp (z). The amplitude Ap , width Tp , and phase φp of the pulse depend on various fiber parameters as Ap (z) = 12 (gE0 )1/3 (γβ2 /2)−1/6 exp(gz/3), −1

Tp (z) = 6g (γβ2 /2)

1/2

Ap (z), 2 φp (z, T ) = φ0 + (3γ /2g)Ap (z) − (g/6β2 )T 2 ,

(4.2.11) (4.2.12) (4.2.13)

where E0 is the input pulse energy. An important feature of this solution is that the pulse width Tp (z) scales linearly with the amplitude Ap (z), as evident from Eq. (4.2.12). Such a solution is referred to as being self-similar [72]. Because of self-similarity, the pulse maintains its parabolic shape even though its width and amplitude increase with z in an exponential fashion. The most noteworthy feature of the preceding self-similar solution is that the timedependence of the phase is quadratic in Eq. (4.2.13), indicating that the amplified pulses are linearly chirped across their entire temporal profile. It follows from the discussion in Section 4.1 that a purely linear chirp is possible through SPM only when pulse shape is parabolic. A somewhat surprising feature of the solution in Eq. (4.2.10) is the total absence of optical wave breaking in the normal-GVD region. It was discovered in a 1993 study that a parabolic shape represent the only shape for which optical wave breaking can be suppressed [54]. Optical amplifiers facilitate the production of a parabolic shape since new frequency components are generated continuously through SPM in such a way that the pulse maintains a linear chirp as it is amplified and its width increases through dispersion. An important property of the self-similar solution given in Eqs. (4.2.10) through (4.2.13) is that the amplified pulse depends on the energy of the input pulse, but not on its other properties such as its shape and width. Parabolic pulses has been observed in several experiments in which picosecond or femtosecond pulses were amplified in the normal-GVD region of a fiber amplifier. Fig. 4.17(A) shows the intensity and phase profiles (deduced from FROG traces) observed at the output end of a 3.6-m-long Ybdoped fiber amplifier with 30-dB gain when a 200-fs pulse with 12-pJ energy was launched into it [65]. Part (B) shows the two profiles after further propagation in a 2-m-long passive fiber. The experimental results agree well with both the numerical results obtained by solving the NLS equation (circles) and the asymptotic solution corresponding to a parabolic pulse (dotted lines). As shown by the dashed lines in Fig. 4.17, the observed intensity profile is far from that of a “sech” pulse. Self-similar evolution toward a parabolic pulse has also been observed in fiber-based Raman amplifiers [68]. In such amplifiers the nonlinear phenomenon of stimulated Raman scattering is used to amplify the input pulse (see Section 8.3.6). Mode-locked Ybdoped fiber lasers can also emit parabolic-shape pulses under suitable conditions [69]. The self-similar nature of parabolic pulses is helpful for generating high-energy pulses

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Figure 4.17 Measured (solid lines) intensity and chirp profiles (A) at the output of a Yb-doped fiber amplifier with 30-dB gain and (B) after propagation in a 2-m-long passive fiber. The circles show numerical results based on the NLS equation, dotted lines show the asymptotic solution, and dashed lines show the intensity profile of a sech pulse. (After Ref. [65]; ©2000 American Physical Society.)

from lasers and amplifiers based on Yb-doped fibers [73–76]. It was found in a 2011 experiment that a parabolic shape is optimum for SPM-induced spectral compression [77]. Indeed, the spectrum of negatively chirped parabolic pulses could be compressed through SPM within 20% of the Fourier-transform limit.

4.3. Semianalytic techniques The results of the preceding section are based on the numerical solutions of the NLS equation with the split-step Fourier method of Section 2.4.1. Although a numerical solution is often necessary, considerable physical insight is gained if the NLS equation can be solved approximately in a semianalytic fashion. In this section we employ two √ techniques for solving the NLS equation (2.3.46). Using A = P0 e−αz/2 U, this equation can be written in the form i

∂ U β2 ∂ 2 U − + γ P0 e−αz |U |2 U = 0. ∂z 2 ∂T 2

(4.3.1)

4.3.1 Moment method The moment method was used as early as 1971 in the context of nonlinear optics [78]. It can be employed for solving Eq. (4.3.1) approximately, provided one can assume that the pulse maintains a specific shape as it propagates down a fiber, even though its amplitude,

Self-phase modulation

width, and chirp change in a continuous fashion [79–81]. This assumption may hold in some cases. For example, it was seen in Section 3.2 that a Gaussian pulse maintains its shape in a linear dispersive medium even though its amplitude, width, and chirp change during propagation. If the nonlinear effects are relatively weak (LNL  LD ), the assumption of a Gaussian shape may hold. Similarly, it was seen in Section 4.1 that a pulse maintains its shape even when nonlinear effects are strong, provided GVD effects are negligible (LNL  LD ). Here we assume that a Gaussian pulse maintains its shape inside the fiber. The moment method focuses on the evolution of three specific parameters of an optical pulse (its energy Ep , RMS width σp , and chirp Cp ) that are related to U (z, T ) as

Ep = Cp =

∞



1 ∞ 2 2 T |U | dT , Ep −∞  ∂U ∂U∗ T U∗ −U dT . ∂T ∂T −∞

|U |2 dT , −∞ ∞ 

i Ep

σp2 =

(4.3.2) (4.3.3)

As the pulse propagates inside the fiber, these three moments change. To find how they evolve with z, we differentiate them with respect to z and use Eq. (4.3.1). After some algebra, we find that dEp /dz = 0 but σp2 and Cp satisfy 



dσp2 β2 ∞ 2 ∂ 2U = T Im U ∗ 2 dT , dz Ep −∞ ∂T dCp 2β2 = dz Ep



∞

−∞

(4.3.4)

   ∂ U 2   dT + e−αz γ P0  ∂T  E p

∞

−∞

|U |4 dT .

(4.3.5)

In the case of a chirped Gaussian pulse, the field U (z, T ) at any distance z has the form U (z, T ) = ap exp[− 12 (1 + iCp )(T /Tp )2 + iφp ],

(4.3.6)

where all four pulse parameters, ap , Cp , Tp , and φp , are functions of z. The phase φp does not appear in Eqs. (4.3.4) and (4.3.5). Even though φp changes with z, it does not affect other pulse parameters and can be ignored. The peak amplitude ap is related to √ the energy as Ep = π a2p Tp . Since Ep does not change with z, we can replace it with √ = π T0 . The width parameter Tp is related to the RMS width σp of its initial value E0 √ the pulse as Tp = 2σp . Using Eq. (4.3.6) and performing the integrals in Eqs. (4.3.4) and (4.3.5), the width Tp and chirp Cp are found to change with z as dTp β2 Cp = , dz Tp

(4.3.7)

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Nonlinear Fiber Optics

dCp β2 γ P0 T0 −αz = (1 + Cp2 ) 2 + √ e . dz Tp 2Tp

(4.3.8)

This set of two first-order differential equations can be used to find how the nonlinear effects modify the width and chirp of the pulse. Considerable physical insight can be gained from Eqs. (4.3.7) and (4.3.8). The SPM does not affect the pulse width directly as the nonlinear parameter γ appears only in the chirp equation (4.3.8). The two terms on the right side of this equation originate from dispersive and nonlinear effects, respectively. They have the same sign for normal GVD (β2 > 0). Since SPM-induced chirp in this case adds to the GVD-induced chirp, we expect SPM to increase the rate of pulse broadening. In contrast, when GVD is anomalous (β2 < 0), the two terms on the right side of Eq. (4.3.8) have opposite signs, and pulse broadening should be reduced in the presence of SPM because of smaller values of Cp in Eq. (4.3.7). In fact, this equation can be integrated to obtain the following general relation between pulse width and chirp:

Tp2 (z) = T02 + 2

z

β2 (z)Cp (z) dz.

(4.3.9)

0

The equation shows explicitly that a pulse is compressed if the product β2 Cp remains negative along the fiber, a result obtained earlier in Section 3.2.

4.3.2 Variational method The variational method is well known from classical mechanics and is used in many different contexts [82–84]. It was applied as early as 1983 to the problem of pulse propagation inside optical fibers [36]. Mathematically, it makes use of the Lagrangian L defined as

L =

∞

−∞

Ld (q, q∗ ) dT ,

(4.3.10)

where the Lagrangian density Ld is a function of the generalized coordinate q(z) and q ∗ (z), both of which evolve with z. Minimization of the “action” functional, S = L dz, requires that Ld satisfy the Euler–Lagrange equation ∂ ∂T



∂ Ld ∂ qt

 +

∂ ∂z



∂ Ld ∂ qz

 −

∂ Ld = 0, ∂q

(4.3.11)

where qt and qz denote the derivative of q with respect to T and z, respectively. The variational method makes use of the fact that the NLS equation (4.3.1) can be derived from the Lagrangian density 







i ∂U ∂U∗ β2  ∂ U 2 1 Ld = U∗ −U + γ P0 e−αz |U |4 , + 2 ∂z ∂z 2  ∂T  2

(4.3.12)

Self-phase modulation

with U ∗ acting as the generalized coordinate q in Eq. (4.3.11). If we assume that the pulse shape is known in advance in terms of a few parameters, the time integration in Eq. (4.3.10) can be performed analytically to obtain the Lagrangian L in terms of these pulse parameters. In the case of a chirped Gaussian pulse of the form given in Eq. (4.3.6), we obtain L =

γ e−αz Ep2 Ep 2 ( 1 + C ) + + √ p 4Tp2 4 8π Tp

β2 Ep





dCp 2Cp dTp dφp − , − Ep dz Tp dz dz

(4.3.13)

√

where Ep = π a2p Tp is the pulse energy.

The final step is to minimize the action S = L (z) dz with respect to the four pulse parameters. This step results in the reduced Euler–Lagrange equation 

d ∂L dz ∂ qz

 −

∂L = 0, ∂q

(4.3.14)

where qz = dq/dz and q represents one of the pulse parameters. If we use q = φp in Eq. (4.3.14), we obtain dEp /dz = 0. This equation indicates that the energy Ep remains constant, as expected. Using q = Ep in Eq. (4.3.14), we obtain the following equation for the phase φp : dφp 5γ e−αz Ep β2 = + . √ dz 2Tp2 4 2π Tp

(4.3.15)

We can follow the same procedure to obtain equations for Tp and Cp . In fact, using q = Cp and Tp in Eq. (4.3.14), pulse width and chirp are found to satisfy the same two equations, namely Eqs. (4.3.7) and (4.3.8), obtained earlier with the moment method. Thus, the two approximate methods lead to identical results in the case of the NLS equation.

4.3.3 Specific analytic solutions As a simple application of the moment or variational method, consider first the case of a low-energy pulse propagating in a constant-dispersion fiber with negligible nonlinear effects. Recalling that (1 + Cp2 )/Tp2 is related to the spectral width of the pulse that does not change in a linear medium, we can replace this quantity with its initial value (1 + C02 )/T02 , where T0 and C0 are input values at z = 0. Since the second term is negligible in Eq. (4.3.8), it can be integrated easily and provides the solution Cp (z) = C0 + sgn(β2 )(1 + C02 )z/LD ,

(4.3.16)

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Nonlinear Fiber Optics

where LD is the dispersion length. Using this solution in Eq. (4.3.9), we find that the pulse width changes as Tp2 (z) = T02 [1 + 2sgn(β2 )C0 (z/LD ) + (1 + C02 )(z/LD )2 ].

(4.3.17)

It is easy to verify that these expressions agree with those obtained in Section 3.2 by solving the pulse propagation equation directly. In the absence of dispersive effects, Eqs. (4.3.7) and (4.3.8) are easily solved by setting β2 = 0. The pulse width Tp remains fixed at its input value T0 , as expected. However, the chirp parameter changes because of SPM and is given by √

Cp (z) = C0 + γ P0 (1 − e−αz )/( 2α).

(4.3.18)

If the√input pulse is unchirped (C0 = 0), SPM chirps the pulse such that Cp (L ) = φmax / 2, where φmax is given in Eq. (4.1.7). Eq. (4.3.18) shows that the SPM-induced chirp is always positive. As a result, when the input pulse is negatively chirped, SPM can reduce the net chirp and produce spectral narrowing, as found earlier in Section 4.1. To solve Eqs. (4.3.7) and (4.3.8) in the weakly nonlinear case, we make two approximations. First, we assume that fiber losses are negligible and set α = 0. Second, the nonlinear effects are weak enough that the chirp at any distance z can be written as Cp = CL + C , where the nonlinear part C  CL . It is easy to see that the linear part is given by Eq. (4.3.16), while the nonlinear part satisfies dC γ P0 T0 = √ . dz 2 Tp

(4.3.19)

Dividing Eqs. (4.3.7) and (4.3.19), we obtain γ P0 T0 γ P0 T0 dC =√ ≈√ , dTp 2β2 Cp 2β2 CL

(4.3.20)

where we replaced C with CL as C  CL . This equation is now easy to solve, and the result can be written as γ P0 T0

C (z) = √

2β2 CL

(T − T0 ).

(4.3.21)

Once C = CL + C is known, the pulse width can be found from Eq. (4.3.9). The preceding analytic solution can only be used when the parameter N 2 = LD /LNL is less than 0.3 or so. However, one can easily solve Eqs. (4.3.7) and (4.3.8) numerically for any value of N. Fig. 4.18 shows changes in Tp /T0 and Cp as a function of z/LD for several values of N assuming that input pulses are unchirped (C0 = 0) and propagate in the region of anomalous GVD region (β2 < 0). In the linear case (N = 0), pulse broadens

Self-phase modulation

Figure 4.18 Evolution of pulse width Tp and chirp parameter Cp as a function of z/LD for several values of N for an unchirped Gaussian pulse propagating with β2 < 0.

rapidly and develops considerable chirp. However, as the nonlinear effects increase and N becomes larger, pulse broadens less and less. Eventually, it even begins to compress, as seen in Fig. 4.18 for N 2 = 1.5. The behavior seen in Fig. 4.18 can be understood in terms of the SPM-induced chirp as follows. As seen from Eq. (4.3.8), the two terms on its right side have opposite signs when β2 < 0. As a result, SPM tends to cancel the dispersion-induced chirp, and reduces pulse broadening. For a certain value of the nonlinear parameter N, the two terms nearly cancel, and pulse width does not change much with propagation. As discussed in Section 5.2, this feature points to the possibility of soliton formation. For even larger values of N, pulse may even compress, at least initially. In the case of normal dispersion, the two terms on the right side have the same sign. Since SPM enhances the dispersion-induced chirp, pulse broadens even faster than that expected in the absence of SPM.

4.4. Higher-order nonlinear effects The discussion of SPM so far is based on the simplified propagation equation (2.3.46). For ultrashort optical pulses (T0 < 1 ps), it is necessary to include the higher-order nonlinear effects through Eq. (2.3.43). If Eq. (3.1.3) is used to define the normalized amplitude U, this equation takes the form ∂U sgn(β2 ) ∂ 2 U sgn(β3 ) ∂ 3 U +i − ∂z 2LD ∂τ 2 6 LD ∂τ 3   ∂ ∂|U |2 e−αz 2 2 =i , |U | U + is (|U | U ) − τR U LNL ∂τ ∂τ

(4.4.1)

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Nonlinear Fiber Optics

where LD , LD , and LNL are the three length scales defined as LD =

T02 |β2 |

LD =

,

T03 |β3 |

,

LNL =

1 . γ P0

(4.4.2)

The parameters s and τR govern the effects of self-steepening and intrapulse Raman scattering, respectively, and are defined as s=

1 , ω0 T0

τR =

TR . T0

(4.4.3)

Both of these effects are quite small for picosecond pulses but must be considered for ultrashort pulses (T0 < 1 ps).

4.4.1 Self-steepening Self-steepening results from the intensity dependence of the group velocity [85–88]. Its effects on SPM were first considered in liquid nonlinear media [2] and later extended to optical fibers [89–93]. Self-steepening leads to an asymmetry in the SPM-broadened spectra of ultrashort pulses [94–99]. Before solving Eq. (4.4.1) numerically, it is instructive to consider the dispersion-free case by setting β2 = β3 = 0. Eq. (4.4.1) can be solved analytically in this specific case if we also set τR = 0 [87]. Defining a normalized distance as Z = z/LNL and neglecting fiber losses (α = 0), Eq. (4.4.1) becomes ∂U ∂ + s (|U |2 U ) = i|U |2 U . ∂Z ∂τ

(4.4.4)

√

Using U = I exp(iφ) in Eq. (4.4.4) and separating the real and imaginary parts, we obtain the following two equations: ∂I ∂I + 3sI = 0, ∂Z ∂τ ∂φ ∂φ + sI = I. ∂Z ∂τ

(4.4.5) (4.4.6)

Since the intensity equation (4.4.5) is decoupled from the phase equation (4.4.6), it can be solved easily using the method of characteristics. Its general solution is given by [89] I (Z , τ ) = f (τ − 3sIZ ),

(4.4.7)

where we used the initial condition I (0, τ ) = f (τ ) and f (τ ) describes the pulse shape at z = 0. Eq. (4.4.7) shows that each point in τ shifts along a straight line from its initial value, and the slope of the line is intensity dependent. This feature leads to pulse

Self-phase modulation

Figure 4.19 Self-steepening of a Gaussian pulse in the dispersionless case. The dashed curve shows the pulse shape at z = 0.

distortion. As an example, consider the case of a Gaussian pulse for which I (0, τ ) ≡ f (τ ) = exp(−τ 2 ).

(4.4.8)

From Eq. (4.4.7), the pulse shape at a distance Z is obtained by using I (Z , τ ) = exp[−(τ − 3sIZ )2 ].

(4.4.9)

The implicit relation for I (Z , τ ) should be solved for each τ to obtain the pulse shape at a given value of Z. Fig. 4.19 shows the calculated pulse shapes at Z = 10 and 20 for s = 0.01. As the pulse propagates inside the fiber, it becomes asymmetric, with its peak shifting toward the trailing edge. As a result, the trailing edge becomes steeper and steeper with increasing Z. Physically, the group velocity of the pulse is intensity dependent such that its peak moves at a lower speed than its wings. Self-steepening of the pulse eventually creates an optical shock, analogous to the development of an acoustic shock on the leading edge of a sound wave [87]. The distance at which the shock is formed is obtained from Eq. (4.4.9) by requiring that ∂ I /∂τ be infinite at the shock location. It is given by [90] zs =

e 1/2 L NL

2

3s

≈ 0.39(LNL /s).

(4.4.10)

A similar relation holds for a “sech” pulse with only a slight change in the numerical coefficient (0.43 in place of 0.39). For picosecond pulses with T0 = 1 ps and P0 ∼ 1 W, the shock occurs at a distance zs ∼ 100 km. However, for femtosecond pulses with T0
1 kW, zs becomes 1 in the dispersion-free case, the shock does not develop at all [90]. Self-steepening also affects SPM-induced spectral broadening. In the dispersion-free case, φ(z, τ ) is obtained by solving Eq. (4.4.6) and the spectrum is found using   S(ω) = 

∞

−∞

[I (z, τ )]

1/2

2  exp[iφ(z, τ ) + i(ω − ω0 )τ ] dτ  .

(4.4.11)

Fig. 4.20 shows the predicted spectrum at z/LNL = 20 using s = 0.01. The most notable feature is the spectral asymmetry—the red-shifted peaks are more intense than blue-shifted peaks. The other notable feature is that SPM-induced spectral broadening is larger on the blue side than the red side (or the Stokes side). Both of these features can be understood qualitatively from the changes in the pulse shape induced by self-steepening. The spectrum is asymmetric simply because pulse shape is asymmetric. A steeper trailing edge of the pulse implies larger spectral broadening on the blue side as SPM generates blue components near the trailing edge (see Fig. 4.1). In the absence of self-steepening (s = 0), a symmetric six-peak spectrum is expected because φmax ≈ 6.4π for the parameter values used in Fig. 4.20. Self-steepening stretches the blue portion. The amplitude of the high-frequency peaks decreases because the same energy is distributed over a wider spectral range.

Self-phase modulation

Figure 4.21 Pulse shapes and spectra at z/LD = 0.2 (upper row) and 0.4 (lower row) for a Gaussian pulse propagating in the normal-GVD region of the fiber. The other parameters are α = 0, β3 = 0, s = 0.01, and N = 10.

4.4.2 Effect of GVD on optical shocks The spectral features seen in Fig. 4.20 are considerably affected by GVD, which cannot be ignored when short optical pulses propagate inside silica fibers [100–107]. The pulse evolution in this case is studied by solving Eq. (4.4.1) numerically. Fig. 4.21 shows the pulse shapes and the spectra at z/LD = 0.2 and 0.4 in the case of an initially unchirped Gaussian pulse propagating with normal GVD (β2 > 0 and β3 = 0). The parameter N, defined in Eq. (4.2.3), is taken to be 10 and corresponds to LD = 100LNL . In the absence of GVD (β2 = 0), the pulse shape and the spectrum shown in the upper row of Fig. 4.21 reduce to those shown in Figs. 4.19 and 4.20 in the case of sz/LNL = 0.2. A direct comparison shows that both the shape and spectrum are significantly affected by GVD even though the propagation distance is only a fraction of the dispersion length (z/LD = 0.2). The lower row of Fig. 4.21 shows the pulse shape and spectrum at z/LD = 0.4; the qualitative changes induced by GVD are self-evident. For this value of z/LD , the propagation distance z exceeds the shock distance zs given by Eq. (4.4.10). It is the GVD that dissipates the shock by broadening the steepened trailing edge, a feature clearly seen in the asymmetric pulse shapes of Fig. 4.21. Although the pulse spectra do not exhibit deep oscillations (seen in Fig. 4.20 for the dispersionless case), the longer

117

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Nonlinear Fiber Optics

Figure 4.22 Experimentally observed spectra of 40-fs input pulses at the output of a 7-mm-long fiber. Spectra are labeled by the peak intensity of input pulses. The top spectrum corresponds to N ≈ 7.7. (After Ref. [100]; ©1985 AIP.)

tail on the blue side is a manifestation of self-steepening. With a further increase in the propagation distance z, the pulse continues to broaden while the spectrum remains nearly unchanged. The effect of self-steepening on pulse evolution has been seen experimentally in liquids and solids as a larger spectral broadening on the blue side compared with that on the red side [4]. In these early experiments, GVD played a relatively minor role, and the spectral structure similar to that of Fig. 4.20 was observed. In the case of optical fibers, the GVD effects are strong enough that the spectra similar to those of Fig. 4.21 are expected to occur in practice. In an experiment on pulse compression [100], 40-fs optical pulses at 620 nm were propagated over a 7-mm-long fiber. Fig. 4.22 shows the experimentally observed spectra at the fiber output for several values of peak intensities. The spectrum broadens asymmetrically with a longer tail on the blue side than on the red side. This feature is due to self-steepening. In this experiment, the self-steepening parameter s ≈ 0.026, and the dispersion length LD ≈ 1 cm if we use T0 = 24 fs (corresponding to a FWHM of 40 fs for a Gaussian pulse). Assuming an effective mode area of 10 µm2 , the peak power for the top trace in Fig. 4.22 is about 200 kW. This value results in a nonlinear length LNL ≈ 0.16 mm and N ≈ 7.7. Eq. (4.4.1) can be used to simulate the experiment by using these parameter values. Inclusion of the β3 term is generally necessary to reproduce the detailed features seen in the experimental spectra of Fig. 4.22 [92]. Similar conclusions were reached in another experiment in which asymmetric spectral broadening of 55-fs pulses from a 620-nm dye laser was observed in a 11-mm-long optical fiber [101].

Self-phase modulation

Figure 4.23 (A) Temporal and (B) spectral evolution of an unchirped Gaussian pulse over 8 dispersion lengths in the cases of normal (left) and anomalous (right) GVD. Parameters used were N = 2, τR = 0.03, s = 0, and β3 = 0. The gray intensity scale is over a 20-dB range.

4.4.3 Intrapulse Raman scattering The discussion so far has neglected the last term in Eq. (4.4.1) that is responsible for intrapulse Raman scattering. In the case of optical fibers, this term becomes quite important for ultrashort optical pulses (T0 < 1 ps) and should be included in modeling pulse evolution of such short pulses in optical fibers [103–109]. Eq. (4.4.1) is solved numerically, typically with the split-step Fourier method, to study the impact of intrapulse Raman scattering on ultrashort pulses. Fig. 4.23 shows the temporal and spectral evolution of an unchirped Gaussian pulse over 8 dispersion lengths in the cases of normal (left) and anomalous (right) GVD by solving Eq. (4.4.1) with N = 2, τR = 0.03, s = 0, and β3 = 0. One can see immediately the dramatic effect of the nature of the fiber dispersion. In the case of normal GVD, pulse broadens rapidly while its spectrum is broadened through SPM by a factor of 2 or so. In contrast, in the case of anomalous GVD, the optical pulse undergoes an initial narrowing stage and then slows down, as apparent from its bent trajectory. The most noteworthy feature of Fig. 4.23 is the Raman-induced frequency shift (RIFS) of the pulse spectrum toward longer wavelengths. This is a direct consequence of intrapulse Raman scattering. As discussed in Section 2.3.2, when the input pulse is relatively short, its high-frequency components can pump the low-frequency components of the same pulse through stimulated Raman scattering, thereby transferring energy to the red side. As the pulse spectrum shifts through the RIFS, pulse slows down because the group velocity of a pulse is lower at longer wavelengths in the β2 < 0 case. This deacceleration is responsible for the bent trajectory of the pulse seen in Fig. 4.23. The RIFS occurs in the case of normal GVD as well, but its magnitude depends on the pulse width and is reduced drastically because of a rapid dispersion-induced broad-

119

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Nonlinear Fiber Optics

Figure 4.24 Evolution of (A) Raman-induced frequency shift and (B) pulse width when an unchirped Gaussian pulse with T0 = 50 fs propagates in the normal-dispersion region of a fiber.

ening of the input pulse. It is possible to extend the moment method of Section 4.3 to obtain an approximate semianalytic expression of the spectral shift induced by intrapulse Raman scattering [109]. Its use requires the introduction of two new moments, a temporal shift qp (z) and a spectral shift p (z), defined as

1 ∞ qp (z) = T |U (z, T )|2 dT , Ep −∞  ∞ i ∂U∗ ∗ ∂U p (z) = U −U dT . 2Ep −∞ ∂T ∂T

(4.4.12) (4.4.13)

Following the technique described in Section 4.3.1, these two moments are found to satisfy dqp = β2 p , dz

d p γ P0 T0 = −TR e−αz √ . dz 2Tp3

(4.4.14)

Physically speaking, the Raman term shifts the carrier frequency at which pulse spectrum is centered. This spectral shift p in turn shifts the pulse position qp in the time domain because of changes in the group velocity through fiber dispersion. Eq. (4.4.14) should be solved together with Eqs. (4.3.7) and (4.3.8) to study how p evolves along the fiber length. Fig. 4.24 shows the evolution of the RIFS, νR ≡ p /2π , when a chirped Gaussian pulse with T0 = 50 fs is launched into a fiber exhibiting normal GVD [D = −4 ps/(km-nm)]. In the case of unchirped pulses, νR saturates at a value of about 0.5 THz. This saturation is related to pulse broadening. Indeed, the spectral shift can be increased by chirping Gaussian pulses such that β2 C < 0. The reason is related to the discussion in Section 3.2.2, where it was shown that such chirped Gaussian pulses go through an initial compression phase before they broaden rapidly. Fig. 4.25(A) shows the experimentally recorded pulse spectrum when 109-fs pulses (T0 ≈ 60 fs) with 7.4 kW peak power were sent through a 6-m-long fiber [107]. The

Self-phase modulation

Figure 4.25 (A) Experimental spectrum of 109-fs input pulses at the output of a 6-m-long fiber and predictions of the generalized NLS equation with (B) s = τR = 0, (C) s = 0, and (D) both s and τR nonzero. Letters (A)–(E) mark different spectral features observed experimentally. (After Ref. [107]; ©1999 OSA.)

fiber had β2 ≈ 4 ps2 /km and β3 ≈ 0.06 ps3 /km at the 1260-nm wavelength used in this experiment. The traces (B) to (D) show the prediction of Eq. (4.4.1) under three different conditions. Both self-steepening and intrapulse Raman scattering were neglected in the trace (B) and included in the trace (D), while only the latter was included in the trace (C). All experimental features, marked as (A)–(E), were reproduced only when both higher-order nonlinear effects were included in the numerical model. Inclusion of the fourth-order dispersion was also necessary for a good agreement. Even the predicted pulse shapes were in agreement with the cross-correlation traces obtained experimentally. We shall see in Chapter 13 that the combination of SPM and other nonlinear effects such as intrapulse Raman scattering and four-wave mixing can broaden the spectrum of an ultrashort pulse inside optical fibers so much that it forms a supercontinuum extending from visible to the infrared regions [110–112]. Pulse spectra extending over 1000 nm or more have been generated using the so-called highly nonlinear fibers (see Chapter 11).

Problems 4.1 A 1.06-µm Q-switched Nd:YAG laser emits unchirped Gaussian pulse with 1-nJ energy and 100-ps width (FWHM). Pulses are transmitted through a 1-km-long

121

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Nonlinear Fiber Optics

4.2

4.3

4.4

4.5

4.6 4.7

4.8

4.9 4.10

4.11 4.12 4.13

fiber having a loss of 3 dB/km and an effective mode area of 20 µm2 . Calculate the maximum values of the nonlinear phase shift and the frequency chirp at the fiber output. Plot the spectrum of the chirped output pulse obtained in the preceding problem by using Eq. (4.1.12). Does the number of spectral peaks agree with the prediction of Eq. (4.1.13)? Repeat Problem 4.1 for a hyperbolic-secant pulse. Plot the spectrum of the chirped output pulse using Eq. (4.1.12). Comment on the impact of the pulse shape on SPM-induced spectral broadening. Determine the shape, width, and peak power of the optical pulse that will produce a linear chirp at the rate of 1 GHz/ps over a 100-ps region when transmitted through a 1-km-long fiber with a loss of 1 dB/km and an effective mode area of 50 µm2 . Calculate numerically the SPM-broadened spectra of a super-Gaussian pulse (m = 3) for C = −15, 0, and 15. Assume a peak power such that φmax = 4.5π . Compare your spectra with those shown in Fig. 4.5 and comment on the main qualitative differences. Perform the ensemble average in Eq. (4.1.18) for a thermal field with Gaussian statistics and prove that the coherence function is indeed given by Eq. (4.1.19). Solve Eq. (4.2.1) numerically using the split-step Fourier method of Section 2.4. Generate curves similar to those shown in Figs. 4.8 and 4.9 for an input pulse with U (0, τ ) = sech(τ ) using N = 1 and α = 0. Compare your results with the Gaussian-pulse case and discuss the differences qualitatively. Use the computer program developed for the preceding problem to study numerically optical wave breaking for an unchirped super-Gaussian pulse with m = 3 by using N = 30 and α = 0. Compare your results with those shown in Figs. 4.11 and 4.12. Perform the integrals appearing in Eqs. (4.3.4) and (4.3.5) using U (z, T ) from Eq. (4.3.6) and reproduce Eqs. (4.3.7) and (4.3.8). Perform the integrals appearing in Eqs. (4.3.4) and (4.3.5) using the field amplitude in the form U (z, T ) = ap sech(T /Tp ) exp(−iCp T 2 /2Tp2 ) and derive equations for Tp and Cp . Prove that the Euler–Lagrange equation (4.3.11) with Ld given in Eq. (4.3.12) reproduces the NLS equation (4.3.1). Perform the integral appearing in Eqs. (4.3.10) using Ld from Eq. (4.3.12) and U (z, T ) from Eq. (4.3.6) and reproduce Eq. (4.3.13). Show that the solution given in Eq. (4.4.9) is indeed the solution of Eq. (4.4.4) for a Gaussian pulse. Calculate the phase profile φ(Z , τ ) at sZ = 0.2 analytically (if possible) or numerically.

Self-phase modulation

4.14 Use the moment method and Ref. [109] to derive equations for the derivatives dq/dz and dp /dz for a Gaussian pulse. Use them to find an approximate expression for the Raman-induced frequency shift p .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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Self-phase modulation

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CHAPTER 5

Optical solitons A fascinating manifestation of the fiber nonlinearity occurs through optical solitons, formed as a result of the interplay between the dispersive and nonlinear effects. The word soliton refers to special kinds of wave packets that can propagate undistorted over long distances. Solitons have been discovered in many branches of physics. This chapter focuses on pulse propagation inside optical fibers in the regime in which both the groupvelocity dispersion (GVD) and self-phase modulation (SPM) are equally important and must be considered simultaneously. It is organized as follows. Section 5.1 considers the phenomenon of modulation instability and shows that propagation of a continuouswave (CW) beam inside optical fibers is inherently unstable and may convert the CW beam into pulse train under appropriate conditions. The inverse-scattering method is discussed in Section 5.2 together with the soliton solutions. The properties of the fundamental and higher-order solitons are also discussed in this section. Section 5.3 is devoted to other kinds of solitons, with emphasis on dark solitons. Section 5.4 considers the effects of external perturbations on solitons. Perturbations discussed include fiber losses, amplification of solitons, and noise introduced by optical amplifiers. Higher-order nonlinear effects such as self-steepening and intrapulse Raman scattering are the focus of Sections 5.5 and 5.6.

5.1. Modulation instability Many nonlinear systems exhibit an instability that leads to modulation of the steady state of that system. This phenomenon is referred to as the modulation instability and it was studied during the 1960s in such diverse fields as fluid dynamics, nonlinear optics, and plasma physics; see a 2009 historical review on this topic [1]. In the context of optical fibers, modulation instability was studied during the 1980s [2–8], and it has remained of continuing interest [9–17]. This section discusses modulation instability in optical fibers as an introduction to soliton theory.

5.1.1 Linear stability analysis Consider propagation of CW light inside an optical fiber. If fiber losses are ignored, Eq. (2.3.46) takes the form i

∂ A β2 ∂ 2 A = − γ | A |2 A , ∂z 2 ∂T 2

Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00012-9

(5.1.1) Copyright © 2019 Elsevier Inc. All rights reserved.

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and is referred to as the nonlinear Schrödinger (NLS) equation in the soliton literature. As discussed in Section 2.3, A(z, T ) represents the amplitude of the field envelope, β2 is the GVD parameter, and the nonlinear parameter γ is responsible for SPM. In the case of CW radiation, the amplitude A is independent of T at the input end of the fiber at z = 0. Neglecting the time derivative, Eq. (5.1.1) is readily solved to obtain the steady-state CW solution 

¯ = P0 exp(iφNL ), A

(5.1.2)

where P0 is the incident power and φNL = γ P0 z is the nonlinear phase shift induced by SPM. Eq. (5.1.2) implies that CW light should propagate through the fiber unchanged except for acquiring a power-dependent phase shift. Before reaching this conclusion, however, we must ask whether the steady-state solution (5.1.2) is stable against small perturbations. To answer this question, we perturb the steady state slightly such that A=





P0 + a) exp(iφNL )

(5.1.3)

and examine evolution of the perturbation a(z, T ) using a linear stability analysis. Substituting Eq. (5.1.3) in Eq. (5.1.1) and linearizing in a, we obtain i

∂ a β2 ∂ 2 a = − γ P0 (a + a∗ ). ∂z 2 ∂T 2

(5.1.4)

This linear equation can be solved easily in the frequency domain. However, because of the a∗ term, the Fourier components at frequencies  and − are coupled. Thus, we should consider its solution in the form a(z, T ) = a1 exp[i(Kz − T )] + a2 exp[−i(Kz − T )],

(5.1.5)

where K and  are the wave number and the frequency of perturbation, respectively. Eqs. (5.1.4) and (5.1.5) provide two coupled equations for a1 and a2 : [K − (γ P0 + β2 2 /2)]a1 − γ P0 a∗2 = 0, ∗

[K + (γ P0 + β2  /2)]a2 + γ P0 a1 = 0. 2

(5.1.6) (5.1.7)

This set of two equations has a nontrivial solution only when K and  satisfy the following dispersion relation K = ± 12 |β2 |[2 + sgn(β2 )2c ]1/2 ,

(5.1.8)

where sgn(β2 ) = ±1 depending on the sign of β2 , c is defined as 2c =

4γ P0 |β2 |

=

4 , |β2 |LNL

(5.1.9)

Optical solitons

and the nonlinear length LNL = (γ P0 )−1 . Because of the factor exp[i(β0 z − ω0 t)] that has been factored out in Eq. (2.3.21), the actual wave number and the frequency of perturbation are β0 ± K and ω0 ± , respectively. With this factor in mind, the two terms in Eq. (5.1.5) represent two different frequency components, ω0 +  and ω0 − , that are present simultaneously. It will be seen later that these frequency components correspond to the two spectral sidebands that are generated when modulation instability occurs. The dispersion relation (5.1.8) shows that steady-state stability depends critically on whether light experiences normal or anomalous GVD inside the fiber. In the case of normal GVD (β2 > 0), the wave number K is real for all , and the steady state is stable against small perturbations. By contrast, K becomes imaginary for || < c in the case of anomalous GVD (β2 < 0), and the perturbation a(z, T ) grows exponentially with z as seen from Eq. (5.1.5). As a result, the CW solution (5.1.2) is inherently unstable for β2 < 0. This instability is called the modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train.

5.1.2 Gain spectrum The gain spectrum of modulation instability is obtained from Eq. (5.1.8) by setting sgn(β2 ) = −1 and g() = 2 Im(K ), where the factor of 2 converts g to power gain. The gain exists only if || < c and is given by 

g() = |β2 | 2c − 2 .

(5.1.10)

Fig. 5.1 shows the gain spectra for three values of the nonlinear length (LNL = 1, 2, and 5 km) for an optical fiber with β2 = −5 ps2 /km. As an example, LNL = 5 km at a power level of 100 mW if we use γ = 2 W−1 /km in the wavelength region near 1.55 µm. The gain spectrum is symmetric with respect to  = 0 such that g() vanishes at  = 0. The gain becomes maximum at two frequencies given by   c 2γ P0 1/2 max = ± √ = ± , |β2 | 2

(5.1.11)

gmax ≡ g(max ) = 12 |β2 |2c = 2γ P0 ,

(5.1.12)

with a peak value

where Eq. (5.1.9) was used to relate c to P0 . The peak gain does not depend on β2 , but it increases linearly with the incident power such that gmax LNL = 2. As discussed in Chapter 10, modulation instability can be interpreted in terms of a four-wave-mixing (FWM) process that is phase-matched by SPM. If a probe wave at the frequency ω1 = ω0 +  is launched into the fiber together with the CW beam at

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Figure 5.1 Gain spectra of modulation instability for three values of the nonlinear length, LNL when a CW beam launched into a fiber with β2 = −5 ps2 /km.

ω0 , it experiences a net power gain given by Eq. (5.1.10) as long as || < c . Physically, the energy of two photons from the intense pump beam is used to create two new photons, one at the probe frequency ω1 and the other at the idler frequency 2ω0 − ω1 . This situation is referred to as the induced modulation instability. Even when the pump wave propagates by itself, modulation instability can lead to spontaneous breakup of a CW beam into a periodic pulse train. Noise photons (vacuum fluctuations) act as a probe in this situation and are amplified by the gain provided by modulation instability [13]. As the largest gain occurs for frequencies ω0 ± max , where max is given by Eq. (5.1.11), these frequency components are amplified most. Thus, a clear-cut evidence of spontaneous modulation instability is provided by the appearance of two spectral sidebands at the fiber output, shifted symmetrically by ±max on each side of the pump located at ω0 . In the time domain, the CW beam is converted into a periodic pulse train with a period Tm = 2π/ max . One may wonder whether modulation instability can occur in the normal-GVD region of optical fibers under certain conditions. It turns out that cross-phase modulation, occurring when two optical beams at different wavelengths or with orthogonal polarizations propagate simultaneously, can lead to modulation instability even in normally dispersive fibers. This situation is discussed in Chapters 6 and 7. Even a single CW beam can become unstable in a normally dispersive medium that responds slowly to the optical field [10]. Modulation instability can also occur for β2 > 0 if the fiber has two zero-dispersion wavelengths [11]. This is the case for dispersion-flattened fibers. This is also the case for tapered and other microstructured fibers in which a narrow core changes dispersive properties of the fiber considerably [12]. In such fibers, the gain

Optical solitons

Figure 5.2 Autocorrelation trace and optical spectrum of 100-ps input pulses showing evidence of modulation instability at a peak power of 7.1 W. (After Ref. [4]; ©1986 APS.)

spectrum of modulation instability exhibits a second peak even in the case of anomalous GVD.

5.1.3 Experimental observation Modulation instability in the anomalous-GVD region of optical fibers was first observed in a 1986 experiment in which 100-ps (FWHM) pulses at a wavelength of 1.319 µm were transmitted through a 1-km-long fiber with β2 ≈ −3 ps2 /km [4]. Fig. 5.2 shows the autocorrelation trace and the optical spectrum measured at the fiber output for a peak power P0 = 7.1 W. The location of spectral sidebands is in agreement with the prediction of Eq. (5.1.11). Furthermore, the interval between the oscillation peaks in the autocorrelation trace is inversely related to max as predicted by theory. The secondary sidebands seen in Fig. 5.2 are also expected when pump depletion is included. In this experiment, it was necessary to use 100-ps pulses rather than CW radiation to avoid stimulated Brillouin scattering (see Chapter 9). However, as the modulation period is ∼1 ps, the 100-ps pulses provide a quasi-CW environment for the observation of modulation instability. In a related experiment, modulation instability was induced by sending a weak CW probe wave together with the intense pump pulses [5]. The probe was obtained from a single-mode semiconductor laser whose wavelength could be tuned over a few nanometers in the vicinity of the pump wavelength. The CW probe power of 0.5 mW was much smaller compared with the pump-pulse peak power of P0 = 3 W. However, its presence led to breakup of each pump pulse into a periodic pulse train whose period was inversely related to the frequency difference between the pump and probe waves. Moreover, the period could be adjusted by tuning the wavelength of the probe laser. Fig. 5.3 shows the autocorrelation traces for two different probe wavelengths. As the observed pulse width is 100 fs in all cases. This study included the higher-order nonlinear effects by solving Eq. (2.3.36) because they become relevant for pulses shorter than 1 ps (see Section 5.5).

5.1.6 Spatial modulation of fiber parameters It has been assumed so far that the fiber parameters β2 and γ appearing in Eq. (5.1.1) are constants along the fiber’s length. As discussed in Section 3.4.3, these parameters can be made to vary with z using DDFs or multiple cascaded fibers. They also become z dependent in fibers whose core size is modulated along the fiber’s length. Considerable attention has been paid to studying modulation instability [35] when β2 and γ are periodic functions of z in Eq. (5.1.1). An early example was provided by optical communication systems that compensated fiber losses through periodic amplification of its channels. The average power in such systems exhibits sawtooth-like variations along the fiber length. This is equivalent to making γ a periodic function with a period equal to the amplifier spacing LA . It was found in 1993 that modulation instability in this situation can produce an infinite number of sidebands, both in the normal and anomalous GVD regions of optical fibers [33]. The analysis of Section 5.1.1 can be extended to include periodic power variations by replacing P0 with P0 f (z), where f (z) is a periodic function, defined such that it decreases as e−αz in each period but jumps to the value 1 at the end of that period. It is easy to show that Eq. (5.1.4) for the perturbation a(z, T ) is modified as i

∂ a β2 ∂ 2 a = − γ P0 f (z)(a + a∗ ). ∂z 2 ∂T 2

(5.1.16)

This equation is solved by expanding f (z) in a Fourier series as f (z) =  cm exp(2π imz/LA ), where the Fourier coefficient cm is found to be cm =

1 − exp(−α LA ) . α LA + 2imπ

(5.1.17)

Optical solitons

The frequencies at which the instability gain peaks are found to satisfy [33] 2m =

2π m 2c0 − , β2 LA β2 LNL

(5.1.18)

where m can be any positive or negative integer. The peak value of the gain for the mth side band depends on the Fourier coefficient cm . The origin of multiple sidebands can be understood as follows. Periodic spatial modulation of P0 or γ is equivalent to the creation of a nonlinear index grating because the refractive index itself becomes periodic in z through the Kerr effect. The period of this grating is equal to the amplifier spacing LA and typically exceeds 50 km. This long-period grating provides a new coupling mechanism among the Fourier components of a(z, T ) and allows them to grow when the perturbation frequency  satisfies the Bragg condition. In other words, the Kerr-induced grating helps in satisfying the phase-matching condition required for FWM to occur when m = 0. In the absence of this grating [f (z) = 1], only one pair of sidebands survives for the choice m = 0 in Eq. (5.1.18), and we recover the earlier result given in Eq. (5.1.11). A new feature is that sidebands can form even in the case of normal GVD (β2 > 0). Their frequencies are found from Eq. (5.1.18) by choosing m > 0. If we neglect the second term assuming √ LA < LNL , the sideband frequencies scale with the integer m as m and are not equally spaced. The GVD parameter can be made periodic by alternating two fibers of constants lengths with different values of β2 [34]. Another way to produce the same effect is to modulate the core diameter of the fiber along its length. Modulation instability in such dispersion-oscillating fibers has been covered in a 2018 review [35]. Mathematically, the situation is similar to that in Eq. (5.1.16), except that f (z) appears in the β2 term. The gain spectrum can still gain be found by expanding f (z) in a Fourier series. Similar to the case of a periodic γ , the β2 grating generates a multitude of sidebands even in the normal-GVD region at frequencies shifted from the pump frequency by 2m =

2 β2



 (γ P0 )2 + (mπ/Lp )2 − γ P0 ,

(5.1.19)

where Lp is the period of the β2 grating and β 2 is the average value of β2 (z). As expected, 0 = 0 because no modulation instability can occur for a constant positive β2 . Fig. 5.6 shows the impact of the shape of the β2 grating on the gain spectrum of modulation instability for a fiber whose GVD oscillates every 5 km such that β2 (z) = β 2 + βM f (z) using β 2 = 1 ps2 /km and βM = 0.9 ps2 /km. The nonlinear parameter of the fiber has a value γ = 7.5 W−1 /km. The β2 grating is shown in part (A) when f (z) has a rectangular, triangular, or sinusoidal shape. The gain spectra in all three cases exhibit multiple sidebands located at frequencies given in Eq. (5.1.19). Even though

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Figure 5.6 (A) GVD variations along the fiber length for three choices of the periodic shape function f (z). (B) Corresponding gain spectra of the modulation instability in three fibers when a CW beam with 20 W power is launched into it. Inset shows a close-up of the first two sidebands. (After Ref. [35]; ©2018 OSA.)

the location of these sidebands does not depend on the shape of the β2 grating, the gain at the peak of each side band depends considerably on it and is the largest for a rectangular-shape grating.

5.2. Fiber solitons The occurrence of modulation instability in the anomalous-GVD region of optical fibers is an indication of a fundamentally different character of Eq. (5.1.1) when β2 < 0. It turns out that this equation has specific solutions that either do not change along the fiber or follow a periodic evolution pattern—such solutions are known as optical solitons. The history of solitons, in fact, dates back to 1834, the year in which Scott Russell observed a heap of water in a canal that propagated undistorted over several kilometers [36]: Such waves were later called solitary waves. However, their properties were not understood completely until the inverse scattering method was developed [37]. The term soliton was coined in 1965 to reflect the particle-like nature of those solitary waves that remained intact even after mutual collisions [38]. Since then, solitons have been discovered and studied in many branches of physics including optics [39–44]. The formation of solitons inside optical fibers was first suggested in 1973 [45]. The word “soliton” has

Optical solitons

become so popular by now that a search on the Internet returns thousands of hits. Similarly, scientific databases reveal that hundreds of research papers are published every year with the word “soliton” in their title. It should be stressed that the distinction between a soliton and a solitary wave is not always made in modern optics literature, and it is common to refer to all solitary waves as solitons.

5.2.1 Inverse scattering method Only certain nonlinear wave equations can be solved with the inverse scattering method [39]. The NLS equation (5.1.1) belongs to this special class of equations. Zakharov and Shabat used the inverse scattering method in 1971 to solve the NLS equation [46]. This method is similar in spirit to the Fourier-transform method used commonly for solving linear partial differential equations. The approach consists of identifying a suitable scattering problem whose potential is the solution sought. The incident field at z = 0 is used to find the initial scattering data, whose evolution along z is then determined by solving the linear scattering problem. The propagated field is reconstructed from the evolved scattering data. Since details of the inverse scattering method are available in many texts [39], only a brief description is given here. Similar to Chapter 4, it is useful to normalize Eq. (5.1.1) by introducing three dimensionless variables A U=√ , P0

ξ=

z , LD

τ=

T , T0

(5.2.1)

and write it in the form i

∂U 1 ∂ 2U − N 2 |U |2 U , = sgn(β2 ) ∂ξ 2 ∂τ 2

(5.2.2)

where P0 is the peak power of the incident pulse, T0 is a measure of its width, and the parameter N is introduced as N2 =

γ P0 T02 LD = . LNL |β2 |

(5.2.3)

The dispersion length LD and the nonlinear length LNL are defined as in Eq. (3.1.5). Fiber losses are neglected in this section but will be included later. The parameter N can be eliminated from Eq. (5.2.2) by introducing 

u = NU = γ LD A.

(5.2.4)

Eq. (5.2.2) then takes the standard form of the NLS equation: i

∂ u 1 ∂ 2u + |u|2 u = 0, + ∂ξ 2 ∂τ 2

(5.2.5)

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where the choice sgn(β2 ) = −1 has been made to focus on the case of anomalous GVD. Note that an important scaling relation holds for Eq. (5.2.5). If u(ξ, τ ) is a solution of this equation, then  u( 2 ξ, τ ) is also a solution, where  is an arbitrary scaling factor. The importance of this scaling will become clear later. In the inverse scattering method, the scattering problem associated with Eq. (5.2.5) is found to be [40] ∂ v1 + uv2 = ζ v1 , ∂τ ∂ v2 i + u∗ v1 = −ζ v2 , ∂τ

i

(5.2.6) (5.2.7)

where v1 and v2 are the amplitudes of the two waves scattered by the potential u(ξ, τ ). The eigenvalue ζ plays a role similar to that played by the frequency in the standard Fourier analysis except that ζ can take complex values when u = 0. This feature can be identified by noting that, in the absence of potential (u = 0), v1 and v2 vary as exp(±iζ τ ). Eqs. (5.2.6) and (5.2.7) apply for all values of ξ . In the inverse scattering method, they are first solved at ξ = 0. For a given initial form of u(0, τ ), Eqs. (5.2.6) and (5.2.7) are solved to obtain the initial scattering data. The direct scattering problem is characterized by a reflection coefficient r (ζ ) that plays a role analogous to the Fourier coefficient. Formation of the bound states (solitons) corresponds to the poles of r (ζ ) in the complex ζ plane. Thus, the initial scattering data consist of the reflection coefficient r (ζ ), the complex poles ζj , and their residues cj , where j = 1 to N if N such poles exist. Although the parameter N of Eq. (5.2.3) is not necessarily an integer, the same notation is used for the number of poles to stress that its integer values determine the number of poles. Evolution of the scattering data along the fiber length is determined by using wellknown techniques [39]. The desired solution u(ξ, τ ) is reconstructed from the evolved scattering data using the inverse scattering method. This step is quite cumbersome mathematically because it requires the solution of a complicated linear integral equation. However, in the specific case in which r (ζ ) vanishes for the initial potential u(0, τ ), the solution u(ξ, τ ) can be determined by solving a set of algebraic equations. This case corresponds to solitons. The soliton order is characterized by the number N of poles, or eigenvalues ζj (j = 1 to N). The general solution can be written as [46] u(ξ, τ ) = −2

N

λ∗j ψ2j∗ ,

(5.2.8)

j=1

where λj =

√

cj exp(iζj τ + iζj2 ξ ),

(5.2.9)

Optical solitons

and ψ2j∗ is obtained by solving the following set of algebraic linear equations: ψ1j + ψ2j∗ −

N λj λ∗k ψ ∗ = 0, ζj − ζk∗ 2k

k=1 N

λ∗j λk

k=1

ζj∗ − ζk

ψ1k = λ∗j .

(5.2.10) (5.2.11)

The eigenvalues ζj are generally complex (2ζj = δj + iηj ). Physically, the real part δj produces a change in the group velocity associated with the jth component of the soliton. For the Nth-order soliton to remain bound, it is necessary that all of its components travel at the same speed. Thus, all eigenvalues ζj should lie on a line parallel to the imaginary axis, i.e., δj = δ for all j. This feature simplifies the general solution in Eq. (5.2.8) considerably. It will be seen later that the parameter δ represents a frequency shift of the soliton from the carrier frequency ω0 .

5.2.2 Fundamental soliton The first-order soliton (N = 1) corresponds to the case of a single eigenvalue. It is referred to as the fundamental soliton because its shape does not change on propagation. We find its amplitude from Eqs. (5.2.8) to (5.2.11) after setting j = k = 1. Noting that ψ21 = λ1 (1 + |λ1 |4 /η2 )−1 and substituting it in Eq. (5.2.8), we obtain u(ξ, τ ) = −2(λ∗1 )2 (1 + |λ1 |4 /η2 )−1 .

(5.2.12)

After using Eq. (5.2.9) for λ1 together with ζ1 = (δ + iη)/2 and introducing the parameters τs and φs through −c1 /η = exp(ητs − iφs ), we obtain the following general form of the fundamental soliton: u(ξ, τ ) = η sech[η(τ − τs + δξ )] exp[i(η2 − δ 2 )ξ/2 − iδτ + iφs ],

(5.2.13)

where η, δ , τs , and φs are four arbitrary parameters that characterize the soliton. Thus, an optical fiber supports a four-parameter family of fundamental solitons, all sharing the condition N = 1. Physically, the four parameters η, δ , τs , and φs represent amplitude, frequency, position, and phase of the soliton, respectively. The phase φs can be dropped from the discussion because a constant absolute phase has no physical significance. It will become relevant later when nonlinear interaction between a pair of solitons is considered. The parameter τs can also be dropped because it denotes the position of the soliton peak: If the origin of time is chosen such that the peak occurs at τ = 0 at ξ = 0, one can set τs = 0. It is clear from the phase factor in Eq. (5.2.13) that the parameter δ represents a frequency shift of the soliton from the carrier frequency ω0 . Using the carrier part,

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exp(−iω0 t), the new frequency becomes ω0 = ω0 + δ/T0 . Note that a frequency shift

also changes the soliton speed from its original value vg . This can be seen more clearly by using τ = (t − β1 z)/T0 in Eq. (5.2.13) and writing it as |u(ξ, t)| = η sech[η(t − β1 z)/T0 ],

(5.2.14)

where β1 = β1 + δ|β2 |/T0 . As expected on physical grounds, the change in group velocity (vg = 1/β1 ) is a consequence of fiber’s dispersive nature. The frequency shift δ can also be removed from Eq. (5.2.13) by choosing the carrier frequency appropriately and setting δ = 0. Fundamental solitons then form a singleparameter family described by u(ξ, τ ) = η sech(ητ ) exp(iη2 ξ/2).

(5.2.15)

The parameter η determines not only the soliton amplitude but also its width. In real units, the soliton width changes with η as T0 /η, i.e., it scales inversely with the soliton amplitude. This inverse relationship between the amplitude and the width of a soliton is the most crucial property of solitons. Its relevance will become clear later. The canonical form of the fundamental soliton is obtained by choosing u(0, 0) = 1 so that η = 1. With this choice, Eq. (5.2.15) becomes u(ξ, τ ) = sech(τ ) exp(iξ/2).

(5.2.16)

One can verify by direct substitution in Eq. (5.2.5) that this solution is indeed a solution of the NLS equation. The solution in Eq. (5.2.16) can also be obtained by solving the NLS equation directly (without employing the inverse scattering method). The approach consists of assuming that a shape-preserving solution of the NLS equation exists and has the form u(ξ, τ ) = V (τ ) exp[iφ(ξ, τ )],

(5.2.17)

where V is independent of ξ for Eq. (5.2.17) to represent a fundamental soliton that maintains its shape during propagation. The phase φ can depend on both ξ and τ . If Eq. (5.2.17) is substituted in Eq. (5.2.5) and the real and imaginary parts are separated, one obtains two equations for V and φ . The phase equation shows that φ should be of the form φ(ξ, τ ) = K ξ − δτ , where K and δ are constants. Choosing δ = 0 (no frequency shift), V (τ ) is found to satisfy d2 V = 2V (K − V 2 ). dτ 2

(5.2.18)

Optical solitons

This nonlinear equation can be solved by multiplying it by 2 (dV /dτ ) and integrating over τ . The result is (dV /dτ )2 = 2KV 2 − V 4 + C ,

(5.2.19)

where C is a constant of integration. Using the boundary condition that both V and dV /dτ vanish as |τ | → ∞, C is found to be 0. The constant K is determined from the condition that V = 1 and dV /dτ = 0 at the soliton peak, assumed to occur at τ = 0. Its use provides K = 12 , resulting in φ = ξ/2. Eq. (5.2.19) is easily integrated to obtain V (τ ) = sech(τ ). We have thus recovered the solution in Eq. (5.2.16) using a simple technique. In the context of optical fibers, the solution (5.2.16) indicates that if a hyperbolic secant pulse, whose width T0 and the peak power P0 are chosen such that N = 1 in Eq. (5.2.3), is launched inside an ideal lossless fiber, the pulse will propagate undistorted for arbitrarily long distances, without change in its shape. It is this feature of the solitons that makes them attractive for practical applications [45]. The peak power P0 required to support a fundamental soliton is obtained from Eq. (5.2.3) by setting N = 1 and is given by P0 =

3.11 |β2 | |β2 | ≈ , 2 2 γ T0 γ TFWHM

(5.2.20)

where the FWHM of the soliton is defined using TFWHM ≈ 1.76 T0 from Eq. (3.2.22). Using typical parameter values, β2 = −1 ps2 /km and γ = 3 W−1 /km, for dispersionshifted fibers, the required P0 is ∼1 W for T0 = 1 ps but reduces to only 10 mW when T0 = 10 ps. Thus, fundamental solitons can form in optical fibers at power levels available from semiconductor lasers.

5.2.3 Second and higher-order solitons Higher-order solitons are also described by the general solution in Eq. (5.2.8). Various combinations of the eigenvalues ηj and the residues cj generally lead to an infinite variety of soliton forms. If the soliton is assumed to be symmetric about τ = 0, the residues are related to the eigenvalues by the relation [47]

N

cj = kN=1

(ηj + ηk )

k=j |ηj

− ηk |

.

(5.2.21)

This condition selects a subset of all possible solitons. Among this subset, a special role is played by solitons whose initial shape at ξ = 0 is given by u(0, τ ) = N sech(τ ),

(5.2.22)

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Figure 5.7 (A) Temporal and (B) spectral evolution of the third-order soliton over one soliton period. Note splitting of the pulse near z/LD = 0.5 and its recovery beyond that.

where the soliton order N is an integer. The peak power necessary to launch the Nth-order soliton is obtained from Eq. (5.2.3) and is N 2 times that required for the fundamental soliton. For the second-order soliton (N = 2), the field distribution is obtained from Eqs. (5.2.8) through (5.2.11). Using ζ1 = i/2 and ζ2 = 3i/2 for the two eigenvalues, the field associated with the second-order soliton is given by [48] u(ξ, τ ) =

4[cosh(3τ ) + 3 exp(4iξ ) cosh(τ )] exp(iξ/2) . cosh(4τ ) + 4 cosh(2τ ) + 3 cos(4ξ )

(5.2.23)

An interesting property of the solution (5.2.23) is that |u(ξ, τ )|2 is periodic in ξ with the period ξ0 = π/2. In fact, this periodicity occurs for all solitons with N ≥ 2. Using the definition ξ = z/LD from Eq. (5.2.1), the soliton period z0 in real units becomes z0 =

π

2

LD =

T2 π T02 ≈ FWHM . 2 |β2 | 2|β2 |

(5.2.24)

Periodic evolution of a third-order soliton over one soliton period is shown in Fig. 5.7(A). As the pulse propagates along the fiber, it first contracts to a fraction of its initial width, splits into two distinct pulses at z0 /2, and then merges again to recover the original shape at the end of the soliton period at z = z0 . This pattern is repeated over each section of length z0 . To understand the origin of periodic evolution for higher-order solitons, it is helpful to look at changes in the pulse spectra shown in Fig. 5.7(B) for the N = 3 soliton. The

Optical solitons

Figure 5.8 (A) Temporal and (B) spectral evolution of a fourth-order soliton over one soliton period. The gray intensity scale is over a 20-dB range.

temporal and spectral changes result from an interplay between the SPM and GVD effects. Initially, SPM generates a frequency chirp and broadens the pulse spectrum. This spectral broadening is clearly seen in Fig. 5.7(B) near z/LD = 0.3 with its typical oscillatory structure. In the absence of GVD, the pulse shape would have remained unchanged. However, anomalous GVD contracts the pulse as the pulse is positively chirped (see Section 3.2). Only the central portion of the pulse contracts because the chirp is nearly linear only over that part. However, as a result of a substantial increase in the pulse intensity near the central part of the pulse, the spectrum changes significantly as seen in Fig. 5.7(B) near z/LD = 0.5. It is this mutual interaction between the GVD and SPM effects that is responsible for the periodic evolution pattern seen in Fig. 5.7. Evolution becomes more complicated for larger soliton orders. As an example, Fig. 5.8 shows the evolution of a fourth-order (N = 4) soliton as a surface plot. One can see in part (A), an initial narrowing of the pulse near ξ = 0.2, followed with a splitting into two and three pulses, and then a subsequent reversal to the original pulse shape at ξ = pi/2. The spectrum in part (B) also exhibits a complicated evolution pattern. In the case of a fundamental soliton (N = 1), GVD and SPM balance each other in such a way that neither the pulse shape nor the pulse spectrum changes along the fiber length. In the case of higher-order solitons, SPM dominates initially but GVD soon catches up and leads to pulse contraction seen in Figs. 5.7 and 5.8. Soliton theory shows that for pulses with a hyperbolic-secant shape and with peak powers determined

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Figure 5.9 Autocorrelation traces (lower row) and pulse spectra (upper row) for several values of input peak power P0 . The corresponding traces for the input pulse are shown inside the rectangular box. (After Ref. [49]; ©1980 APS.)

from Eq. (5.2.3), the two effects can cooperate in such a way that the pulse follows a periodic evolution pattern with original shape recurring at multiples of the soliton period z0 given by Eq. (5.2.24). Near the 1.55-µm wavelength, if we use β2 = −20 ps2 /km for standard silica fibers, the soliton period is ∼80 m for T0 = 1 ps and scales as T02 , becoming 8 km when T0 = 10 ps. For dispersion-shifted fibers β2 is smaller by a factor of 10 or so, and z0 increases by the same factor for a given value of T0 .

5.2.4 Experimental confirmation Although the possibility of soliton formation in optical fibers was suggested in 1973 [45], the lack of a suitable source of picosecond pulses at wavelengths >1.3 µm delayed their experimental observation until 1980. Solitons in optical fibers were first observed in an experiment [49] that used a mode-locked, color-center laser emitting short optical pulses (TFWHM ≈ 7 ps) near 1.55 µm, a wavelength near which optical fibers exhibit anomalous GVD as well as low losses. The pulses were propagated inside a 700-m-long silica fiber with a core diameter of 9.3 µm. The fiber parameters for this experiment were estimated to be β2 ≈ −20 ps2 /km and γ ≈ 1.3 W−1 /km. Using T0 = 4 ps in Eq. (5.2.20), the peak power required for forming a fundamental soliton was about 1 W. In the experiment, the peak power of optical pulses was varied over a range 0.3–25 W, and their pulse shapes and spectra were monitored at the fiber output. Fig. 5.9 shows autocorrelation traces and pulse spectra at several power levels and compares them with those of the input pulse. The measured spectral width of 25 GHz of the input pulse is nearly transform limited, indicating that mode-locked pulses used in the experiment were unchirped. At a low power level of 0.3 W, optical pulses experienced dispersioninduced broadening inside the fiber, as expected from Section 3.2. However, as the power was increased, output pulses steadily narrowed, and their width became the same as the input width at P0 = 1.2 W. This power level corresponds to the formation of a

Optical solitons

fundamental soliton and should be compared with the theoretical value of 1 W obtained from Eq. (5.2.20). The agreement is quite good in spite of many uncertainties inherent in the experiment. At higher power levels, output pulses exhibited dramatic changes in their shape and developed a multipeak structure. For example, the autocorrelation trace for 11.4 W contained three peaks. This three-peak structure indicated a two-fold splitting of the pulse, similar to that seen in Fig. 5.7 near z/z0 = 0.5 for the third-order soliton. The observed spectrum also exhibited the specific features seen in Fig. 5.10 near z/z0 = 0.5. The estimated soliton period for this experiment was 1.26 km, and the ratio z/z0 was 0.55 at the output end of the 700-m-long fiber used in the experiment. As the power level of 11.4 W was also nearly nine times the fundamental soliton power, the trace in Fig. 5.9 at this power level indeed corresponds to a N = 3 soliton. This conclusion is further corroborated by the autocorrelation trace for P0 = 22.5 W. The observed five-peak structure implies a three-fold splitting of the laser pulse, in agreement with the prediction of soliton theory for a fourth-order soliton (see Fig. 5.8). The periodic nature of a high-order soliton implies that the pulse should restore its original shape and spectrum at distances that are multiples of the soliton period. This feature was observed for second- and third-order solitons in a 1983 experiment in which the fiber length of 1.3 km corresponded to nearly one soliton period [50]. In a different experiment, initial narrowing of high-order solitons, seen in Fig. 5.7 for N = 3, was observed for values of N up to 13 [51]. High-order solitons were also found to form inside the cavity of a mode-locked dye laser operating in the visible region near 620 nm by incorporating an optical element with negative GVD inside the laser cavity [52]. Such a laser emitted asymmetric second-order solitons, under certain operating conditions, as predicted by inverse scattering theory.

5.2.5 Soliton stability A natural question is what happens if the initial pulse shape or the peak power is not matched to that required by Eq. (5.2.22), and the input pulse does not correspond to an optical soliton. Similarly, one may ask how the soliton is affected if it is perturbed during its propagation inside the fiber. Such questions are answered by using perturbation methods developed for solitons and are discussed later in Section 5.4. In general, one must solve the NLS equation (5.2.2) numerically to study the pulse evolution inside optical fibers (see Appendix B for a sample numerical code). Consider first the case when the peak power is not exactly matched and the value of N obtained from Eq. (5.2.3) is not an integer. Fig. 5.10 shows the evolution of a “sech” pulse launched with N = 1.2 by solving the NLS equation numerically. Even though pulse width and peak power change initially, the pulse eventually evolves toward a fundamental soliton of narrower width for which N = 1 asymptotically. Perturbation theory has been used to study this behavior analytically [48]. Because details are cum-

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Figure 5.10 Temporal evolution over 10 dispersion lengths when N = 1.2 at z = 0. The pulse evolves into a fundamental soliton of narrower width for which N approaches 1 asymptotically.

bersome, only results are summarized here. In physical terms, the pulse adjusts its shape and width as it propagates along the fiber and evolves into a soliton. A part of the pulse energy is dispersed away in the process. This part is known as the continuum radiation. It separates from the soliton as ξ increases and its amplitude decays as ξ −1/2 . For ξ 1, the pulse evolves asymptotically into a soliton whose order is an integer N˜ closest to the launched value of N. Mathematically, if N = N˜ +  , where || < 1/2, the soliton part corresponds to an initial pulse shape of the form u(0, τ ) = (N˜ + 2) sech[(1 + 2/N˜ )τ ].

(5.2.25)

The pulse broadens if  < 0 and narrows if  > 0. No soliton is formed when N ≤ 12 . The effect of pulse shape on soliton formation can also be investigated solving Eq. (5.2.5) numerically. Fig. 4.8 of Chapter 4 shows evolution of a Gaussian pulse using the initial field u(0, τ ) = exp(−τ 2 /2). Even though N = 1, pulse shape changes along the fiber because of deviations from the “sech” shape required for a fundamental soliton. The interesting feature of Fig. 4.8 is that the pulse adjusts its width and evolves asymptotically into a fundamental soliton. In fact, the evolution appears to be complete by z/LD = 5, a distance that corresponds to about three soliton periods. An essentially similar evolution pattern occurs for other pulse shapes such as a super-Gaussian shape. The final width of the soliton and the distance needed to evolve into a fundamental soliton depend on the exact shape but the qualitative behavior remains the same. As pulses emitted from laser sources are often chirped, we should also consider the effect of initial frequency chirp on soliton formation [53–58]. The chirp can be detrimental simply because it superimposes on the SPM-induced chirp and disturbs the exact balance between the GVD and SPM effects necessary for solitons. Its effect on soliton formation can be studied by solving Eq. (5.2.5) numerically with an input amplitude u(0, τ ) = N sech(τ ) exp(−iC τ 2 /2),

(5.2.26)

Optical solitons

Figure 5.11 Soliton formation in the presence of an initial linear chirp for the case N = 1 and C = 0.5.

where C is the chirp parameter introduced in Section 3.2. The quadratic form of phase variation corresponds to a linear chirp such that the optical frequency increases with time (up-chirp) for positive values of C. Fig. 5.11 shows evolution of a fundamental soliton (N = 1) in the case of a relatively low chirp (C = 0.5). The pulse compresses initially mainly because of the positive chirp; initial compression occurs even in the absence of nonlinear effects. The pulse then broadens but is eventually compressed a second time with the tail separating from the main peak gradually. The main peak evolves into a soliton over a propagation distance ξ > 15. A similar behavior occurs for negative values of C. Formation of a soliton is expected for small values of |C | because solitons are generally stable under weak perturbations. However, a soliton is destroyed if |C | exceeds a critical value Ccr . For N = 1, a soliton does not form if C is increased from 0.5 to 2. The critical value of the chirp parameter can be obtained with the inverse scattering method [55–57]. More specifically, Eqs. (5.2.6) and (5.2.7) are solved to obtain the eigenvalue ζ using u from Eq. (5.2.26). Solitons exist as long as the imaginary part of ζ is positive. The critical value depends on N and is found to be about 1.64 for N = 1. It also depends on the form of the phase factor in Eq. (5.2.26). From a practical standpoint, initial chirp should be minimized as much as possible. This is necessary because, even if the chirp is not detrimental for |C | < Ccr , a part of the pulse energy is shed as dispersive waves (also called continuum radiation) during the process of soliton formation [55]. For example, only 83% of the input energy is converted into a soliton in the case of C = 0.5 shown in Fig. 5.11, and this fraction reduces to 62% for C = 0.8. It is clear from the preceding discussion that the exact shape of the input pulse used to launch a fundamental (N = 1) soliton is not critical. Moreover, as solitons can form for values of N in the range 0.5 < N < 1.5, even the width and peak power of the input pulse can vary over a wide range [see Eq. (5.2.3)] without hindering soliton formation. It is this relative insensitivity to the exact values of input parameters that makes the use of solitons feasible in practical applications [44,59]. However, it is important to realize that, when input parameters deviate substantially from their ideal values, a part of the pulse energy is invariably shed away in the form of dispersive waves as the pulse

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evolves to form a fundamental soliton [60]. Such dispersive waves can interfere with the soliton itself and modify its characteristics [61]. In a practical situation, solitons can be subjected to many types of perturbations as they propagate inside an optical fiber. Examples of perturbations include fiber losses, amplifier noise (if amplifiers are used to compensate fiber losses), third-order dispersion, and intrapulse Raman scattering. Such perturbations are discussed in Sections 5.4 and 5.5.

5.3. Other types of solitons The soliton solution given in Eq. (5.2.8) is not the only possible solution of the NLS equation. Many other kinds of solitons have been discovered depending on the dispersive and nonlinear properties of fibers. This section describes several of them, focusing mainly on dark and bistable solitons.

5.3.1 Dark solitons Dark solitons correspond to the solutions of Eq. (5.2.2) with sgn(β2 ) = 1 and occur in the normal-GVD region of fibers. They were discovered in 1973 and observed nearly 15 years later [62–69]. The intensity profile associated with such solitons exhibits a dip in a uniform background, hence the name dark soliton. Pulse-like solutions obtained in Section 5.2 are called bright solitons to make the distinction clear. The NLS equation describing dark solitons is obtained from Eq. (5.2.5) by changing the sign of the second-derivative term and is given by i

∂ u 1 ∂ 2u + |u|2 u = 0. − ∂ξ 2 ∂τ 2

(5.3.1)

Similar to the case of bright solitons, the inverse scattering method can be used [63] to find dark-soliton solutions of Eq. (5.3.1) by imposing the boundary condition that |u(ξ, τ )| tends toward a constant nonzero value for large values of |τ |. Dark solitons can also be obtained by assuming a solution of the form u(ξ, τ ) = V (τ ) exp[iφ(ξ, τ )], and then solving the ordinary differential equations satisfied by V and φ . The main difference compared with the case of bright solitons is that V (τ ) becomes a constant (rather than being zero) as |τ | → ∞. The general solution can be written as [69] 

u(ξ, τ ) = η[B tanh(ζ ) − i 1 − B2 ] exp(iη2 ξ ), √

(5.3.2)

where ζ = ηB(τ − τs − ηB 1 − B2 ). The parameters η and τs represent the background amplitude and the dip location, respectively. Similar to the bright-soliton case, τs can be chosen to be zero without loss of generality. In contrast with the bright-soliton case, the dark soliton has a new parameter B. Physically, B governs the depth of the dip (|B| ≤ 1). For |B| = 1, the intensity at the dip center falls to zero. For other values of B, the dip

Optical solitons

Figure 5.12 Intensity and phase profiles of dark solitons for several values of the blackness parameter B.

does not go to zero. Dark solitons for which |B| < 1 are called gray solitons to emphasize this feature. The |B| = 1 case corresponds to a black soliton. For a given value of η, Eq. (5.3.2) describes a family of dark solitons whose width increases inversely with B. Fig. 5.12 shows the intensity and phase profiles of such dark solitons for three values of B. Whereas the phase of bright solitons [Eq. (5.2.15)] remains constant across the entire pulse, the phase of dark soliton changes with a total phase shift of 2 sin−1 B, i.e., dark solitons are chirped. For the black soliton (|B| = 1), the chirp is such that the phase changes abruptly by π in the center. This phase change becomes more gradual and smaller for smaller values of |B|. The time-dependent phase or frequency chirp of dark solitons represents a major difference between the bright and dark solitons. One consequence of this difference is that higher-order dark solitons neither form a bound state nor follow a periodic evolution pattern discussed in Section 5.2.3 in the case of bright solitons. Dark solitons exhibit several interesting features [69]. Consider a black soliton whose canonical form, obtained from Eq. (5.3.2) after choosing η = 1 and B = 1, is given by u(ξ, τ ) = tanh(τ ) exp(iξ ),

(5.3.3)

where the phase jump of π at τ = 0 is included in the amplitude part. Thus, an input pulse with “tanh” amplitude, exhibiting an intensity “hole” at the center, would propagate unchanged in the normal-GVD region of optical fibers. One may ask, in analogy with the case of bright solitons, what happens when the input power exceeds the N = 1 limit. This question can be answered by solving Eq. (5.3.1) numerically with an input of the form u(0, τ ) = N tanh(τ ). Fig. 5.13 shows the evolution pattern for N = 3; it should be compared with Fig. 5.7 where the evolution of a third-order bright soliton is

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Figure 5.13 Evolution of a third-order dark soliton showing narrowing of the central dip and creation of two pairs of gray solitons.

shown. Two pairs of gray solitons appear and move away from the central black soliton as the propagation distance increases. At the same time, the width of the black soliton decreases [68]. This behavior can be understood by noting that an input pulse of the form N tanh(τ ) can form a fundamental black soliton of amplitude N tanh(N τ ) provided its width decreases by a factor of N. It sheds part of its energy in the process that appears in the form of gray solitons. These gray solitons move away from the central black soliton because of their different group velocities. The number of pairs of gray solitons is N  − 1, where N  is the next integer closest to N when N is not an integer. The important feature is that a fundamental dark soliton always forms for N > 1. Experimental realization of dark solitons is possible only with a finite background instead of the infinite background associated with ideal dark solitons. In practice, a pulse with a narrow dip at its center is used to excite a dark soliton. Numerical calculations show that dark solitons with a finite background pulse exhibit propagation properties nearly identical to those with infinite background if the background pulse is wider by a factor of 10 or more compared with the soliton width [67]. Several techniques have been used to generate optical pulses with a narrow dip in the center [64–66]. Such pulses have been used for observing dark solitons in optical fibers. In one experiment [64], 26-ps input pulses (at 595 nm) with a 5-ps-wide central hole were launched along a 52-m fiber. In another experiment [65], the input to a 10-m fiber was a relatively wide 100-ps pulse (at 532 nm) with a 0.3-ps-wide hole that served as a dark pulse. However, as the phase was relatively constant over the hole width, such even-symmetry input pulses did not have the chirp appropriate for a dark soliton. Nonetheless, output pulses exhibited features that were in agreement with the predictions of Eq. (5.3.1). The odd-symmetry input pulses appropriate for launching a dark soliton were used in a 1988 experiment [66]. A spatial mask, in combination with a grating pair, was used to modify the pulse spectrum such that the input pulse had a phase profile appropriate for forming the dark soliton represented by Eq. (5.3.3). The input pulses obtained from a 620-nm dye laser were ≈2-ps wide with a 185-fs hole in their center. The central hole widened at low powers but narrowed down to its original width when the peak power

Optical solitons

was high enough to sustain a dark soliton of that width. The experimental results agreed with the theoretical predictions of Eq. (5.3.1) quite well. In this experiment, the optical fiber was only 1.2 m long. In a 1993 experiment [70], 5.3-ps dark solitons, formed on a 36-ps wide pulse obtained from a 850-nm Ti:sapphire laser, were propagated over 1 km of fiber. The same technique was later extended to transmit dark-soliton pulse trains at a repetition rate of up to 60 GHz over 2 km of fiber. These results show that dark solitons can be generated and maintained over considerable fiber lengths. During the 1990s, several practical techniques were introduced for generating dark solitons. In one method, a Mach–Zehnder modulator, driven by nearly rectangular electrical pulses, modulates the CW output of a semiconductor laser [71]. In an extension of this method, electric modulation is performed in one of the two arms of a Mach–Zehnder interferometer. A simple all-optical technique consists of propagating two optical pulses, with a relative time delay between them, in the normal-GVD region of the fiber [72]. The two pulses broaden, become chirped, and acquire a nearly rectangular shape as they propagate inside the fiber. As these chirped pulses merge into each other, they interfere. The result at the fiber output is a train of isolated dark solitons. In another all-optical technique, nonlinear conversion of a beat signal in a dispersiondecreasing fiber is used to generate a train of dark solitons [73]. The technique is similar to that discussed in Section 5.1 for generating a regular pulse train except that fiber GVD is chosen to be in the normal-GVD region everywhere along the fiber length. A 100-GHz train of 1.6-ps dark solitons was generated by using this technique and propagated over 2.2 km (two soliton periods) of a dispersion-shifted fiber. Optical switching using a fiber-loop mirror, in which a phase modulator is placed asymmetrically, can also be used to generate dark solitons [74]. In another variation, a fiber with comblike dispersion profile was used to generate dark soliton pulses with a width of 3.8 ps at the 48-GHz repetition rate [75]. An interesting scheme uses electronic circuitry to generate a coded train of dark solitons directly from the NRZ data in electric form [76]. First, the NRZ data and its clock at the bit rate are passed through an AND gate. The resulting signal is then sent to a flip-flop circuit in which all rising slopes flip the signal. The resulting electrical signal drives a Mach–Zehnder LiNbO3 modulator and converts the CW output from a semiconductor laser into a coded train of dark solitons. This technique was used for data transmission, and a 10-Gb/s signal was transmitted over 1200 km by using dark solitons [77]. Another relatively simple method uses spectral filtering of a mode-locked pulse train by using a fiber grating [78]. This scheme has also been used to generate a 6.1-GHz train and propagate it over a 7-km-long fiber [79]. Dark solitons remain a subject of continuing interest [41]. Numerical simulations show that they are more stable in the presence of noise and spread more slowly in the presence of fiber loss compared with bright solitons. However, similar to the case of bright solitons, when perturbed, they emit radiation in the form of dispersive waves [80].

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5.3.2 Bistable solitons The discussion in this chapter is based on the Kerr nonlinearity for which the refractive index varies with the intensity I as n˜ (I ) = n + n2 I. The nonlinear response of any material begins to saturate at high intensity levels, and it becomes necessary to modify the form of n(I ). For silica fibers, saturation of the Kerr nonlinearity occurs at very high intensity levels. However, if fibers are made with other materials (such as chalcogenide glasses), or if a silica fiber is doped with nonlinear materials such as an organic dye, the nonlinear response can saturate at practical intensity levels. In that case, the refractive index changes with intensity as n˜ (I ) = n + n2 f (I ),

(5.3.4)

where f (I ) is some known function of the mode intensity I. The NLS equation (5.2.5) can be generalized to accommodate Eq. (5.3.4) and takes the form [81] i

∂ u 1 ∂ 2u + f (|u|2 )u = 0. + ∂ξ 2 ∂τ 2

(5.3.5)

This equation is not generally integrable by the inverse scattering method. However, it can be solved to find shape-preserving solutions by the method outlined in Section 5.2. The approach consists of assuming a solution of the form u(ξ, τ ) = V (τ ) exp(iK ξ ), where K is a constant and V is independent of ξ . Using this form in Eq. (5.3.5), V (τ ) is found to satisfy d2 V = 2V [K − f (V 2 )]. dτ 2

(5.3.6)

This equation can be solved by multiplying it by 2 (dV /dτ ) and integrating over τ . Using the boundary condition V = 0 as |τ | → ∞, we obtain  (dV /dτ ) = 4 2

V

[K − f (V 2 )]V dV .

(5.3.7)

0

The preceding equation can be integrated to yield 

V

2τ = 0



V2

−1/2 [K − f (P )] dP

dV ,

(5.3.8)

0

where P = V 2 . For a given functional form of f (P ), we can determine the soliton shape V (τ ) from Eq. (5.3.8) if K is known. The parameter K is related to the soliton energy

Optical solitons

defined as Es =

∞

−∞ V

Es (K ) =

1 2

2 dτ .



Pm

Using Eq. (5.3.7), Es depends on the wave number K as [81]

[K − F (P )]−1/2 dP ,

F (P ) =

0

1 P



P

f (P ) dP ,

(5.3.9)

0

where F (0) = 0 and Pm is the smallest positive root of F (P ) = K. Depending on the function f (P ), this relation can have more than one solution, each having the same energy Es but different values of K and Pm . Typically, only two solutions correspond to stable solitons. Such solitons are called bistable solitons and have been of interest since their discovery in 1985 [81–86]. For a given amount of pulse energy, bistable solitons propagate in two different stable states and can be made to switch from one state to another [82]. An analytic form of the bistable soliton has also been found for a specific form of the saturable nonlinearity [84]. Bistable behavior has not yet been observed in optical fibers as the peak-power requirements are extremely high. Other nonlinear media with easily saturable nonlinearity may be more suitable for this purpose.

5.3.3 Dispersion-managed solitons The NLS equation (5.2.5) is based on the assumption that the GVD parameter β2 is constant along the fiber. As discussed in Section 3.4, one can employ several fibers with different β2 values to create a periodically varying β2 (z) such that the average GVD in each period is quite low, while the local GVD at every point along the fiber link is relatively large. In practice, just two kinds of fibers with opposite signs of β2 are combined to reduce the average GVD to a small value. Mathematically, Eq. (5.2.5) is replaced with i

∂ u d(ξ ) ∂ 2 u + |u|2 u = 0, + ∂ξ 2 ∂τ 2

(5.3.10)

where d(ξ ) is a periodic function of ξ with the period ξp = Lp /LD . Here Lp is the length of one period and is typically in the range of 50 to 100 km. Eq. (5.3.10) does not appear to be integrable with the inverse scattering method. However, it has been found to have periodic solutions representing a pulse train. These solutions are referred to as dispersion-managed solitons [87–90]. It should be stressed that the term soliton is used loosely in this context because the properties of dispersionmanaged solitons are quite different from those of the bright solitons discussed in Section 5.2. Not only the amplitude and the width of dispersion-managed solitons oscillate in a periodic manner, but their frequency also varies across the pulse, i.e., such solitons are chirped. Pulse shape is also close to being Gaussian, although it contains considerable oscillatory structure in the pulse wings. Even more surprisingly, such solitons can exist even when average GVD along the fiber link is positive. Dispersion-managed solitons were studied during the 1990s in the context of optical communication systems and later found applications in fiber lasers [91–95].

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5.3.4 Optical similaritons Eq. (5.3.10) corresponds to the situation in which the dispersion parameter β2 changes along the fiber length but the nonlinear parameter γ remains constant. In the most general case, all fiber parameters appearing in Eq. (2.3.46) may vary with z, resulting in the so-called inhomogeneous NLS equation, i

∂ A β2 (z) ∂ 2 A ig(z) − + γ (z)|A|2 A = A, 2 ∂z 2 ∂T 2

(5.3.11)

where the loss parameter α has been replaced with −g(z) to allow for distributed amplification. The solutions of this equation have been studied since the year 2000 [96–102]. Some of the solutions describe pulses that evolve in a self-similar fashion and are known as optical similaritons. The situation is similar to that encountered in Section 4.2.5 in the context of pulse propagation in fiber amplifiers, with the difference that those similaritons form only in an asymptotic limit. Similariton solutions of Eq. (5.3.11) do not require the asymptotic limit and thus are exact solutions. It turns out that a self-similar transformation reduces Eq. (5.3.11) to the standard homogeneous NLS equation, provided a specific relation exists among g(z), β(z), and γ (z). Thus, any soliton solution of the NLS equation can be mapped to a similariton using this transformation. More specifically, the transformation has the form [101] A(z, T ) = a(z)U (ζ, χ ) exp[iφ(z, T )],

(5.3.12)

where the similarity variable χ depends on both z and T as χ (z, T ) = [T − Tc (z)]/Tp (z),

(5.3.13)

and the quantities a(z), Tp (z), and Tc (z) represent the amplitude, width, and position of the similariton, respectively. Here, ζ (z) is yet to be determined. The phase in Eq. (5.3.12) is assumed to vary quadratically with time and has the general form φ(z, T ) = C (z)T 2 /(2T02 ) + b(z)T + d(z).

(5.3.14)

This phase dependence corresponds to a pulse that is linearly chirped with the zdependent chirp parameter C (z). The linear term with b(z) allows for a frequency shift. The input pulse width T0 has been introduced to make the chirp parameter C (z) dimensionless. The use of Eqs. (5.3.12) and (5.3.14) into Eq. (5.3.11) results in a set of differential equations for the parameters describing the evolution of the pulse. These equations can

Optical solitons

be solved if the pulse and fiber parameters satisfy the compatibility condition [96,101] g(z) = C (z)

β2 (z)

T02





β2 (z) d + ln , dz γ (z)

(5.3.15)

and U (ζ, χ ) obeys the standard NLS equation i

∂U 1 ∂ 2U + |U |2 U = 0. − sgn(β2 ) ∂ζ 2 ∂χ 2

(5.3.16)

As long as the compatibility condition (5.3.15) is satisfied, the effective propagation distance and the amplitude, width, and position of the pulse are given by ζ (z) = D(z)[1 − C0 D(z)]−1 ,

Tp (z) = T0 [1 − C0 D(z)],



A(z) = |β2 (z)|/[Tp2 (z)γ (z)],

(5.3.17)

Tc (z) = Tc0 − (C0 Tc0 + b0 )D(z),

(5.3.18)



where the dimensionless parameter D(z) = T0−2 0z β2 (z) dz represents the total dispersion accumulated over the distance z. The parameters related to the phase evolve with z as C (z) =

C0 , 1 − C0 D(z)

b(z) =

b0 , 1 − C0 D(z)

d(z) =

(b20 /2)D(z) , 1 − C0 D(z)

(5.3.19)

where C0 is the input chirp parameter. Eqs. (5.3.16)–(5.3.19) show that any soliton of the standards NLS equation can be mapped into to a similariton obeying the inhomogeneous NLS equation with the compatibility condition given in Eq. (5.3.15). It follows from the integrability of the NLS equation [39] that all such similaritons must be stable, even though their width, amplitude, and chirp change continuously as the pulse propagates down the fiber. As a simple example, consider a fiber with z-dependent gain but constant values of β2 and γ . This case was first studied in 1996 [103], and it leads to a simple compatibility condition g(z) = β2 C (z)/T02 . If the initial chirp C0 = 0, the fiber should exhibit no gain or loss to satisfy this condition. Since all pulse parameters then become independent of z, we recover the standard solitons. On the other hand, if C0 = 0, the compatibility condition is satisfied if the gain (or loss) of the fiber varies with z as [103] g(z) = (C0 β2 /T02 )[1 − C0 D(z)]−1 ,

(5.3.20)

where D(z) = (β2 /T02 )z varies linearly with z. The gain is needed when C0 β2 > 0. Thus, a fiber amplifier whose gain increases along the fiber length, as indicated in Eq. (5.3.20), supports bright similaritons when β2 < 0, and dark similaritons when β2 > 0, with the opposite types of chirps. In both cases, the temporal width decreases

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Figure 5.14 Evolution of a bright similariton over 8 dispersion lengths in the anomalous-GVD region of a lossy fiber for when the input pulse has an initial chirp with (A) C0 = 0.1 and (B) C0 = 0.2.

with z as indicated in Eq. (5.3.18). It is important to stress that the limit C0 D(z) → 1 cannot be attained in practice not only because it requires infinite gain but also because the higher-order effects not included in Eq. (5.3.11) would stop the compression process well before this limit is reached. The opposite case of C0 β2 < 0 in Eq. (5.3.20) is even more interesting because it shows that similaritons can form even in a lossy fiber (g < 0). However, fiber’s loss must decrease with z. Fig. 5.14 shows the evolution of a bright similariton over eight dispersion lengths for two values of the initial chirp C0 , after choosing b0 = 0 and Tc0 = 0. As the pulse is propagating as a similariton in the anomalous-GVD region of a lossy fiber, its width and chirp increase continuously. It is remarkable that stable similaritons can form in a lossy, constant-dispersion fiber. In practice, fiber losses can be made z-dependent by pumping the fiber suitably so that the gain compensates for a part of the total loss. Eq. (5.3.11) applies to fiber lasers, which employ doped fibers to realize amplification and implement dispersion management to control the average GVD inside the laser cavity. Indeed, similariton-like evolution of pulses inside the doped fiber have been seen in such lasers [104–109]. As early as 2011, mode-locked pulses shorter than 100 fs were generated from a Yb-doped fiber lasers employing dispersion management [106]. In a 2015 experiment, an erbium-doped fiber laser produced 89-fs pulses with 2.9-nJ energy when a fiber grating was used inside the laser cavity [109].

Optical solitons

5.4. Perturbation of solitons As we saw in Section 5.2, a soliton can form inside optical fibers provided the input pulse has the right shape and the peak power such that the condition N = 1 is satisfied in Eq. (5.2.3). In practice, pulses do not propagate as an ideal soliton because they are perturbed by a variety of effects that are not included in the standard NLS equation (5.2.2). The impact of perturbations on a soliton can be studied using a suitable perturbation technique. In this section, we first discuss the technique and then apply it to perturbations resulting from fiber losses, periodic amplification, amplifier noise, and soliton interaction.

5.4.1 Perturbation methods Let us write the perturbed NLS equation in the form i

∂ u 1 ∂ 2u + |u|2 u = i(u), + ∂ξ 2 ∂τ 2

(5.4.1)

where (u) is a small perturbation that can depend on u, u∗ , and their derivatives. In the absence of perturbation ( = 0), the soliton solution of the NLS equation is known and is given by Eq. (5.2.13). The question then becomes what happens to the soliton when  = 0. Several perturbation techniques have been developed for answering this question [110–117]. They all assume that the functional form of the soliton remains intact in the presence of a small perturbation but its four parameters change with ξ as the soliton propagates down the fiber, i.e., a solution of the perturbed NLS equation can be written as u(ξ, τ ) = η(ξ )sech[η(ξ )(τ − q(ξ ))] exp[iφ(ξ ) − iδ(ξ )τ ].

(5.4.2)

The ξ dependence of η, δ , q, and φ is yet to be determined. We apply the variational method discussed in Section 4.3.2 to study the impact of a small perturbation on solitons [112]. In this approach, Eq. (5.4.1) is obtained from the Euler–Lagrange equation using the Lagrangian density [118] Ld =

i  ∗ du du∗  1  4  du 2  u |u| −   + i( ∗ u −  u∗ ). −u + 2 dξ dξ 2 dτ

(5.4.3)

This Lagrangian density has the same form as Eq. (4.3.12), except for the last two terms resulting from the perturbation. As in Section 4.3.2, we integrate the Lagrangian density over τ and then use the reduced Euler–Lagrange equation to determine how the four soliton parameters evolve with ξ . After considerable algebra, this procedure leads to the following set of four

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ordinary differential equations [40]: dη dξ dδ dξ dq dξ dφ dξ

 = Re

∞

(u)u∗ (τ ) dτ,

(5.4.4)

−∞  ∞

(u) tanh[η(τ − q)]u∗ (τ ) dτ, −∞  ∞ 1 (u)(τ − q)u∗ (τ ) dτ, = −δ + Re η −∞  ∞ (u){1/η − (τ − q) tanh[η(τ − q)]}u∗ (τ ) dτ = Im = −Im

(5.4.5) (5.4.6)

−∞

+

1 2 2 (η

− δ 2 ) + q(dδ/dξ ),

(5.4.7)

where Re and Im stand for the real and imaginary parts, respectively. This set of four equations can also be obtained by using adiabatic perturbation theory or perturbation theory based on the inverse scattering method.

5.4.2 Fiber loss Because solitons result from a balance between the nonlinear and dispersive effects, the pulse must maintain its peak power if it has to preserve its soliton character. Fiber’s loss is detrimental simply because it reduces the peak power of a soliton along the fiber’s length. The question is whether solitons can survive this power reduction [119–125]. Mathematically, the loss is accounted for by adding a loss term to Eq. (5.1.1) so that it takes the form of Eq. (2.3.46). In terms of the soliton units used in Section 5.2, the modified NLS equation becomes i

∂ u 1 ∂ 2u i + |u|2 u = −  u, + 2 ∂ξ 2 ∂τ 2

(5.4.8)

where  = α LD . Comparing it with Eq. (5.4.1), the perturbation is given by (u) = − u/2. Using it in Eqs. (5.4.4) through (5.4.7) and performing the integrations, we find that only the amplitude η and the phase φ of a soliton are affected by fiber’s loss. They are found to vary along the fiber’s length as [119] η(ξ ) = exp(−ξ ),

φ(ξ ) = φ(0) + [1 − exp(−2ξ )]/(4),

(5.4.9)

where we assumed that η(0) = 1, δ(0) = 0, and q(0) = 0 at ξ = 0. Both δ and q remain zero along the fiber. Recalling that the amplitude and width of a soliton are related inversely, a decrease in soliton amplitude leads to broadening of the soliton. Indeed, if we write η(τ − q) in Eq. (5.4.2) as T /T1 and use τ = T /T0 , T1 increases along the fiber as T1 (z) = T0 exp(ξ ) ≡ T0 exp(α z).

(5.4.10)

Optical solitons

Figure 5.15 Variation of pulse width with distance in a lossy fiber for the fundamental soliton. The prediction of perturbation theory is also shown. Dashed curve shows the behavior expected in the absence of nonlinear effects. (After Ref. [126]; ©1985 Elsevier.)

An exponential increase in the soliton width with z cannot be expected to continue for arbitrarily large distances. Numerical solutions of Eq. (5.4.8) show that the perturbative solution is accurate only for values of z such that α z  1 [126]. Fig. 5.15 shows the broadening factor T1 /T0 as a function of ξ when a fundamental soliton is launched into a fiber with  = 0.07. The perturbative result is acceptable for up to ξ ≈ 1. In the region (ξ 1), pulse width increases linearly with a rate slower than that of a linear medium [127]. Higher-order solitons show a qualitatively similar asymptotic behavior. However, their pulse width oscillates a few times before increasing monotonically [126]. The origin of such oscillations lies in the periodic evolution of higher-order solitons. How can a soliton survive inside lossy optical fibers? An interesting scheme restores the balance between the GVD and SPM in a lossy fiber [128] by using a dispersiondecreasing fiber (DDF). Its GVD must decrease in such a way that it compensates for the reduced SPM experienced by the soliton, as its peak power is reduced inside the fiber. To see which GVD profile is needed, we modify Eq. (5.4.8) to allow for GVD variations along the fiber length and eliminate the loss term using u = v exp(−ξ/2), resulting in the following equation: i

∂ v d(ξ ) ∂ 2 v + e−ξ |v|2 v = 0, + ∂ξ 2 ∂τ 2

(5.4.11)

where d(ξ ) = |β2 (ξ )/β2 (0)| is the normalized local GVD. The distance ξ is normalized to the dispersion length LD = T02 /|β2 (0)|, defined using the GVD value at the input end of the fiber.  If we rescale ξ using the transformation ξ  = 0ξ d(ξ ) dξ , Eq. (5.4.11) becomes i

∂v 1 ∂ 2 v e−ξ 2 + + |v| v = 0, ∂ξ  2 ∂τ 2 d(ξ )

(5.4.12)

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Nonlinear Fiber Optics

If the GVD profile is chosen such that d(ξ ) = e−ξ , Eq. (5.4.12) reduces to the standard NLS equation. Thus, fiber losses have no effect on soliton propagation if the GVD of a fiber decreases exponentially along its length as |β2 (z)| = |β2 (0)| exp(−α z).

(5.4.13)

This result can be easily understood from Eq. (5.2.3). If the soliton peak power P0 decreases exponentially with z, the requirement N = 1 can still be maintained at every point along the fiber if |β2 | were also to reduce exponentially. Fibers with a nearly exponential GVD profile have been fabricated [129]. A practical technique for making DDFs consists of reducing the core diameter along fiber length in a controlled manner during the fiber-drawing process. Variations in the core diameter change the waveguide contribution to β2 and reduce its magnitude. Typically, GVD can be changed by a factor of 10 over a length of 20–40 km. The accuracy realized by the use of this technique is estimated to be better than 0.1 ps2 /km [130]. Among other applications, DDFs can produce parabolic-shape pulses when the input pulse propagates in the normal-GVD region and β2 decreases with z as (1 + az)−1 , where a is a constant [131].

5.4.3 Soliton amplification As already discussed, fiber losses lead to broadening of solitons. Such loss-induced broadening is unacceptable for many applications. One approach to overcoming the impact of fiber losses amplifies the pulse propagating as a soliton periodically so that its energy is restored to its initial value. Two techniques have been used for this purpose [119–122]. They are known as lumped and distributed amplification schemes and are shown in Fig. 5.16 schematically. In the lumped scheme [120], an optical amplifier boosts soliton’s energy to its input level after it has propagated a certain distance. The soliton then readjusts its parameters to their input values. However, it also sheds a part of its energy as dispersive waves during this adjustment phase. The dispersive part is undesirable and can accumulate to significant levels over a large number of amplification stages. This problem can be solved by reducing the spacing LA between neighboring amplifiers such that LA  LD . The reason is that the dispersion length LD sets the scale over which a soliton responds to external perturbations. If the amplifier spacing is much smaller than this length scale, soliton width is hardly affected over one amplifier spacing in spite of energy variations. In practice, the condition LA  LD restricts LA typically in the range of 20–40 km when LD is close to 100 km [120]. The distributed-amplification scheme employs stimulated Raman scattering [122–124] to provide gain (see Chapter 8). In this approach, CW power from a pump laser, operating at a suitable wavelength, is injected periodically into the fiber, as shown

Optical solitons

Figure 5.16 (A) Lumped and (B) distributed amplification schemes used for compensating fiber losses. Tx and Rx stand for transmitters and receivers, respectively.

in Fig. 5.16(B). In the 1.55-µm wavelength region, one needs a pump laser operating near 1.45 µm with power levels above 100 mW. As the optical gain is distributed over the entire fiber length, solitons can be amplified adiabatically, while maintaining N close to 1, a feature that reduces the dispersive part almost entirely [123]. The Raman-amplification scheme was first used in 1985 in an experiment in which soliton pulses of 10-ps width were propagated over a 10-km-long fiber [122]. In the absence of Raman gain, the width of solitons increased by 50% because of loss-induced broadening, in agreement with Eq. (5.4.10), which predicts T1 /T0 = 1.51 for z = 10 km and α = 0.18 dB/km. The soliton width did not change much when a CW pump beam at 1.46 µm was injected, and its power was adjusted close to 125 mW such that the total fiber loss of 1.8 dB was exactly balanced by the Raman gain. In a 1988 experiment [124], 55-ps solitons could be circulated up to 96 times inside a 42-km fiber loop without significant increase in their width, resulting in an total distance of >4000 km. The lumped-amplification scheme was used starting in 1989 with the advent of erbium-doped fiber amplifiers [125]. The main shortcoming of lumped amplifiers is that soliton energy can vary by as much as a factor of 100 between two neighboring amplifiers. To understand how solitons can survive in spite of such large energy variations, we include the gain provided by lumped amplifiers in Eq. (5.4.8) by replacing  ˜ ) such that (ξ ˜ ) =  everywhere except at the location of with a periodic function (ξ amplifiers where it changes abruptly. If we make the transformation 

u(ξ, τ ) = exp −

1 2

 0

ξ

 ˜ ) dξ v(ξ, τ ) ≡ a(ξ ) v(ξ, τ ), (ξ

(5.4.14)

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Nonlinear Fiber Optics

where a(ξ ) contains rapid variations and v(ξ, τ ) is a slowly varying function of ξ , and use it in Eq. (5.4.8), v(ξ, τ ) is found to satisfy i

∂ v 1 ∂ 2v + a2 (ξ )|v|2 v = 0. + ∂ξ 2 ∂τ 2

(5.4.15)

In this equation, a(ξ ) is a periodic function of ξ with the period ξA = LA /LD , where LA is the amplifier spacing. In each period, a(ξ ) ≡ a0 exp(−ξ/2) decreases exponentially and jumps to its initial value a0 at the end of the period. The concept of the guiding-center (or path-averaged) soliton [132] makes use of the fact that a2 (ξ ) in Eq. (5.4.15) varies rapidly in a periodic fashion. If the period ξA satisfies the condition ξA  1, solitons evolve little over a distance short compared with the dispersion length LD . As a result, we can replace a2 (ξ ) in Eq. (5.4.15) by its average to obtain i

∂ v 1 ∂ 2v + a2 (ξ )|v|2 v = 0. + ∂ξ 2 ∂τ 2

(5.4.16)

The practical importance of the averaging concept stems from the fact that Eq. (5.4.16) describes soliton propagation quite accurately when ξA  1 [40]. In practice, this approximation works reasonably well for ξA as large as 0.25. Eq. (5.4.16) shows the input peak power Ps of the path-averaged soliton should be chosen such that a2 (ξ ) = 1. Noting that the amplifier gain is G = exp(ξA ), the peak power is given by Ps =

G ln G ξA P0 P0 , = 1 − exp(−ξA ) G − 1

(5.4.17)

where P0 is the required peak power when  = 0. Thus, soliton evolution in lossy fibers with periodic lumped amplification is identical to that in lossless fibers provided: (i) amplifiers are spaced such that LA  LD ; and (ii) the launched peak power is larger by a factor G ln G/(G − 1). As an example, G = 10 and Pin ≈ 2.56 P0 for 50-km amplifier spacing and a fiber loss of 0.2 dB/km. Fig. 5.17 shows pulse evolution in the average-soliton regime over a distance of 10,000 km, assuming solitons are amplified every 50 km. When the soliton width corresponds to a dispersion length of 200 km, the soliton is well preserved even after 200 lumped amplifiers because the condition ξA  1 is reasonably well satisfied. However, if the dispersion length reduces to 25 km, the soliton is destroyed because of relatively large loss-induced perturbations. Optical amplifiers, needed to restore the soliton energy, also add noise originating from spontaneous emission. The effect of spontaneous emission is to change randomly the four soliton parameters, η, δ , q, and φ in Eq. (5.4.2), at the output of each amplifier [114]. Amplitude fluctuations, as one might expect, degrade the signal-to-noise

Optical solitons

Figure 5.17 Evolution of loss-managed solitons over 10,000 km for LD = 200 (left) and LD = 25 km (right) when LA = 50 km, α = 0.22 dB/km, and β2 = 0.5 ps2 /km.

ratio. However, phase fluctuations also affect the soliton by inducing random frequency shifts. As seen from Eq. (5.4.2), a change in the soliton frequency by δ affects the speed at which the soliton propagates through the fiber. If δ fluctuates because of amplifier noise, soliton’s transit time through the fiber also becomes random. Fluctuations in the arrival time of a soliton are referred to as the Gordon–Haus timing jitter [133]. It can be reduced in practice through a variety of techniques [44].

5.4.4 Soliton interaction Most applications of solitons send a pulse train through an optical fiber. It is thus important to determine how close two solitons can come without affecting each other. The nonlinear interaction of two solitons was studied extensively during the 1980s [134–140]. It is clear on physical grounds that two solitons would affect each other only when they are close enough that their tails overlap. Consider two solitons whose peaks are separated in time. To be as general as possible we assume that their amplitudes, phases, and frequencies differ such that uj (ξ, τ ) = ηj sech[ηj (τ − qj )] exp(iφj − iδj τ ),

(5.4.18)

with j = 1 or 2. It is the total field u = u1 + u2 that satisfies the NLS equation, rather than u1 and u2 individually. By using u = u1 + u2 in Eq. (5.2.5), we obtain the following perturbed NLS equation satisfied by each soliton: i

∂ uj 1 ∂ 2 uj + |uj |2 uj = −2|uj |2 u3−j − u2j u∗3−j , + ∂ξ 2 ∂τ 2

(5.4.19)

where j = 1 or 2. The two terms on the right side act as a perturbation and are responsible for nonlinear interaction between two neighboring solitons.

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Eqs. (5.4.4)–(5.4.7) can now be used to study how the four soliton parameters, ηj , qj , δj , and φj , are affected by the perturbation. Introducing new variables as η ± = η1 ± η 2 ,

q± = q1 ± q2 ,

δ± = δ1 ± δ2 ,

φ± = φ1 ± φ2 ,

(5.4.20) (5.4.21)

we obtain, after considerable algebra, the following set of equations [116]: d η+ = 0, dξ dδ+ = 0, dξ dq− = −δ− , dξ

d η− 3 exp(−q− ) sin φ− , = η+ dξ dδ− 3 exp(−q− ) cos φ− , = η+ dξ dφ− 1 = 2 η+ η− . dξ

(5.4.22) (5.4.23) (5.4.24)

Equations for q+ and φ+ are omitted since they do not affect the soliton interaction. Also, η+ and δ+ remain constant during the interaction. Using η+ = 2 for two fundamental solitons, the remaining four equations can be combined to yield: d2 q = −4e−2q cos(2ψ), dξ 2

d2 ψ = 4e−2q sin(2ψ), dξ 2

(5.4.25)

where the notation is simplified using q = q− /2 and ψ = φ− /2. The same equations are obtained with the inverse scattering method [135]. They show that the relative separation q between two solitons depends only on their relative phase difference. Two solitons may attract (come closer) or repel (move apart) each other depending on the initial value of ψ . Eqs. (5.4.25) can be solved analytically [137]. When both solitons initially have the same amplitudes and frequencies, the solution takes the form [40] q(ξ ) = q0 + 12 ln[cosh2 (2ξ e−q0 sin ψ0 ) + cos2 (2ξ e−q0 cos ψ0 ) − 1],

(5.4.26)

where q0 and ψ0 are the initial values of q and ψ , respectively. Fig. 5.18 shows how the relative separation q(ξ ) changes along the fiber length for two solitons with different phases. For ψ0 below a certain value, q becomes zero periodically. This is referred to as a collision resulting from an attractive force between the two solitons. Collisions do not occur for ψ0 > π/8, and q increases with ξ . This is interpreted in terms of a repulsive force between the two solitons. The specific cases ψ0 = 0 and π/2 correspond, respectively, to two solitons that are initially in phase or out of phase. In the case of two in-phase solitons, we set ψ0 = 0 in Eq. (5.4.26) and find that q changes with distance in a periodic fashion as q(ξ ) = q0 + ln | cos(2ξ e−q0 )|.

(5.4.27)

Optical solitons

Figure 5.18 Relative spacing q between two interacting solitons as a function of fiber length for several values of initial phase difference ψ0 (in degrees) when q0 = 4.

Because q(ξ ) ≤ q0 for all values of ξ , two in-phase solitons attract each other. In fact, q becomes zero after a distance 1 2

ξc = eq0 cos−1 (e−q0 ) ≈

π

4

exp(q0 ),

(5.4.28)

where the approximate form is valid for q0 > 5. At this distance, two solitons collide for the first time. After this, they separate from each other, recover their original spacing at a distance 2ξc , and the whole process repeats periodically. The oscillation period is called the collision length. In real units, the collision length is given by Lcol =

π

2

LD exp(q0 ) ≡ z0 exp(q0 ),

(5.4.29)

where z0 is the soliton period given in Eq. (5.2.24). This expression is accurate for q0 > 3, as also found numerically [136]. A more accurate expression, valid for arbitrary values of q0 , is obtained using inverse scattering theory and is given by [139] Lcol π sinh(2q0 ) cosh(q0 ) = . LD 2q0 + sinh(2q0 )

(5.4.30)

In the case of two out-of-phase solitons (ψ0 = π/2), the separation q changes with distance as q(ξ ) = q0 + ln[cosh(2ξ e−q0 )]. As cosh(x) > 1 for all values of x, q increases monotonically with ξ .

(5.4.31)

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Figure 5.19 Evolution of a soliton pair over 90 dispersion lengths showing the effects of soliton interaction for four different choices of amplitude ratio r and relative phase θ . Initial separation q0 = 3.5 in all four cases.

Numerical solutions of the NLS equation are quite instructive and allow exploration of different amplitudes and different phases associated with a soliton pair by using the following form at the input end of the fiber: u(0, τ ) = sech(τ + q0 ) + r sech[r (τ − q0 )]eiθ ,

(5.4.32)

where r is relative amplitude, θ is the initial phase difference, and 2q0 is the initial separation between the two solitons. Fig. 5.19 shows evolution of a soliton pair when q0 = 3.5 for several values of parameters r and θ . In the case of equal-amplitude solitons (r = 1), the two solitons attract each other in the in-phase case (θ = 0) and collide periodically along the fiber length, just as predicted by perturbation theory. For θ = π/4, the solitons separate from each other after an initial attraction stage, in agreement with the results shown in Fig. 5.18. For θ = π/2, the solitons repel each other, and their spacing increases with distance monotonically. The last case shows the effect of slightly different soliton amplitudes by choosing r = 1.1. Two in-phase solitons oscillate periodically but never collide or move far away from each other. The periodic collapse of neighboring solitons is undesirable from a practical standpoint. One way to avoid the collapse is to increase soliton separation such that Lcol LT , where LT is the transmission distance. Because Lcol ≈ 3000 z0 for q0 = 8, and z0 ∼ 100 km typically, a value of q0 = 8 is large enough for any communication system. Several schemes can be used to reduce q0 without inducing the collapse. If the

Optical solitons

two solitons have the same phase (θ = 0) but different amplitudes, the interaction is still periodic but without collapse [139]. Even for r = 1.1, the separation does not change by more than 10% during each period if q0 > 4. Soliton interaction can also be modified by other factors such as bandwidth-limited amplification [141], timing jitter [142], and Raman scattering [143].

5.5. Higher-order effects The properties of optical solitons considered so far are based on the NLS equation (5.1.1). As discussed in Section 2.3, when input pulses are so short that T0 < 1 ps, it is necessary to include several higher-order effects through the generalized NLS equation (2.3.36), or its simpler version in Eq. (2.3.43). Eq. (2.3.36) is used in Chapter 13 in the context of supercontinuum generation. Here, we focus on Eq. (2.3.43). If we use Eq. (3.1.3) to define the normalized amplitude U, this equation takes the form   ∂ U iβ2 ∂ 2 U β3 ∂ 3 U i ∂ ∂|U |2 −α z 2 2 + − = i γ P e U | U + (| U | U ) − T U | . 0 R ∂z 2 ∂T 2 6 ∂T 3 ω0 ∂ T ∂T

(5.5.1)

5.5.1 Moment equations for pulse parameters In general, Eq. (5.5.1) equation should be solved numerically. However, if we assume that higher-order effects are weak enough that the pulse maintains its shape, even though its parameters change, we can employ the moment method of Section 4.3.1 to gain some physical insight. In the anomalous-GVD region, we use the following form for U (z, T ): 







T − qp (T − qp )2 + iφp , U (z, T ) = ap sech exp −ip (T − qp ) − iCp Tp 2Tp2

(5.5.2)

where ap , Tp , Cp , and φp represent the amplitude, width, chirp, and phase of the pulse. We also allow for the temporal shift qp of the pulse envelope and the frequency shift p of the pulse spectrum. All six parameters may change with z as the pulse propagates through the fiber. Using the definitions of moments in Sections 4.3.1 and 4.4.3 and following the moment method, we obtain the following equations for the four main pulse parameters [144,145]: Cp dTp = (β2 + β3 p ) , dz Tp   (β2 + β3 p ) dCp 4 4T0 = + Cp2 + 2 (γ¯ + p /ω0 )P0 , dz π2 Tp2 π Tp

(5.5.3) (5.5.4)

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dqp β3 β3 π2 2 γ¯ P0 T0 = β2 p + 2p + 1 + , C + dz 2 6Tp2 4 p ω0 Tp d p 8TR γ¯ P0 T0 2γ¯ P0 T0 Cp =− + , dz 15 Tp3 3ω0 Tp3

(5.5.5) (5.5.6)

where γ¯ = γ e−αz . As in Section 4.3.1, we have ignored the phase equation. The amplitude ap (z) can be determined from the relation E0 = 2P0 T0 = 2a2p (z)Tp (z), where E0 is the input pulse energy. It is evident from Eqs. (5.5.3) through (5.5.6) that the pulse parameters are affected considerably by the three higher-order terms in Eq. (5.5.1). Before considering their impact, we use these equations to determine the conditions under which a fundamental soliton can form. Eq. (5.5.3) shows that pulse width will not change if the chirp Cp remains zero for all z. The chirp equation (5.5.4) is quite complicated. However, if we neglect the higher-order terms and fiber losses (α = 0), this equation reduces to 



dCp β2 4 4γ P0 T0 = + Cp2 + . dz π2 Tp2 π 2 Tp

(5.5.7)

If β2 > 0, both terms on the right side are positive, and Cp cannot remain zero even if Cp = 0 initially. However, in the case of anomalous GVD (β2 < 0), the two terms cancel precisely when Cp = 0 initially and input parameters satisfy the condition γ P0 T02 = |β2 |. It follows from Eq. (5.2.3) that this condition is equivalent to setting N = 1. It is useful in the following discussion to normalize Eq. (5.5.1) using dimensionless variables ξ and τ defined in Eq. (5.2.1). The normalized NLS equation takes the form i

∂ u 1 ∂ 2u ∂ 3u ∂ ∂|u|2 2 2 + | u | u = i δ − is u | u ) + τ u + (| , 3 R ∂ξ 2 ∂τ 2 ∂τ 3 ∂τ ∂τ

(5.5.8)

where the pulse is assumed to propagate in the region of anomalous GVD (β2 < 0) and fiber losses are neglected (α = 0). The parameters δ3 , s, and τR govern, respectively, the effects of third-order dispersion (TOD), self-steepening, and intrapulse Raman scattering. Their explicit expressions are δ3 =

β3 , 6|β2 |T0

s=

1 , ω0 T0

τR =

TR . T0

(5.5.9)

All three parameters vary inversely with pulse width and are negligible for T0 1 ps. They become appreciable for femtosecond pulses. As an example, if we take TR = 3 fs and consider a 50-fs pulse (T0 ≈ 30 fs) at wavelengths near 1.55 µm, we find that δ3 ≈ 0.03, s ≈ 0.03, and τR ≈ 0.1 for a silica fiber with β3 = 0.01 ps3 /km and β2 = −20 ps2 /km.

Optical solitons

5.5.2 Third-order dispersion When optical pulses propagate relatively far from the zero-dispersion wavelength of an optical fiber, the TOD effects on the solitons are small and can be treated perturbatively. To study such effects as simply as possible, let us set s = 0 and τR = 0 in Eq. (5.5.8) and treat the δ3 term as a small perturbation. It follows from Eqs. (5.5.3) through (5.5.6) that p = 0 under such conditions. Also, Cp = 0 and Tp = T0 if the pulse forms a N = 1 soliton. However, the temporal position of the soliton changes linearly with z as qp (z) = (β3 /6T02 )z ≡ δ3 (z/LD ).

(5.5.10)

Thus the main effect of TOD is to shift the soliton peak from its original position. Whether the peak is delayed or advanced depends on the sign of β3 . When β3 > 0, the TOD slows down the soliton, and the soliton peak is delayed by an amount that increases linearly with distance. This TOD-induced delay is negligible in practice for picosecond pulses. If we use a typical value of β3 = 0.1 ps3 /km, the temporal shift is only 17 fs for T0 = 10 ps even after a distance of 100 km. However, the shift becomes relatively large for femtosecond pulses. For example, the peak shifts at a rate of 1.7 ps/km for T0 = 100 fs. What happens if an optical pulse propagates at or near the zero-dispersion wavelength of an optical fiber such that β2 is nearly zero. Considerable work has been done to understand the evolution features in this regime [146–154]. The case β2 = 0 has been discussed in Section 4.2.5 for Gaussian pulses by solving Eq. (4.2.7) numerically. We can use the same equation for a soliton by using U (0, τ ) = sech(τ ) as the input at z = 0. Fig. 5.20 shows the (A) temporal and (B) spectral evolution of a “sech” pulse for z/LD in the range of 0 to 4 for a peak power such that N˜ = 2. The most striking feature of Fig. 5.20 is the splitting of the spectrum into two wellresolved spectral peaks [146]. These peaks correspond to the outermost peaks of the SPM-broadened spectrum (see Fig. 4.2). As the red-shifted peak lies in the anomalousGVD region, pulse energy in that spectral band can form a soliton. The energy in the other spectral band disperses away simply because that part of the pulse experiences normal GVD. It is the trailing part of the pulse that disperses away with propagation because SPM generates blue-shifted components near the trailing edge. The pulse shapes in Fig. 5.20 show a long trailing edge with oscillations that continues to separate from the leading part with increasing ξ  . The important point to note is that, because of SPM-induced spectral broadening, the input pulse does not really propagate at the zero-dispersion wavelength even if β2 = 0 initially. In effect, the pulse creates its own |β2 | through SPM. The effective value of |β2 | is given by Eq. (4.2.9) and is larger for pulses with higher peak powers. An interesting question is whether soliton-like solutions exist near the zerodispersion wavelength of an optical fiber. In the case of β2 = 0, numerical solutions

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Figure 5.20 (A) Temporal and (B) spectral evolutions of a “sech” pulse propagating at the zerodispersion wavelength with a peak power such that N˜ = 2.

show [148] that a sech-shape pulse evolves over a length ξ  ∼ 10/N˜ 2 into a soliton that contains about half of the pulse energy. The remaining energy is carried by an oscillatory structure near the trailing edge that disperses away with propagation. Similar features are found by solving Eq. (5.5.8) with s = τR = 0 [148–151]. The dispersing oscillatory structure, similar to that seen in Fig. 5.20, represents the radiation shed by the perturbed soliton; it is often referred to as a dispersive wave. The generation of dispersive waves will be discussed in detail in Section 12.1.2. The effect of TOD on the second and higher-order solitons is even more remarkable as it leads to breakup of such solitons into multiple fundamental solitons of different widths, a phenomenon referred to as soliton fission [148]. It can be understood from the inverse scattering method of Section 5.2. In the absence of any perturbation, a Nhorder soliton represents a bound state of N first-order solitons of different widths, all propagating at the same speed (the eigenvalues ζj in Section 5.2.1 have the same real part). The effect of TOD is to break this degeneracy so that individual solitons propagate at different speeds. As a result, they separate from each other, and the separation increases linearly with the distance [155]. Soliton fission can also be initiated by other perturbations such as higher-order nonlinear effects discussed later in this section. More details on soliton fission can be found in Section 12.1. With the advent of microstructured fibers (see Chapter 11), fibers can be made such that β3 is nearly zero over a certain wavelength range, while |β2 | remains finite. Such fibers are called dispersion-flattened fibers. Their use requires consideration of the effects of fourth-order dispersion on solitons. The NLS equation in this case takes the

Optical solitons

form i

∂ u 1 ∂ 2u ∂ 4u 2 + | u | u = −δ , + 4 ∂ξ 2 ∂τ 2 ∂τ 4

(5.5.11)

where δ4 = β4 /(24|β2 |T02 ). This parameter is relatively small for T0 > 1 ps, and its effect can be treated perturbatively. However, δ4 may become large enough for shorter pulses that a perturbative solution is not appropriate. A solitary-wave solution of Eq. (5.5.11) can be found by assuming u(ξ, τ ) = V (τ ) exp(iK ξ ) and solving the resulting ordinary differential equation for V (τ ). This solution is given by [156] u(ξ, τ ) = 3b2 sech2 (bτ ) exp(8ib2 ξ/5),

(5.5.12)

where b = (40δ4 )−1/2 . Note the sech2 -type form of the pulse amplitude rather than the usual “sech” form required for standard bright solitons. It should be stressed that both the amplitude and the width of the soliton are determined uniquely by the fiber parameters. Such fixed-parameter solitons are sometimes called autosolitons.

5.5.3 Self-steepening The phenomenon of self-steepening has already been discussed in Section 4.4.1. It introduces several new features for solitons forming in the anomalous-GVD region [157–160]. The effects of self-steepening appear in Eqs. (5.5.3) to (5.5.6) through the terms containing ω0 . The most important feature is that self-steepening can produce spectral and temporal shifts of the soliton even when TR = 0. In fact, we can integrate Eq. (5.5.6) to obtain the spectral shift in the form γ E0 p (z) = 3ω0

 0

z

Cp (z) −αz e dz, Tp3 (z)

(5.5.13)

where E0 = 2P0 T0 is the input pulse energy. As soon as the pulse becomes chirped, its spectrum shifts because of self-steepening. If the chirp is negligible, spectral shift is relatively small. Even when p = 0, self-steepening produces a temporal shift because of the last term in Eq. (5.5.5). If we assume that soliton maintains its width to the first order, the temporal shift of the soliton peak for a fiber of length L given by qp (L ) = γ P0 Leff /ω0 = φmax /ω0 ,

(5.5.14)

where Leff is the effective length and φmax is the maximum SPM-induced phase shift introduced in Section 4.1.1. Noting that ω0 = 2π/Topt , where Topt is the optical period, the shift is only 5Topt for φmax = 10π . However, when p = 0, the temporal shift is enhanced considerably.

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Figure 5.21 Pulse shapes at z/LD = 5 and 10 for a fundamental soliton in the presence of selfsteepening (s = 0.2). The dotted curve shows the initial pulse shape at z = 0 for comparison. The solid and dashed curves coincide with the dotted curve when s = 0.

The preceding analysis is valid for relatively small values of the parameter s. When pulses are so short that s exceeds 0.1, one must use a numerical approach. To isolate the effects of self-steepening governed by the parameter s, it is useful to set δ3 = 0 and τR = 0 in Eq. (5.5.8). Pulse evolution inside fibers is then governed by i

∂ u 1 ∂ 2u ∂ + |u|2 u + is (|u|2 u) = 0. + ∂ξ 2 ∂τ 2 ∂τ

(5.5.15)

The self-steepening-induced temporal shift is shown in Fig. 5.21, where pulse shapes at ξ = 0, 5, and 10 are plotted for s = 0.2 and N = 1 by solving Eq. (5.5.15) numerically with the input u(0, τ ) = sech(τ ). As the peak moves slower than the wings for s = 0, it is delayed and appears shifted toward the trailing side. Although the pulse broadens slightly with propagation (by about 20% at ξ = 10), it nonetheless maintains its soliton nature. This feature suggests that Eq. (5.5.15) has a solitary-wave solution toward which the input pulse is evolving asymptotically. Such a solution indeed exists and has the form [118] u(ξ, τ ) = V (τ + M ξ ) exp[i(K ξ − M τ )],

(5.5.16)

where M is related to the shift p of the carrier frequency. The group velocity changes as a result of this spectral shift, and the delay of the peak seen in Fig. 5.21 is related to this change in the group velocity. The explicit form of V (τ ) depends on M and s [160]. In the limit s = 0, it reduces to the ‘sech’ form given in Eq. (5.2.16). Note also that Eq. (5.5.15) can be transformed into a so-called derivative NLS equation that is

Optical solitons

Figure 5.22 (A) Temporal and (B) spectral evolutions of a second-order soliton (N = 2) over five dispersion lengths for s = 0.2.

integrable by the inverse scattering method and whose solutions have been studied in plasma physics [161]. Self-steepening also leads to breakup of high-order solitons through the phenomenon of soliton fission if the parameter s is relatively large [158]. Fig. 5.22 shows this behavior for a second-order soliton (N = 2) by displaying the temporal and spectral evolutions for s = 0.2. For this value of s, the two solitons separate from each other within a distance of 2LD and continue to move apart with further propagation inside the fiber. The ratio of the peak heights in Fig. 5.22 is about 9 and is in agreement with the expected ratio (η2 /η1 )2 , where η1 and η2 are the imaginary parts of the eigenvalues introduced in Section 5.2.1. The third-order soliton (N = 3) decays into three solitons whose peak heights are again in agreement with inverse scattering theory.

5.5.4 Intrapulse Raman scattering Intrapulse Raman scattering plays a very important role among all higher-order nonlinear effects. Its effects on solitons are governed by the last term in Eq. (5.5.8) and were observed as early as 1985 in one experiment [162]. The need to include this term became apparent when a phenomenon, called the soliton self-frequency shift, was observed [163] and explained using the delayed nature of the Raman response [164]. It was studied extensively soon after [165–176]. We discuss this nonlinear effect fully in Section 12.2. Here we consider its main features using Eq. (5.5.8) and the moment method of Section 5.5.1.

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Let us focus first on a fundamental soliton and consider the prediction of the moment method. We isolate the effects of intrapulse Raman scattering by using β3 = 0 and let ω0 → ∞ in Eqs. (5.5.3) through (5.5.6). The main effect of the Raman term is to shift the soliton frequency p along the fiber length through Eq. (5.5.6). Integrating this equation, we obtain [144] 8TR p (z) = − γ P0 T0 15

 0

z

e−αz dz. Tp3 (z)

(5.5.17)

The evolution of the pulse width Tp (z) along the fiber is governed by Eq. (5.5.3). If fiber length is short enough that losses are negligible, and p remains small enough that the p terms in Eq. (5.5.4) remain negligible, the soliton can maintain Cp ≈ 0 with its width fixed at its input value T0 . Only under such restrictive conditions, we can replace the integral in Eq. (5.5.17) with z/T03 (assuming negligible losses) and find that the Raman-induced frequency shift (RIFS) grows linearly with distance as p (z) = −

8TR γ P0 8TR |β2 | z≡− z, 2 15T0 15T04

(5.5.18)

where we used the soliton condition N = γ P0 T02 /|β2 | = 1. The negative sign shows that the carrier frequency is reduced, i.e., the soliton spectrum shifts toward longer wavelengths (the “red” side). The T0−4 scaling of p with the width T0 was first found in 1986 using soliton perturbation theory [164]. It shows why the RIFS becomes important for only short pulses (T0 ∼1 ps or less). However, one should keep in mind that such a dependence holds only over relatively short distances over which the soliton remains unchirped. Physically, the red shift of fundamental solitons can be understood in terms of Raman scattering (see Chapter 8) as follows. For pulse widths ∼1 ps or shorter, the spectral width of the pulse is large enough that the Raman gain can amplify the low-frequency components of a pulse, with the high-frequency components of the same pulse acting as a pump. This process continues along the fiber, and energy from the blue-side of the pulse spectrum is continuously transferred to its red side. Such an energy transfer appears as a red shift of the soliton spectrum, with shift increasing with distance. As seen from Eq. (5.5.18), the frequency shift increases linearly along the fiber. However, as it scales with the pulse width as T0−4 , it can become quite large for short pulses. As an example, soliton frequency shifts at a rate of 51 GHz/m for T0 = 0.1 ps (FWHM about 175 fs) if we use β2 = −20 ps2 /km and TR = 3 fs in Eq. (5.5.18). The spectrum of such pulses will shift by 5.1 THz after only 100 m of propagation. This is a large shift if we note that the initial spectral width (FWHM) of such a soliton is 0. For Cp < 0, pulse begins to broaden immediately, and RIFS is reduced considerably. The RIFS of solitons was observed in 1986 using 0.5-ps pulses obtained at a wavelength near 1550 nm [163]. The pulse spectrum was found to shift as much as 8 THz over a 400-m-long fiber. This shift was called the soliton self-frequency shift because it was induced by the soliton itself [164]. However, as discussed in Section 4.4.3, RIFS is a general phenomenon and occurs for all short pulses, irrespective of whether they propagate as a soliton or not [144]. The shift is relatively large if a pulse maintains its

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Figure 5.24 (A) Temporal and (B) spectral evolutions of a second-order (N = 2) soliton when τR = 0.01, depicting soliton’s fission induced by intrapulse Raman scattering.

width along the fiber. In recent years, RIFS has attracted considerable attention for producing femtosecond pulses whose wavelength can be tuned over a wide range by simply propagating them through a suitable optical fiber [177–183]. This scheme is discussed in Section 12.2. In the case of higher-order solitons, we can solve the generalized NLS equation (5.5.8) numerically. To focus on the effects of intrapulse Raman scattering, we set δ3 = 0 and s = 0 in this equation. Pulse evolution inside fibers is then governed by i

∂ u 1 ∂ 2u ∂|u|2 2 + | u | u = τ u + . R ∂ξ 2 ∂τ 2 ∂τ

(5.5.19)

Fig. 5.24 shows the spectral and temporal evolution of a second-order soliton (N = 2) by solving Eq. (5.5.19) numerically with τR = 0.01. The effect of intrapulse Raman scattering on higher-order solitons is similar to the case of self-steepening. In particular, even relatively small values of τR lead to the fission of higher-order solitons into its constituents [170]. A comparison of Figs. 5.22 and 5.24 shows the similarity and the differences for two different higher-order nonlinear mechanisms. An important difference is that relatively smaller values of τR compared with s can induce soliton fission over a given distance. For example, if s = 0.01 is chosen in Fig. 5.22, the soliton does not split over the distance of 5LD . This feature indicates that the effects of τR are likely to dominate in practice over those of self-steepening. Another difference is that both solitons are delayed in the

Optical solitons

case of self-steepening, while in the Raman case the low-intensity soliton appears not to have shifted in both the spectral and temporal domains. This feature is related to the T0−4 dependence of the RIFS on the soliton width in Eq. (5.5.18). The second soliton is much wider than the first one, and thus its spectrum shifts at a much smaller rate. A question one may ask is whether Eq. (5.5.19) has solitary-wave solutions. An approximate solution was found in 1990 using a group-theory approach [20]. It exhibits both the RIFS and the corresponding temporal delay seen in Fig. 5.24 for the fundamental soliton formed after the fission of a second-order soliton. This solution has the form u(ξ, τ ) ≈ [1 + (τ  )]sech(τ  ) exp[iξ( 12 + bτ − b2 ξ 2 /3)],

(5.5.20)

where the parameter b depends on the Raman parameter as b = 8τR /15 and  represents a small correction to the amplitude. The new temporal variable τ  is in a reference frame moving with the soliton and is defined as τ  = τ − bξ 2 /2. In terms of the original variables, z and T, this solution can be written as (neglecting the small  term) 



T − Td (z) u(z, T ) ≈ sech exp[iφ(z) − ip (z)T ], T0

(5.5.21)

where p (z) is the RIFS given in Eq. (5.5.18) and Td (z) is the delay experienced by the soliton:  4T |β |2  b R 2 Td (z) = ξ 2 T0 = z2 . 2 15T04

(5.5.22)

In the case of a second- or higher-order soliton, this solution applies to each fundamental soliton created after the soliton fission, and T0 is the width of this soliton (and not the input pulse width). The RIFS also affects considerably the interaction of two solitons discussed in Section 5.4.4. As an example, Fig. 5.25 compares the evolution of two in-phase solitons, separated by 2q0 = 7, over a distance of 100LD with and without including the Raman term in the NLS equation [143]. As expected, in the absence of RIFS, the Kerr nonlinearity leads to their periodic collisions. However, when the Raman effects are included, this classical interaction behavior changes drastically. Both solitons slow down and their trajectories tilt toward the right side, as their spectra red-shift because of intrapulse Raman scattering. Temporal separation between the pulses decreases initially owing to the Kerr effect. The two solitons come closest near ξ = 40, after which they begin to separate. Moreover, the trailing pulse becomes narrower and its RIFS increases considerably (because it scales with the local pulse width as Ts−4 ). The origin of temporal narrowing is due to a Raman-induced transfer of energy from the leading pulse to the trailing pulse, which must reduce its width to maintain N = 1, as required for

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Figure 5.25 Evolution over 100LD of two temporally separated in-phase solitons when the Raman term is (A) excluded or (B) included in the NLS equation. Notice the drastic changes induced by intrapulse Raman scattering.

fundamental solitons. This energy transfer is related to the delayed nature of the Raman response and can exceed 40% in some conditions. Physically speaking, the leading pulse excites molecular vibrations that are seen by the trailing pulse arriving later. The net result of energy transfer is that the two pulses separate from each other owing to the different RIFS experienced by each pulse.

5.6. Propagation of femtosecond pulses For femtosecond pulses, it becomes necessary to include all the higher-order terms in Eq. (5.5.8) because all three parameters δ3 , s, and τR become non-negligible. Evolution of such ultrashort pulses in optical fibers can be studied by solving Eq. (5.5.8) numerically [184]. As an example, Fig. 5.26 shows the evolutions of pulse shape and spectrum (parts a and b) when a fourth-order soliton is launched at the input end of a fiber, after choosing δ3 = 0.03, s = 0.03, and τR = 0.1. These values are appropriate for a 50-fs pulse (T0 ≈ 28 fs) propagating in the 1.55-µm region of a silica fiber (LD ≈ 4 cm). Soliton fission occurs within 0.4LD , and the shortest fundamental soliton shifts toward the trailing side in part (A) as it slows down because of the RIFS, seen as a red-shift in part (B). Both the temporal and spectral shifts agree with the analytical prediction in Eq. (5.5.21). In physical units, the 50-fs pulse has shifted by almost 40 THz, or 20% of its carrier frequency after propagating a distance of only 8 cm. We also see the evidence of a dispersive wave at a blue-shifted frequency [vertical line near (ν − ν0 )T0 = 4 in part (B)] that spread rapidly in the time domain. Eq. (5.5.8) should be used with care because of its approximate nature [185]. As discussed in Section 2.3, an accurate approach employs Eq. (2.3.36), where R(t) takes

Optical solitons

Figure 5.26 Temporal and spectral evolution over a distance of 2LD when a 50-fs input pulse propagates as a fourth-order soliton (N = 4) inside a standard fiber. These numerical results were obtained using the approximate (a and b) and more accurate (c and d) generalized NLS equations.

into account the time-dependent response of the fiber nonlinearity. The form of R(t) given in Eq. (2.3.33) includes both the electronic and Raman contributions [171–174]. Eq. (2.3.36) was used as early as 1992 to study how intrapulse Raman scattering affects the evolution of femtosecond optical pulses in optical fibers [175]. As an example, parts (C) and (D) of Fig. 5.26 show the evolution of the same 50-fs pulse using the form of Raman Response given in Eq. (2.3.40). A comparison of the temporal and spectral evolutions in the two cases reveals the similarities and the differences between the predictions of Eq. (2.3.36) and Eq. (5.5.8). Although qualitative features are produced by both equations, there are major quantitative differences. One must use Eq. (2.3.36) for short optical pulses to get accurate results. As seen in Fig. 5.26, when the input peak power is large enough to excite a highorder soliton (N > 1), the pulse spectrum evolves into several bands, each corresponding to fundamental solitons of different widths produced after the fission process. Such an evolution pattern was observed in 1987 when 830-fs pulses with peak powers up to 530 W were propagated in fibers up to 1 km long [184]. The spectral peak at the extreme red end was associated with a soliton whose width was the shortest (≈55 fs) after 12 m and then increased with a further increase in the fiber length. These experimental results are discussed further in Section 12.1. In highly nonlinear fibers (see Chapter 11), the soliton order N can easily exceed 10 for intense pulses. The output pulse spectrum can extend over more than 100 THz in such fibers. Such an extreme spectral broadening is referred to as supercontinuum and is the topic of Chapter 13. An interesting question is whether Eq. (5.5.8) permits solitary-wave solutions under certain conditions. Several such solutions have been found using a variety of techniques

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[186–196]. In most cases, the solution exists only for a specific choice of parameter combinations. For example, fundamental and higher-order solitons have been found when τR = 0 and s = −2δ3 or −6δ3 [190]. From a practical standpoint, such solutions of Eq. (5.5.8) are rarely useful because it is hard to find fibers whose parameters satisfy the required constraints.

Problems 5.1 Solve Eq. (5.1.4) and derive an expression for the gain of modulation instability. What is the peak value of the gain and at what frequency does this gain occur? 5.2 Extend Eq. (5.1.4) such that it includes dispersive effects up to fourth-order. Derive an expression for the gain produced by modulation instability. Comment on the new features associated with this instability. 5.3 Consider a lightwave system in which fiber losses are compensated periodically using optical amplifiers. Solve Eq. (5.1.4) for this case and derive an expression for the gain produced by modulation instability. Prove that the gain peaks at frequencies given in Eq. (5.1.18). 5.4 Consider a dispersion-managed fiber link for which β2 in Eq. (5.1.4) is a periodic function. Derive an expression for the gain produced by modulation instability for such a link. You are allowed to consult Ref. [34]. 5.5 A 1.55-µm soliton communication system is operating at 10 Gb/s using dispersion-shifted fibers with D = 2 ps/(km-nm). The effective core area of the fiber is 50 µm2 . Calculate the peak power and the pulse energy required for launching fundamental solitons of 30-ps width (FWHM) into the fiber. 5.6 Verify by direct substitution that the soliton solution given in Eq. (5.2.16) satisfies Eq. (5.2.5). 5.7 Develop a numerical code using Matlab software for solving Eq. (5.2.5) with the split-step Fourier method of Section 2.4.1. Test it by comparing its output with the analytical solution in Eq. (5.2.16) when a fundamental soliton is launched into the fiber. 5.8 Use the computer code developed in the preceding problem to study the case of an input pulse of the form given in Eq. (5.2.22) for N = 0.2, 0.6, 1.0, and 1.4. Explain the different behavior occurring in each case. 5.9 Solve numerically the NLS equation (5.2.5) for an input pulse with u(0, τ ) = 4 sech(τ ). Plot the pulse shapes and spectra over one soliton period and compare your results with those shown in Fig. 5.7. Comment on new qualitative features of the fourth-order bright soliton. 5.10 Repeat the preceding problem for a fourth-order dark soliton by solving Eq. (5.2.5) with the input u(0, τ ) = 4 tanh(τ ). Plot the pulse shapes and spectra over three dispersion lengths and compare your results with those shown in Fig. 5.13. Comment on new qualitative features.

Optical solitons

5.11 A soliton communication system is designed with an amplifier spacing of 50 km. What should the input value of the soliton parameter N be to ensure that a fundamental soliton is maintained in spite of 0.2-dB/km fiber losses? What should the amplifier gain be? Is there any limit on the bit rate of such a system? 5.12 Study the soliton interaction numerically using an input pulse profile given in Eq. (5.4.32). Choose r = 1, q0 = 3, and four values of θ = 0, π/4, pi/2, and π . Compare your results with those shown in Fig. 5.19. 5.13 A soliton system is designed to transmit a signal over 5000 km at B = 5 Gb/s. What should the pulse width (FWHM) be to ensure that the neighboring solitons do not interact during transmission? The dispersion parameter D = 2 ps/(km-nm) at the operating wavelength. 5.14 Follow Ref. [144] and derive the moment equations for the pulse parameters given as Eqs. (5.5.3)–(5.5.6). 5.15 Verify by direct substitution that the solution given in Eq. (5.5.12) is indeed the solution of Eq. (5.5.11) with (b = 40δ4 )−1/2 . 5.16 What is intrapulse Raman scattering? Why does it lead to a shift in the carrier frequency of solitons? Derive an expression for the frequency shift of a fundamental from Eqs. (5.5.3)–(5.5.6) assuming that α = 0, β3 = 0 and Cp = 0. Neglect also the self-steepening terms containing ω0 .

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CHAPTER 6

Polarization effects A major simplification was made in Section 2.3 while deriving the nonlinear Schrödinger (NLS) equation. It consisted of assuming that the polarization state of the incident light is preserved during its propagating inside an optical fiber. This is not really the case in practice. In this chapter we focus on the polarization effects and consider the coupling between the two orthogonally polarized components of an optical field induced by the nonlinear phenomenon known as cross-phase modulation (XPM). The XPM is always accompanied with self-phase modulation (SPM) and can also occur between two optical fields of different wavelengths, a situation covered in Chapter 7. The chapter is organized as follows. The origin of nonlinear birefringence is discussed first in Section 6.1. It is followed with the derivation of a set of two coupled NLS equations that describes evolution of the two orthogonally polarized components of an optical pulse inside a fiber. The XPM-induced nonlinear birefringence has several practical applications discussed in Section 6.2. The next section considers nonlinear polarization changes with focus on polarization instability. Section 6.4 is devoted to the vector modulation instability occurring in birefringent fibers. In contrast with the scalar case discussed in Section 5.1, the vector modulation instability can occur even in the normal-GVD region of a birefringent fiber. We consider the effects of birefringence on solitons in Section 6.5. The last section focuses on polarization-mode dispersion (PMD) in fibers with randomly varying birefringence along their length and its implications for optical communication systems.

6.1. Nonlinear birefringence As mentioned in Section 2.2, even a single-mode fiber, in fact, supports two orthogonally polarized modes with the same spatial distribution. The two modes are degenerate in an ideal fiber (with perfect cylindrical symmetry along its entire length) in the sense that their effective refractive indices, nx and ny , are identical. In practice, all fibers exhibit some modal birefringence (nx = ny ) because of unintentional variations in the core shape and anisotropic stresses along the fiber length. Moreover, the degree of modal birefringence, Bm = |nx − ny |, and the orientation of x and y axes change randomly over a length scale ∼10 m, unless special precautions are taken. In polarization-maintaining fibers, the built-in birefringence is made much larger than random changes occurring due to stress and core-shape variations. As a result, such fibers exhibit nearly constant birefringence along their entire length. This kind of birefringence is called linear birefringence. When the nonlinear effects in optical fibers Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00013-0

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become important, a sufficiently intense optical field can induce nonlinear birefringence whose magnitude is intensity dependent. Such self-induced polarization effects were observed as early as 1964 in bulk nonlinear media [1–6]. In this section, we discuss the origin of nonlinear birefringence and develop mathematical tools that are needed for studying the polarization effects in optical fibers, assuming a constant modal birefringence.

6.1.1 Origin of nonlinear birefringence A fiber with constant modal birefringence has two principal axes along which the fiber is capable of maintaining the state of linear polarization of the incident light. These axes are called the slow and fast axes based on the speed at which light polarized along them travels inside the fiber. Let nx and ny be the mode indices along the slow and fast axes of the fiber, respectively (nx > ny ). When low-power, continuous-wave (CW) light is launched with its polarization vector oriented at an angle from the slow axis, the state of polarization (SOP) of the CW light changes along the fiber from linear to elliptical, and then back to linear, in a periodic manner (see Fig. 1.9) over a distance known as the beat length. The beat length is defined as LB = λ/Bm for a given wavelength λ, and it can be as small as 1 cm in high-birefringence fibers with Bm ∼ 10−4 . In low-birefringence fibers with Bm < 10−6 , the beat length exceeds 1 meter. If we assume that the longitudinal (or axial) component Ez of the electromagnetic field remains small enough that it can be ignored in comparison with the transverse components, the electric field associated with an arbitrarily polarized optical wave can be written as E(r, t) = 12 (ˆxEx + yˆ Ey ) exp(−iω0 t) + c.c.,

(6.1.1)

where xˆ and yˆ stand for unit vectors, Ex and Ey are the complex amplitudes of the polarization components of the field oscillating at the carrier frequency ω0 , and c.c. stands for the complex conjugate of the preceding terms. To keep the following analysis simple, we obtain the nonlinear part PNL by substituting Eq. (6.1.1) in Eq. (2.3.6). This amounts to considering pulses wider that the nonlinear response time of the fiber; the delayed nature of the Raman contribution will be considered in Section 6.6. In general, the third-order susceptibility is a fourth-rank tensor with 81 elements. In an isotropic medium, such as silica glass, it can be written in the form [7,8] (3) (3) (3) (3) χijkl = χxxyy δij δkl + χxyxy δik δjl + χxyyx δil δjk ,

(6.1.2)

where δij is the Kronecker delta function defined such that δij = 1 when i = j and zero otherwise. Using this result in Eq. (2.3.6), PNL can be written as PNL (r, t) = 12 (ˆxPx + yˆ Py ) exp(−iω0 t) + c.c.,

(6.1.3)

Polarization effects

with Px and Py given by Pi =

 30   (3) (3) (3) Ej Ei Ej∗ + χxyyx Ej Ej Ei∗ , χxxyy Ei Ej Ej∗ + χxyxy 4 j

(6.1.4)

where i, j = x or y. From the rotational symmetry of an isotropic medium, we also have the relation [8] (3) (3) (3) (3) χxxxx = χxxyy + χxyxy + χxyyx ,

(6.1.5)

(3) is the element appearing in the scalar theory of Section 2.3, and used in where χxxxx Eq. (2.3.13) to define the Kerr coefficient n2 . The relative magnitudes of the three components in Eq. (6.1.5) depend on the physical mechanisms that contribute to χ (3) . In the case of silica fibers, the dominant contribution is of electronic origin [4], and the three components have nearly the same magnitude. If they are assumed to be identical, the polarization components Px and Py in Eq. (6.1.4) take the form







30 (3) 2 1 χxxxx |Ex |2 + |Ey |2 Ex + (Ex∗ Ey )Ey , Px = 4 3 3    30 (3) 2 1 2 2 ∗ Py = χ |Ey | + |Ex | Ey + (Ey Ex )Ex . 4 xxxx 3 3

(6.1.6) (6.1.7)

The nonlinear contribution nx to the refractive index is governed by the term proportional to Ex in Eq. (6.1.6). Writing Pj = 0 jNL Ej and using j = jL + jNL = (nj + nj )2 ,

(6.1.8)

where nj is the linear part of the refractive index (j = x, y), the nonlinear contributions nx and ny are given by   2 nx = n¯ 2 |Ex |2 + |Ey |2 ,

3

  2 ny = n¯ 2 |Ey |2 + |Ex |2 ,

3

(6.1.9)

where n¯ 2 is the Kerr coefficient defined in Eq. (2.3.13). The physical meaning of the two terms in these equations is as follows. The first term is responsible for SPM. The second term results in XPM because the nonlinear phase shift acquired by one polarization component depends on the intensity of the other polarization component. The presence of this term induces a nonlinear coupling between the field components Ex and Ey . The nonlinear contributions, nx and ny , are in general unequal and thus create nonlinear birefringence whose magnitude depends on the intensity and the SOP of the incident light. In practice, nonlinear birefringence manifests as a rotation of the polarization ellipse [1], a phenomenon referred to as nonlinear polarization rotation.

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6.1.2 Coupled-mode equations The propagation equations governing evolution of the two polarization components along a fiber can be obtained following the method of Section 2.3. Assuming that the nonlinear effects do not affect the fiber mode significantly, the transverse dependence of Ex and Ey can be factored out using Ej (r, t) = F (x, y)Aj (z, t) exp(iβ0j z),

(6.1.10)

where F (x, y) is the spatial distribution of the single mode supported by the fiber, Aj (z, t) is the slowly varying amplitude, and β0j is the corresponding propagation constant (j = x, y). The dispersive effects are included by expanding the frequencydependent propagation constant in a manner similar to Eq. (2.3.23). The slowly varying amplitudes, Ax and Ay , are found to satisfy the following set of two coupled-mode equations: ∂ Ax ∂ Ax iβ2 ∂ 2 Ax α + β1x + + Ax ∂z 2 ∂ t2  2  ∂t

2 3

= i γ | Ax |2 + |Ay |2 Ax +

iγ ∗ 2 A A exp(−2iβ z), 3 x y

(6.1.11)

∂ Ay ∂ Ay iβ2 ∂ 2 Ay α + β1y + + Ay ∂z 2 ∂ t2  2 ∂t

2 3

= i γ | Ay |2 + |Ax |2 Ay +

iγ ∗ 2 A A exp(2iβ z), 3 y x

(6.1.12)

where β = β0x − β0y = (2π/λ)Bm = 2π/LB

(6.1.13)

is related to the linear birefringence of the fiber. The group velocities for the two polarization components are different because β1x = β1y in general. In contrast, the parameters β2 and γ are the same for both polarization components because they are evaluated the same wavelength λ. The last term in Eqs. (6.1.11) and (6.1.12) represent phase-sensitive nonlinear coupling between Ax and Ay and leads to degenerate four-wave mixing (FWM). If the fiber is much longer than the beat length LB , the last term in Eqs. (6.1.11) and (6.1.12) changes sign often and its contribution averages out to zero. In highly birefringent fibers (LB ∼ 1 cm), the FWM term can often be neglected for this reason. In contrast, it should be retained for weakly birefringent fibers, especially those with short lengths. In that case, it is often convenient to rewrite Eqs. (6.1.11) and (6.1.12) using circularly polarized components defined as √

A+ = (Ax + iAy )/ 2,

√

A− = (Ax − iAy )/ 2,

(6.1.14)

Polarization effects

where Ax = Ax exp(iβ z/2) and Ay = Ay exp(−iβ z/2). The variables A+ and A− represent the right and left circularly polarized states (often denoted as σ+ and σ− ), respectively, and they satisfy somewhat simpler equations: ∂ A+ ∂ A+ iβ2 ∂ 2 A+ α + β1 + + A+ = ∂z ∂t 2 ∂ t2 2 2 ∂ A− ∂ A− iβ2 ∂ A− α + β1 + + A− = ∂z ∂t 2 ∂ t2 2

iβ 2iγ A− + 2 3 iβ 2iγ A+ + 2 3



|A+ |2 + 2|A− |2 A+ ,

(6.1.15)



|A− |2 + 2|A+ |2 A− ,

(6.1.16)

where we assumed that β1x ≈ β1y = β1 for fibers with relatively low birefringence. Notice that the FWM terms appearing in Eqs. (6.1.11) and (6.1.12) are replaced with a linear coupling term containing β . At the same time, the relative strength of XPM changes from 23 to 2 when circularly polarized components are used to describe wave propagation.

6.1.3 Elliptically birefringent fibers The derivation of Eqs. (6.1.11) and (6.1.12) assumes that the fiber is linearly birefringent, i.e., it has two principal axes along which linearly polarized light remains linearly polarized in the absence of nonlinear effects. Although this is the case for most polarization-maintaining fibers, elliptically birefringent fibers can be made by twisting a fiber preform during the draw stage [9]. The coupled-mode equations are modified considerably for elliptically birefringent fibers. This case can be treated by replacing Eq. (6.1.1) with E(r, t) = 12 (ˆex Ex + eˆy Ey ) exp(−iω0 t) + c.c.,

(6.1.17)

where eˆx and eˆy are orthonormal polarization eigenvectors related to the unit vectors xˆ and yˆ used before as [10] xˆ + ir yˆ eˆx = √ , 1 + r2

r xˆ − iyˆ eˆy = √ . 1 + r2

(6.1.18)

The parameter r represents the ellipticity introduced by twisting the preform. It is common to introduce the ellipticity angle θ as r = tan(θ/2). The cases θ = 0 and π/2 correspond to linearly and circularly birefringent fibers, respectively. Following a procedure similar to that outlined earlier for linearly birefringent fibers, the slowly varying amplitudes Ax and Ay are found to satisfy the following set of coupled-mode equations [10]: ∂ Ax ∂ Ax iβ2 ∂ 2 Ax α + β1x + + Ax = iγ [(|Ax |2 + B|Ay |2 )Ax + CA∗x A2y e−2iβ z ] ∂z ∂t 2 ∂ t2 2 + iγ D[A∗y A2x eiβ z + (|Ay |2 + 2|Ax |2 )Ay e−iβ z ], (6.1.19)

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∂ Ay ∂ Ay iβ2 ∂ 2 Ay α + β1y + + Ay = iγ [(|Ay |2 + B|Ax |2 )Ay + CA∗y A2x e2iβ z ] ∂z ∂t 2 ∂ t2 2 ∗ 2 −iβ z + iγ D[Ax Ay e + (|Ax |2 + 2|Ay |2 )Ax eiβ z ], (6.1.20)

where the parameters B, C, and D are related to the ellipticity angle θ as B=

2 + 2 sin2 θ , 2 + cos2 θ

C=

cos2 θ , 2 + cos2 θ

D=

sin θ cos θ . 2 + cos2 θ

(6.1.21)

For a linearly birefringent fiber (θ = 0), B = 23 , C = 13 , D = 0, and Eqs. (6.1.19) and (6.1.20) reduce to Eqs. (6.1.11) and (6.1.12), respectively. Eqs. (6.1.19) and (6.1.20) can be simplified considerably for optical fibers with large birefringence for which the beat length LB is much smaller than the fiber’s length. As a result, the exponential factors in the last three terms of Eqs. (6.1.19) and (6.1.20) oscillate rapidly, contributing little to the pulse evolution process on average. If these terms are neglected, we obtain ∂ Ax ∂ Ax iβ2 ∂ 2 Ax α + β1x + + Ax = iγ (|Ax |2 + B|Ay |2 )Ax , ∂z ∂t 2 ∂ t2 2 ∂ Ay ∂ Ay iβ2 ∂ 2 Ay α + β1y + + Ay = iγ (|Ay |2 + B|Ax |2 )Ay . ∂z ∂t 2 ∂ t2 2

(6.1.22) (6.1.23)

These equations represent an extension of the scalar NLS equation (2.3.27), derived without the polarization effects, to the vector case and are referred to as the coupled NLS equations. The coupling parameter B depends on the ellipticity angle θ and can vary from 23 to 2 for values of θ in the range 0 to π /2. For a linearly birefringent fiber, θ = 0, and B = 23 . In contrast, B = 2 for a circularly birefringent fiber (θ = π /2).

6.2. Nonlinear phase shift As seen in Section 6.1, a nonlinear coupling between the two polarization components of an optical wave changes the refractive index by different amounts for the two components. As a result, the nonlinear effects in birefringent fibers are polarization dependent. In this section we use the coupled NLS equations obtained in the case of high-birefringence fibers to study the XPM-induced nonlinear phase shift.

6.2.1 Nondispersive XPM Eqs. (6.1.22) and (6.1.23) need to be solved numerically when ultrashort optical pulses propagate inside birefringent fibers. However, they can be solved analytically in the case of CW radiation. The CW solution is also applicable for pulses whenever the fiber length L is much shorter than both the dispersion length LD = T02 /|β2 | and the walk-off

Polarization effects

length LW = T0 /|β|, where T0 is a measure of the pulse width. As this case is applicable to pulses as short as 100 ps and sheds considerable physical insight, we discuss it first. Neglecting the terms with time derivatives in Eqs. (6.1.22) and (6.1.23), we obtain the following two ordinary differential equations: dAx α + Ax = iγ (|Ax |2 + B|Ay |2 )Ax , dz 2 dAy α + Ay = iγ (|Ay |2 + B|Ax |2 )Ay . dz 2

(6.2.1) (6.2.2)

These equations describe nondispersive XPM in birefringent fibers and extend the scalar theory of SPM in Section 4.1 to the vector case. They can be solved by using 

Ax = Px e−αz/2 eiφx ,



Ay = Py e−αz/2 eiφy ,

(6.2.3)

where Px and Py are the powers and φx and φy are the phases associated with the two polarization components. It is easy to deduce that Px and Py do not change with z. However, the phases φx and φy do change and evolve as dφx = γ e−αz (Px + BPy ), dz

dφy = γ e−αz (Py + BPx ). dz

(6.2.4)

Since Px and Py are constants, the phase equations can be solved easily with the result φx = γ (Px + BPy )Leff ,

φy = γ (Py + BPx )Leff ,

(6.2.5)

where the effective fiber length Leff = [1 − exp(−α L )]/α is defined in the same way as in the SPM case [see Eq. (4.1.5)]. It is clear from Eq. (6.2.5) that both polarization components develop a nonlinear phase shift whose magnitude is the sum of the SPM and XPM contributions. In practice, the quantity of practical interest is the relative phase difference given by φNL ≡ φx − φy = γ Leff (1 − B)(Px − Py ).

(6.2.6)

No relative phase shift occurs when B = 1. However, when B = 1, a relative nonlinear phase shift occurs if input light is launched such that Px = Py . As an example, consider a linearly birefringent fiber for which B = 23 . If CW light with power P0 is launched such that it is linearly polarized at an angle θ from the slow axis, Px = P0 cos2 θ , Py = P0 sin2 θ , and the relative phase shift becomes φNL = (γ P0 Leff /3) cos(2θ ).

This θ -dependent phase shift has several applications discussed next.

(6.2.7)

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Figure 6.1 Schematic illustration of a Kerr shutter. Pump and probe beams are linearly polarized at 45◦ to each other at the input end. Polarizer blocks probe transmission in the absence of pump.

6.2.2 Optical Kerr effect In the optical Kerr effect, the nonlinear phase shift induced by an intense pump beam is used to change the transmission of a weak probe through a nonlinear medium [4]. This effect can be used to make an optical shutter with fast response times [6]. It was first observed in optical fibers in 1973 [11] and attracted considerable attention during the 1980s [12–16]. The operating principle of a Kerr shutter can be understood from Fig. 6.1. The pump and probe beams are linearly polarized at the fiber input with a 45◦ angle between their directions of polarization. A crossed polarizer at the fiber output blocks probe transmission in the absence of the pump beam. When the pump is turned on, the refractive indices for the parallel and perpendicular components of the probe (with respect to the direction of pump polarization) become slightly different because of pump-induced birefringence. The resulting phase difference manifests as a change in the probe polarization at the fiber output, and a portion of the probe intensity is transmitted through the polarizer. The probe transmissivity depends on the pump intensity and can be controlled simply by changing it. In particular, a pulse at the pump wavelength opens the Kerr shutter only during its passage through the fiber. As the probe output can be modulated through a pump at a different wavelength, this device is also referred to as the Kerr modulator. Eq. (6.2.6) cannot be used to calculate the phase difference between the x and y components of the probe because the pump and probe beams have different wavelengths in Kerr shutters. We follow a slightly different approach and neglect fiber losses for the moment; they can be included later by replacing L with Leff . The relative phase difference for the probe at the output of a fiber of length L can always be written as φ = (2π/λ)(˜nx − n˜ y )L ,

(6.2.8)

where λ is the probe wavelength and n˜ x = nx + nx ,

n˜ y = ny + ny .

(6.2.9)

As discussed earlier, the nonlinear parts nx and ny are different because of pumpinduced birefringence.

Polarization effects

Consider the case of a pump polarized linearly along the x axis. The x component of the probe is polarized parallel to the pump but its wavelength is different. For this reason, the corresponding index change nx must be obtained from Section 7.1. If the SPM contribution is neglected, nx = 2n2 Ip ,

(6.2.10)

where |Ep |2 is the pump intensity. When the pump and probe are orthogonally polarized, only the first term in Eq. (6.1.4) contributes to ny because of their different wavelengths. Again neglecting the SPM term, ny becomes ny = 2n2 bIp ,

(3) (3) b = χxxyy /χxxxx .

(6.2.11)

If the origin of χ (3) is purely electronic, b = 13 . Combining Eqs. (6.2.8) through (6.2.11), the phase difference becomes φ ≡ φL + φNL = (2π L /λ)(nL + n2B Ip ),

(6.2.12)

where nL = nx − ny accounts for linear birefringence, and the Kerr coefficient n2B is given by n2B = 2n2 (1 − b).

(6.2.13)

The probe transmissivity Tp can now be obtained noting that probe light is blocked by the polarizer when φ = 0 (see Fig. 6.1). When φ = 0, fiber acts as a birefringent phase plate, and some probe light passes through the polarizer. The probe transmissivity is related to φ by the simple relation Tp = 14 |1 − exp(iφ)|2 = sin2 (φ/2).

(6.2.14)

It becomes 100% when φ = π or any odd multiple of π . On the other hand, a phase shift by an even multiple of π blocks the probe completely. To observe the optical Kerr effect, a polarization-maintaining fiber is generally used to ensure that the pump maintains its state of polarization. The constant phase shift φL resulting from linear birefringence can be compensated by inserting a quarter-wave plate before the polarizer in Fig. 6.1. However, in practice, φL fluctuates because of temperature and pressure variations, making it necessary to adjust the wave plate continuously. An alternative approach is to use two identical pieces of polarization-maintaining fibers, spliced together such that their fast (or slow) axes are at right angles to each other [15]. As nL changes sign in the second fiber, the net phase shift resulting from linear birefringence is canceled. Under ideal conditions, the response time of a Kerr shutter would be limited only by the response time of the Kerr nonlinearity ( 0 and P0 > Pcr (lower row). Left and right spheres in the bottom row show the front and back of the Poincaré sphere. (After Ref. [27]; ©1986 OSA.)

S1 axis, it remains fixed. This can also be seen from the steady-state (z-invariant) solution of Eqs. (6.3.16) and (6.3.17) because (S0 , 0, 0) and (−S0 , 0, 0) represent their fixed points. These two locations of the Stokes vector correspond to the linearly polarized incident light oriented along the slow and fast axes, respectively. In the purely nonlinear case of isotropic fibers (β = 0), WL = 0. The Stokes vector now rotates around the S3 axis with an angular velocity 2γ S3 /3 (upper right sphere in Fig. 6.6). This rotation is referred to as self-induced ellipse rotation, or as nonlinear polarization rotation, because it has its origin in the nonlinear birefringence. Two fixed points in this case correspond to the north and south poles of the Poincaré sphere and represent right and left circular polarizations, respectively. In the mixed case, the behavior depends on the power of incident light. As long as P0 < Pcr , nonlinear effects play a minor role, and the situation is similar to the linear case. At higher power levels, the motion of the Stokes vector on the Poincaré sphere becomes quite complicated because WL is oriented along the S1 axis while WNL is oriented along the S3 axis. Moreover, the nonlinear rotation of the Stokes vector along the S3 axis depends on the magnitude of S3 itself. The bottom row in Fig. 6.6 shows motion of the Stokes vector on the front and back of the Poincaré sphere in the case P0 > Pcr . When input light is polarized close to the slow axis (left sphere), the situation is similar to the linear case. However, the behavior is qualitatively different when input light is polarized close to the fast axis (right sphere).

Polarization effects

To understand this asymmetry, let us find the fixed points of Eqs. (6.3.16) and (6.3.17) by setting the z derivatives to zero. The location and number of fixed points depend on the optical power P0 launched inside the fiber. More specifically, the number of fixed points changes from two to four at the critical power level Pcr defined in Eq. (6.3.10). For P0 < Pcr , only two fixed points, (S0 , 0, 0) and (−S0 , 0, 0), are found; these are identical to the low-power case. In contrast, when P0 exceeds Pcr , two new fixed points emerge. The components of the Stokes vector at the location of the new fixed points on the Poincaré sphere are given by [37] S1 = −Pcr ,

S2 = 0,



S3 = ± P02 − Pcr2 .

(6.3.20)

These two fixed points correspond to elliptically polarized light and occur on the back of the Poincaré sphere in Fig. 6.6 (lower right). At the same time, the fixed point (−S0 , 0, 0), corresponding to light polarized linearly along the fast axis, becomes unstable. This is equivalent to the pitchfork bifurcation discussed earlier. If an input beam is polarized elliptically with its Stokes vector oriented as indicated in Eq. (6.3.20), its SOP will not change inside the fiber. When the SOP is close to the new fixed points, the Stokes vector forms a closed loop around the elliptically polarized fixed point. This behavior corresponds to the analytic solution discussed earlier. However, if the SOP is close to the unstable fixed point (−S0 , 0, 0), small changes in input polarization can induce large changes at the output. This phenomenon is discussed next.

6.3.3 Polarization instability The polarization instability manifests as large changes in the output SOP when the input power or the input SOP of a CW beam is changed slightly [27–29]. The presence of polarization instability shows that slow and fast axes of a polarization-preserving fiber are not entirely equivalent. The origin of polarization instability can be understood from the following qualitative argument [28]. When the input beam is polarized close to the slow axis (x axis if nx > ny ), nonlinear birefringence adds to intrinsic linear birefringence, making the fiber more birefringent. By contrast, when the input beam is polarized close to the fast axis, nonlinear effects decrease the net birefringence by an amount that depends on the input power. As a result, the fiber becomes less birefringent, and the effective beat length LBeff increases. At a critical value of the input power, nonlinear birefringence cancels the linear birefringence completely, and LBeff becomes infinite. With a further increase in the input power, the fiber again becomes birefringent but the roles of the slow and fast axes are reversed. Clearly large changes in the output polarization state can occur when the input power is close to the critical power required to balance the linear and nonlinear birefringences. Roughly speaking, polarization instability occurs when the input peak power is large enough to make the nonlinear length LNL comparable to the intrinsic beat length LB .

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Figure 6.7 Effective beat length as a function of input power for beams polarized along the fast (solid line) and slow (dashed line) axes. (After Ref. [28]; ©1986 OSA.)

The period of the elliptic function in Eq. (6.3.11) determines the effective beat length as [28] LBeff =

2K (m)  LB , π |q|

(6.3.21)

where LB is the low-power beat length, K (m) is the quarter-period of the elliptic function, and m and q are given by Eq. (6.3.12) with p defined as p = P0 /Pcr . In the absence of nonlinear effects, p = 0, q = 1, K = π/2, and we recover LBeff = LB = 2π/|β|. Fig. 6.7 shows how LBeff varies with p for θ = 0◦ and θ = 90◦ . The effective beat length becomes infinite when P0 = Pcr and θ = 90◦ because of complete cancellation of fiber’s birefringence [29]. This is the origin of polarization instability. The critical power level Pcr at which LBeff becomes infinite is the same at which the number of fixed points on the Poincaré sphere changes from 2 to 4. Thus, polarization instability can be interpreted in terms of the emergence of elliptically polarized fixed points on the Poincaré sphere. The two viewpoints are identical. As a result of large changes in LBeff , the output polarization state can change drastically when P0 is close to Pcr and the input beam is polarized close to the fast axis. Fig. 6.8 shows the transmissivity Tp as a function of the input power for several values of θ after assuming that a crossed polarizer at the fiber output blocks the light at low intensities (see Fig. 6.1). When θ = 0◦ or 90◦ , Tp remains zero at all power levels. Small changes in θ near the slow axis still keep Tp near zero. However, Tp changes dramatically when θ is changed slightly near the fast axis. Note the extreme sensitivity of Tp to the input polarization angle as θ is varied from 89◦ to 90◦ . Fig. 6.8 is drawn for the case (β)L = 2π or L = LB . However, the qualitative behavior remains the same for other fiber lengths. Polarization instability was first observed in 1986 by transmitting 80-ps pulses (at 532 nm) through a 53-cm-long fiber with a measured intrinsic beat length

Polarization effects

Figure 6.8 Transmissivity of a birefringent fiber of length L = LB as a function of input power for different input angles. (After Ref. [28]; ©1986 OSA.)

LB ≈ 50 cm [31]. The input pulses were right-circularly polarized and passed through a circular analyzer at the fiber output that transmitted only left-circularly polarized light. The shape of output pulses was found to change dramatically when the peak power exceeded a critical value. The measured critical power and the output pulse shapes were in agreement with the theoretical predictions. In a later experiment, polarization instability led to significant enhancement of weak intensity modulations when the input signal was polarized near the fast axis of a low-birefringence fiber [38]. The 200-ns input pulses were obtained from a Q-switched Nd:YAG laser operating at 1.06 µm. The intensity of these pulses exhibited 76-MHz modulation because of longitudinalmode beating inside the laser. These small-amplitude modulations were unaffected when the signal was polarized near the slow axis of the fiber, but became amplified by as much as a factor of 6 when input pulses were polarized near the fast axis. The experimental results were in good qualitative agreement with theory, especially when the theory was generalized to include twisting of the fiber that resulted in elliptical birefringence [38]. The power-dependent transmissivity seen in Fig. 6.8 can be useful for optical switching. Self-switching of linearly polarized beams induced by polarization instability has been demonstrated in silica fibers [39]. It can also be used to switch the polarization state of an intense beam through a weak pulse. Polarization switching can also occur for solitons [40]. In all cases, the input power required for switching is quite large unless fibers with low modal birefringence are used. A fiber with a beat length LB = 1 m requires P0 ∼ 1 kW if we use γ = 10 W−1 /km in Eq. (6.3.10). This value becomes larger by a factor of 100 or more when high-birefringence fibers are used. For this reason, polarization instability is not of concern when highly birefringent fibers are used because P0 remains 0) provided C2 < 0. This condition is satisfied for frequencies in the range 0 < || < c1 , where  c1 = (4γ /3β2 )1/2 P0 − Pcr .

(6.4.7)

Thus, P0 > Pcr is required for modulation instability to occur in the normal-GVD regime of the fiber. When this condition is satisfied, the gain is given by

g() = |β2 | (2 + 2c2 )(2c1 − 2 ),

(6.4.8)

c2 = (2β/β2 )1/2 .

(6.4.9)

with

Consider now the case in which the CW beam is polarized along the slow axis (Ay = 0). We can follow essentially the same steps to find the dispersion relation K (). In fact, Eqs. (6.4.4)–(6.4.6) remain applicable if we change the sign of β . Modulation instability can still occur in the normal-GVD region of the fiber, but the gain exists only for frequencies in the range c2 < || < c3 , where  c3 = (4γ /3β2 )1/2 P0 + Pcr .

(6.4.10)

The instability gain is now given by

g() = |β2 | (2 − 2c2 )(2c3 − 2 ).

(6.4.11)

Fig. 6.9 compares the gain spectra for light polarized along the slow and fast axes using β2 = 60 ps2 /km and γ = 25 W−1 /km for a fiber with a beat length LB = 5 m. For these parameter values, p = 1 at an input power of 75.4 W. At a power level of 100 W, p = 1.33 (left part) while p > 2 at 200 W (right part). The most noteworthy

Polarization effects

Figure 6.9 Gain spectra of modulation instability in the case of normal dispersion at power levels of 100 W (left) and 200 W (right) for a CW beam polarized along the slow (solid curves) or fast (dashed curves) axis of a low-birefringence fiber (LB = 5 m).

feature of Fig. 6.9 is that, in contrast with the gain spectra of Fig. 5.1, the gain does not vanish near  = 0 when light is polarized along the fast axis such that p > 1. This is a manifestation of the polarization instability discussed in Section 6.3, occurring only when the input beam is polarized along the fast axis. When p < 2, the gain is larger for CW light polarized along the slow axis. However, the three peaks become comparable as p approaches 2. When p exceeds 2, the fast-axis gain spectrum develops a dip at  = 0, and the gain peak occurs for a finite value of . In that case, the CW beam would develop spectral sidebands irrespective of whether it is polarized along the slow or fast axis. This situation is similar to the scalar case of Section 5.1. The new feature is that the sidebands can develop even in the normal-GVD regime of a birefringent fiber. All of these features have been observed experimentally [51].

6.4.2 High-birefringence fibers For high-birefringence fibers, the last term can be neglected in Eqs. (6.1.11) and (6.1.12). These equations then reduce to Eqs. (6.1.22) and (6.1.23) with B = 23 and exhibit a different kind of modulation instability [44–47]. This case is mathematically similar to the two-wavelength case discussed in Section 7.2. To obtain the steady-state solution, the time derivatives in Eqs. (6.1.22) and (6.1.23) can be set to zero. If fiber losses are neglected by setting α = 0, the steady-state solution is given by (see Section 6.2.1) 

Ax (z) = Px exp[iφx (z)],



Ay (z) = Py exp[iφy (z)],

(6.4.12)

where Px and Py are the constant mode powers and φx (z) = γ (Px + BPy )z,

φy (z) = γ (Py + BPx )z.

(6.4.13)

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The phase shifts depend on the powers of both polarization components. In contrast with the case of weakly birefringent fibers, this solution is valid for a CW beam polarized at an arbitrary angle with respect to the slow axis. Stability of the steady state is examined assuming a time-dependent solution of the form Aj =





Pj + aj exp(iφj ),

(6.4.14)

where aj (z, t) is a weak perturbation with j = x, y. We substitute Eq. (6.4.14) in Eqs. (6.1.22) and (6.1.23) and linearize them with respect to ax and ay . The resulting linear equations can again be solved in the form aj = uj exp[i(Kz − t) + ivj exp[−i(Kz − t)],

(6.4.15)

where j = x, y, K is the wave number, and  is the frequency of perturbation. To simplify the algebra, let us focus on the case in which the input CW beam is polarized at 45◦ from the slow axis. As a result, both polarization modes have equal powers (Px = Py = P). The dispersion relation in this case can be written as [44] [(K − b)2 − H ][(K + b)2 − H ] = CX2 ,

(6.4.16)

where b = (β1x − β1y )/2 takes into account the group-velocity mismatch, H = β2 2 (β2 2 /4 + γ P ),

(6.4.17)

and the XPM coupling parameter CX is defined as CX = Bβ2 γ P 2 . As before, modulation instability occurs when K becomes complex for some values of . Its gain is obtained from g = 2 Im(K ). The most important conclusion drawn from Eq. (6.4.16) is that modulation instability can occur irrespective of the sign of the GVD parameter [44]. In the case of normal GVD (β2 > 0), the gain exists only if CX > |H − b|2 . Fig. 6.10 shows the gain spectra at three power levels using parameter values β2 = 60 ps2 /km, γ = 25 W−1 /km, and a group-velocity mismatch of 1.5 ps/m. At low power levels, the gain spectrum is relatively narrow and its peak is located near m = |β1x − β1y |/β2 . As input power increases, gain spectrum widens and its peak shifts to lower frequencies. In all three cases shown in Fig. 6.10, the CW beam develops temporal modulations at frequencies >2.5 THz as it propagates through the fiber. As m depends on birefringence of the fiber, it can be changed easily and provides a tuning mechanism for the modulation frequency. An unexpected feature is that modulation instability ceases to occur when the input power exceeds a critical value Pc = 3(β1x − β1y )2 /(4β2 γ ).

(6.4.18)

Polarization effects

Figure 6.10 Gain spectra of modulation instability in the normal-GVD region of a high-birefringence fiber when the input beam is linearly polarized at 45◦ from the slow axis.

Another surprising feature is that modulation instability does not occur when input light is polarized close to a principal axis of the fiber [45]. Both of these features can be understood qualitatively if we interpret modulation instability in terms of a FWM process that is phase matched by the modal birefringence of the fiber (see Chapter 10). In the case of normal GVD, the SPM- and XPM-induced phase shifts actually add to the GVD-induced phase mismatch. It is the fiber birefringence that cancels the phase mismatch. Thus, for a given value of birefringence, the phase-matching condition can only be satisfied if the nonlinear phase shifts remain below a certain level. This is the origin of the critical power level in Eq. (6.4.18). An interesting feature of the FWM process is that spectral sidebands are generated such that the low-frequency sideband at ω0 −  is polarized along the slow axis, whereas the sideband at ω0 +  is polarized along the fast axis. This can also be understood in terms of the phase-matching condition of Section 10.3.3.

6.4.3 Isotropic fibers It is clear that modal birefringence of fibers plays an important role for modulation instability to occur. A natural question is whether modulation instability can occur in isotropic fibers with no birefringence (nx = ny ). Even though such fibers are hard to fabricate, fibers with extremely low birefringence (|nx − ny | < 10−8 ) can be made by spinning the preform during the drawing stage. The question is also interesting from a fundamental standpoint and was discussed as early as 1970 [41]. The theory developed for high-birefringence fibers cannot be used in the limit β = 0 because the coherent-coupling term has been neglected. In contrast, theory developed for low-birefringence fibers remains valid in that limit. The main difference is that Pcr = 0 as polarization instability does not occur for isotropic fibers. As a result,

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c2 = 0 while c1 = c3 ≡ c . The gain spectrum of modulation instability in Eq. (6.4.8)

reduces to

g() = |β2 | 2c − 2 ,

(6.4.19)

irrespective of whether the input beam is polarized along the slow or fast axis. This is the same result obtained in Section 5.1 for the scalar case. It shows that the temporal and spectral features of modulation instability should not depend on the direction in which the input beam is linearly polarized. This is expected for any isotropic nonlinear medium on physical grounds. The situation changes when the input beam is circularly or elliptically polarized. We can consider this case by setting β = 0 in Eqs. (6.1.15) and (6.1.16). Using α = 0 for simplicity, these equations reduce to the following set of two coupled NLS equations [41]:

∂ A+ iβ2 ∂ 2 A+ + + iγ  |A+ |2 + 2|A− |2 A+ = 0, 2 ∂z 2 ∂T

∂ A− iβ2 ∂ 2 A− + + iγ  |A− |2 + 2|A+ |2 A− = 0, ∂z 2 ∂T 2

(6.4.20) (6.4.21)

where T = t − β1 z is the reduced time and γ  = 2γ /3. The steady-state solution of these equations is obtained easily and is given by 

A± (z) = P± exp(iφ± ),

(6.4.22)

where P± is the input power in the two circularly polarized components and φ± (z) = γ  (P∓ + 2P± )z is the nonlinear phase shift. As before, we perturb the steady-state solution using 

A± (z, t) = [ P± + a± (z, t)] exp(iφ± ),

(6.4.23)

where a± (z, t) is a small perturbation. By using Eq. (6.4.23) in Eqs. (6.4.20) and (6.4.21) and linearizing in a+ and a− , we obtain a set of two coupled linear equations. These equations can be solved assuming a solution in the form of Eq. (6.4.3). We then obtain a set of four algebraic equations for u± and v± . This set has a nontrivial solution only when the perturbation satisfies the following dispersion relation [41] (K − H+ )(K − H− ) = CX2 ,

(6.4.24)

H± = 12 β2 2 ( 12 β2 2 + γ P± ),

(6.4.25)

where

Polarization effects

and the XPM coupling parameter CX is now defined as 

CX = 2β2 γ 2 P+ P− .

(6.4.26)

A necessary condition for modulation instability to occur is CX2 > H+ H− . As CX de√ pends on P+ P− and vanishes for a circularly polarized beam, we can conclude that no instability occurs in that case. For an elliptically polarized beam, the instability gain depends on the ellipticity ep defined in Eq. (6.3.4).

6.4.4 Experimental results The vector modulation instability was first observed in the normal-dispersion region of a high-birefringence fiber [44–46]. In one experiment, 30-ps pulses at the 514-nm wavelength with 250-W peak power were launched into a 10-m fiber with a 45◦ -polarization angle [45]. At the fiber output, the pulse spectrum exhibited modulation sidebands with a 2.1-THz spacing, and the autocorrelation trace showed 480-fs intensity modulation. The observed sideband spacing was in good agreement with the value calculated theoretically. In another experiment, 600-nm input pulses were of only 9-ps duration [44]. As the 18-m-long fiber had a group-velocity mismatch of ≈1.6 ps/m, the two polarization components would separate from each other after only 6 m of fiber. The walk-off problem was solved by delaying the faster-moving polarization component by 25 ps at the fiber input. The temporal and spectral measurements indicated that both polarization components developed high-frequency (∼3 THz) modulations, as expected from theory. Moreover, the modulation frequency decreased with an increase in the peak power. This experiment also revealed that each polarization component of the beam develops only one sideband, in agreement with theory. In a later experiment [46], modulation instability developed from temporal oscillations induced by optical wave breaking (see Section 4.2.3). This behavior can be understood from Fig. 4.13, noting that optical wave breaking manifests as spectral sidebands. If these sidebands fall within the bandwidth of the modulation-instability gain curve, their energy can seed the instability process. Although vector modulation instability is predicted to occur even in the anomalousGVD regime of a high-birefringence fiber, its experimental observation turned out to be more difficult [54]. The reason is that the scalar modulation instability (discussed in Section 5.1) also occurs in this regime. If the input field is polarized along a principal axis, the scalar one dominates. In a 2005 experiment, clear-cut evidence of vector modulation instability was seen when the input field was polarized at a 45◦ angle from a principal axis [55]. Fig. 6.11 shows the observed spectrum at the output of a 51-mlong fiber, with a birefringence-induced differential group delay of 286 fs/m, when 3.55-ns pulses at a 2.5-kHz repetition rate (with an average power of about 1 mW) were launched into it. The central multipeak structure is due to scalar modulation instability, but the two extreme peaks result from vector modulation instability. These two

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Figure 6.11 Spectral sidebands observed at the output of a 51-m-long high-birefringence fiber when the pump is polarized at 45◦ from the principal axes. Black and gray traces correspond to light polarized along the fast and slow axes, respectively. The central multipeak structure results from the scalar modulation instability. (After Ref. [55]; ©2005 OSA.)

peaks are orthogonally polarized along the fast and slow axes, a feature unique to the vector modulation instability. Modulation instability in low-birefringence fibers was observed in 1995 using 60-ps pulses (peak power >1 kW) obtained from a krypton-ion laser operating at 647 nm [48]. Fibers used in the experiment were a few meters long, and their birefringence was controlled through stress induced by winding the fiber on a spool with a relatively small diameter. When input pulses were polarized along the slow axis, the two sidebands indicative of modulation instability had the same polarization and were polarized along the fast axis. Their spacing could be varied over a range of 20 nm or so by simply changing the spool size (a smaller spool diameter produces more stress-induced birefringence), resulting in larger sideband spacing. In a variation of this idea, fibers with periodically varying birefringence along their length were produced by wrapping the fiber around two spools [49]. Such a periodic variation can create new sidebands through quasi-phase matching, similar to the periodic variation of dispersion and nonlinearity discussed in Section 5.1. A systematic study of induced modulation instability in low-birefringence fibers was performed in 1998 using a pump-probe configuration [51]. The probe beam was used to seed the process. In a series of experiments, the pump beam was obtained from a dye laser operating near 575 nm and consisted of 4-ns pulses that were wide enough to realize quasi-CW operation. The pump-probe wavelength separation was tunable; tuning allowed different regimes of modulation instability to be investigated. The fiber was drawn using a rapidly rotating preform so that its intrinsic birefringence averaged out to zero. A controlled amount of weak birefringence was introduced by winding the fiber onto a 14.5-cm-diameter spool. The measured beat length of 5.8 m for the fiber

Polarization effects

Figure 6.12 Modulation-instability sidebands observed in a low-birefringence fiber. Pump is polarized along the fast (top row) or slow axis (bottom row). Pump-probe detuning is 0.3 THz for left and 1.2 THz for right columns. (After Ref. [51]; ©1998 OSA.)

corresponded to a modal birefringence of only 10−7 . The critical power Pcr required for the onset of polarization instability [see Eq. (6.3.10)] was estimated to be 70 W for this fiber. Fig. 6.12 shows the pump spectra measured under several different experimental conditions. In all cases, the pump power was 112 W (1.6Pcr ) while the probe power was kept low (∼1 W). Consider first the case of a pump polarized along the fast axis (top row). For a pump-probe detuning of 0.3 THz, the probe frequency falls within the gain spectrum of modulation instability (see Fig. 6.9). As a result, the pump spectrum develops a series of sidebands spaced apart by 0.3 THz. In contrast, the probe frequency falls outside the gain spectrum for a detuning of 1.2 THz, and modulation instability does not occur. When the pump is polarized along the slow axis (bottom row), the situation is reversed. Now the 0.3-THz detuning falls outside the gain spectrum, and modulation-instability sidebands form only when the detuning is 1.2 THz. These experimental results are in agreement with the theory given earlier. In the time domain, the pump pulse develops deep modulations that correspond to a train of dark solitons with repetition rates in the terahertz regime [51]. A dark-soliton is also formed when modulation instability occurs in high-birefringence fibers [52]. The formation of dark solitons is not surprising if we recall from Chapter 5 that optical fibers support only dark solitons in their normal-GVD regime. In all of these experiments, fiber birefringence plays an important role. As discussed before, vector modulation instability can occur in isotropic fibers (nx = ny ) such that the gain spectrum depends on the polarization state of the input CW beam. Unfortunately, it is difficult to make birefringence-free fibers. As an alternative, modulation instability

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was observed in a bimodal fiber in which the input beam excited two fiber modes (the LP01 and LP11 modes) with nearly equal power levels, and the two modes had the same group velocity [50]. In a 1999 experiment [53], a nearly isotropic fiber was realized by winding 50 m of “spun” fiber with a large radius of curvature of 25 cm. The beat length for this fiber was ∼1 km, indicating a birefringence level Pcr becomes (β)LNL < 23 , where LNL = (γ P0 )−1 is the nonlinear length. By using β = 2π/LB , N 2 = LD /LNL and z0 = (π/2)LD , the condition can be written as z0 < N 2 LB /6, and the numerical results agree with it [56]. Typically, LB ∼ 1 m for weakly birefringent fibers. Thus, polarization instability affects a fundamental soliton (N = 1) only if z0 1 m. Such values of z0 are realized in practice only for femtosecond pulses (T0 < 100 fs).

6.5.2 High-birefringence fibers In high-birefringence fibers, the group-velocity mismatch between the fast and slow components of the input pulse cannot be neglected. Such a mismatch would split a pulse into its two components if it is not polarized along a principal axis of the fiber. The interesting question is whether such a splitting also occurs for solitons. The effects of group-velocity mismatch are studied by solving Eqs. (6.1.22) and (6.1.23) numerically. If we assume anomalous GVD (β2 < 0) and use the soliton units

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of Section 5.2, these equations become  ∂u 1 ∂ 2u ∂u + (|u|2 + B|v|2 )u = 0, i +δ + ∂ξ ∂τ 2 ∂τ 2   ∂v 1 ∂ 2v ∂v + (|v|2 + B|u|2 )v = 0, i −δ + ∂ξ ∂τ 2 ∂τ 2 

(6.5.4) (6.5.5)

where u and v are the normalized amplitudes of the pulse components along the x and y axes, respectively, and δ = (β1x − β1y )T0 /(2|β2 |)

(6.5.6)

governs the group-velocity mismatch between the two polarization components. The normalized time τ = (t − β1 z)/T0 , where β1 = 12 (β1x + β1y ). Fiber losses are ignored for simplicity but can be included easily. The XPM coupling parameter B = 23 for linearly birefringent fibers. When an input pulse is launched with a polarization angle θ (measured from the slow axis), Eqs. (6.5.4) and (6.5.5) should be solved with the input u(0, τ ) = N cos θ sech(τ ),

v(0, τ ) = N sin θ sech(τ ),

(6.5.7)

where N is the soliton order. In the absence of XPM-induced coupling, the two polarization components evolve independently and separate from each other because of their different group velocities. The central question is how this behavior is affected by the XPM. We can answer it by solving Eqs. (6.5.4) and (6.5.5) numerically with B = 2/3 for various values of N, θ , and δ [57–59]. The numerical results can be summarized as follows. When both polarization components are equally excited (θ = 45◦ ), they remain bound together if N exceeds a critical value Nth that depends on δ ; Nth ≈ 0.7 for δ = 0.15, but Nth ≈ 1 for δ = 0.5. As an example, Fig. 6.13 shows the temporal and spectral evolutions of the slow and fast components of the pulse over 10 dispersion lengths by choosing N = 1 and δ = 0.2. The two components move initially at different speeds but they trap each other soon after and move at a common speed. This trapping is a consequence of the XPM-induced spectral shifts occurring in opposite directions for the two components. This trapping behavior is clearly evident in Fig. 6.14 where the shapes and spectra of the two components are compared at a distance of z = 10LD . As seen there, the temporal profiles almost overlap, but the spectra are shifted in opposite directions. A similar behavior is observed for larger values of δ as long as N > Nth . Solitons can form even when N < Nth , but the two components travel at their own group velocities and become widely separated. When θ = 45◦ , the two modes have unequal amplitudes initially. In this case, if N exceeds Nth , a qualitatively different evolution scenario occurs depending on the value

Polarization effects

Figure 6.13 Temporal and spectral evolution of the two components of a vector soliton over 10 dispersion lengths for N = 1, θ = 45◦ , and δ = 0.2. The gray scale is over a 20-dB intensity range.

of δ . Fig. 6.15 is obtained under conditions identical to those of Fig. 6.14 except that θ is reduced to 30◦ so that the slow-axis component of the vector soliton dominates. As seen there, the weaker pulse (polarized along the fast axis) is still captured by the stronger one, and the two move together. However, the trapped pulses are shifted from the center because the stronger pulse moves slower initially and has shifted toward the right before it captures the fast-axis component. Notice also how the pulse spectra in Fig. 6.15(B) are affected differently for the two pulse components because of the asymmetric nature of the XPM-induced coupling. The numerical results shown in Figs. 6.13–6.15 clearly indicate that, under certain conditions, the two orthogonally polarized components of a pulse move with a common group velocity in spite of their different speeds in the absence of XPM-induced coupling (B = 0). This phenomenon is called soliton trapping and, as discussed later, it can be used for optical switching. It owes its existence solely to XPM. It is the XPMinduced nonlinear coupling that allows the two solitons to propagate at the same speed. Physically, the two solitons shift their carrier frequencies in the opposite directions to realize temporal synchronization. Specifically, the soliton along the fast axis slows down while the one along the slow axis speeds up. Because soliton trapping requires a balance between XPM and linear birefringence, it can occur only when the peak power of the input pulse, or equivalently the soliton order N, exceeds a threshold value Nth . As Nth depends on both the polarization angle θ and δ , attempts have been made to estimate Nth analytically by solving Eqs. (6.5.4)

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Figure 6.14 Comparison of the temporal and spectral profiles at z = 10LD for the slow (solid line) and fast (dashed line) components for N = 1, θ = 45◦ , and δ = 0.2.

Figure 6.15 Comparison of the temporal and spectral profiles at z = 10LD for the slow (solid line) and fast (dashed line) components for N = 1, θ = 30◦ , and δ = 0.2.

and (6.5.5) approximately [61–67]. In a simple approach, the XPM term is treated as a perturbation within the Lagrangian formulation. In the case of equal amplitudes, realized by choosing θ = 45◦ in Eq. (6.5.7), the threshold value for soliton trapping is found to be [61] Nth = [2(1 + B)]−1/2 + (3/8B)1/2 δ.

(6.5.8)

For B = 23 , the predictions of Eq. (6.5.8) are in good agreement with the numerical results for small values of δ (up to 0.5). For large values of δ , the threshold value is well approximated by [67] Nth = [(1 + 3δ 2 )/(1 + B)]1/2 . Soliton trapping was first observed [68] in 1989 by launching 300-fs pulses (obtained from a color-center laser) into a 20-m single-mode fiber with modal birefringence n ≈ 2.4 × 10−5 , a value leading to polarization dispersion of 80 ps/km. In this experiment, the soliton period z0 was 3.45 m with δ = 0.517. The measured pulse spectra for the

Polarization effects

orthogonally polarized components were found to be separated by about 1 THz when the polarization angle was 45◦ . The autocorrelation trace indicated that the two pulses at the fiber output were synchronized temporally as expected for soliton trapping.

6.5.3 Soliton-dragging logic gates An important application of XPM interaction in birefringent fibers has led to the realization of all-optical, ultrafast logic gates, first demonstrated in 1989 [69]. The performance of such logic gates was studied during the 1990s, both theoretically and experimentally [70–76]. The basic idea behind the operation of fiber-optic logic gates has its origin in the nonlinear phenomenon of soliton trapping discussed earlier. It can be understood as follows. In digital logic, each optical pulse is assigned a time slot whose duration is determined by the clock speed. If a signal pulse is launched together with an orthogonally polarized control pulse, and the control pulse is intense enough to trap the signal pulse during a collision, then both pulses can be dragged out of their assigned time slot because of the XPM-induced change in their group velocity. In other words, the absence or presence of a signal pulse at the fiber input dictates whether the control pulse ends up arriving within the assigned time slot or not. This temporal shift forms the basic logic element and can be used to perform more complex logic operations. Because the control pulse propagating as a soliton is dragged out of its time slot through the XPM interaction, such devices are referred to as soliton-dragging logic gates. In a network configuration, output signal pulse can be discarded while control pulse becomes the signal pulse for the next gate. This strategy makes the switching operation cascadable. In effect, each control pulse is used for switching only once irrespective of the number of gates in the network. The experimental demonstration of various logic gates (such as exclusive OR, AND and NOR gates), based on the concept of soliton trapping, used femtosecond optical pulses (pulse width ∼300 fs) from a mode-locked color-center laser operating at 1.685 µm [69–71]. In these experiments, orthogonally polarized signal and control pulses were launched into a highly birefringent fiber. In the implementation of a NOR gate, the experimental conditions were arranged such that the control pulse arrived in its assigned time slot of 1-ps duration in the absence of signal pulses (logical “1” state). In the presence of one or both signal pulses, the control pulse shifted by 2–4 ps because of soliton dragging and missed the assigned time slot (logical “0” state). The energy of each signal pulse was 5.8 pJ. The energy of the control pulse was 54 pJ at the fiber input but reduced to 35 pJ at the output, resulting in an energy gain by a factor of six. The experimental results can be explained quite well by solving Eqs. (6.5.4) and (6.5.5) numerically [75]. Since the first demonstration of such logic gates in 1989, considerable progress has been made. The use of soliton-dragging logic gates for soliton ring networks has also been proposed [76].

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6.5.4 Vector solitons The phenomenon of soliton trapping suggests that the coupled NLS equations may possess exact solitary-wave solutions with the property that the orthogonally polarized components propagate in a birefringent fiber without change in shape. Such solitary waves are referred to as vector solitons to emphasize the fact that an input pulse maintains not only its intensity profile but also its state of polarization even when it is not launched along one of the principal axes of the fiber [77]. A more general question one may ask is whether conditions exist under which two orthogonally polarized pulses of different widths and different peak powers can propagate undistorted in spite of the XPM-induced nonlinear coupling between them. Consider the case of high-birefringence fibers. To obtain soliton solutions of Eqs. (6.5.4) and (6.5.5), it is useful to simplify them with the transformation u = u˜ exp(iδ 2 ξ/2 − iδτ ),

v = v˜ exp(iδ 2 ξ/2 + iδτ ).

(6.5.9)

The resulting equations are independent of δ and take the form ∂ u˜ 1 ∂ 2 u˜ + (|˜u|2 + B|˜v|2 )˜u = 0, + ∂ξ 2 ∂τ 2 ∂ v˜ 1 ∂ 2 v˜ i + + (|˜v|2 + B|˜u|2 )˜v = 0. ∂ξ 2 ∂τ 2

i

(6.5.10) (6.5.11)

In the absence of XPM-induced coupling (B = 0), the two NLS equations become decoupled and have independent soliton solutions of the form discussed in Section 5.2. When B = 0, Eqs. (6.5.10) and (6.5.11) can be solved by the inverse scattering method only for a specific value of parameter B, namely B = 1. Manakov obtained in 1973 such a solution [78]. In its simplest form, the solution can be written as u˜ (ξ, τ ) = cos θ sech(τ ) exp(iξ/2),

(6.5.12)

v˜ (ξ, τ ) = sin θ sech(τ ) exp(iξ/2),

(6.5.13)

where θ is an arbitrary angle. A comparison with Eq. (5.2.15) shows that this solution corresponds to a vector soliton that is identical with the fundamental soliton (N = 1) of Section 5.2 in all respects. The angle θ can be identified as the polarization angle. More complex forms of vector solitons have also been found in the B = 1 case [79]. The preceding solution predicts that a “sech” pulse, launched with N = 1 and linearly polarized at an arbitrary angle from a principal axis, can maintain both its shape and polarization, provided the fiber is birefringent such that the XPM parameter B = 1. However, as discussed in Section 6.1, unless the fiber is especially designed, B = 1 in practice. In particular, B = 23 for linearly birefringent fibers. For this reason, solitarywave solutions of Eqs. (6.5.10) and (6.5.11) for B = 1 have been studied in several

Polarization effects

different contexts [80–88]. Such solutions represent solitary waves with the shapepreserving property of solitons. In the specific case of equal amplitudes (θ = 45◦ ), a solitary-wave solution of Eqs. (6.5.10) and (6.5.11) is given by [84] u˜ = v˜ = η sech[(1 + B)1/2 ητ ] exp[i(1 + B)η2 ξ/2],

(6.5.14)

where η represents the soliton amplitude. For B = 0, this solution reduces to the scalar soliton of Section 5.2. For B = 0, it represents a vector soliton polarized at 45◦ with respect to the principal axes of the fiber. Because of the XPM interaction, the vector soliton is narrower by a factor of (1 + B)1/2 compared with the scalar soliton. For such a soliton, the combination of SPM and XPM compensates for the GVD. At the same time, the carrier frequencies of two polarization components must be different for compensating the PMD. This can be seen by substituting Eq. (6.5.14) in Eq. (6.5.9). The canonical form of the vector soliton, obtained by setting η = 1 is then given by u(ξ, τ ) = sech[(1 + B)1/2 τ ] exp[i(1 + B + δ 2 )ξ/2 − iδτ ],

(6.5.15)

v(ξ, τ ) = sech[(1 + B)

(6.5.16)

1/2

τ ] exp[i(1 + B + δ )ξ/2 + iδτ ]. 2

The only difference between u(ξ, τ ) and v(ξ, τ ) is the sign of the last phase term involving the product δτ . This sign change reflects the shift of the carrier frequency of the u and v components in the opposite directions. The solution given in Eq. (6.5.14) represents only one of the several solitary-wave solutions that have been discovered in birefringent fibers by solving Eqs. (6.1.19) and (6.1.20) under various approximations. In one case, the two components not only have an asymmetric shape but they can also have a double-peak structure [82]. In another interesting class of solutions, two solitary waves form bound states such that the SOP is not constant over the entire pulse but changes with time [83]. In some cases the SOP evolves periodically along the fiber length. During the 1990s, several other solitary-wave solutions were found [89–99]. We refer the reader to Ref. [77] for further details. Similar to the case of modulation instability, we should ask whether vector solitons exist in isotropic fibers with no birefringence. In this case, we should use Eqs. (6.5.1) and (6.5.2), written in terms of the circularly polarized components defined in Eq. (6.5.3), and set b = 0. The solitary-wave solutions of these equations are found assuming u+ = U exp(ipξ ) and u− = V exp(iqξ ), where U and V do not vary with ξ and satisfy d2 U 2 = 2pU − (U 2 + 2V 2 )U , 2 dτ 3 2 2 d2 V = 2qV − (V + 2U 2 )V . dτ 2 3

(6.5.17) (6.5.18)

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These equations have a variety of solutions representing vector solitary waves with different polarization properties that can exist in isotropic fibers [96–98]. The simplest solution corresponds to a circularly polarized vector soliton of the form 



U (τ ) = 6p sech( 2p τ ),

V (τ ) = 0.

(6.5.19)

A linearly polarized vector soliton is obtained when q = p and has the form 



U (τ ) = V (τ ) = 2p sech( 2p τ ).

(6.5.20)

It corresponds to a linearly polarized pulse whose electric field vector may be oriented at any angle in the plane transverse to the fiber axis. Elliptically polarized vector solitons with U (τ ) > V (τ ) > 0 can also form in isotropic fibers [96]. However, it is not possible to write an analytic form for them. Their origin is related to the phenomenon of soliton trapping and can be understood as follows. In the case of V U, the solution of U (τ ) given in Eq. (6.5.19) can be used in Eq. (6.5.18). The resulting linear equation for V (τ ) has two types of bound solutions. These symmetric and asymmetric solution are given by [97] 

Vs (τ ) = V0 sechs ( 2pτ ),   Va (τ ) = V0 sechs−1 ( 2pτ ) tanh( 2pτ ),

qs = (4 − s)p, qa = (5 − 3s)p,

(6.5.21) (6.5.22)

√

where s = 12 ( 17 − 1) ≈ 1.562. These solutions are valid as long as V02 2p. As long as this condition is satisfied, the fast-axis component is trapped by the much stronger slow-axis component of the same pulse, and the two form a bound vector soliton. Notice that qs exceeds p for the symmetric branch but qa < p for the asymmetric branch. Also, the state of polarization is not uniform across the pulse for such vector solitons and has an ellipticity that varies with τ . Moreover, the axes of the ellipse rotate at a fixed rate q − p as the pulse propagates down the fiber. Numerical simulations show that elliptically polarized vector solitons with comparable magnitudes of U and V also exist [96]. Although their stability over long distances is not guaranteed, such solitons can propagate stably over hundreds of dispersion lengths when they correspond to the symmetric branch. In contrast, the double-hump vector solitons on the asymmetric branch are much less stable and they loose their stability over distances z < 100LD [97]. In addition to these pulse-like solitons, dark vector solitons also form in the case of normal GVD [96]. Vector solitons in isotropic Kerr media have continued to attract attention in recent years [100–104].

6.6. Higher-order effects As discussed in Section 2.3, several higher-order effects such as third-order dispersion (TOD) and intrapulse Raman scattering must be included for pulses shorter than 1 ps.

Polarization effects

These were considered in Section 5.5 for the scalar NLS equation. Here we extend the coupled mode equations (6.1.11) and (6.1.12) by including the higher-order effects so that they govern the evolution of ultrashort pulses inside birefringent fibers.

6.6.1 Extended coupled-mode equations The TOD effects can be included easily by adding a third derivative term containing β3 on the left side of Eqs. (6.1.11) and (6.1.12). The right side of these equations is also modified when we include the delayed nature of the Raman nonlinearity (see Appendix B). Denoting this contribution by Rx , Eq. (6.1.11) takes the form [105–107] ∂ Ax ∂ Ax iβ2 ∂ 2 Ax iβ3 ∂ 3 Ax α + β1x + − + Ax ∂z ∂t 2 ∂ t2 6 ∂ t3 2

   2 i γ = iγ (1 − fR ) |Ax |2 + |Ay |2 Ax + A∗x A2y e−2iβ z + iγ fR Rx ,

3

3

(6.6.1)

where Rx is given by 

∞

Rx = A x 0

fb + Ay 2





hR (s)|Ax (t − s)|2 ds + fa Ax ∞

0

∞

ha (s)|Ay (t − s)|2 ds

0



hb (s) Ax (t − s)A∗y (t − s) + A∗x (t − s)Ay (t − s)e−2iβ z ds.

(6.6.2)

The equation for Ay is obtained by interchanging the subscripts x and y in the preceding two equations. Here hR = fa ha + fb hb is the Raman response function introduced in Section 2.3, fa and fb represent the relative weights of ha (t) and hb (t) such that fa + fb = 1 (see Appendix B); for silica fibers fb = 0.21. The physical meaning of the four terms in Eq. (6.6.2) is clear. The first two terms lead to the Raman-induced, selfand cross-spectral shifts and the last two are the FWM-like terms, only one of which is phase-matched. If the input pulse is polarized along the slow or the fast axis, only the first term survives and we recover the scalar result. Eq. (6.6.1), together with its partner equation for Ay , can be used to investigate the propagation of femtosecond pulses inside optical fibers, while retaining all polarizations effects. In the case of isotropic fibers, we can set β = 0. When β is relatively large, the terms containing it can be neglected. The impact of the Raman terms on modulation instability was investigated in a 1994 study [107]. More recent work has considered the phenomena such as perturbation of vector solitons by the TOD and the impact of Raman terms on the SOP of ultrashort pulses [108–111]. As before, it is useful to convert the coupled NLS equations into a dimensionless form by using τ = (t − β¯1 z)/T0 ,

ξ = z/LD ,



Ax = u P0 e−iβ z/2 ,



Ay = v P0 eiβ z/2 , (6.6.3)

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where T0 and P0 are the width and the peak power of input pulses and β¯1 = (β1x +β1y )/2 is the average group delay. The resulting normalized NLS equations take the form  ∂u ∂u i +δ + bu + ∂ξ ∂τ   ∂v ∂v i −δ − bv + ∂ξ ∂τ 

1 ∂ 2u ∂ 3u + i δ = −N 2 Qx , 3 2 ∂τ 2 ∂τ 3 1 ∂ 2v ∂ 3v + i δ = −N 2 Qy , 3 2 ∂τ 2 ∂τ 3

(6.6.4) (6.6.5)

where we assumed β2 < 0 and introduced three dimensionless parameters b=

T02 (β) , 2|β2 |

δ=

T0 (β1x − β1y ), 2|β2 |

δ3 =

β3 . 6|β2 |T0

(6.6.6)

The nonlinear terms in Eqs. (6.6.4) and (6.6.5) include both the Kerr and Raman contributions and are given by

  2 1 Qx = (1 − fR ) |u|2 u + |v|2 u + v2 u∗ 3 3   fb + fR u[hR ⊗ |u|2 + fa ha ⊗ |v|2 ] + v [hb ⊗ (uv∗ + vu∗ )] , 2

 2 2 1 2 ∗  2 Qy = (1 − fR ) |v| v + |u| v + u v 3 3   fb 2 2 + fR v[hR ⊗ |v| + fa ha ⊗ |u| ] + u [hb ⊗ (uv∗ + vu∗ )] , 2

(6.6.7)

(6.6.8)

where a ⊗ represents the convolution in Eq. (6.6.2). Before solving Eqs. (6.6.4) and (6.6.5), one must specify the two Raman response functions. It is common to employ the forms [110] ha (t) =

 t t τ12 + τ22 exp − sin , τ2 τ1 τ1 τ22

hb (t) =

 t (2τb − t) exp − , τb τb2

(6.6.9)

where ha has the form given in Eq. (2.3.39) and hb is from Ref. [112]. The appropriate parameter values for silica fibers are: fa = 0.79, fb = 0.21, fR = 0.245, τ1 = 12.2 fs, τ2 = 32 fs, and τb = 96 fs. In a more accurate treatment, fa is split into two parts as fa + fc with fc = 0.04 and the last term in Eqs. (6.6.7) and (6.6.8) is modified by replacing fb hb with fb hb + fc ha [110].

6.6.2 Impact of TOD and Raman nonlinearity As a simple example of the impact of TOD on vector solitons, we consider a short pulse (T0 = 100 fs) that is launched into a birefringent fiber with b = 0.1, δ = 0.1 and δ3 = 0.1. The pulse can have any SOP initially. We solve Eqs. (6.6.4) and (6.6.5) with an input in

Polarization effects

Figure 6.16 Temporal (top) and spectral (bottom) evolution of a vector soliton inside a birefringent fiber over 25LD for its x (left) and y (right) polarized components. Birefringence and TOD of the fiber are included using b = 0.1, δ = 0.1 and δ3 = 0.1. The Raman contribution to fiber’s nonlinearity is ignored. (After Ref. [110]; ©2018 OSA.)

the form u(0, τ ) = cos θ sech(τ ),

v(0, τ ) = sin θ sech(τ )eiψ ,

(6.6.10)

where the angles θ and ψ characterize the initial SOP of the pulse. As an example, we choose the initial SOP of the input pulse to be linear (ψ = 0) and take θ = 45◦ so that the x and the y components of the soliton are equally intense. The parameter N is taken to be 1.4 so that the input pulse would evolve toward a vector soliton in the absence of the higher-order effects. To isolate the impact of TOD on this vector soliton, we set fR = 0 and ignore the Raman effects. Fig. 6.16 shows the temporal (top row) and spectral (bottom row) evolution of the vector soliton inside the fiber over 25 dispersion lengths [110]. As expected, the two components of the vector soliton move at the same speed, in spite of a group-velocity mismatch resulting from birefringence. Ad discussed in Section 6.5.2, this is a consequence of the XPM that shifts the pulse spectra in the opposite directions and allows the two components to trap each other and move at a common speed. Tilting of the temporal trajectories toward the right is due to the TOD effects included through δ3 = 0.1; no tilt occurs for δ3 = 0, and a left tilt occurs for negative values of δ3 . The fringe-like pattern (spreading beyond τ = 100) represents the radiation in the form of a dispersive wave that is shed by the vector soliton in response to the TOD-induced perturbations. This interpretation is supported by the appearance of a new spectral peak in the pulse spectra near (ν − ν0 )T0 = 1 after a few dispersion lengths. However, this peak is more intense for Ax compared to Ay , which is also apparent from the larger amplitude of the

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Figure 6.17 SOP Evolution for the vector soliton (dark trace), dispersive wave (dotted trace), and pulse remnants (gray) (A) without and (B) with the Raman effects. In the Hammer projection, the longitude and latitude lines are 15◦ apart, with H and V marking the horizontal and vertical SOPs and the poles representing circular SOPs. (After Ref. [110]; ©2018 OSA.)

dispersive wave in the time domain. The reason behind this asymmetry is related is to the nonlinear rotation of the SOP of the vector soliton, a topic discussed next. To characterize the SOP evolution, it is common to trace the trajectory of the Stokes vector on the Poincaré sphere, as in Fig. 6.6. The components of the Stokes vector can be calculated using Eq. (6.3.15). However, they depend on time for a pulse because the SOP can be nonuniform temporally at a given distance, while also varying along the fiber length. To focus on length variations, the SOP is calculated at the intensity peak for both the dispersive wave and the vector soliton. Also, the Hammer projection is employed for the Poincaré sphere to display its front and back in a single plot. Fig. 6.17(A) shows how the SOP evolves along the fiber for the vector soliton, dispersive wave, and pulse remnants. Even though the SOP of all three entities changes with propagation, the resulting evolution patterns are quite distinct for them. The SOP of the vector soliton starts linearly polarized at θ = 45◦ (marked by an arrow in Fig. 6.17). The SOP soon becomes slightly elliptical and moves toward the center as energy is transferred from the y-polarized component to the x-polarized component of the pulse. We can understand this behavior qualitatively from Eq. (6.3.18). In the absence of the nonlinear effects, the SOP will move in a circular pattern around

Polarization effects

the center. However, the Kerr nonlinearity forces it to rotate around the S3 axis once the SOP becomes elliptical. As the linear and nonlinear rotations are comparable in magnitude, their combination leads to the pattern seen in Fig. 6.17(A). In contrast, the nonlinear effects are negligible for both the dispersive wave and the pulse remnants. As a result, the SOPs of both of them move in a circular fashion around the center, as expected from linear birefringence. The reason why the SOP of dispersive wave does not trace a perfect circle is related to the XPM-induced phase shifts produced by the vector soliton. Fig. 6.17(B) shows the drastic changes occurring in the SOP of the vector soliton when the Raman effects are turned on. Rather than moving in a loop-like pattern as seen in part (A), the SOP traces a curve that covers the entire Poincaré sphere. The initial evolution of the soliton’s SOP is similar to that in part (A) in the sense that it becomes elliptically polarized and moves toward the center in a circular pattern. However, the Raman-induced frequency shifts change the SOP in such a fashion that the SOP trajectory soon drifts far from its original location. By contrast, the SOPs of dispersive wave and pulse remnants still move in a circular fashion around the center. This is expected because these two evolve linearly, as dictated by the linear birefringence, and are not affected by the Raman effects. The temporal and spectral evolutions of the vector soliton look similar to those seen in Fig. 6.17, except for the Raman-induced red-shift of its spectrum and a corresponding slowing down of the soliton’s speed.

6.6.3 Interaction of two vector solitons The Raman nonlinearity and TOD also affect considerably the interaction of two vector solitons [111]. We can use Eqs. (6.6.4) and (6.6.5) to study such an interaction by modifying the input profiles in Eq. (6.6.10) as u(0, τ ) = cos θ [sech(τ + q0 ) + sech(τ − q0 )],

(6.6.11)

v(0, τ ) = sin θ [sech(τ + q0 ) + sech(τ − q0 )]e ,

(6.6.12)

iψ

where θ and ψ are the angles used to indicate the SOP of two input pulses separated by Ts = 2q0 T0 initially. This input corresponds to two arbitrarily polarized pulses that are separated in time but have the same temporal shape and optical spectra. We set ψ = 0 and choose θ = 45◦ with N = 1 for which the x and y components of the soliton have the same amplitude. Focusing on an isotropic fiber (b = δ = 0), we choose q0 = 3.5 to ensure that tails of two solitons overlap to some extent for them to interact nonlinearly. The relative phase φ between two pulses is set to zero to focus on the in-phase case that leads to an attractive force and periodic collisions of the two vector solitons. Fig. 6.18 shows the temporal evolution of the x (left) and y (right) components of the two solitons over 60 dispersion lengths (ξ = 0–60). In the top row, all higher-order effects are turned off by setting fR = 0 and δ3 = 0 so that the interaction of two vector

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Figure 6.18 Evolution of the x (left) and y (right) components of two temporally separated vector solitons in isotropic fibers. (top row) The ideal case with no higher-order effects; (middle row) Raman nonlinearity included; (bottom row) TOD is also included. (After Ref. [111]; ©2018 APS.)

solitons is governed by only the fast-responding Kerr nonlinearity. Similar to the scalar case discussed in Section 5.5.4, the two vector solitons attract each other and collide at a distance of about ξ = 25, after which they separate and recover their original spacing around ξ = 50. This process repeats periodically just as it does for two scalar in-phase solitons. It is remarkable that both vector solitons survive multiple collisions in spite of the fact that the underlying equations are not integrable in the sense of inverse scattering theory. Inclusion of the Raman nonlinearity (middle row in Fig. 6.18) changes this interaction scenario drastically. First, the spectrum of each pulse undergoes a Raman-induced red shift that slows down both solitons. As a result, their trajectories bend toward the right side. The Kerr nonlinearity still provides an attractive force that brings the two solitons closer initially. However, two pulses never collide and begin to separate from each other at a distance of about 40LD . Moreover, energy transfer (of about 18%) takes place from the leading pulse to the trailing pulse. Since a soliton must conserve N = 1, the soliton gaining energy becomes narrower, as the one losing energy broadens, as seen clearly in the middle row. The same behavior also occurs for scalar solitons and is a consequence of the delayed nature of the Raman nonlinearity [113]. The bottom row of Fig. 6.18 shows what happens when the TOD is also included using δ3 = 0.1. As seen earlier, TOD perturbs each soliton and forces it to shed some energy in the form of dispersive waves that spread rapidly. The amount of energy lost

Polarization effects

Figure 6.19 Evolution of SOP for the leading (S1, near center) and trailing (S2, dotted) solitons over 400LD inside a fiber with medium birefringence (b = 0.1, δ = 0.1): (top) Kerr nonlinearity alone; (bottom) Raman effects included. (After Ref. [111]; ©2018 APS.)

through dispersive waves is relatively small (< 2%). However, the presence of TOD also reduces the extent of energy transfer between two vector solitons. This can be deduced from the bottom row of Fig. 6.18 by noting that the right soliton is not as narrow as in the middle row. One may wonder what happens to the SOP of the two solitons as they propagate inside the fiber. The SOP changes do occur but they are relatively small in the absence of birefringence. However, if we include a relatively small birefringence by using b = 0.1 and δ = 0.1, we find that SOP changes drastically even when the TOD and Raman effects are ignored. The top part of Fig. 6.19 shows the evolution of the SOP of the leading soliton S1 and the trailing soliton S2 over a distance of 400LD . The input SOP of both solitons is marked by an arrow. The SOP changes initially in an identical fashion for both solitons but, after a few dispersion lengths, the SOPs of two solitons follow different trajectories on the Poincaré sphere. The reason that two trajectories are so different is related to the Kerr-induced energy transfer from S1 to S2. The bottom of Fig. 6.19 shows SOP evolution after the Raman contributions is turned on. Clearly, the SOPs of the two solitons evolve quite differently when the Raman effects are included. The Raman-induced spectral red shifts modify the SOP in such a fashion that trajectory S1 of the leading soliton moves toward the center (near H), while that of the trailing

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soliton S2 moves away from center (near V), in sharp contrast with the case of purely Kerr nonlinearity. As H and V denote the x and y polarized SOPs, the SOP of two solitons become nearly orthogonal in a sufficiently long fiber. The preceding discussion indicates that the higher-order effects can induce qualitatively new features and affect the SOP drastically when pulses shorter than 1 ps are propagated through a birefringent fiber. They should be included for an accurate modeling of the experiments making use of such short optical pulses.

6.7. Random birefringence As mentioned in Section 6.1, conventional optical fibers exhibit small birefringence that changes randomly over a length scale ∼10 m, unless polarization-maintaining fibers are used. Because fibers with randomly varying birefringence are often used in practice, it is important to study how optical pulses are affected by random birefringence changes. Indeed, this issue has been investigated extensively since 1986 in the context of optical communication systems [114–120]. In this section we focus on the effects of random birefringence.

6.7.1 Polarization-mode dispersion It is intuitively clear that the SOP of CW light propagating inside fibers with randomly varying birefringence will generally be elliptical and would change randomly along the fiber during propagation. In the case of optical pulses, the SOP can also be different for different parts of the pulse, unless the pulse propagates as a soliton. Such random polarization changes typically are not of concern for telecommunication systems because an optical receiver is insensitive to the SOP of the light incident on it (unless a coherentdetection scheme is employed). What affects such systems is not the random polarization state but pulse broadening induced by random changes in fiber’s birefringence. This is referred to as PMD-induced pulse broadening. The analytical treatment of PMD is quite complex in general because of its statistical nature. A simple model [114], first introduced in 1986, divides the fiber into a large number of sections. Both the degree of birefringence and the orientation of the principal axes remain constant in each section but change randomly from section to section. In effect, each fiber section can be treated as a phase plate, and a Jones matrix can be used for it [119]. Propagation of each frequency component associated with an optical pulse is then governed by a composite Jones matrix obtained by multiplying individual Jones matrices for each fiber section. This matrix shows that two principal SOPs exist for any fiber such that, when a pulse is polarized along them, the SOP at fiber’s output is frequency independent to first order, in spite of random changes in fiber birefringence. These states are analogs of the slow and fast axes associated with polarization-maintaining fibers. Indeed, the differential group delay T (relative time

Polarization effects

delay in the arrival time of the two polarization components) is largest for these two states [116]. The principal SOPs provide a convenient basis for calculating the moments of T [121]. The PMD-induced pulse broadening is characterized by the root-meansquare (RMS) value of T, obtained after averaging over random birefringence changes. Several approaches have been used to calculate this average [122–125]. The variance σT2 ≡ (T )2  turns out to be the same in all cases and is given by [126] σT2 (z) = 2(β1 )2 lc2 [exp(−z/lc ) + z/lc − 1],

(6.7.1)

where the intrinsic modal dispersion β1 = d(β)/dω is related to the difference in group velocities along the two principal SOPs. The parameter lc is the correlation length, defined as the length over which two polarization components remain correlated; its typical values are ∼10 m. For short distances such that z lc , σT = (β1 )z from Eq. (6.7.1), as expected for a polarization-maintaining fiber. For distances z > 1 km, a good estimate of pulse broadening is obtained using z lc . For a fiber of length L, σT in this approximation becomes  √ σT ≈ β1 2lc L ≡ Dp L ,

(6.7.2)

Measured values of√Dp vary from fiber to fiber in where Dp is the PMD parameter. √ the range Dp = 0.1 to 2 ps/ km [127]. Because of the L dependence, PMD-induced pulse √ broadening is relatively small compared with the GVD effects. If we use Dp = 0.1 ps/ km as a typical value, σT ∼ 1 ps for fiber lengths ∼100 km and can be ignored for pulse widths >10 ps. However, PMD becomes a limiting factor for systems designed to operate over long distances at high bit rates. Several other factors need to be considered in practice. The derivation of Eq. (6.7.1) assumes that the fiber link has no elements exhibiting polarization-dependent loss or gain. The presence of polarization-dependent losses along a fiber link can modify the PMD effects considerably [128–131]. Similarly, the effects of second-order PMD should be considered for fibers with relatively low values of Dp [118].

6.7.2 Vector form of the NLS equation As mentioned earlier, the SOP of a pulse becomes nonuniform across the pulse because of random changes in fiber birefringence. At the same time, PMD leads to pulse broadening. Such effects can be studied by generalizing the coupled NLS equations derived in Section 6.1, namely Eqs. (6.1.11) and (6.1.12), to the case in which a fiber exhibits random birefringence along its length. It is more convenient to write these equations in terms of the normalized amplitudes u and v defined as 

u = Ax γ LD eiβ z/2 ,



v = Ay γ LD e−iβ z/2 .

(6.7.3)

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If we also use soliton units and introduce normalized distance and time as ξ = z/LD ,

τ = (t − β 1 z)/T0 ,

(6.7.4)

where β 1 = 12 (β1x + β1y ), Eqs. (6.1.11) and (6.1.12) take the form   ∂u 1 ∂ 2u ∂u i + |u|2 + +δ + bu + ∂ξ ∂τ 2 ∂τ 2    ∂v 1 ∂ 2v ∂v + | v |2 + i −δ − bv + ∂ξ ∂τ 2 ∂τ 2 



2 2 1 |v| u + v2 u∗ = 0, 3 3  2 2 1 2 ∗ |u| v + u v = 0, 3 3

(6.7.5) (6.7.6)

where b and δ are defined in Eq. (6.6.6). These equations can also be obtained from Eqs. (6.6.4) and (6.6.5) if we neglect the TOD term containing δ3 and set fR = 0 in Eqs. (6.6.7) and (6.6.8). The new feature in this section is that both b and δ vary randomly along the fiber’s length. Eqs. (6.7.5) and (6.7.6) can be written in a compact form using the Jones-matrix formalism. We introduce the Jones vector |U  and the Pauli matrices as [119]  

u , |U  = v





1 0 σ1 = , 0 −1





0 1 σ2 = , 1 0





0 −i σ3 = . i 0

(6.7.7)

In terms of the Jones vector |U , the coupled NLS equations become [133]   ∂|U  1 ∂ 2 |U  1 ∂|U  + s0 |U  − s3 σ3 |U  = 0, + σ1 b|U  + iδ + i ∂ξ ∂τ 2 ∂τ 2 3

(6.7.8)

where the Stokes parameters are defined in terms of |U  as [119] s1 = U |σ1 |U  = |u|2 − |v|2 , s0 = U |U  = |u|2 + |v|2 , s2 = U |σ2 |U  = 2 Re(u∗ v), s3 = U |σ3 |U  = 2 Im(u∗ v).

(6.7.9) (6.7.10)

These Stokes parameters are analogous to those introduced in Section 6.3.2 for describing the SOP of CW light on the Poincaré sphere. The main difference is that they are time dependent and describe the SOP of a pulse.They can be reduced to those of ∞ Section 6.3.2 by integrating over time such that Sj = −∞ sj (t) dt for j = 0 to 3. In the case of random birefringence, it is not enough to treat the parameters b and δ in Eq. (6.7.8) as random variables because the slow and fast axes themselves rotate along the fiber in a random fashion. To include such random rotations, it is common in numerical simulations to divide the fiber into sections of length lc or shorter and rotate the Jones vector at the end of each section through the transformation |U   = R |U , where R is a random rotation matrix of the form 

R=

cos θ sin θ eiφ −iφ − sin θ e cos θ



,

(6.7.11)

Polarization effects

and the angles θ and φ are distributed uniformly in the range [−π , π ] and [−π/2, π/2], respectively. Of course, b and δ also vary randomly from section to section. In a simple but reasonably accurate model, β is treated as a Gaussian stochastic process whose first two moments are given by β(z) = 0,

β(z)β(z ) = σβ2 exp(−|z − z |/lc ),

(6.7.12)

where σβ2 is the variance and lc is the correlation length of birefringence fluctuations.

6.7.3 Effects of PMD on solitons In the absence of nonlinear effects or low-energy pulses, one neglects the last two terms in Eq. (6.7.8). The resulting linear equation is solved in the frequency domain to study how the PMD affects an optical pulse [116–120]. The Fourier-domain approach cannot be used for pulses that propagate as solitons because the nonlinear effects play an essential role for solitons. An interesting question is how the XPM-induced coupling modifies the birefringence-induced PMD effects. This issue has been investigated in the context of long-haul telecom systems [132–135]. In the case of constant birefringence, it was seen in Section 6.5 that the orthogonally polarized components of a soliton can travel at the same speed, in spite of different group velocities associated with them at low powers. Solitons realize this synchronization by shifting their frequencies appropriately. It is thus not hard to imagine that solitons may avoid splitting and PMD-induced pulse broadening through the same mechanism. Indeed, numerical simulations based on Eqs. (6.1.11) and (6.1.12) indicate this to be the case [132] as long as the PMD parameter is small enough to satisfy the condition √ Dp < 0.3 |β2 |. It thus appears that solitons are relatively immune to random changes because of their particle-like nature. As in the CW case, the Stokes vector with components s1 , s2 , and s3 moves on the surface of the Poincaré sphere of radius s0 . When birefringence of the fiber varies randomly along the fiber, the tip of the Stokes vector moves randomly over the Poincaré sphere. The important question is on what length scale random variations in b occur compared with the soliton period or the dispersion length. For most fibers, random changes in b occur on a length scale ∼10 m. As they only affect the phases of u and v, it is clear that such changes leave s1 unchanged. As a result, the Stokes vector rotates rapidly around the s1 axis. In contrast, changes in the orientation of the birefringence axes leave s3 unchanged and thus rotate the Stokes vector around that axis. The combination of these two types of rotations forces the Stokes vector to fill the entire surface of the Poincaré sphere over a length scale ∼1 km. As this distance is typically much shorter than the dispersion length, a soliton cannot respond to random changes in birefringence. The situation is similar to the case of power variations occurring when fiber losses are compensated periodically using

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optical amplifiers (see Section 5.4). We can thus follow a similar approach and average Eq. (6.7.8) over random birefringence changes. The last term in this equation requires the average of s3 σ3 |U . This average turns out to be s0 |U /3 if we make use of the identity |U U | = s1 σ1 + s2 σ2 + s3 σ3 [119]. As b = 0 from Eq. (6.7.12), the two terms containing σ1 change sign often along the fiber length. If we ignore them to the first order and average over birefringence fluctuations, Eq. (6.7.8) reduces to i

∂|U  1 ∂ 2 |U  8 + s0 |U  = 0. + ∂ξ 2 ∂τ 2 9

(6.7.13)

The factor of 89 can be absorbed in the normalization used for |U  and amounts to reducing the nonlinear parameter γ , or increasing the input peak power P0 , by this factor. Recalling from Eq. (6.7.10) that s0 = |u|2 + |v|2 , Eq. (6.7.13) can be written in terms of the components u and v as ∂ u 1 ∂ 2u + (|u|2 + |v|2 )u = 0, + ∂ξ 2 ∂τ 2 ∂ v 1 ∂ 2v i + + (|v|2 + |u|2 )v = 0. ∂ξ 2 ∂τ 2

i

(6.7.14) (6.7.15)

As discussed in Section 6.5.3, this set of two coupled NLS equations is integrable by the inverse scattering method [78] and has the solution in the form of a fundamental vector soliton given in Eqs. (6.5.12) and (6.5.13). This solution shows that a fundamental soliton maintains the same polarization across the entire pulse “on average” in spite of random birefringence changes along the fiber. This is an extraordinary result and is indicative of the particle-like nature of solitons. In effect, solitons maintain uniform polarization across the entire pulse and resist small random changes in fiber birefringence [132]. Extensive numerical simulations based on Eqs. (6.7.5) and (6.7.6) confirm that solitons can maintain a uniform polarization state approximately over long fiber lengths even when optical amplifiers are used periodically for compensating fiber losses [133]. It is important to note that the vector soliton associated with Eqs. (6.7.14) and (6.7.15) represents the average behavior. The five parameters associated with this soliton (amplitude, frequency, position, phase, and polarization angle) will generally fluctuate along fiber length in response to random birefringence changes. Perturbation theory, similar to that used for scalar solitons in Section 5.4, can be used to study birefringenceinduced changes in soliton parameters [136–139]. For example, the amplitude of the soliton decreases and its width increases because of perturbations produced by random birefringence. The reason behind soliton broadening is related to the generation of dispersive waves and the resulting energy loss. From a practical standpoint, uniformity of soliton polarization is useful for polarization-division multiplexing.

Polarization effects

Problems 6.1 Derive an expression for the nonlinear part of the refractive index when an optical beam propagates inside a high-birefringence optical fiber. 6.2 Prove that Eqs. (6.1.15) and (6.1.16) indeed follow from Eqs. (6.1.11) and (6.1.12). 6.3 Prove that a high-birefringence fiber of length L introduces a relative phase shift of φNL = (γ P0 L /3) cos(2θ ) between the two linearly polarized components when a CW beam with peak power P0 and polarization angle θ propagates through it. Neglect fiber losses. 6.4 Explain the operation of a Kerr shutter. What factors limit the response time of such a shutter when optical fibers are used as the Kerr medium? 6.5 How can fiber birefringence be used to remove the low-intensity pedestal associated with an optical pulse? 6.6 Solve Eqs. (6.3.1) and (6.3.2) in terms of the elliptic functions. You can consult Reference [38]. 6.7 Prove that Eqs. (6.3.13) and (6.3.14) can be written in the form of Eq. (6.3.18) after introducing the Stokes parameters through Eq. (6.3.15). 6.8 What is meant by polarization instability in birefringent optical fibers? Explain the origin of this instability. 6.9 Derive the dispersion relation K () for modulation instability to occur in low-birefringence fibers starting from Eqs. (6.1.15) and (6.1.16). Discuss the frequency range over which the gain exists when β2 > 0. 6.10 Derive the dispersion relation K () for modulation instability to occur in highbirefringence fibers starting from Eqs. (6.1.22) and (6.1.23). Discuss the frequency range over which the gain exists when β2 > 0. 6.11 Solve Eqs. (6.5.4) and (6.5.5) numerically by using the split-step Fourier method. Reproduce the results shown in Figs. 6.14 and 6.15. Check the accuracy of Eq. (6.5.8) for δ = 0.2 and B = 23 . 6.12 Verify by direct substitution that the solution given by Eq. (6.5.14) satisfies Eqs. (6.5.4) and (6.5.5). 6.13 Explain the operation of soliton-dragging logic gates. How would you design a NOR gate by using such a technique? 6.14 Explain the origin of PMD in optical fibers. Why does PMD lead to pulse broadening. Do you expect PMD-induced broadening to occur for solitons?

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[94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139]

D.J. Kaup, B.A. Malomed, R.S. Tasgal, Phys. Rev. E 48 (1993) 3049. J.C. Bhakta, Phys. Rev. E 49 (1994) 5731. M. Haelterman, A.P. Sheppard, Phys. Rev. E 49 (1994) 3389. Y. Silberberg, Y. Barad, Opt. Lett. 20 (1995) 246. N.N. Akhmediev, A.V. Buryak, J.M. Soto-Crespo, D.R. Andersen, J. Opt. Soc. Am. B 12 (1995) 434. Y. Chen, J. Atai, Phys. Rev. E 55 (1997) 3652. F. Lu, Q. Lin, W.H. Knox, G.P. Agrawal, Phys. Rev. Lett. 93 (2004) 183901. M. Delqué, T. Sylvestre, H. Maillotte, C. Cambournac, P. Kockaert, M. Haelterman, Opt. Lett. 30 (2005) 3383. M. Delqué, G. Fanjoux, T. Sylvestre, Phys. Rev. E 75 (2007) 016611. Z.-B. Liu, X.-Q. Yan, W.-Y. Zhou, J.-G. Tian, Opt. Express 16 (2008) 8144. A.V. Kim, S.A. Skobelev, Phys. Rev. A 83 (2011) 063832. C.R. Menyuk, M.N. Islam, J.P. Gordon, Opt. Lett. 16 (1991) 566. S. Trillo, S. Wabnitz, J. Opt. Soc. Am. B 9 (1992) 1061. E.A. Golovchenko, A.N. Pilipetskii, J. Opt. Soc. Am. B 11 (1994) 92. R.C. Herrera, C.T. Law, Opt. Eng. 52 (2013) 015007. J. Szczepanek, T.M. Kardas, C. Radzewicz, Y. Stepanenko, Opt. Express 26 (2018) 13590. P. Balla, G.P. Agrawal, J. Opt. Soc. Am. B 35 (2018) 2302. P. Balla, G.P. Agrawal, Phys. Rev. A 98 (2018) 023822. Q. Lin, G.P. Agrawal, Opt. Lett. 31 (2006) 3086. P. Balla, G.P. Agrawal, J. Opt. Soc. Am. B 34 (2017) 1247. C.D. Poole, R.E. Wagnar, Electron. Lett. 22 (1986) 1029. F. Bruyère, Opt. Fiber Technol. 2 (1996) 269. C.D. Poole, J. Nagel, in: I.P. Kaminow, T.L. Koch (Eds.), Optical Fiber Telecommunications, Vol. 3A, Academic Press, 1997, Chap. 6. J.P. Gordon, H. Kogelnik, Proc. Natl. Acad. Sci. USA 97 (2000) 4541. H. Kogelnik, R.M. Jopson, L.E. Nelson, in: I.P. Kaminow, T. Li (Eds.), Optical Fiber Telecommunications, Vol. 4A, Academic Press, 2002, Chap. 15. J.N. Damask, Polarization Optics in Telecommunications, Springer, 2005. G.P. Agrawal, Lightwave Technology: Telecommunication Systems, Wiley, 2005. C.D. Poole, Opt. Lett. 13 (1988) 687. F. Curti, B. Diano, G. De Marchis, F. Matera, J. Lightwave Technol. 8 (1990) 1162. C.D. Poole, J.H. Winters, J.A. Nagel, Opt. Lett. 16 (1991) 372. N. Gisin, J.P. von der Weid, J.-P. Pellaux, J. Lightwave Technol. 9 (1991) 821. G.J. Foschini, C.D. Poole, J. Lightwave Technol. 9 (1991) 1439. P.K.A. Wai, C.R. Menyuk, J. Lightwave Technol. 14 (1996) 148. M.C. de Lignie, H.G. Nagel, M.O. van Deventer, J. Lightwave Technol. 12 (1994) 1325. B. Huttner, C. Geiser, N. Gisin, IEEE J. Sel. Top. Quantum Electron. 6 (2000) 317. Y. Li, A. Yariv, J. Opt. Soc. Am. B 17 (2000) 1821. D. Wang, C.R. Menyuk, J. Lightwave Technol. 19 (2001) 487. C. Xie, L.F. Mollenauer, J. Lightwave Technol. 21 (2003) 1953. L.F. Mollenauer, K. Smith, J.P. Gordon, C.R. Menyuk, Opt. Lett. 14 (1989) 1219. S.G. Evangelides, L.F. Mollenauer, J.P. Gordon, N.S. Bergano, J. Lightwave Technol. 10 (1992) 28. D. Marcuse, C.R. Menyuk, P.K.A. Wai, J. Lightwave Technol. 15 (1997) 1735. A. Levent, S.G. Rajeev, F. Yaman, G.P. Agrawal, Phys. Rev. Lett. 90 (2003) 013902. T. Ueda, W.L. Kath, Physica D 55 (1992) 166. M. Matsumoto, Y. Akagi, A. Hasegawa, J. Lightwave Technol. 15 (1997) 584. T.L. Lakoba, D.J. Kaup, Phys. Rev. E 56 (1997) 6147. C. de Angelis, P. Franco, M. Romagnoli, Opt. Commun. 157 (1998) 161.

CHAPTER 7

Cross-phase modulation So far we have focused on optical pulses whose spectrum is centered at a single wavelength. When two or more pulses, launched at different wavelengths, propagate simultaneously inside a fiber, they interact with each other through the fiber’s nonlinearity. In general, such an interaction can generate new waves under appropriate conditions through a variety of nonlinear phenomena such as stimulated Raman or Brillouin scattering and four-wave mixing; these are discussed in Chapters 8 to 10. The nonlinearity can also couple two optical fields through cross-phase modulation (XPM), without inducing any energy transfer between them [1]. The XPM phenomenon is discussed in this chapter. A set of two coupled nonlinear Schrödinger (NLS) equations is derived in Section 7.1, assuming that each wave maintains its state of polarization. These equations are used in Section 7.2 to discuss how the XPM affects the phenomenon of modulation instability. Section 7.3 focuses on the soliton pairs whose members support each other through their XPM-mediated nonlinear interaction. The effects of XPM on the shape and the spectrum of ultrashort pulses are described in Section 7.4 by solving the coupled NLS equations. Several applications of XPM are discussed in Section 7.5. A vector theory of XPM is developed in Section 7.6 to account for the polarization effects. In Section 7.7 we extend this theory to the case of birefringent fibers. The case of two counterpropagating waves is discussed in Section 7.8.

7.1. XPM-induced nonlinear coupling In this section we extend the theory of Section 2.3 to the case of two optical pulses of different wavelengths that overlap temporally as they propagate inside a single-mode fiber. In general, the two optical pulses may also differ in their states of polarization. We first focus on the case in which the two optical fields have different wavelengths but are polarized linearly along the same direction and maintain their state of polarization inside the fiber. The more general case of arbitrarily polarized pulses is discussed in Sections 7.6 to 7.8.

7.1.1 Nonlinear refractive index Similar to the single-wavelength case discussed in earlier chapters, it is useful to separate the rapidly varying part of the electric field by writing it in the form 



E(r, t) = 12 xˆ E1 exp(−iω1 t) + E2 exp(−iω2 t) + c.c., Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00014-2

(7.1.1)

Copyright © 2019 Elsevier Inc. All rights reserved.

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where xˆ is the polarization unit vector, ω1 and ω2 are the carrier frequencies of the two pulses, and the corresponding amplitudes E1 and E2 are slowly varying functions of time compared with an optical period. This requirement is equivalent to assuming that the spectral width of each pulse satisfies the condition ωj  ωj (j = 1, 2). Assuming pulses are wider than the response time of the nonlinearity, evolution of the slowly varying amplitudes E1 and E2 is governed by the wave equation (2.3.1), with the linear and nonlinear parts of the induced polarization given in Eqs. (2.3.5) and (2.3.6). To see the origin of XPM, we substitute Eq. (7.1.1) in Eq. (2.3.6) and find that the nonlinear polarization can be written as PNL (r, t) = 12 xˆ [PNL (ω1 )e−iω1 t + PNL (ω2 )e−iω2 t + PNL (2ω1 − ω2 )e−i(2ω1 −ω2 )t + PNL (2ω2 − ω1 )e−i(2ω2 −ω1 )t ] + c.c., (7.1.2) where the four terms depend on E1 and E2 as PNL (ω1 ) = χeff (|E1 |2 + 2|E2 |2 )E1 , PNL (ω2 ) = χeff (|E2 |2 + 2|E1 |2 )E2 ,

PNL (2ω1 − ω2 ) = χeff E12 E2∗ , PNL (2ω2 − ω1 ) = χeff E22 E1∗ ,

(7.1.3) (7.1.4)

(3) acting as an effective nonlinear parameter. with χeff = (30 /4)χxxxx Eq. (7.1.2) has terms oscillating at the new frequencies 2ω1 − ω2 and 2ω2 − ω1 . These terms are responsible for four-wave mixing (FWM) discussed in Chapter 10. It is necessary to satisfy a phase-matching condition for the new frequency components to build up significantly. The FWM terms are neglected in this chapter, assuming that phase matching does not occur. The remaining two terms provide a nonlinear contribution to the refractive index. This can be seen writing PNL (ωj ) in the form PNL (ωj ) = 0 jNL Ej and combining it with the linear part so that the total induced polarization is given by P (ωj ) = 0 j Ej , where j = 1 or 2,

j = jL + jNL = (nj + nj )2 ,

(7.1.5)

nj is the linear part of the refractive index, and nj is the change induced by the thirdorder nonlinear effects. With the approximation nj  nLj , the nonlinear part of the refractive index is given by (j = 1, 2) nj ≈ jNL /2nj ≈ n¯ 2 (|Ej |2 + 2|E3−j |2 ),

(7.1.6)

where the nonlinear parameter n¯ 2 is defined in Eq. (2.3.13). Eq. (7.1.6) shows that the refractive index seen by an optical field inside an optical fiber depends not only on the intensity of that field but also on the intensity of other copropagating fields [2–4]. As the optical field propagates inside the fiber, it acquires an intensity-dependent nonlinear phase shift φjNL (z) = (ωj /c )nj z = n2 (ωj /c )(|Ej |2 + 2|E3−j |2 )z,

(7.1.7)

Cross-phase modulation

where j = 1 or 2. The first term is responsible for SPM discussed in Chapter 4. The second term results from phase modulation of one wave by the second copropagating wave; it represents XPM between the two waves. The factor of 2 in this term shows that XPM is twice as effective as SPM for the same intensity [1]. Its origin can be traced back to the number of terms that contribute to the triple sum implied in Eq. (2.3.6). Qualitatively speaking, the number of terms doubles when the two optical frequencies are distinct compared with that when the frequencies are degenerate. The XPM-induced phase shift in optical fibers was measured in 1984 by injecting two continuous-wave (CW) beams into a 15-km-long fiber [3]. Soon after, picosecond pulses were used to observe the XPM-induced spectral changes [4–6].

7.1.2 Coupled NLS equations The pulse-propagation equations for the two optical fields can be obtained by following the procedure of Section 2.3. Assuming that the nonlinear effects do not affect significantly the fiber modes, the transverse dependence can be factored out writing Ej (r, t) in the form Ej (r, t) = Fj (x, y)Aj (z, t) exp(iβ0j z),

(7.1.8)

where Fj (x, y) is the transverse distribution of the fiber mode for the jth field (j = 1, 2), Aj (z, t) is the slowly varying amplitude, and β0j is the corresponding propagation constant. The dispersive effects are included by expanding the frequency-dependent propagation constant βj (ω) for each wave in a way similar to Eq. (2.3.23) and retaining only up to the quadratic term. The resulting propagation equation for Aj (z, t) becomes  ∂ Aj ∂ Aj iβ2j ∂ 2 Aj αj in2 ωj  + β1j + + Aj = fjj |Aj |2 + 2fjk |Ak |2 , 2 ∂z ∂t 2 ∂t 2 c

(7.1.9)

where k = j, β1j = 1/vgj , vgj is the group velocity, β2j is the GVD parameter, and αj is the loss coefficient. The overlap integral fjk is defined as (j, k = 1, 2)  ∞

2 |F (x, y)|2 dx dy k   ∞ . 2 2 −∞ |Fj (x, y)| dx dy −∞ |Fk (x, y)| dx dy

fjk =  ∞

−∞ |Fj (x, y)|

(7.1.10)

The differences among the overlap integrals can be significant if the two waves propagate in different fiber modes (see Chapter 14). Even in single-mode fibers, f11 , f22 , and f12 may differ from each other because of the frequency dependence of Fj (x, y). The difference is small, however, and can be neglected in practice. In that case, Eq. (7.1.9) can be written as the following set of two coupled NLS equations [7–9]

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∂ A1 1 ∂ A1 iβ21 ∂ 2 A1 α1 + + + A1 = iγ1 (|A1 |2 + 2|A2 |2 )A1 , ∂z vg1 ∂ t 2 ∂ t2 2

(7.1.11)

∂ A2 1 ∂ A2 iβ22 ∂ 2 A2 α2 + + + A2 = iγ2 (|A2 |2 + 2|A1 |2 )A2 , ∂z vg2 ∂ t 2 ∂ t2 2

(7.1.12)

where the nonlinear parameter γj is defined similar to Eq. (2.3.29) as γj = n2 ωj /(cAeff ),

(j = 1, 2),

(7.1.13)

and Aeff is the effective mode area (Aeff = 1/f11 ), assumed to be the same for both optical waves. It is important to realize that the two pulses propagate at different speeds because of the difference in their group velocities. The group-velocity mismatch depends on the wavelength difference between two pulses, and it limits their XPM interaction by separating pulses from each other. We can define the walk-off length LW through Eq. (1.2.13). This length is a measure of the distance over which two initially overlapping pulses separate from each other as a result of their different speeds.

7.2. XPM-induced modulation instability This section extends the analysis of Section 5.1 to the case in which two CW beams of different wavelengths propagate inside a fiber simultaneously [10–16]. Similar to the single-beam case, modulation instability is expected to occur in the anomalous-GVD region of the fiber. The main issue is whether XPM-induced coupling can destabilize the CW state even when one or both beams experience normal GVD.

7.2.1 Linear stability analysis The following analysis is similar to that of Section 6.4.2. The two differences are that XPM-induced coupling is stronger and the parameters β2 and γ are different for the two beams because of their different wavelengths. As usual, the steady-state solution is obtained by setting the time derivatives in Eqs. (7.1.11) and (7.1.12) to zero. If fiber losses are neglected, the solution is of the form 

¯ j (z) = Pj exp[iφj (z)], A

φj (z) = γj (Pj + 2P3−j )z,

(7.2.1)

where j = 1 or 2, Pj is the incident optical power, and φj is the nonlinear phase shift acquired by the jth field. Following the procedure of Section 5.1, stability of the steady state is examined assuming a time-dependent solution of the form Aj =





Pj + aj exp(iφj ),

(7.2.2)

Cross-phase modulation

where aj (z, t) is a small perturbation. By using Eq. (7.2.2) in Eqs. (7.1.11) and (7.1.12) and linearizing in a1 and a2 , the perturbations a1 and a2 satisfy the following set of two coupled linear equations:  ∂ a1 1 ∂ a1 iβ21 ∂ 2 a1 ∗ + + = i γ P ( a + a ) + 2i γ P1 P2 (a2 + a∗2 ), 1 1 1 1 1 ∂z vg1 ∂ t 2 ∂ t2  ∂ a2 1 ∂ a2 iβ22 ∂ 2 a2 + + = iγ2 P2 (a2 + a∗2 ) + 2iγ2 P1 P2 (a1 + a∗1 ). 2 ∂z vg2 ∂ t 2 ∂t

(7.2.3) (7.2.4)

The preceding set of linear equations has the following general solution: aj = uj exp[i(Kz − t)] + ivj exp[−i(Kz − t)],

(7.2.5)

where  is the frequency of perturbation, and K is its wave number. Eqs. (7.2.3) to (7.2.5) provide a set of four homogeneous equations for u1 , u2 , v1 , and v2 . This set has a nontrivial solution only when the perturbation satisfies the dispersion relation [(K − /vg1 )2 − f1 ][(K − /vg2 )2 − f2 ) = CXPM ,

(7.2.6)

fj = 12 β2j 2 ( 12 β2j 2 + 2γj Pj )

(7.2.7)

where

for j = 1 or 2. The coupling parameter CXPM is defined as CXPM = 4β21 β22 γ1 γ2 P1 P2 4 .

(7.2.8)

The steady-state solution becomes unstable if the wave number K has an imaginary part for some values of . The perturbations a1 and a2 then experience an exponential growth along the fiber length. In the absence of XPM coupling (CXPM = 0), Eq. (7.2.6) shows that the analysis of Section 5.1 applies to each wave independently. In the presence of XPM coupling, Eq. (7.2.6) provides a fourth-degree polynomial in K whose roots determine the conditions under which K becomes complex. In general, these roots are obtained numerically. If the wavelengths of the two optical beams are close to each other, or they are located on opposite sides of the zero-dispersion wavelength (ZDWL) such that vg1 ≈ vg2 , the four roots are given by [10] K = /vg1 ± { 12 (f1 + f2 ) ± [(f1 − f2 )2 /4 + CXPM ]1/2 }1/2 .

(7.2.9)

It is easy to verify that K can become complex only if CXPM > f1 f2 . Using Eqs. (7.2.7) and (7.2.8), the condition for modulation instability to occur can be written as [2 / 2c1 + sgn(β21 )][2 / 2c2 + sgn(β22 )] < 4,

(7.2.10)

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Figure 7.1 Gain spectra of XPM-induced modulation instability in the normal-dispersion regime of a fiber for (i) different power ratios with δ = 0 and (ii) different values of δ with equal beam powers. (After Ref. [10]; ©1987 APS.)

where cj = (4γj Pj /|β2j |)1/2 with j = 1 or 2. When the condition (7.2.10) is satisfied, the gain spectrum of modulation instability is obtained using g() = 2 Im(K ). The condition (7.2.10) shows that there is a range of  over which the gain g() exists. It also shows that modulation instability can occur irrespective of the signs of the GVD coefficients. Thus, whereas modulation instability requires anomalous GVD in the case of a single beam (see Section 5.1), it can occur in the two-beam case even if both beams experience normal GVD. The frequency range over which g() > 0 depends on whether β21 and β22 are both positive, both negative, or one positive and the other negative. The smallest frequency range corresponds to the case in which both beams are in the normal-GVD regime of the fiber. Because modulation instability in that case is due solely to XPM, only this case is discussed further. Fig. 7.1 shows the gain spectra of XPM-induced modulation instability for silica fibers in the visible region near 0.53 µm choosing β2j = 60 ps2 /km and γj = 15 W−1 /km in Eq. (7.2.7). The group-velocity mismatch is neglected in the left graph where different curves correspond to values of the power ratio P2 /P1 in the range 0–2. The right graph shows the effect of group-velocity mismatch for equal beam powers by −1 −1 varying the parameter δ = |vg1 − vg2 | in the range 0–3 ps/m. These results show that XPM-induced modulation instability can occur in the normal-GVD regime for relatively small values of δ . The peak gain of about 5 m−1 at the 100-W power level implies that the instability can develop inside fibers a few meters long.

7.2.2 Experimental results The experimental attempts to observe the XPM-induced modulation instability in the normal-GVD region of an optical fiber have focused mostly on the case of two polarization components of a single beam (see Section 6.4). This instability is difficult

Cross-phase modulation

to observe in the case of two beams of different wavelengths. The reason is related to the fact that Eqs. (7.1.11) and (7.1.12) neglect FWM. The neglect of the FWM terms can be justified when the wavelength difference is so large that phase matching cannot occur [13–16]. However, to observe modulation instability, the wavelength difference needs to be reduced to ∼1 nm or less. Four-wave mixing then becomes nearly phase matched and cannot be ignored. Indeed, a careful analysis that includes dispersion to all orders shows that XPM-induced modulation instability is not likely to occur in the normal-GVD region of conventional silica fibers [16]. It can occur in especially designed, dispersion-flattened fibers in which two normal-GVD regions are separated by an intermediate wavelength region of anomalous dispersion. In such fibers, it is possible to match the group velocities even when the wavelengths of two beams differ by 100 nm or more. The XPM-induced modulation instability has been observed when only one of the beams propagates in the normal-GVD region. In a 1988 experiment [17], a pump-probe configuration was used such that the 1.06-µm pump pulses experienced normal GVD, while 1.32-µm probe pulses propagated in the anomalous-GVD region of the fiber. When the pump and probe pulses were launched simultaneously, the probe developed modulation sidebands, with a spacing of 260 GHz at the 0.4-W peak-power level of pump pulses. This configuration can be used to advantage if the pump beam is in the form of intense pulses whereas the other beam forms a weak CW signal. The XPM-induced modulation instability in this situation converts the weak CW beam into a train of ultrashort pulses because it occurs only when the two waves are present simultaneously [8]. In a 1990 realization of the preceding scheme [18], 100-ps pump pulses were obtained from a Nd:YAG laser operating at 1.06-µm, while a semiconductor laser provided the weak CW prove (power 0. If fiber losses are ignored (α1 = α2 = 0) and group velocities are assumed to be equal by setting vg1 = vg2 = vg in Eqs. (7.1.11) and (7.1.12), the bright–dark soliton pair is given by [21] A1 (z, t) = B1 tanh[W (t − z/V )] exp[i(K1 z − 1 t)], A2 (z, t) = B2 sech[W (t − z/V )] exp[i(K2 z − 2 t)],

(7.3.1) (7.3.2)

where the soliton amplitudes are determined from B12 = (2γ1 β22 + γ2 |β21 |)W 2 /(3γ1 γ2 ), B22 = (2γ2 |β21 | + γ1 β22 )W 2 /(3γ1 γ2 ),

(7.3.3) (7.3.4)

the wave numbers K1 and K2 are given by K1 = γ1 B12 − |β21 |21 /2,

K2 = β22 (22 − W 2 )/2,

(7.3.5)

and the effective group velocity of the soliton pair is obtained from V −1 = vg−1 − |β21 |1 = vg−1 + β22 2 .

(7.3.6)

As seen from Eq. (7.3.6), the frequency shifts 1 and 2 must have opposite signs and cannot be chosen independently. The parameter W governs the pulse width and determines the soliton amplitudes through Eqs. (7.3.3) and (7.3.4). Thus, two members of the soliton pair have the same width, the same group velocity, but different shapes

Cross-phase modulation

and amplitudes such that they support each other through the XPM coupling. In fact, their shapes correspond to bright and dark solitons discussed in Chapter 5. The most striking feature of this soliton pair is that the dark soliton propagates in the anomalousGVD region, whereas the bright soliton propagates in the normal-GVD region, exactly opposite of the behavior expected in the absence of XPM. The physical mechanism behind such an unusual pairing can be understood as follows. Because XPM is twice as strong as SPM, it can counteract the temporal spreading of an optical pulse induced by the combination of SPM and normal dispersion, provided the XPM-induced chirp is of the opposite kind than that produced by SPM. A dark soliton can generate this kind of chirp. At the same time, the XPM-induced chirp on the dark soliton is such that the pair of bright and dark solitons can support each other in a symbiotic manner.

7.3.2 Bright–gray soliton pair A more general form of an XPM-coupled soliton pair can be obtained by solving Eqs. (7.1.11) and (7.1.12) with the postulate Aj (z, t) = Qj (t − z/V ) exp[i(Kj z − j t + φj )],

(7.3.7)

where V is the common velocity of the soliton pair, Qj governs the soliton shape, Kj and j represent changes in the propagation constant and the frequency of two solitons, and φj is the phase (j = 1, 2). The resulting solution has the form [25] Q1 (τ ) = B1 [1 − b2 sech2 (W τ )],

Q2 (τ ) = B2 sech(W τ ),

(7.3.8)

where τ = t − z/V . The parameters W and b depend on the soliton amplitudes, B1 and B2 , and on fiber parameters through the relations 

W=

3γ1 γ2 2γ1 β22 − 4γ2 β21



1/2

B2 ,

b=

2γ1 β22 − γ2 β21 γ1 β22 − 2γ2 β21

1/2

B2 . B1

(7.3.9)

The constants K1 and K2 are also fixed by various fiber parameters and soliton amplitudes. The phase of the bright soliton is constant but the dark-soliton phase φ1 is time-dependent. The frequency shifts 1 and 2 are related to the soliton-pair speed as in Eq. (7.3.6). The new feature of the XPM-coupled soliton pair in Eq. (7.3.8) is that the dark soliton is of “gray” type. The parameter b controls the depth of the intensity dip associated with a gray soliton. Both solitons have the same width W but different amplitudes. A new feature is that the two GVD parameters can be positive or negative. However, the soliton pair exists only under certain conditions. The solution is always possible if β21 < 0 and β22 > 0 and does not exist when β21 > 0 and β22 < 0. As discussed before, this behavior is opposite to what would normally be expected and is due solely to

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XPM. If both solitons experience normal GVD, the bright–gray soliton pair can exist if γ1 β22 > 2γ2 β21 . Similarly, if both solitons experience anomalous GVD, the soliton pair can exist if 2γ1 |β22 | < γ2 |β21 |. The preceding soliton-pair solutions are not the only possible solutions of Eqs. (7.1.11) and (7.1.12). These equations also support pairs with two bright or two dark solitons depending on various parameter values [22]. Moreover, the XPMsupported soliton pairs can exist even when group velocities are not equal because, similar to soliton trapping in birefringent fibers (see Section 6.5), two pulses can shift their carrier frequencies to equalize their group velocities. A simple way to find the conditions under which XPM-paired solitons can exist is to postulate an appropriate solution, substitute it in Eqs. (7.1.11) and (7.1.12), and then investigate whether soliton parameters (amplitude, width, group velocity, frequency shift, and wave number) can be determined with physically possible values [25–28]. As an example, consider the case when Eqs. (7.3.1) and (7.3.2) describe the postulated solution. Assume also that K1 = K2 and 1 = 2 so that the frequency shifts are equal. It turns out that the postulated solution is always possible if β21 < 0 and β22 > 0, but exists only under certain conditions if β21 and β22 have the same sign [27]. Further, the assumption 1 = 2 can be relaxed to obtain another set of solitary-wave solutions. Stability of the XPM-paired solitons is not always guaranteed and should be checked through numerical simulations.

7.3.3 Periodic solutions The coupled NLS equations (7.1.11) and (7.1.12) also have periodic solutions that represent two pulse trains that propagate undistorted through an optical fiber because of the XPM-induced coupling between them. One such periodic solution in terms of the elliptic functions was found in 1989 in the specific case in which both pulse trains have the same group velocity and experienced anomalous GVD inside the fiber [29]. By 1998, nine periodic solutions, written as different combinations of the elliptic functions, have been found [30]. All of these solutions assume equal group velocities and anomalous GVD for the two pulse trains. The case in which one wave experiences anomalous GVD, while the other propagates in the normal-dispersion region, is of practical interest because the group velocities for the two waves can then be matched with a proper choice of the zero-dispersion wavelength of the fiber. If we assume this to be the case, neglect fiber losses, and introduce normalized variables using ξ = z/LD ,

τ = (t − z/vg1 )/T0 ,

Aj = γ1 LD uj ,

(7.3.10)

Eqs. (7.1.11) and (7.1.12) can be written in the form i

∂ u1 d1 ∂ 2 u1 + (|u1 |2 + σ |u2 |2 )u1 = 0, − ∂ξ 2 ∂τ 2

(7.3.11)

Cross-phase modulation

i

∂ u2 d2 ∂ 2 u2 + (|u2 |2 + σ |u1 |2 )u2 = 0, + ∂ξ 2 ∂τ 2

(7.3.12)

where dj = |β2j /β20 |, β20 is a reference value used to define the dispersion length, and u1 is assumed to propagate in the normal-dispersion region. We have also assumed γ2 ≈ γ1 . The parameter σ has a value of 2 when all waves are linearly polarized but becomes 1, the solution has the following form:

u1 (ξ, τ ) = r

u2 (ξ, τ ) = r

σ d2 + d1 dn(r τ, p)[q dn−2 (r τ, p) ∓ 1] exp(iQ1± ξ ), σ2 − 1

(7.3.13)

d2 + σ d1 p2 sn(r τ, p)cn(r τ, p) exp(iQ2± ξ ), σ2 − 1 dn(r τ, p)

(7.3.14)

where the propagation constants Q1 and Q2 are given by Q1± = Q2± =

r2 σ2

−1

r2 σ2 − 1

[σ (d2 + σ d1 )(1 + q2 ) ∓ 2q(σ d2 + d1 )] −

r 2 d1 (1 ∓ q)2 , 2

[(d2 + σ d1 )(1 + q2 ) ∓ 2qσ (σ d2 + d1 )].

(7.3.15) (7.3.16)

In these equations, sn, cn, and dn are the standard Jacobi elliptic functions with the modulus p (0 < p < 1) and the period 2K (p)/r, where K (p) is the complete elliptic integral of the first kind, q = (1 − p2 )1/2 , and r is an arbitrary scaling constant. Two families of periodic solutions correspond to the choice of upper and lower signs. For each family, p can take any value in the range of 0 to 1. The preceding solution exists only for σ > 1. When σ < 1, the following single family of periodic solutions is found [31]:

u1 (ξ, τ ) = r

u2 (ξ, τ ) = r

σ d2 + d1 p sn(r τ, p) exp(iQ1 ξ ), 1 − σ2

(7.3.17)

d2 + σ d1 p dn(r τ, p) exp(iQ2 ξ ), 1 − σ2

(7.3.18)

where the propagation constants Q1 and Q2 are given by Q1 = − 12 r 2 d1 q2 + r 2 (d1 + σ d2 )/(1 − σ 2 ), Q2 =

2 2 1 2 2 r d2 (1 + q ) + r σ (d1

(7.3.19)

+ σ d2 )/(1 − σ ). 2

(7.3.20)

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It is well known that period of any Jacobi elliptic function becomes infinite in the limit p = 1. In this limit, the preceding periodic solutions reduce to the bright–dark soliton pairs discussed earlier in this section. We should stress that the mere existence of a periodic or solitary-wave solution does not guarantee that it can be observed experimentally. The stability of such solutions must be studied by perturbing them and then propagating the perturbed field over long distances. Numerical simulations show that all periodic solutions are unstable in principle, but the distance after which instability sets in depends on the strength of perturbation [31]. In particular, a periodic solution can survive over tens of dispersion lengths for relatively weak perturbations.

7.3.4 Multiple coupled NLS equations The concept of XPM-paired solitons can be easily generalized to multicomponent solitons in which multiple pulses at different carrier frequencies are transmitted over the same fiber. In practice, such a situation occurs naturally in wavelength-divisionmultiplexed systems [32]. In this case, in place of Eqs. (7.1.11) and (7.1.12), one needs to solve a set of multiple coupled NLS equations of the form 

∂ Aj 1 ∂ Aj iβ2j ∂ 2 Aj 2 2 + + = i | A | + σ γ | A | Aj , γ j j k k ∂z vgj ∂ t 2 ∂ t2

(7.3.21)

k=j

where j = −M to M and 2M + 1 is the total number of components. The dimensionless parameter σ indicates the XPM strength and has a value of 2 when all waves are linearly polarized. These equations have periodic as well as soliton solutions for certain combinations of parameter values [33–37]. In this section, we focus on the soliton solutions with multiple components. Such solitons are often referred to as multicomponent vector solitons [38]. We normalize Eq. (7.3.21) using the central j = 0 component as a reference and introduce the normalized variables ξ , τ , and uj as indicated in Eq. (7.3.10), where the dispersion length LD = T02 /|β20 | and β20 is taken to be negative. The set (7.3.21) can then be written as i

 ∂u

j

∂ξ

+ δj

∂ uj dj ∂ 2 uj  + γ j |Aj |2 + σ γk |Ak |2 Aj = 0, + 2 ∂τ 2 ∂τ

(7.3.22)

k=j

−1 where δj = vgj−1 − vg0 represents group-velocity mismatch with respect to the central component, dj = β2j /β20 , and the parameter γj is now dimensionless as it has been normalized with γ0 . The solitary-wave solution of Eq. (7.3.22) is found by seeking a solution in the form

uj (ξ, τ ) = Uj (τ ) exp[i(Kj ξ − j τ )],

(7.3.23)

Cross-phase modulation

where Kj is the propagation constant and j represents a frequency shift from the carrier frequency. It is easy to show that Uj satisfies the ordinary differential equation



d j d 2 Uj  1 2 2 2 + γ | U | + σ γ | U | U = λ U K − β  Uj , j j k k j j j j 2j j 2 dτ 2 2 k=j

(7.3.24)

provided frequency shifts are such that j = (β2j vgj )−1 and λj = Kj − δj2 /(2dj ). For the central j = 0 component, d0 = γ0 = 1. In the absence of XPM terms, this component has the standard soliton solution U0 (x) = sech(τ ). We assume that, in the presence of XPM, all components have the same “sech” shape but different amplitudes, i.e., Un (τ ) = an sech(τ ). Substituting this solution in Eq. (7.3.24), the amplitudes an are found to satisfy the following set of algebraic equations: a20 + σ



γn a2n = 1,

γn a2n + σ



γm a2m = dn .

(7.3.25)

m=n

n=0

As long as parameters dn and γn are close to 1 for all components, this solution describes a vector soliton with N components of nearly equal intensity. In the degenerate case, dn = γn = 1, the amplitudes can be found analytically [33] in the form Un = [1 + σ (N − 1)]−1/2 , where N is the total number of components. The stability of any multicomponent vector soliton is not guaranteed and should be studied carefully. Consult Ref. [38] for a detailed discussion of the stability issue.

7.4. Spectral and temporal effects In this section we focus on the spectral and temporal changes occurring as a result of XPM-induced nonlinear coupling between two optical pulses with non-overlapping spectra [39–45]. For simplicity, the polarization effects are ignored assuming that the input beams preserve their polarization during propagation. Eqs. (7.1.11) and (7.1.12) then govern evolution of two pulses along the fiber length and include the effects of group-velocity mismatch, GVD, SPM, and XPM. If fiber losses are neglected for simplicity, these equations become ∂ A1 iβ21 ∂ 2 A1 + = iγ1 (|A1 |2 + 2|A2 |2 )A1 , ∂z 2 ∂T 2 ∂ A2 ∂ A2 iβ22 ∂ 2 A2 +d + = iγ2 (|A2 |2 + 2|A1 |2 )A2 , ∂z ∂T 2 ∂T 2

(7.4.1) (7.4.2)

where T is measured in a reference frame moving at the speed vg1 as T =t−

z , vg1

d=

vg1 − vg2 , vg1 vg2

(7.4.3)

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Nonlinear Fiber Optics

and d is a measure of the differential group delay between the two pulses. In general, two pulses can have different widths. Using the width T0 of the pulse at the wavelength λ1 as a reference, we introduce the walk-off length LW and the dispersion length LD as LW = T0 /|d|,

LD = T02 /|β21 |.

(7.4.4)

Depending on the relative magnitudes of LW , LD , and the fiber length L, the two pulses can evolve very differently. If L is small compared to both LW and LD , the dispersive effects do not play a significant role and can be neglected. For example, this can occur for T0 > 100 ps and L < 10 m if the center wavelengths of the two pulses are within 10 nm of each other (|d| < 1 ps/m). In this quasi-CW situation, the steady-state solution of Section 7.3 is applicable. If LW < L but LD L, the second derivatives in Eqs. (7.4.1) and (7.4.2) can be neglected, but the first derivatives must be retained. Even though the pulse shape does not change under such conditions, the combination of group-velocity mismatch and the nonlinearity-induced frequency chirp can affect the spectrum drastically. This is generally the case for T0 ∼ 100 ps, L ∼ 10 m, and |d| < 10 ps/m. Finally, for short pulses (T0 < 10 ps), the GVD terms should also be included; XPM then affects both the shape and the spectrum of two pulses. Both of these cases are discussed in what follows.

7.4.1 Asymmetric spectral broadening Consider first the simple case L  LD for which the second-derivative terms in Eqs. (7.4.1) and (7.4.2) can be neglected. The group-velocity mismatch is included through the parameter d assuming LW < L. As the pulse shapes do not change in the absence of GVD, Eqs. (7.4.1) and (7.4.2) can be solved analytically. The general solution at z = L is given by [41] A1 (L , T ) = A1 (0, T )eiφ1 ,

A2 (L , T ) = A2 (0, T − dL )eiφ2 ,

(7.4.5)

where the time-dependent nonlinear phase shifts are obtained from 





L

φ1 (T ) = γ1 L |A1 (0, T )| + 2 |A2 (0, T − zd)| dz , 0   L 2 2 φ2 (T ) = γ2 L |A2 (0, T )| + 2 |A1 (0, T + zd)| dz . 2

2

(7.4.6) (7.4.7)

0

The physical interpretation of Eqs. (7.4.5) through (7.4.7) is clear. As the pulse propagates through the fiber, its phase is modulated because of the intensity dependence of the refractive index. The modulated phase has two contributions. The first term in Eqs. (7.4.6) and (7.4.7) is due to SPM (see Section 4.1) but the second term has

Cross-phase modulation

Figure 7.2 Spectra of two pulses exhibiting XPM-induced asymmetric spectral broadening. The parameters are γ1 P1 L = 40, P2 /P1 = 0.5, γ2 /γ1 = 1.2, τd = 0, and L/LW = 5.

its origin in XPM. The XPM contribution changes along the fiber length because of the group-velocity mismatch. The total XPM contribution to the phase is obtained by integrating over the fiber length. The integration in Eqs. (7.4.6) and (7.4.7) can be performed analytically for some specific pulse shapes. As an illustration, consider the case of two unchirped Gaussian pulses of the same width T0 with the initial amplitudes 





T2 A1 (0, T ) = P1 exp − 2 , 2T0

A2 (0, T ) =



 (T − Td )2 P2 exp − , (7.4.8) 2

2T0

where P1 and P2 are the peak powers and Td is the initial time delay between the two pulses. Substituting Eq. (7.4.8) in Eq. (7.4.6), φ1 is given by √  π 2 φ1 (τ ) = γ1 L P1 e−τ + P2 [erf(τ − τd ) − erf(τ − τd − δ)] , δ

(7.4.9)

where erf(x) stands for the error function and τ = T /T0 ,

τd = Td /T0 ,

δ = dL /T0 .

(7.4.10)

A similar expression can be obtained for φ2 (T ) using Eq. (7.4.7). As discussed in Section 4.1, the time dependence of the phase manifests as spectral broadening. Similar to the case of pure SPM, the spectrum of each pulse is expected to broaden and develop a multipeak structure. However, the spectral shape is now governed by the combined contributions of SPM and XPM to the pulse phase. Fig. 7.2 shows the spectra of two pulses using γ1 P1 L = 40, P2 /P1 = 0.5, γ2 /γ1 = 1.2, τd = 0, and δ = 5. These parameters correspond to an experimental situation in which a pulse at 630 nm, with 100-W peak power, is launched inside a fiber together with another pulse at

259

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Nonlinear Fiber Optics

530 nm with 50-W peak power such that Td = 0, T0 = 10 ps, and L = 5 m. The most noteworthy feature of Fig. 7.2 is spectral asymmetry that is due solely to XPM. In the absence of XPM interaction the two spectra would be symmetric and would exhibit less broadening. The spectrum of pulse 2 is more asymmetric because the XPM contribution is larger for this pulse (P1 = 2P2 ). A qualitative understanding of the spectral features seen in Fig. 7.2 can be developed from the XPM-induced frequency chirp using 



1 ∂φ1 γ1 L P2  −(τ −τd )2 2 2 ν1 (τ ) = − = P1 τ e−τ − e − e−(τ −τd −δ) , 2π ∂ T π T0 δ

(7.4.11)

where Eq. (7.4.9) was used. For τd = 0 and |δ|  1 (L  LW ), the chirp is given by the simple relation ν1 (τ ) ≈

γ1 L −τ 2 e [P1 τ + P2 (2τ − δ)]. π T0

(7.4.12)

The chirp for pulse 2 is obtained following the same procedure and is given by ν2 (τ ) ≈

γ2 L −τ 2 e [P2 τ + P1 (2τ + δ)]. π T0

(7.4.13)

For positive values of δ , the chirp is larger near the leading edge for pulse 1 while the opposite occurs for pulse 2. Because the leading and trailing edges carry red- and blue-shifted components, respectively, the spectrum of pulse 1 is shifted toward red while that of pulse 2 is shifted toward blue. This is precisely what occurs in Fig. 7.2. The spectrum of pulse 2 shifts less on the red side because the SPM contribution is much smaller for it. When P1 = P2 and γ1 ≈ γ2 , the spectra of two pulses would be the mirror images of each other. The qualitative features of spectral broadening can be quite different if the two pulses do not overlap initially but have a relative time delay [41]. To isolate the effects of XPM, it is useful to consider a pump-probe configuration assuming P1  P2 . The pump-induced chirp imposed on the probe pulse is obtained from Eq. (7.4.11) by neglecting the SPM contribution and is of the form ν1 (τ ) = sgn(δ)νmax exp[−(τ − τd )2 ] − exp[−(τ − τd − δ)2 ],

(7.4.14)

where νmax is the maximum XPM-induced chirp given by νmax =

γ1 P 2 L γ1 P 2 L W . = π T0 |δ| π T0

(7.4.15)

Note that νmax is set by the walk-off length LW , rather than the actual fiber length L. This is expected because the XPM interaction occurs as long as the two pulses overlap.

Cross-phase modulation

Figure 7.3 Optical spectra (left column) and XPM-induced phase and chirp (right column) for a probe pulse copropagating with a faster-moving pump pulse. The shape of probe pulse is shown by a dashed line. Three rows correspond to τd = 0, 2, and 4, respectively. (After Ref. [41]; ©1989 APS.)

Eq. (7.4.14) shows that the XPM-induced chirp can vary significantly along the probe pulse if τd and δ are of opposite signs. As a result, the probe spectrum can have qualitatively different features depending on the relative values of τd and δ . Consider, for example, the case in which the pump pulse travels faster than the probe pulse (δ < 0) and is delayed initially (τd ≥ 0). Fig. 7.3 shows the probe spectrum together with the phase φ1 and the chirp ν1 for δ = −4 and τd = 0, 2, and 4. The fiber length L and the pump peak power P2 are chosen such that γ1 P2 L = 40 and L /LW = 4. For reference, a 10-ps pump pulse with a group-velocity mismatch d = 10 ps/m has LW = 1 m. The probe spectrum in Fig. 7.3 is shifted toward blue with strong asymmetry for τd = 0. For τd = 2, it becomes symmetric while for τd = 4 it is again asymmetric with a shift toward red. In fact, the spectra for τd = 0 and τd = 4 are mirror images of each other about the central frequency ν1 = ω1 /(2π). The probe spectra can be understood physically by considering the XPM-induced chirp shown in the right column of Fig. 7.3. For τd = 0, the chirp is positive across the entire probe pulse, and the maximum chirp occurs at the pulse center. This is in contrast to the SPM case (shown in Fig. 4.1) where the chirp is negative near the leading edge, zero at the pulse center, and positive near the trailing edge. The differences

261

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Nonlinear Fiber Optics

in the SPM and XPM cases are due to group-velocity mismatch. When τd = 0, the slow-moving probe pulse interacts mainly with the trailing edge of the pump pulse. As a result, the XPM-induced chirp is positive and the probe spectrum has only blue-shifted components. When τd = 4, the pump pulse just catches up with the probe pulse at the fiber output. Its leading edge interacts with the probe; the chirp is therefore negative and the spectrum is shifted toward red. When τd = 2, the pump pulse has time not only to the catch up but pass through the probe pulse in a symmetric manner. The chirp is zero at the pulse center similar to the case of SPM. However, its magnitude is considerably small across the entire pulse. As a result, the probe spectrum is symmetrically broadened but its tails carry a relatively small amount of pulse energy. The probe spectrum in this symmetric case depends quite strongly on the ratio L /LW . If L /LW = 2 with τd = 1, the spectrum is broader with considerably more structure. By contrast, if L LW , the probe spectrum remains virtually unchanged. The XPM-induced spectral broadening has been observed experimentally in the pump-probe configuration. In one experiment [5], the 10-ps pump pulses were obtained from a color-center laser operating at 1.51 µm while the probe pulses at 1.61 µm were generated using a fiber-Raman laser (see Section 8.2). The walk-off length was about 80 m while the dispersion length exceeded 10 km. Both the symmetric and asymmetric probe spectra were observed as the fiber length was increased from 50 to 400 m and the effective delay between the pulses was varied using time-dispersion tuning. In a different experiment, a Nd:YAG laser was used to provide 33-ps pump pulses at 1.06 µm and 25-ps probe pulses at 0.53 µm [40]. The delay between two pulses was adjusted using a Mach–Zehnder interferometer. Because of a relatively large groupvelocity mismatch (d ≈ 80 ps/m), the walk-off length was only about 25 cm. For a 1-m-long fiber used in the experiment, L /LW = 4. The probe spectra were recorded by varying the delay Td and the peak power of the pump pulse. The spectra exhibited a shift toward the red or the blue side with some broadening as the multiple peaks could not be resolved. Fig. 7.4 shows the measured wavelength shift as a function of time delay Td . The solid curve is the theoretical prediction of Eq. (7.4.14). The frequency shift for a given time delay is obtained by maximizing ν1 (τ ). The maximum occurs near τ = 0, and the frequency shift is given by ν1 = νmax {exp(−τd2 ) − exp[−(τd + δ)2 ]},

(7.4.16)

where δ ≈ −4 for the experimental values of the parameters and τd = Td /T0 with T0 ≈ 20 ps. Eq. (7.4.16) shows that the maximum shift νmax occurs for τd = 0 and τd = 4, but the shift vanishes for τd = 2. These features are in agreement with the experiment. According to Eq. (7.4.15) the maximum shift should increase linearly with the peak power of the pump pulse. This behavior is indeed observed experimentally, as seen in Fig. 7.4. The XPM-induced shift of the probe wavelength is about 0.1 nm/kW. It is

Cross-phase modulation

Figure 7.4 XPM-induced wavelength shift of a 0.53-μm probe pulse as a function of (A) the time delay Td and (B) peak power of the 1.06-μm pump pulse for Td = 0. Open circles show the experimental data and the solid lines show theoretical predictions. (After Ref. [40]; ©1988 AIP.)

limited by the walk-off length and can be increased by an order of magnitude or more if the wavelength difference between the pump and probe is reduced to a few nanometers.

7.4.2 Asymmetric temporal changes In the preceding discussion, the dispersion length LD was assumed to be much larger than the fiber length L. As a result, both pulses maintained their shape during propagation through the fiber. If LD becomes comparable to L or the walk-off length LW , the combined effects of XPM, SPM, and GVD can lead to qualitatively new temporal changes that accompany the spectral changes discussed earlier. These temporal changes can be studied by solving Eqs. (7.1.11) and (7.1.12) numerically. It is useful to introduce the normalization scheme of Section 4.2 by defining ξ=

z , LD

τ=

t − z/vg1 , T0

Aj Uj = √ , P1

(7.4.17)

and write the coupled amplitude equations in the form [41] ∂ U1 i ∂ 2 U1 = iN 2 (|U1 |2 + 2|U2 |2 )U1 , + sgn(β21 ) ∂ξ 2 ∂τ 2 ∂ U2 LD ∂ U2 i β22 ∂ 2 U2 ω2 = iN 2 (|U2 |2 + 2|U1 |2 )U2 , ± + ∂ξ LW ∂τ 2 β21 ∂τ 2 ω1

(7.4.18) (7.4.19)

where the parameter N is introduced as N2 =

γ1 P1 T02 LD = . LNL |β21 |

(7.4.20)

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Nonlinear Fiber Optics

Figure 7.5 Shapes (upper row) and spectra (lower row) of probe and pump pulses at ξ = 0.4. Dashed line shows location of input pulses. Both pulses are Gaussian with the same width and overlap entirely at ξ = 0. (After Ref. [41]; ©1989 APS.)

Fiber losses have been neglected assuming that αj L  1 for j = 1 and 2. The second term in Eq. (7.4.19) accounts for the group-velocity mismatch between the two pulses. The choice of plus or minus depends on the sign of the parameter d defined in Eq. (7.4.3). To isolate the XPM effects, it is useful to consider a pump-probe configuration. If we assume |U2 |2  |U1 |2 , we can neglect the term containing |U2 |2 in Eqs. (7.4.18) and (7.4.19). Evolution of the pump pulse, governed by Eq. (7.4.18), is then unaffected by the probe pulse. Evolution of the probe pulse is, however, affected considerably by the presence of the pump pulse because of XPM. Eq. (7.4.19) governs the combined effects of XPM and GVD on the shape and the spectrum of the probe pulse. These equations can be solved numerically using the split-step Fourier method discussed in Section 2.4. Fig. 7.5 shows the shapes and the spectra of the pump and probe pulses at ξ = 0.4 for the case N = 10, LD /LW = 10, ω2 /ω1 = 1.2, and β22 ≈ β21 > 0. Both pulses at the fiber input are taken to be Gaussian of the same width with no initial time delay between them. The pump pulse is assumed to travel faster than the probe pulse (d > 0). The shape and the spectrum of the pump pulse have features resulting from the combined effects of SPM and GVD (see Section 4.2). In contrast, the shape and the spectrum of the probe pulse are governed by the combined effects of XPM and GVD. For comparison, Fig. 7.6 shows the probe and pump spectra in the absence of GVD; asymmetric

Cross-phase modulation

Figure 7.6 Spectra of probe and pump pulses under conditions identical to those of Fig. 7.5 except that the GVD effects are ignored. Pulse shapes are not shown as they remain unchanged.

spectral broadening of the probe spectrum toward the blue side in the absence of GVD is discussed in Section 7.4.1. The effect of GVD is to reduce the extent of asymmetry; a part of the pulse energy is now carried by the red-shifted spectral components (see Fig. 7.5). The most notable effect of GVD is seen in the shape of the probe pulse in Fig. 7.5. In the absence of GVD, the pulse shape remains unchanged as XPM only affects the optical phase. However, when GVD is present, different parts of the probe pulse propagate at different speeds because of the XPM-induced chirp imposed on the probe pulse. This results in an asymmetric shape with considerable structure [41]. The probe pulse develops rapid oscillations near the trailing edge while the leading edge is largely unaffected. These oscillations are related to the phenomenon of optical wave breaking discussed in Section 4.2.3. There, the combination of SPM and GVD led to oscillations in the pulse wings (see Fig. 4.12). Here, it is the combination of XPM and GVD that results in oscillations over the entire trailing half of the probe pulse. The features seen in Fig. 7.5 can be understood qualitatively noting that the XPMinduced chirp is maximum at the pulse center (top row of Fig. 7.3). The combined effect of frequency chirp and positive GVD is to slow down the peak of the probe pulse with respect to its tails. The XPM-induced optical wave breaking occurs because the peak lags behind and interferes with the trailing edge. This can also be understood by noting that a faster moving pump pulse interacts mainly with the trailing edge of a probe pulse. In fact, if the probe and pump wavelengths were reversed so that the slower moving pump pulse interacts mainly with the leading edge, oscillations would develop near the leading edge because the XPM-induced chirp would speed up the peak of the probe pulse with respect to its tails. An initial delay between the pump and probe pulses can lead to qualitative features quite different for the dispersive XPM compared with those shown in Fig. 7.3. For example, even if the pump pulse walks through the probe

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pulse in an asymmetric manner, the probe spectrum is no longer symmetric when the GVD effects are included. The experimental observation of the XPM-induced asymmetric temporal effects requires the use of femtosecond pulses. This is so because LD > 1 km for T0 > 5 ps while LW ∼ 1 m for typical values of |d| ∼ 10 ps/m. Because XPM occurs only during a few walk-off lengths, the interplay between the XPM and GVD effects can occur only if LD and LW become comparable. For example, if T0 = 100 fs, LD and LW both become ∼10 cm, and the temporal effects discussed earlier can occur in a fiber less than one meter long. For such short pulses, however, it becomes necessary to include the higher-order nonlinear effects.

7.4.3 Higher-order nonlinear effects As discussed in Sections 2.3 and 5.5, several higher-order nonlinear effects should be considered for femtosecond optical pulses. The most important among them in practice is the Raman effect. In the case of a single pulse, its inclusion leads to intrapulse Raman scattering that manifests through a Raman-induced frequency shift. Even for two pulses at different wavelengths that maintain their linear SOPs, energy transfer between them can occur during the XPM interaction [46–48]. When the Raman contribution to the nonlinear polarization PNL is included, one must use Eq. (2.3.32) in place of Eq. (2.3.6). The coupled NLS equations can still be obtained by following the procedure of Section 7.1 but mathematical details become quite cumbersome. By using Eq. (2.3.33) for the functional form of the Raman response function, the resulting equations can be written as [47] ∂ Aj 1 ∂ Aj iβ2j ∂ 2 Aj αj + + + Aj = iγj (1 − fR )(|Aj |2 + 2|Am |2 )Aj ∂z vgj ∂ t 2 ∂ t2 2  ∞ + iγj fR ds hR (s){[|Aj (z, t − s)|2 + |Am (z, t − s)|2 ]Aj (z, t) 0

+ Aj (z, t − s)A∗m (z, t − s) exp[i(ωj − ωm )s]Am (z, t)},

(7.4.21)

where j = 1 or 2 and m = 3 − j. The Raman response function hR (t) and the parameter fR appeared first in Eq. (2.3.33) of Section 2.3. In spite of the complexity of Eq. (7.4.21), the physical meaning of various nonlinear terms is quite clear. On the right side of Eq. (7.4.21), the first two terms represent the SPM and XPM contributions from the electronic response, the next two terms provide the SPM and XPM contributions from the Raman response, and the last term governs the energy transfer between the two pulses due to Raman amplification (see Chapter 8). When the delayed nature of the Raman contribution is ignored by setting fR = 0, Eq. (7.4.21) reduces to the set of Eqs. (7.1.11) and (7.1.12). Eq. (7.4.21) shows that the XPM-induced coupling affects ultrashort optical pulses in several different ways when the Raman contribution is included. Energy transfer

Cross-phase modulation

represented by the last term is discussed in Chapter 8 in the context of stimulated Raman scattering. The novel aspect of Eq. (7.4.21) is the SPM and XPM contributions from molecular vibrations. Similar to the single-pulse case, these contributions lead to a shift of the carrier frequency. The most interesting feature is that such a shift results from both intrapulse and interpulse Raman scattering. In the context of solitons, the self-frequency shift is accompanied with a cross-frequency shift [47], occurring whenever the two pulses overlap temporally. The self- and cross-frequency shifts may have the same or the opposite signs, depending on whether the difference ω1 − ω2 in the carrier frequencies is smaller or larger than the frequency at which the Raman gain is maximum (see Chapter 8). As a result, the XPM interaction between two pulses of different carrier frequencies can either enhance or suppress the self-frequency shift expected when each pulse propagates alone [7].

7.5. Applications of XPM The nonlinear phenomenon of XPM can be both beneficial and harmful. Its most direct impact is related to multichannel telecommunication systems whose performance is invariably limited by the XPM interaction among neighboring channels. Such systems are also affected by the so-called intrachannel XPM, resulting from overlapping of neighboring pulses belonging to the same channel [32]. This section is devoted to the beneficial applications of XPM such as pulse compression and optical switching.

7.5.1 XPM-induced pulse compression It is well known that the SPM-induced frequency chirp can be used to compress optical pulses [49]. Because XPM also imposes a frequency chirp on an optical pulse, it can be used for pulse compression as well [50–55]. An obvious advantage of XPM-induced pulse compression is that, in contrast to the SPM technique requiring the input pulse to be intense and energetic, XPM can compress weak input pulses because the frequency chirp is produced by a copropagating intense pump pulse. However, the XPM-induced chirp is affected by the walk-off effects and depends critically on the initial relative pump–probe delay. As a result, the practical use of XPM-induced pulse compression requires a careful control of the pump-pulse parameters such as its width, peak power, wavelength, and initial delay relative to the probe pulse. Two cases must be distinguished depending on the relative magnitudes of the walkoff length LW and the dispersion length LD . If LD LW throughout the fiber, the GVD effects are negligible. In that case, an optical fiber generates the chirp through XPM, and a grating pair is needed to compress the chirped pulse. Eq. (7.4.11) can be used to analyze the magnitude and the form of the chirp. A nearly linear chirp can be imposed across the signal pulse when the pump pulse is much wider compared with it [51]. The compression factor depends on the pump-pulse energy and can easily exceed 10.

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Nonlinear Fiber Optics

Figure 7.7 Pulse shapes for the pump and probe at a distance z/LD = 0.2. The dashed curves show for comparison input pulse shapes at z = 0. XPM-induced pulse compression is realized using pump pulses of peak power such that N = 10.

Another pulse-compression mechanism can be used when LD and LW are comparable. In this case, the same piece of fiber that is used to impose the XPM-induced chirp also compresses the pulse through the GVD. Interestingly, in contrast to the SPM case where such compression can occur only in the anomalous-GVD region, the XPM offers the possibility of pulse compression even in the visible region (normal GVD) without the need of a grating pair. The performance of such a compressor can be studied by solving Eqs. (7.4.18) and (7.4.19) numerically for a given set of pump and signal pulses [41]. It is generally necessary to introduce a relative time delay Td between the pump and probe pulses such that the faster moving pulse overtakes the slower pulse and passes through it. The maximum compression occurs at a distance |τd |LW although the pulse quality is not necessarily the best at the point of maximum compression. In general, a trade-off exists between the magnitude and the quality of compression. As an example, Fig. 7.7 compares the pulse shapes for both the pump and probe at a distance of z/LD = 0.2 (solid curves). Both pulses are initially Gaussian of the same width (dashed curves) but have different wavelengths such that λ1 /λ2 = 1.2. However, the probe is advanced by τd = −2.5. Other parameters are N = 10 and LD /LW = 10. As expected, the pump pulse broadens considerably in the normal-dispersion regime of the fiber. However, the probe pulse is compressed by about a factor of 4 and, except for some ringing on the leading edge, is pedestal-free. Even this ringing can be suppressed if the pump pulse is initially made wider than the signal pulse, although only at the expense of reduction in the amount of compression achievable at a given pump power. Of course, larger compression factors can be realized by increasing the pump power. XPM-induced pulse compression in the normal-GVD region of a fiber can also occur when the XPM coupling is due to interaction between two orthogonally polarized components of a single beam [53]. An experiment in 1990 demonstrated pulse com-

Cross-phase modulation

Figure 7.8 Temporal evolution of (A) signal and (B) pump pulses when the pump pulse propagates in the normal-dispersion regime with the same group velocity and has a peak power such that N equals 30. (After Ref. [41]; ©1989 APS.)

pression using just such a technique [52]. A polarizing Michelson interferometer was used to launch 2-ps pulses in a 1.4-m fiber (with a 2.1-mm beat length) such that the peak power and the relative delay of the two polarization components were adjustable. For a relative delay of 1.2 ps, the weak component was compressed by a factor of about 6.7 when the peak power of the other polarization component was 1.5 kW. When both the pump and signal pulses propagate in the normal-GVD region of the fiber, the compressed pulse is necessarily asymmetric because of the group-velocity mismatch and the associated walk-off effects. The group velocities can be made nearly equal when wavelengths of the two pulses lie on opposite sides of the zero-dispersion wavelength. One possibility consists of compressing 1.55-µm pulses by using 1.06-µm pump pulses. The signal pulse by itself is too weak to form an optical soliton. However, the XPM-induced chirp imposed on it by a copropagating pump pulse can be made strong enough that the signal pulse goes through an initial compression phase associated with higher-order solitons [8]. Fig. 7.8 shows evolution of signal and pump pulses when the pump pulse has the same width as the signal pulse but is intense enough that N = 30 in Eq. (7.4.18). Because of the XPM-induced chirp, the signal pulse compresses by about a factor of 10 before its quality degrades. Both the compression factor and the pulse quality depend on the width and the energy of the pump pulse and can be controlled by optimizing pump-pulse parameters. This method of pulse compression is similar to that provided by higher-order solitons even though, strictly speaking, the signal pulse never forms

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a soliton. With the use of dispersion-shifted fibers, the technique can be used even when both pump and signal wavelengths are in the 1.55-µm region as long as the zerodispersion wavelength of the fiber lies in the middle. In a 1993 experiment, 10.6-ps signal pulses were compressed to 4.6 ps by using 12-ps pump pulses [54]. Pump and signal pulses were obtained from mode-locked semiconductor lasers operating at 1.56 and 1.54 µm, respectively, but pump pulses were amplified using a fiber amplifier. This experiment demonstrated that XPM-induced pulse compression can occur at power levels achievable with semiconductor lasers.

7.5.2 XPM-induced optical switching The XPM-induced phase shift can also be used for optical switching. Several interferometric schemes were used during the 1990s to take advantage of XPM for ultrafast optical switching [56–64]. The physics behind XPM-induced switching can be understood by considering a generic interferometer designed such that a weak signal pulse, divided equally between its two arms, experiences identical phase shifts in each arm and is transmitted through constructive interference. If a pump pulse at a different wavelength is injected into one of the arms of the interferometer, it would change the signal phase through XPM in that arm. If the XPM-induced phase shift is large enough (close to π ), the signal pulse will not be transmitted because of the destructive interference occurring at the output. Thus, an intense pump pulse can switch the signal pulse through the XPM-induced phase shift. XPM-induced optical switching was demonstrated in 1990 using a fiber-loop mirror acting as a Sagnac interferometer [58]. Such interferometers are covered in a separate book devoted to the applications of nonlinear fiber optics. In the 1990 experiment, a dichroic fiber coupler, with 50:50 splitting ratio at 1.53 µm and 100:0 splitting ratio at 1.3 µm, was used to allow for dual-wavelength operation. A 1.53-µm color-center laser provided a low-power (∼5 mW) CW signal. As expected, the counterpropagating signal beams experienced identical phase shifts, and the 500-m-long fiber loop acted as a perfect mirror, in the absence of a pump beam. When 130-ps pump pulses, obtained from a 1.3-µm Nd:YAG laser, were injected into the clockwise direction, the XPM interaction between the pump and the signal introduced a phase difference between the counterpropagating signal beams. Most of the signal power was transmitted when the peak power of the pump pulse was large enough to introduce a π phase shift. The XPM-induced phase shift depends not only on the width and the shape of the pump pulse but also on the group-velocity mismatch. In the case in which both the pump and signal beams are pulsed, the phase shift also depends on the initial relative time delay between the pump and signal pulses. In fact, the magnitude and the duration of the XPM-induced phase shift can be controlled through the initial delay (see Fig. 7.3). The main point to note is that phase shift can be quite uniform over most of the signal pulse when the two pulses are allowed to completely pass through each other, resulting

Cross-phase modulation

in complete switching of the signal pulse. The pump power required to produce π phase shift is generally quite large because of the group-velocity mismatch. The group-velocity mismatch can be reduced significantly if the pump and signal pulses are orthogonally polarized but have the same wavelength. Moreover, even if the XPM-induced phase shift is less than π because of the birefringence-related pulse walkoff, the technique of cross-splicing can be used to accumulate it over long lengths [61]. In this technique, the fiber loop consists of multiple sections of polarization-maintaining fibers that are spliced together in such a way that the fast and slow axes are rotated by 90◦ in successive sections. As a result, the pump and signal pulses are forced to pass through each other in each section of the fiber loop, and the XPM-induced phase shift is enhanced by a factor equal to the number of sections.

7.5.3 XPM-induced wavelength conversion XPM can be employed for converting the wavelength of a pulse train such as those emitted from a mode-locked laser. The pulse train does not even have to be periodic. Thus, XPM can be used for converting the wavelength of a data channel in optical communication systems in the form of a coded sequence of optical pulses. The channel is launched into a fiber of appropriate length together with a CW seed at the wavelength at which the signal needs to be converted. The signal is amplified considerably before launching so that it can act as a pump and thus impose a large enough XPM-induced phase shift in the time slots containing an optical pulse. This phase shift is then converted into amplitude modulation using an interferometer. Although a Mach–Zehnder interferometer can be used in principle, its use is not practical because it is hard to maintain the linear phase shifts constant in two arms of a fiber-based Mach–Zehnder interferometer. The use of a Sagnac interferometer in the form of a nonlinear fiber-loop mirror solves this problem. Its operation is similar to the optical switching discussed in the preceding subsection. The CW seed is split such that it propagates in both directions within the fiber loop. The channel whose wavelength need to be converted acts as a pump and is launched such that it propagates in one direction only. The XPM-induced phase shift is much larger in the direction of the co-propagating seed compared to the other direction. The resulting differential phase shift can be adjusted such that the Sagnac interferometer acts as a switch that opens only during the time slots in which a pump pulse is present. Such a device was used in 1994 to realize the wavelength conversion of a 10-Gb/s data channel [65]. By 2000 wavelength converters capable of operating at bit rates of 40 Gb/s had been demonstrated using a Sagnac loop was made using 3 km of dispersion-shifted fiber [66]. The on–off ratio between maximum and minimum transmission through the loop was measured to be 25 dB, indicating relatively high quality of the wavelength-conversion process.

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It is not essential to use an optical interferometer for XPM-based wavelength converters. As an alternative, one can pass the phase-modulated output of the fiber through a suitable optical filter. The basic idea behind this scheme is that the phase modulation creates sidebands on each side of the CW seed’s spectrum. If an optical filter blocks the original CW wavelength but lets one of the sidebands pass, the output is a replica of the original bit stream at the seed’s wavelength. Any optical filter with a bandwidth of ∼1 nm can be used for this purpose. In a 2000 experiment [67], the wavelength of a 40-Gb/s signal was shifted by several nanometers through XPM inside a 10-km-long fiber. The required fiber length can be reduced considerably by employing highly nonlinear fibers (see Chapter 11). In a 2001 experiment [68], wavelength conversion at a bit rate of 80 Gb/s was realized by using such a 1-km-long fiber with γ = 10.9 W−1 /km. A tunable optical filter with a 1.5-nm bandwidth was used at the output end to produce the wavelength-converted channel.

7.6. Polarization effects So far, both optical pulses are assumed to maintain their linear state of polarization (SOP). This assumption should be relaxed for fibers exhibiting birefringence. Even in an isotropic fiber, XPM-induced nonlinear birefringence leads to changes in the SOP when the input field is not linearly polarized [69–71]. In this section we develop a vector theory of XPM that applies to optical pulses propagating with different states of polarization (SOPs) inside an optical fiber [70].

7.6.1 Vector theory of XPM As in Section 6.6, it is useful to employ the Jones-matrix formalism to account for the polarization effects in a succinct fashion. We use the form of the third-order nonlinear polarization given in Eq. (2.3.6). In the case of two distinct optical fields propagating simultaneously inside an optical fiber, the total electric field can be written as E(r, t) = 12 [E1 exp(−iω1 t) + E2 exp(−iω2 t)] + c.c.,

(7.6.1)

where Ej is the slowly varying part of the optical field oscillating at the frequency ωj (j = 1, 2). If we follow the method described in Section 6.1.1 and use Eq. (6.1.2) for the third-order susceptibility, PNL can be written in the form PNL = 12 [P1 exp(−iω1 t) + P2 exp(−iω2 t)] + c.c., where P1 and P2 are given by

(7.6.2)

Cross-phase modulation

0

χ (3) [(Ej · Ej )E∗j + 2(E∗j · Ej )Ej 4 xxxx + 2(E∗m · Em )Ej + 2(Em · Ej )E∗m + 2(E∗m · Ej )Em ],

Pj =

(7.6.3)

with j = m. In obtaining this expression, we used the relation (6.1.5) among the components of χ (3) and assumed that the three components have the same magnitude. At this point, it is useful to employ the ket notation for a Jones vector representing light polarized in the x–y plane [72] and write Ej at any point r inside the fiber in the form (j = 1, 2) Ej (r, t) = Fj (x, y)|Aj (z, t) exp(iβj z),

(7.6.4)

where Fj (x, y) represents the fiber-mode profile and βj is the propagation constant at the carrier frequency ωj . The Jones vector |Aj is a two-dimensional column vector representing the two components of the electric field in the x–y plane. In this notation, E∗j · Ej and Ej · Ej are related to Aj |Aj and A∗j |Aj , respectively. We can now follow the method of Section 7.1.2 and obtain the following vector form of the coupled NLS equations [70]: ∂|A1 1 ∂|A1 iβ21 ∂ 2 |A1 α1 i γ1  + + + |A1 = 2A1 |A1 + |A∗1 A∗1 | 2 ∂z vg1 ∂ t 2 ∂t 2 3

+ 2A2 |A2 + 2|A2 A2 | + 2|A∗2 A∗2 | |A1 , (7.6.5) ∂|A2 1 ∂|A2 iβ22 ∂ 2 |A2 α2 i γ2  + + + |A2 = 2A2 |A2 + |A∗2 A∗2 | 2 ∂z vg2 ∂ t 2 ∂t 2 3

+ 2A1 |A1 + 2|A1 A1 | + 2|A∗1 A∗1 | |A2 , (7.6.6)

where γj is defined in Eq. (7.1.13). As usual, A| represents the Hermitian conjugate of |A , i.e., it is a row vector whose elements have been complex-conjugated. The inner product A|A is related to the power associated with the optical field |A . In deriving the coupled NLS equations, the fiber is assumed to be without any birefringence. As shown in Section 7.7, the residual birefringence of realistic fibers can be included in a simple fashion.

7.6.2 Polarization evolution Eqs. (7.6.5) and (7.6.6) are quite complicated and their solution requires a numerical approach. To study the XPM-induced polarization effects as simply as possible, we make two simplifications. First, we assume that the fiber length L is much shorter than the dispersion lengths associated with the two waves and neglect the effects of GVD. Second, we adopt a pump-probe configuration and assume that the probe power A2 |A2 is much smaller than the pump power A1 |A1 . This allows us to neglect the

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nonlinear effects induced by the probe and to simplify the right side of Eqs. (7.6.5) and (7.6.6) considerably. If we work in a frame moving with the probe pulse and introduce a normalized time as τ = (t − z/vg2 )/T0 , where T0 is the width of input pump pulse, Eqs. (7.6.5) and (7.6.6) reduce to

∂|A1 1 ∂|A1 iγ1  + 2A1 |A1 + |A∗1 A∗1 | |A1 , = ∂z LW ∂τ 3

∂|A2 2iγ2  = A1 |A1 + |A1 A1 | + |A∗1 A∗1 | |A2 , ∂z 3

(7.6.7) (7.6.8)

where LW = |vg1 − vg2 |/(T0 vg1 vg2 ) is the walk-off length and fiber losses have been neglected. As was seen in Chapter 6, the evolution of the SOP of an optical field is easier to visualize through the rotation of the Stokes vector on the Poincaré sphere. We thus introduce the normalized Stokes vectors for the pump and probe fields as [72] p = A1 |σ |A1 /P0 ,

s = A2 |σ |A2 /P20 ,

(7.6.9)

where P0 and P20 represent the peak powers of the pump and probe pulses at z = 0. The Pauli spin vector σ is defined using the unit vectors eˆj in the Stokes space as σ = σ1 eˆ1 + σ2 eˆ2 + σ3 eˆ3 , where the three Pauli matrices are given in Eq. (6.7.7). Using Eqs. (7.6.7) through (7.6.9), the two Stokes vectors are found to satisfy [70] ∂p 2 ∂p +μ = p3 × p, ∂ξ ∂τ 3 ∂s 4ω2 (p − p3 ) × s, =− ∂ξ 3ω1

(7.6.10) (7.6.11)

where ξ = z/LNL is the distance normalized to the nonlinear length LNL = (γ1 P0 )−1 , μ = LNL /LW , and p3 = (p · eˆ3 )ˆe3 is the third component of the Stokes vector p. Whenever p3 = 0, p lies entirely in the equatorial plane of the Poincaré sphere, and the pump field is linearly polarized. In deriving Eqs. (7.6.10) and (7.6.11), we made use of the following identities [72] |A A| = 12 [I + A|σ |A · σ ], ∗

∗

|A A | = |A A| − A|σ3 |A σ3 , σ (a · σ ) = aI + ia × σ ,

where I is an identity matrix and a is an arbitrary Stokes vector.

(7.6.12) (7.6.13) (7.6.14)

Cross-phase modulation

The pump equation (7.6.10) is relatively easy to solve and has the solution p(ξ, τ ) = exp[(2ξ/3)p3 (0, τ − μξ )×]p(0, τ − μξ ),

(7.6.15)

where exp(a×) is an operator interpreted in terms of a series expansion [72]. Physically speaking, the Stokes vector p rotates on the Poincaré sphere along the vertical axis at a rate 2p3 /3. As discussed in Section 6.3, this rotation is due to XPM-induced nonlinear birefringence and is known as the nonlinear polarization rotation. If the pump is linearly or circularly polarized initially, its SOP does not change along the fiber. For an elliptically polarized pump, the SOP changes as the pump pulse propagates through the fiber. Moreover, as the rotation rate depends on the optical power, different parts of the pump pulse acquire different SOPs. Such intrapulse polarization effects have a profound effect on the probe evolution. The probe equation (7.6.11) shows that the pump rotates the probe’s Stokes vector around p − p3 , a vector that lies in the equatorial plane of the Poincaré sphere. As a result, if the pump is circularly polarized initially, XPM effect becomes polarization independent since p − p3 = 0. On the other hand, if the pump is linearly polarized, p3 = 0, and p remains fixed in the Stokes space. However, even though the pump SOP does not change in this case, the probe SOP can still change through XPM. Moreover, XPM produces different SOPs for different parts of the probe pulse depending on the local pump power, resulting in nonuniform polarization along the probe pulse shape. The XPM-induced polarization effects become quite complicated when the pump is elliptically polarized because the pump SOP itself changes along the fiber. As an example, consider the case in which the pump pulse is elliptically polarized but the probe pulse is linearly polarized at the input end. Assuming a Gaussian shape for both pulses, the Jones vectors for the two input pulses have the form  |A1 (0, τ ) =

 cos φ i sin φ

P01/2 exp

 2 τ − ,

2

 |A2 (0, τ ) =

 cos θ sin θ

 1/2 P20 exp −

τ2

,

2r 2 (7.6.16)

where φ is the ellipticity angle for the pump, θ is the angle at which probe is linearly polarized from the x axis, and r = T2 /T0 is the relative width of the probe compared to the pump. Fig. 7.9 shows how the pump SOP (solid curve) and the probe SOP (dashed curve) change on the Poincaré sphere with τ at a distance of ξ = 20, assuming φ = 20◦ , θ = 45◦ , r = 1, and μ = 0.1. The pump SOP varies in a simple manner as it rotates around eˆ3 and traces a circle on the Poincaré sphere. In contrast, the probe SOP traces a complicated pattern, indicating that the SOP of the probe is quite different across different parts of the probe pulse [70]. Such polarization changes impact the XPM-induced chirp. As a result, the spectral profile of the probe develops a much more complicated structure compared with the scalar case. We focus next on such spectral effects.

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Figure 7.9 Evolution of pump SOP (solid curve) and probe SOP (dashed curve) on the Poincaré sphere as a function of τ at a distance of ξ = 20. Parts (A) and (B) show the front and back faces of the Poincaré sphere, respectively. Both pulses are assumed to be Gaussian with the same width but are launched with different SOPs.

7.6.3 Polarization-dependent spectral broadening In general, Eq. (7.6.8) should be solved numerically except when the pump maintains its SOP. As discussed earlier, this happens if the pump pulse is linearly or circularly polarized when it is launched into the fiber. To gain some physical insight, we first consider the case in which both the pump and probe fields are linearly polarized at the input end, but the probe is oriented at an angle θ with respect to the pump. The Jones vectors for the two input fields are then obtained from Eq. (7.6.16) after setting φ = 0. As the SPM does not affect the pump SOP in this case, the analytical solution of Eq. (7.6.8) is found to be 

  τ2 cos θ exp(iφn )  |A2 (z, τ = P20 exp − 2 , sin θ exp(iφn /3) 2r 

(7.6.17)

where φn (z, τ ) = 2γ2 0z P0 (z , τ − z/LW ) dz is the XPM-induced nonlinear phase shift. The shape of probe pulse does not change in the absence of GVD, as expected. However, its SOP changes and it acquires a nonlinear time-dependent phase shift that is different for the two polarization components. More specifically, the XPM-induced phase shift for the orthogonally polarized component is one third of the copolarized one because of a reduced XPM coupling efficiency. Clearly spectral broadening for the two components would be different under such conditions. Fig. 7.10 shows the (A) probe shapes and (B) spectra for the x (dashed curve) and y (dotted curve) components for a fiber of length L = 20LNL , assuming φ = 0, θ = 45◦ , r = 1, and μ = 0.1 (LW = L /2). The copolarized component has a broader spectrum and exhibits more oscillations compared with the orthogonally polarized one. If the spectrum is measured without placing a polarizer in front of the photodetector, one

Cross-phase modulation

Figure 7.10 (A) Shapes and (B) spectra at ξ = 20 for the x (dashed curve) and y (dotted curve) components of the probe pulse, linearly polarized with θ = 45◦ , when the pump pulse is polarized along the x axis (φ = 0). Total intensity is shown by the solid line.

would record the total spectral intensity shown by a solid line in Fig. 7.10(B). However, it is important to realize that the SOP is not the same for all spectral peaks. For example, the leftmost peak is x-polarized, whereas the dominant peak in the center is mostly y-polarized. This spectral nonuniformity of the SOP is a direct consequence of the XPM-induced polarization effects. Consider next the case when the pump is elliptically polarized initially with φ = 20◦ but the probe remains linearly polarized. The Jones vectors for these two pulses at ξ = 0 are given in Eq. (7.6.16). Using this form, we solve Eqs. (7.6.7) and (7.6.8) numerically to obtain the probe’s Jones vector |A2 (z, τ ) at a distance ξ = 20. The spectrum for the two orthogonally polarized components of the probe is then obtained by taking the Fourier transform. Fig. 7.11 shows the (A) probe shapes and (B) spectra for the x (solid curve) and y (dashed curve) components under conditions identical to those of Fig. 7.10 except that the pump pulse is polarized elliptically rather than linearly. A comparison of Figs. 7.10(B) and 7.11(B) gives an idea how much probe spectrum can change with a relatively small change in the pump SOP. The spectral asymmetry seen in these figures is a direct consequence of the walk-off effects resulting from the group-velocity mismatch [70].

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Figure 7.11 (A) Shapes and (B) spectra at ξ = 20 for the x (dashed curve) and y (dotted curve) components of the probe pulse, linearly polarized with θ = 45◦ , when the pump pulse is elliptically polarized (φ = 20◦ ). Total intensity is shown by the solid line.

The most dramatic changes occur in the temporal shape of the probe when the pump is elliptically polarized. In contrast with the case of a linearly polarized pump pulse, the temporal profiles for the x- and y-polarized probe components become different and develop considerable internal structure, even though the total power remains unchanged as long as GVD effects are negligible. In Fig. 7.11(A) pulse shapes for the two polarization components exhibit a multipeak structure such that the total power at any time adds up to the input value. The physical origin of this behavior is related to the XPM-induced changes in the SOP of the pump pulse. As the pump SOP evolves, the probe SOP changes in a complex manner, as seen in Fig. 7.9. The temporal structure seen in Fig. 7.11(A) stems from this complex polarization behavior.

7.6.4 Pulse trapping and compression The preceding discussion of the polarization effects ignores the GVD effects and is valid if the fiber is much shorter than the dispersion length. In this section we relax this restriction and focus on the polarization-dependent temporal effects. In the presence of GVD, we should retain the second-derivative term in Eqs. (7.6.5) and (7.6.6). If we consider again the pump-probe configuration and ignore the nonlinear effects induced

Cross-phase modulation

Figure 7.12 Evolution of (A) copolarized and (B) orthogonally polarized components of a probe pulse polarized at 45◦ with respect to a pump pulse polarized linearly along the x axis.

by the weak probe, Eqs. (7.6.7) and (7.6.8) contain an extra term and take the form [70]  ∂|A1 i ∂|A1 iβ21 LNL ∂ 2 |A1 = 3hp − p3 · σ |A1 , +μ + 2 2 ∂ξ ∂τ ∂τ 3 2T0 2  ∂|A2 iβ22 LNL ∂ |A1 2iω2  = 2hp + (p − p3 ) · σ |A2 , + ∂ξ ∂τ 2 3ω1 2T02

(7.6.18) (7.6.19)

where hp (ξ, τ ) = A1 (ξ, τ )|A1 (ξ, τ ) /P0 is the temporal profile of the pump pulse normalized to 1 at the pulse center at ξ = 0. One can introduce two dispersion lengths as Ldj = T02 /|β2j | (j = 1, 2). In the following discussion, we assume for simplicity that Ld1 = Ld2 ≡ Ld . This may the case for a dispersion-flattened fiber, or when the pump and probe wavelengths do not differ by more than a few nanometers. The two vector NLS equations can be solved numerically with a generalized version of the split-step Fourier method introduced in Section 2.4. We stress this that this amounts to solving a set of four coupled NLS equations. Similar to the scalar case studied in Section 7.4, three length scales—nonlinear length LNL , walk-off length LW , and dispersion length LD —govern the interplay between the dispersive and nonlinear effects. The temporal evolution of the probe pulse depends considerably on whether the GVD is normal or anomalous at the wavelengths of the two pulses. Consider first the anomalous-GVD case. Fig. 7.12 shows an example of the probe evolution over a length L = 50LNL , when both the pump and probe pulses

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Figure 7.13 Temporal shapes of (A) pump and (B) probe pulses at ξ = 0 (dotted curves) and ξ = 50 for the x (solid curves) and y (dashed curve) components. The pump pulse is linearly polarized along the x axis (φ = 0) while the probe is oriented at 45◦ with respect to it.

are initially Gaussian, have the same width, and are linearly polarized at a 45◦ angle. The dispersion and walk-off lengths are chosen such that LD = 5LNL and LW = 10LNL , respectively. Since N 2 = LD /LNL = 5, the pump pulse evolves toward a second-order soliton. As seen in Fig. 7.13, the pump pulse is compressed considerably at ξ = 50 and has a shape expected for a second-order soliton. It also shifts toward the right side because of the group-velocity mismatch between the pump and probe pulses. The probe pulse would disperse completely in the absence of the XPM effects because of its low peak power. However, as seen in Fig. 7.12, its x component, which is copolarized with the pump pulse and experiences XPM-induced chirp, shifts with the pump pulse and travels at the same speed as the pump pulse. This feature is similar to the phenomenon of soliton trapping discussed in Section 6.5.2 in the context of highbirefringence fibers. The pump pulse traps the copolarized component and the two move together because of XPM coupling between the two. The XPM also compresses the probe pulse. This feature is seen more clearly in Fig. 7.13, where the copolarized component of the probe appears to be compressed even more than the pump pulse, and the two occupy nearly the same position because of soliton trapping. The enhanced compression of the probe pulse is related to the fact that the XPM-induced chirp is larger by a factor of two when the pump and probe are copolarized. The y component of the probe pulse (dashed curve), in contrast, disperses completely because it is not trapped by the orthogonally polarized pump pulse. If the pump pulse is elliptically polarized, both the x and y components of the probe would experience XPM-induced chirp, but the amount of this chirp would depend on the ellipticity angle. Fig. 7.14 shows, as an example, the case in which the pump pulse is elliptically polarized with φ = 20◦ . All other parameter remains identical to those used for Fig. 7.12. As expected, both components of the probe pulse are now trapped

Cross-phase modulation

Figure 7.14 Temporal evolution of (A) copolarized and (B) orthogonally polarized components of the probe pulse when the pump pulse is elliptically polarized with φ = 20◦ . All other parameters are identical to those used for Fig. 7.12.

by the pump pulse, and both of them move with the pump pulse, while also getting compressed. The two probe components also exhibit a kind of periodic power transfer between them. This behavior is related to the complicated polarization evolution of the probe pulse occurring when the pump pulse is elliptically polarized (see Fig. 7.9).

7.6.5 XPM-induced wave breaking The situation is quite different in the case of normal GVD because the pump pulse then cannot evolve toward a soliton. However, interesting temporal effects still occur if the three length scales are chosen such that LNL  LW  LD . Under such conditions, a relatively weak dispersion can lead to XPM-induced optical wave breaking, a phenomenon discussed in Section 4.2.3 in the context of self-phase modulation (SPM). Similar to the XPM case, XPM-induced wave breaking forces the probe pulse to break up into multiple parts, but in an asymmetric fashion. Fig. 7.15 shows XPM-induced wave breaking, under conditions identical to those used for Fig. 7.12, except that the GVD is normal and its strength is reduced such that LD = 100LNL . As a result, the parameter N ≡ (LD /LNL )1/2 is larger and has a value of 10. The temporal evolution of the x and y component of the probe pulse seen in Fig. 7.15 exhibits different features, and both pulses evolve in an asymmetric fashion. The asymmetry is related to the walk-off effects governed by the walk-off length (LW = 10LNL ). Physically speaking, the leading edge of the probe pulse experiences the maximum

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Nonlinear Fiber Optics

Figure 7.15 Temporal evolution of (A) copolarized and (B) orthogonally polarized components of the probe pulse polarized at 45◦ with respect to a pump pulse that is linearly polarized along the x axis. Both pulses propagate in the normal-dispersion regime.

XPM-induced chirp because the pump and probe pulses overlap initially. As the pump pulse walks off and separates from the probe pulse, it is also considerably broadened. As a result, the XPM-induced chirp is much reduced near the trailing edge. For this reason, an oscillatory structure develops only near the leading or trailing edge of the probe pulse, depending on how the pump walks away from the probe pulse. The details of wave breaking are quite different for the two polarization components of the probe in Fig. 7.15. In the case of a linearly polarized pump, the copolarized component of the probe develops much more oscillatory structure than the orthogonally polarized one because the XPM-induced chirp and spectral broadening is larger for the former. This feature can be verified experimentally by placing a polarizer before the photodetector and noticing that the oscillatory structure changes as the polarizer is rotated. When the pump pulse is elliptically polarized, the polarization dependence of the probe is reduced as both components have some pump power in the same state of polarization. In particular, when φ = 45◦ , the two probe components would evolve in a nearly identical fashion.

7.7. XPM effects in birefringent fibers In a birefringent fiber, the orthogonally polarized components of each wave propagate with different propagation constants because of a slight difference in their effective mode

Cross-phase modulation

indices. In addition, these two components are coupled through XPM, resulting in nonlinear birefringence. In this section we incorporate the effects of linear birefringence in the Jones-matrix formalism developed in the preceding section.

7.7.1 Fibers with low birefringence The induced nonlinear polarization can still be written in the form of Eq. (7.6.3) but Eq. (7.6.4) for the optical field Ej should be modified as Ej (r, t) = Fj (x, y)[ˆxAjx exp(iβjx z) + yˆ Ajy exp(iβjy z)],

(7.7.1)

where βjx and βjy are the propagation constants for the two orthogonally polarized components of the optical field with the carrier frequency ωj . It is useful to write Eq. (7.7.1) in the form Ej (r, t) = Fj (x, y)[ˆxAjx exp(iδβj z/2) + yˆ Ajy exp(−iδβj z/2)] exp(iβ¯j z),

(7.7.2)

where β¯j = 12 (βjx + βjy ) is the average value of the propagation constants and δβj is their difference. One can now introduce the Jones vector |Aj and write Eq. (7.7.2) in the form of Eq. (7.6.4), provided |Aj is defined as 



Ajx exp(iδβj z/2) . |Aj = Ajy exp(−iδβj z/2)

(7.7.3)

At this point, we can follow the method outlined in Section 7.1.2 to obtain the coupled vector NLS equations. It is important to keep in mind that δβj (ω) itself is frequency dependent and it should be expanded in a Taylor series together with β¯j (ω). However, it is sufficient in practice to approximate it as δβj (ω) ≈ δβj0 + δβj1 (ω − ωj ),

(7.7.4)

where δβj1 is evaluated at the carrier frequency ωj . The δβj1 term is responsible for different group velocities of the orthogonally polarized components. It should be retained for fibers with high birefringence but can be neglected when linear birefringence is relatively small. We assume that to be the case in this subsection. The final set of coupled NLS equations for low-birefringence fibers is identical to those given in Eqs. (7.6.5) and (7.6.6) but contains an additional birefringence term on the right side of the form Bj = 2i δβj0 σ1 |Aj , where j = 1 or 2. The Pauli matrix σ1 takes

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into account the different phase shifts acquired by the x and y components of an optical field in a birefringent fiber. It is useful to write the birefringence term as Bj = 12 iωj bσ1 |Aj ,

(7.7.5)

where b = (¯nx − n¯ y )/c is a measure of the modal birefringence of the fiber. As discussed in Section 6.6, b varies randomly in most fibers because of unintentional random variations in the core shape and size along the fiber length. Moreover, the principal axes of the fiber also rotate in a random fashion on a length scale ∼10 m. Such random variations can be included by replacing the matrix bσ1 in Eq. (7.7.5) with b · σ , where σ is the Pauli spin vector and b is a random vector in the Stokes space. It is common to model b as a three-dimensional Markovian process whose first and second-order moments are given by b(z) = 0,

b(z)b(z ) = 13 Dp2 I δ(z − z ),

(7.7.6)

where Dp is the PMD parameter of the fiber and I is the identity matrix. An important question is how the PMD affects the XPM coupling between two waves copropagating simultaneously inside a fiber. To answer this question, it is useful to consider a pump-probe configuration and neglect the nonlinear effects induced by the probe but retain those induced by the pump. Working in a reference frame in which the probe pulse is stationary, and neglecting the β2 terms, Eqs. (7.6.5) and (7.6.6) lead to Eqs. (7.6.7) and (7.6.8) but with an additional birefringence term on the right side. These modified equations can be written as [71]

∂|A1 1 ∂|A1 iω1 i γ1  + 2A1 |A1 + |A∗1 A∗1 | |A1 , b · σ | A1 + = ∂z LW ∂τ 2 3

 ∂|A2 iω2 2iγ2 = A1 |A1 + |A1 A1 | + |A∗1 A∗1 | |A2 . b · σ | A2 + ∂z 2 3

(7.7.7) (7.7.8)

The preceding coupled NLS equations are stochastic in nature because of the random nature of the birefringence vector b. To simplify them, we note that, even though fluctuating residual birefringence of the fiber randomizes the SOP of both the pump and probe, it is only their relative orientation that affects the XPM process. Since randomization of the pump and probe SOPs occurs over a length scale (related to the birefringence correlation length) much shorter than a typical nonlinear length, one can average over rapid random SOP variations by adopting a rotating frame in which the pump SOP remains fixed. Following the averaging procedure discussed in Section 6.6.2

Cross-phase modulation

and Ref. [73], we obtain ∂|A1 1 ∂|A1 + = iγe P1 |A1 , ∂z LW ∂τ

 ∂|A2 i iω2 + b · σ |A2 = γe P1 3 + pˆ · σ |A2 ∂z 2 2ω1

(7.7.9) (7.7.10)

where  = ω1 − ω2 is the pump-probe frequency difference, P1 = A1 |A1 is the pump power and pˆ = A1 |σ |A1 /P1 is the Stokes vector representing the pump SOP on the Poincaré sphere. The effective nonlinear parameter, γe = 8γ /9, is reduced by a factor of 8/9 because of the averaging over rapid variations of the pump SOP. Residual birefringence enters through the vector b that is related to b by a random rotation on the Poincaré sphere. Since such rotations leave the statistics of b unchanged, we drop the prime over b in what follows. The pump equation (7.7.9) can be easily solved. The solution shows that the pump pulse shifts in time as its walks away from the probe pulse but its shape does not change, i.e., P1 (z, τ ) = P1 (0, τ − z/LW ). The total probe power also does not change as it is easy to show from Eq. (7.7.10) that the probe power P2 = A2 |A2 satisfies ∂ P2 /∂ z = 0. However, the orthogonally polarized components of the probe pulse can exhibit complicated dynamics as its state of polarization evolves because of XPM. It is useful to introduce the normalized Stokes vectors for the pump and probe as pˆ = A1 |σ |A1 /P1 ,

ˆs = A2 |σ |A2 /P2 ,

(7.7.11)

and write Eq. (7.7.10) in the Stokes space as [71]  ∂ˆs  = b − γe P1 pˆ × ˆs. ∂z

(7.7.12)

The two terms on the right side of this equation show that the SOP of the probe rotates on the Poincaré sphere along an axis whose direction changes in a random fashion as dictated by b. Moreover, the speed of rotation also changes with z randomly. It is useful to note that Eq. (7.7.12) is isomorphic to the Bloch equation governing the motion of spin density in a solid [71]. In that context, P1 corresponds to a static magnetic field and b has its origin in small magnetic fields associated with nuclei [74]. In a fashion similar to the phenomenon of spin decoherence, the probe SOP evolves along the fiber in a random fashion and leads to intrapulse depolarization, a phenomenon produced by the combination of XPM and PMD. Physically speaking, the XPM interaction between the pump and probe makes the probe SOP vary across the temporal profile of the pump. Fluctuating residual birefringence of the fiber then transfers this spatial randomness into temporal randomness of the signal polarization. The distance over which the SOPs of the pump and probe become decorrelated because of PMD

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Figure 7.16 Intrapulse depolarization di across the probe profile for three walk-off lengths when the probe is launched with a copolarized Gaussian pump pulse. Fiber length is such that γ P0 L = π . (After Ref. [71]; ©2005 OSA.)

is governed by the PMD diffusion length defined as Ldiff = 3/(Dp )2 . This distance is 1 W. This is due to gain saturation occurring because of pump depletion. The solid lines in Fig. 8.8 are obtained by solving Eqs. (8.1.2) and (8.1.3) numerically to include pump depletion. The numerical results are in excellent agreement with the data. An approximate expression for the saturated gain Gs in Raman amplifiers can be obtained by solving Eqs. (8.1.2) and (8.1.3) analytically [16] with the assumption αs = αp ≡ α . Making the transformation Ij = ωj Fj exp(−α z) with j = s or p, we obtain two simple equations: dFs = ωp gR Fp Fs e−αz , dz

dFp = −ωp gR Fp Fs e−αz . dz

(8.2.5)

Noting that Fp (z) + Fs (z) = C, where C is a constant, the differential equation for Fs can be integrated over the amplifier length, and the result is 

Gs =



Fs (L ) C − Fs (L ) = exp(ωp gR CLeff − α L ). Fs (0) C − Fs (0)

(8.2.6)

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Figure 8.9 Gain-saturation characteristics of Raman amplifiers for several values of the unsaturated amplifier gain GA .

Using C = Fp (0) + Fs (0) in this equation the saturated gain of the amplifier is given by Gs =

(1 + r0 )e−αL

r0 + GA−(1+r0 )

,

(8.2.7)

where r0 is related to the signal-to-pump power ratio at the fiber input as r0 =

Fs (0) ωp Ps (0) , = Fp (0) ωs P0

(8.2.8)

and GA = exp(gR P0 Leff /Aeff ) is the small-signal (unsaturated) gain. Fig. 8.9 shows the saturation characteristics by plotting Gs /GA as a function of GA r0 for several values of GA with α = 0. The saturated gain is reduced by a factor of 2 when GA r0 ≈ 1. This condition is satisfied when the power in the amplified signal starts to approach the input pump power P0 that is a good measure of the saturation power of Raman amplifiers. As typically P0 ∼1 W, the saturation power of Raman amplifiers is much larger compared with that of doped-fiber amplifiers or semiconductor optical amplifiers [73]. As seen in Fig. 8.8, Raman amplifiers can easily amplify an input signal by a factor of 1000 (30-dB gain) at a pump power of about 1 W [65]. In a 1983 experiment, a 1.24-µm signal from a semiconductor laser was amplified by 45 dB using a 2.4-km-long fiber [67]. This experiment used the forward-pumping configuration. In a different experiment [66], a 1.4-µm signal was amplified using both the forward- and backwardpumping configurations. The CW light from a Nd:YAG laser operating at 1.32 µm

Stimulated Raman scattering

acted as the pump. Gain levels of up to 21 dB were obtained at a pump power of 1 W. The amplifier gain was nearly the same in both pumping configurations. For optimum performance of Raman amplifiers, the frequency difference between the pump and signal beams should correspond to the peak of the Raman gain in Fig. 8.2. In the near-infrared region, the most practical pump source is the Nd:YAG laser operating at 1.06 or 1.32 µm. For this laser, the maximum gain occurs for signal wavelengths near 1.12 and 1.40 µm, respectively. However, the signal wavelengths of most interest from the standpoint of optical fiber communications are near 1.3 and 1.5 µm. A Nd:YAG laser can still be used if a higher-order Stokes line is used as a pump. For example, the third-order Stokes line at 1.24 µm from a 1.06-µm laser can act as a pump to amplify the signals at 1.3 µm. Similarly, the first-order Stokes line at 1.4 µm from a 1.32-µm laser can act as a pump to amplify the signals near 1.5 µm. As early as 1984, amplification factors of more than 20 dB were realized by using such schemes [68]. These experiments also indicated the importance of matching the polarization directions of the pump and probe waves as SRS nearly ceases to occur in the case of orthogonal polarizations. The use of a polarization-preserving fiber with a high-germania core has resulted in 20-dB gain at 1.52 µm by using only 3.7 W of input power from a Q-switched 1.34-µm laser [69]. From a practical standpoint, the quantity of interest is the so-called on–off ratio, defined as the ratio of the signal power with the pump on to that with the pump off. This ratio can be measured experimentally. The experimental results for a 1.34-µm pump show that the on–off ratio is about 24 dB for the first-order Stokes line at 1.42 µm but degrades to 8 dB when the first-order Stokes line is used to amplify a 1.52-µm signal. The on–off ratio is also found to be smaller in the backward-pumping configuration [72]. It can be improved if the output is passed through an optical filter that passes the amplified signal but reduces the bandwidth of the spontaneous noise. An attractive feature of Raman amplifiers is related to their broad bandwidth. It can be used to amplify several channels simultaneously in a WDM lightwave system. This feature was demonstrated in a 1987 experiment [74] in which signals from three distributed feedback semiconductor lasers operating in the range 1.57–1.58 µm were amplified simultaneously using a 1.47-µm pump. The gain of 5 dB was obtained at a pump power of only 60 mW. A theoretical analysis shows that a trade-off exists between the on–off ratio and channel gains [75]. During the 1980s, several experiments used the Raman gain for improving the performance of optical communication systems [76–79]. This scheme is called distributed amplification as fiber losses accumulated over 100 km or so are compensated in a distributed manner. It was used in 1988 to demonstrate soliton transmission over 4000 km [79]. The main drawback of Raman amplifiers from the standpoint of system applications is that a high-power laser is required for pumping. Early experiments employed tunable color-center lasers that were too bulky for practical applications. With the advent of

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erbium-doped fiber amplifiers in 1989, Raman amplifiers were rarely used for 1.55-µm telecommunication systems. The situation changed with the availability of compact high-power semiconductor lasers in the 1990s. As early as 1992, a Raman amplifier was pumped using a 1.55-µm semiconductor laser [80]. The 140-ns pump pulses had a 1.4-W peak-power level at the 1-kHz repetition rate and were capable of amplifying 1.66-µm signal pulses by more than 23 dB in a 20-km-long dispersion-shifted fiber. The resulting 200-mW peak power of 1.66-µm pulses was large enough for their use for optical time-domain reflection measurements, a technique commonly used for supervising and maintaining fiber-optic networks [81]. The use of Raman amplifiers in the 1.3-µm region also attracted attention during the 1990s [82–87]. In one approach, three pairs of fiber gratings were inserted within the fiber used for Raman amplification [82]. The Bragg wavelengths of these gratings were chosen such that they form three cavities for three Raman lasers operating at wavelengths 1.117, 1.175, and 1.24 µm that correspond to the first three Stokes lines of a 1.06-µm pump. All three lasers were pumped through cascaded SRS using a diodepumped fiber laser. The 1.24-µm laser then pumped the Raman amplifier to amplify a 1.3-µm signal. The same idea of cascaded SRS was used to obtain a 39-dB gain by using WDM couplers in place of fiber gratings [83]. In a different approach, the core of silica fiber is doped heavily with germania. Such a fiber can be pumped to provide 30-dB gain at a pump power of only 350 mW, which can be obtained by using two or more semiconductor lasers [84]. A dual-stage configuration has also been used in which a 2-km-long germania-doped fiber is placed in series with a 6-km-long dispersionshifted fiber in a ring geometry [87]. Such a Raman amplifier, when pumped with a 1.24-µm Raman laser, provided 22-dB gain in the 1.3-µm wavelength region with a noise figure of about 4 dB. Raman amplifiers can be used to extend the bandwidth of WDM systems operating in the 1.55-µm region [88–90]. Erbium-doped fiber amplifiers used commonly in this wavelength regime, have a bandwidth of under 40 nm. Moreover, a gain-flattening technique is needed to use the entire 40-nm bandwidth. Massive WDM systems typically require optical amplifiers capable of providing uniform gain over a 70 to 80-nm wavelength range. Hybrid amplifiers made by combining erbium doping with Raman gain have been developed for this purpose. In one implementation of this idea [90], a nearly 80-nm bandwidth was realized by combining an erbium-doped fiber amplifier with two Raman amplifiers, pumped simultaneously at three different wavelengths (1471, 1495, and 1503 nm) using four pump modules, each module launching more than 150 mW of power into the fiber. The combined gain of 30 dB was nearly uniform over a wavelength range of 1.53 to 1.61 µm. Starting around 2000, distributed Raman amplification was used for compensation of fiber losses in long-haul WDM systems [91–96]. In this configuration, in place of using erbium-doped fiber amplifiers, relatively long spans (80 to 100 km) of the transmission fiber are pumped bidirectionally to provide Raman amplification. In a 2000

Stimulated Raman scattering

demonstration of this technique, 100 WDM channels with 25-GHz channel spacing, each operating at a bit rate of 10 Gb/s, were transmitted over 320 km [91]. All channels were amplified simultaneously by pumping each 80-km fiber span in the backward direction using four semiconductor lasers. By 2004, 128 channels, each operating at 10 Gb/s, could be transmitted over 4000 km with the use of distributed Raman amplification [95]. With the advent of high-power Yb-doped fiber lasers, Raman amplifiers have been made by using such lasers as a pumping source. In a 2008 experiment, 4.8 W of CW power was produced at 1178 nm by pumping a 150-m long single-mode fiber with a Yb-doped fiber laser operating at 1120 nm. The 1178-nm signal obtained from a DFB laser could be amplified by 27 dB while maintaining a 10-MHz line width [97]. By 2010, the output power could be increased to above 60 W by combining coherently the output of three such Raman amplifiers [98]. Moreover, frequency doubling of this output beam inside an external cavity provided 50 W of CW power at a wavelength of 589 nm. In a 2013 experiment, a Raman amplifier produced 22 W of power at 1178 nm, while maintaining a narrow line width, when pumped at 1120 nm in the backward direction [99].

8.2.4 Raman-induced crosstalk The same Raman gain that is beneficial for making fiber amplifiers and lasers is also detrimental to WDM systems. The reason is that a short-wavelength channel can act as a pump for longer-wavelength channels and thus transfer part of the pulse energy to neighboring channels. This leads to Raman-induced crosstalk among channels that can affect the system performance considerably [100–109]. Consider first a two-channel system with the short-wavelength channel acting as a pump. The power transfer between the two channels is governed by Eqs. (8.1.2) and (8.1.3). These equations can be solved analytically if the fiber loss is assumed to be the same for both channels (αs = αp ), an assumption easily justified for typical channel spacings near 1.55 µm. The amplification factor Gs for the longer-wavelength channel is given by Eq. (8.2.7). The associated reduction in the short-wavelength channel power is obtained from the pump-depletion factor Dp =

Ip (L ) 1 + r0 , = Ip (0) exp(−αp L ) 1 + r0 GA1+r 0

(8.2.9)

where GA and r0 are defined by Eqs. (8.2.4) and (8.2.8), respectively. Fig. 8.10 shows the pump-depletion characteristics by plotting Dp as a function of GA for several values of r0 . These curves can be used to obtain the Raman-induced power penalty defined as the relative increase in the pump power necessary to maintain the same level of output power as that in the absence of Raman crosstalk. The power penalty can be written as

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Figure 8.10 Pump-depletion characteristics showing variation of Dp with GA for three values of r0 .

(in decibels) R = 10 log(1/Dp ).

(8.2.10)

A 1-dB penalty corresponds to Dp ≈ 0.8. If we assume equal channel powers at the fiber input (r0 ≈ 1), Dp = 0.8 corresponds to GA ≈ 1.22. The input channel powers corresponding to 1-dB penalty can be obtained from Eq. (8.2.4). If we use the typical values for 1.55-µm optical communication systems, gR = 7 × 10−14 m/W, Aeff = 50 µm2 , and Leff = 1/αp ≈ 20 km, GA = 1.22 corresponds to P0 = 7 mW. If the Raman gain is reduced by a factor of 2 to account for polarization scrambling, this value increases to P0 = 14 mW. The experimental measurements of the power penalty are in agreement with the predictions of Eqs. (8.2.9) and (8.2.10). The situation is more complicated for multichannel WDM systems. The intermediate channels not only transfer energy to the longer-wavelength channels but, at the same time, also receive energy from the shorter-wavelength channels. For an M-channel system one can obtain the output powers for each channel by solving a set of M coupled equations [103] in a form similar to Eqs. (8.1.2) and (8.1.3). The shortest-wavelength channel is most affected by the Raman-induced crosstalk because it transfers a part of its energy to all channels lying within the Raman-gain bandwidth. The transferred amount, however, is different for each channel as it is determined by the amount of the Raman gain corresponding to the relative wavelength spacing. In one approach, the Raman-gain spectrum of Fig. 8.2 is approximated by a triangular profile [100]. The results show that for a 10-channel system with 10-nm separation, the input power of each channel should not exceed 3 mW to keep the power penalty below 0.5 dB. In a

Stimulated Raman scattering

10-channel experiment with 3-nm channel spacing, no power penalty related to Raman crosstalk was observed when the input channel power was kept below 1 mW [101]. The foregoing discussion provides only a rough estimate of the Raman crosstalk as it neglects the fact that signals in different channels consist of different random sequences of 0 and 1 bits. It is intuitively clear that such pattern effects will reduce the level of Raman crosstalk. The GVD effects that were neglected in the preceding analysis also reduce the Raman crosstalk since pulses in different channels travel at different speeds because of the group-velocity mismatch [104]. For a realistic WDM system, one must also consider dispersion management and add the contributions of multiple fiber segments separated by optical amplifiers [109].

8.3. SRS with short pump pulses The quasi-CW regime of SRS considered in Section 8.2 applies for pump pulses of widths >1 ns because the walk-off length LW , defined in Eq. (8.1.22), generally exceeds the fiber length L for such pulses. However, for pulses of widths below 100 ps, typically LW < L. SRS is then limited by the group-velocity mismatch and occurs only over distances z ∼ LW even if the fiber used is considerable longer than LW . At the same time, the nonlinear effects such as SPM and XPM become important because of relatively large peak powers and affect considerably the evolution of both pump and Raman pulses [110–121]. In this section we discuss the experimental and theoretical aspects of SRS in the normal-GVD region of optical fibers. Section 8.4 is devoted to the anomalous-GVD region in which the soliton effects become important. In both cases, pulse widths are assumed to be larger than the Raman response time (∼50 fs) so that transient effects are negligible.

8.3.1 Pulse-propagation equations In the general case in which GVD, SPM, XPM, pulse walk-off, and pump depletion all play an important role, Eqs. (8.1.20) and (8.1.21) should be solved numerically. If we neglect fiber losses because of relatively small fiber lengths used in most experiments, set δR = 0 assuming  = R , and measure time in a frame of reference moving with the pump pulse, these equations take the form

 ∂ Ap iβ2p ∂ 2 Ap gp 2 2 + = i γ A | + ( 2 − f )| A | Ap − |As |2 Ap , | p p R s 2 ∂z 2 ∂T 2

 ∂ As ∂ As iβ2s ∂ 2 As g s −d + = iγs |As |2 + (2 − fR )|Ap |2 As + |Ap |2 As , ∂z ∂T 2 ∂T 2 2

(8.3.1) (8.3.2)

where T = t − z/vgp ,

−1 d = vgp − vgs−1 .

(8.3.3)

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The walk-off parameter d accounts for the group-velocity mismatch between the pump and Raman pulses and is typically 2–6 ps/m. The GVD parameter β2j , the nonlinear parameter γj , and the Raman-gain coefficient gj (j = p or s) are slightly different for the pump and Raman pulses because of the Raman shift of about 13 THz between their carrier frequencies. In terms of the wavelength ratio λp /λs , these parameters are related as β2s =

λp β2p , λs

γs =

λp γp , λs

gs =

λp gp . λs

(8.3.4)

Four length scales can be introduced to determine the relative importance of various terms in Eqs. (8.3.1) and (8.3.2). For pump pulses of duration T0 and peak power P0 , these are defined as LD =

T02 |β2p |

,

LW =

T0 , |d |

LNL =

1 , γ p P0

LG =

1 . gp P0

(8.3.5)

The dispersion length LD , the walk-off length LW , the nonlinear length LNL , and the Raman-gain length LG provide, respectively, the length scales over which the effects of GVD, pulse walk-off, nonlinearity (both SPM and XPM), and Raman gain become important. The shortest length among them plays the dominant role. Typically, LW ∼ 1 m (for T0 < 10 ps) while LNL and LG become smaller or comparable to it for P0 > 100 W. In contrast, LD ∼ 1 km for T0 = 10 ps. Thus, the GVD effects are generally negligible for pulses as short as 10 ps. The situation changes for pulse widths ∼1 ps or less because LD decreases faster than LW with a decrease in the pulse width. The GVD effects can then affect SRS evolution significantly, especially in the anomalous-GVD region.

8.3.2 Nondispersive case When the second-derivative term in Eqs. (8.3.1) and (8.3.2) is neglected, these equations can be solved analytically [122–125]. The analytic solution takes a simple form if pump depletion during SRS is neglected. As this assumption is justified for the initial stages of SRS and permits us to gain physical insight, let us consider it in some detail. The resulting analytic solution includes the effects of both XPM and pulse walk-off. The XPM effects without walk-off effects were considered relatively early [111]. Both of them can be included by solving Eqs. (8.3.1) and (8.3.2) with β2p = β2s = 0 and gp = 0. Eq. (8.3.1) for the pump pulse then yields the solution Ap (z, T ) = Ap (0, T ) exp[iγp |Ap (0, T )|2 z],

(8.3.6)

where the XPM term has been neglected assuming |As |2 |Ap |2 . For the same reason, the SPM term in Eq. (8.3.2) can be neglected. The solution of Eq. (8.3.2) is then given

Stimulated Raman scattering

by [120] As (z, T ) = As (0, T + zd) exp{[gs /2 + iγs (2 − fR )]ψ(z, T )},

(8.3.7)

where 

z

ψ(z, T ) =

|Ap (0, T + zd − z d)|2 dz .

(8.3.8)

0

Eq. (8.3.6) shows that the pump pulse of initial amplitude Ap (0, T ) propagates without change in its shape. The SPM-induced phase shift imposes a frequency chirp on the pump pulse that broadens its spectrum (see Section 4.1). The Raman pulse, by contrast, changes both its shape and spectrum as it propagates through the fiber. Temporal changes occur owing to Raman gain while spectral changes have their origin in XPM. Because of pulse walk-off, both kinds of changes are governed by an overlap factor ψ(z, T ) that takes into account the relative separation between the two pulses along the fiber. This factor depends on the pulse shape. For a Gaussian pump pulse with the input amplitude

Ap (0, T ) = P0 exp(−T 2 /2T02 ),

(8.3.9)

the integral in Eq. (8.3.8) can be performed in terms of error functions with the result √ ψ(z, τ ) = [erf(τ + δ) − erf(τ )]( π P0 z/δ),

(8.3.10)

where τ = T /T0 and δ is the distance in units of the walk-off length, that is, δ = zd/T0 = z/LW .

(8.3.11)

An analytic expression for ψ(z, τ ) can also be obtained for pump pulses having “sech” shape. In both cases, the Raman pulse compresses initially, reaches a minimum width, and then begins to rebroaden as it is amplified through SRS. It also acquires a frequency chirp through XPM. This qualitative behavior persists even when pump depletion is included [122–124]. Eq. (8.3.7) describes Raman amplification when a weak signal pulse is injected together with the pump pulse. The case in which the Raman pulse builds from noise is much more involved from a theoretical viewpoint. A general approach should use a quantum-mechanical treatment, similar to that employed for describing SRS in molecular gases [43]. It can be considerably simplified for optical fibers if transient effects are ignored by assuming that pump pulses are much wider than the Raman response time. In that case, Eqs. (8.3.1) and (8.3.2) can be used, provided a noise term (often called the Langevin force) is added to the right-hand side of these equations. The noise term leads to pulse-to-pulse fluctuations in the amplitude, width, and energy of the Raman pulse,

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similar to those observed for SRS in molecular gases [21]. Its inclusion is essential if the objective is to quantify such fluctuations. The average features of noise-seeded Raman pulses can be described with the theory of Section 8.1.2, where the effective Stokes power at the fiber input is obtained by using one photon per mode at all frequencies within the Raman-gain spectrum. Eq. (8.1.10) then provides the input peak power of the Raman pulse, while its shape remains undetermined. Numerical solutions of Eqs. (8.3.1) and (8.3.2) show that the average pulse shapes and spectra at the fiber output are not dramatically affected by different choices of the shape of the seed pulse. A simple approximation consists of assuming As (0, T ) = (Ps0eff )1/2 ,

(8.3.12)

where Ps0eff is given by Eq. (8.1.10). Alternatively, one may take the seed pulse to be Gaussian with a peak power Ps0eff . As a simple application of the analytic solution (8.3.7), consider the Raman threshold for SRS induced by short pump pulses of width T0 and peak power P0 [115]. The peak power of the Raman pulse at the fiber output (z = L ) is given by √

Ps (L ) = |As (L , 0)|2 = Ps0eff exp( π gs P0 LW ),

(8.3.13)

where Eq. (8.3.10) was used with τ = 0 and L /LW  1. If we define the Raman threshold in the same way as for the CW case, the threshold is achieved when Ps (L ) = P0 . The comparison of Eqs. (8.1.12) and (8.3.13) shows that one can use the CW criterion provided the effective length is taken to be √

Leff = π LW ≈ TFWHM /|d|.

(8.3.14)

In particular, Eq. (8.1.13) can be used to obtain the critical peak power of the pump pulse if Leff is obtained from Eq. (8.3.14). This change is expected because the effective interaction length between the pump and Raman pulses is determined by the length LW —SRS ceases to occur when the two pulses move apart enough that they stop overlapping significantly. Eqs. (8.1.13) and (8.3.14) show that the Raman threshold depends on the width of the pump pulse and increases inversely with TFWHM . For pulse widths ∼10 ps (LW ∼ 1 m), the threshold pump powers are ∼100 W. The analytic solution (8.3.7) can be used to obtain both the shape and the spectrum of the Raman pulse during the initial stages of SRS [120]. The spectral evolution is governed by the XPM-induced frequency chirp. The chirp behavior has been discussed in Section 7.4 in the context of XPM-induced asymmetric spectral broadening (see Fig. 7.3). The qualitative features of the XPM-induced chirp are identical to those shown there as long as the pump remains undepleted. Note, however, that the Raman pulse travels faster than the pump pulse in the normal-GVD regime. As a result, chirp

Stimulated Raman scattering

is induced mainly near the trailing edge. It should be stressed that both pulse shapes and spectra are considerably modified when pump depletion is included [118]. As the energy of the Raman pulse grows, it affects itself through SPM and the pump pulse through XPM.

8.3.3 Effects of GVD When the fiber length is comparable to the dispersion length LD , it is important to include the GVD effects. Such effects cannot be described analytically, and a numerical solution of Eqs. (8.3.1) and (8.3.2) is necessary to understand the SRS evolution. A generalization of the split-step Fourier method of Section 2.4 can be used for this purpose. The method requires specification of the Raman pulse at the fiber input by using Eq. (8.1.10). For numerical purposes, it is useful to introduce the normalized variables. A relevant length scale along the fiber length is provided by the walk-off length LW . By defining z =

z , LW

τ=

T , T0

Aj Uj = √ , P0

(8.3.15)

and using Eq. (8.3.4), Eqs. (8.3.1) and (8.3.2) become  ∂ Up iLW ∂ 2 Up iLW

LW + = |Us |2 Up , (8.3.16) |Up |2 + (2 − fR )|Us |2 Up −  2 ∂z 2LD ∂τ LNL 2LG  ∂ Us ∂ Us irLW ∂ 2 Us irLW

rLW − = |Up |2 Us , (8.3.17) |Us |2 + (2 − fR )|Up |2 Us + +  2 ∂z ∂τ 2LD ∂τ LNL 2LG

where the lengths LD , LW , LNL , and LG are given by Eq. (8.3.5). The parameter r = λp /λs and is about 0.95 at λp = 1.06 µm. Fig. 8.11 shows the evolution of the pump and Raman pulses over three walk-off lengths using LD /LW = 1000, LW /LNL = 24, and LW /LG = 12. The pump pulse is taken to be a Gaussian. The Stokes pulse is seeded at z = 0 as indicated in Eq. (8.3.12) such that its power is initially smaller by a factor of 2 × 10−7 compared with the pump pulse. The results shown in Fig. 8.11 are applicable to a wide variety of input pulse widths and pump wavelengths by using the scaling of Eqs. (8.3.5) and (8.3.15). The choice LW /LG = 12 implies that √

π gs P0 LW ≈ 21,

(8.3.18)

and corresponds to a peak power 30% above the Raman threshold. Several features of Fig. 8.11 are noteworthy. The Raman pulse starts to build up after one walk-off length. Energy transfer to the Raman pulse from the pump pulse is nearly complete by z = 3LW because the two pulses are then physically separated owing to the group-velocity mismatch. As the Raman pulse moves faster than the pump pulse in the

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Figure 8.11 Evolution of (A) pump and (B) Raman pulses over three walk-off lengths in the specific case for which LD /LW = 1000, LW /LNL = 24, and LW /LG = 12.

normal-GVD regime, the energy for SRS comes from the leading edge of the pump pulse. This is clearly apparent at z = 2LW where energy transfer has led to a two-peak structure in the pump pulse as a result of pump depletion—the hole near the leading edge corresponds exactly to the location of the Raman pulse. The small peak near the leading edge disappears with further propagation as the Raman pulse walks through it. The pump pulse at z = 3LW is asymmetric in shape and appears narrower than the input pulse as it consists of the trailing portion of the input pulse. The Raman pulse is also narrower than the input pulse and is asymmetric with a sharp leading edge. The spectra of pump and Raman pulses display many interesting features resulting from the combination of SPM, XPM, group-velocity mismatch, and pump depletion. Fig. 8.12 shows the pump and Raman spectra at z = 2LW (top row) and 3LW (bottom row). The asymmetric nature of these spectra is due to XPM-induced chirp. The highfrequency side of the pump spectra exhibits an oscillatory structure that is characteristic of SPM (see Section 4.1). In the absence of SRS, the spectrum would be symmetric with the same structure appearing on the low-frequency side. As the low-frequency components occur near the leading edge of the pump pulse and because the pump is depleted on the leading side, the energy is transferred mainly from the low-frequency components of the pump pulse. This is clearly seen in the pump spectra of Fig. 8.12. The long tail on the low-frequency side of the Raman spectra is also partly for the same reason. The Raman spectrum is nearly featureless at z = 2LW but develops considerable internal structure at z = 3LW . This is due to the combination of XPM and pump de-

Stimulated Raman scattering

Figure 8.12 Spectra of pump (left column) and Raman (right column) pulses at z = 2LW and 3LW under conditions of Fig. 8.11.

pletion; the frequency chirp across the Raman pulse induced by these effects can vary rapidly both in magnitude and sign and leads to a complicated spectral shape [118]. The temporal and spectral features seen in Figs. 8.11 and 8.12 depend on the peak power of input pulses through the lengths LG and LNL in Eqs. (8.3.16) and (8.3.17). When peak power is increased, both LG and LNL decrease by the same factor. Numerical results show that because of a larger Raman gain, the Raman pulse grows faster and carries more energy than that shown in Fig. 8.11. More importantly, because of a decrease in LNL , both the SPM and XPM contributions to the frequency chirp are enhanced, and the pulse spectra are wider than those shown in Fig. 8.12. An interesting feature is that the Raman-pulse spectrum becomes considerably wider than the pump spectrum. This is due to the stronger effect of XPM on the Raman pulse compared with that of the pump pulse. The XPM-enhanced frequency chirp for the Raman pulse was predicted as early as 1976 [110]. In a theoretical study that included XPM but neglected group-velocity mismatch and pump depletion, the spectrum of the Raman pulse was shown to be wider by a factor of two [111]. Numerical results that include all of these effects show an enhanced broadening by up to a factor of three, in agreement with experiments discussed later. Direct measurements of the frequency chirp also show an enhanced chirp for the Raman pulse compared with that of the pump pulse [121].

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8.3.4 Raman-induced index changes So far we have ignored the impact of the Raman-induced changes in the refractive index of the fiber mode. As seen from Eq. (8.1.19) and Fig. 8.3, the parameter δR should not be ignored because it produces a change in the refractive index seen by the Stokes pulse as it propagates down the fiber. From Eqs. (8.1.19) and (8.1.21), the Raman-induced index change nR can be written in the form nR = (c /ωs )γs δR |Ap |2 =

Re[h˜ R ()] c gs ()|Ap |2 . Im[h˜ R ()] 2ωs

(8.3.19)

This equation shows that Raman-induced index changes scale linearly with the pump power and depend strongly on the frequency dependence of the Raman response. The usefulness of Eq. (8.3.19) is limited because an analytic form of h˜ R () is not known for silica fibers. Considerable physical insight can be gained by using the approximate form of the Raman response function hR (t) given in Eq. (2.3.39). By taking its Fourier transform, we obtain h˜ R () =

2R + R2 /4 R /2 ≈ , 2 2 ( − ) − iR /2 R − ( + iR /2) R

(8.3.20)

where R = 1/τ1 is the vibrational resonance frequency (the Raman shift) and R = 2/τ2 is the damping rate of molecular vibrations. The approximate form is obtained by using R /2 R and assuming that  is close to R . Using this functional form of h˜ R () in Eq. (8.3.19) and taking into account of the frequency dependence of gs (), the Raman-induced index change is found to be nR =

c 2ωs



 δ gs (R )|Ap |2 , 1 + δ2

(8.3.21)

where δ = 2(R − )/ R is a normalized detuning parameter and gs (R ) is the peak value of the Raman gain at δ = 0. Even though nR itself is relatively small, it varies rapidly with  near the Ramangain peak. The group velocity of a pulse is related inversely to the group index as vg = c /ng with ng = nt + ω(dnt /dω), where the total refractive index nt includes nR . For this reason, rapid changes in nR with δ can change ng considerably and affect the group velocity of a pulse. Including the contribution of fiber dispersion as well, the group index is given by 

cgR P0 ng = ng0 + R Aeff



1 − δ2 , (1 + δ 2 )2

(8.3.22)

where ng0 is the group index in the absence of SRS and we used gs (R ) = gR /Aeff together with P0 = |Ap |2 . The contribution of the last term peaks at the Raman-gain

Stimulated Raman scattering

peak (δ = 0). Thus, when the Stokes frequency is tuned to match the Raman shift ( = R ), one can expect the Stokes pulse to travel slower because of Raman-induced changes in the refractive index of the fiber mode. The slowing down of optical pulses in the vicinity of an optical resonance (the socalled slow light) has attracted considerable attention. This phenomenon is discussed in Section 9.4.3 in the context of SBS, because Brillouin gain is exploited more often than the Raman gain for this purpose. In a 2005 experiment, Raman amplification of 430-fs pulses inside a 1-km-long fiber resulted in their relative delay of up to 370 fs [126]. Since the use of the Raman gain for slowing down of optical pulses requires high pump powers together with femtosecond pulses, it is unlikely to be employed in practice. As we saw in Section 8.3.3, any group-velocity mismatch between the pump and Stokes pulses is undesirable for the Raman-amplification process because it leads to the separation of two pulses and hinders transfer of energy from the pump to the Stokes pulse. An interesting question is whether the walk-off effects can be reduced, or even eliminated, by using the slow-light effect. The answer is affirmative when both pulses experience normal GVD inside the fiber providing the Raman gain, because the Stokes pulse travels faster than the pump pulse in that situation [127]. It is evident that any amount of slowing down of the Stokes pulse will reduce the impact of group-velocity mismatch. However, a complete cancellation can occur only for a specific input pump power because it requires the Raman-induced reduction in the Stokes speed to match the speed of the pump pulse. In the case of δ = 0 corresponding to  = R , the relative slowing down of the Stokes pulse over a fiber of length L is found from Eq. (8.3.22) to be T = gR P0 L /(R Aeff ). This should match the relative delay of Ld = L β2 R resulting from the group-velocity mismatch, where the parameter d is taken from Eq. (8.3.3) and β2 is the GVD parameter at the pump wavelength. Equating the two delays, the pump power required for a complete elimination of the walk-off effects is found to be P0 = β2 R R Aeff /gR .

(8.3.23)

Numerical simulations validate this analytical reasoning [127], although a complete elimination of the walk-off effects is difficult to realize in practice because of the pumpdepletion effects. As another application of the slow-light effect, it was found that the XPM-induced frequency chirp can also be tuned optically through Raman-induced changes in the group velocity of the Stokes pulse [128].

8.3.5 Experimental results The spectral and temporal features of ultrafast SRS have been studied in many experiments performed in the visible as well as in the near-infrared region [129–131]. In one experiment, 60-ps pulses from a mode-locked Nd:YAG laser operating at 1.06 µm were propagated through a 10-m-long fiber [112]. When the pump power exceeded

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Figure 8.13 Experimental spectra observed when 25-ps pump pulses at 532 nm were propagated through a 10-m-long fiber. Four spectra correspond to different input pulse energies normalized to E0 . (After Ref. [129]; ©1987 OSA.)

the Raman threshold (∼1 kW), a Raman pulse was generated. Both the pump and Raman pulses were narrower than the input pulse as expected from the results shown in Fig. 8.11. The spectrum of the Raman pulse was much broader (spectral width ≈2 THz) than that of the pump pulse. The XPM-enhanced spectral broadening of the Raman pulse was quantified in an experiment in which 25-ps pulses at 532 nm were propagated through a 10-m-long fiber [129]. Fig. 8.13 shows the observed spectra at four values of pump-pulse energies. The Raman spectral band located at 544.5 mm has a width about three times that of the pump. This is expected from theory and is due to the XPM-induced frequency chirp [111]. In the spectra of Fig. 8.13 the fine structure could not be resolved because of a limited spectrometer resolution. Details of the pump spectrum were resolved in another experiment in which 140-ps input pulses at 1.06 µm were propagated through a 150-m-long fiber [119]. Fig. 8.14 shows the observed pump spectra at several values of input peak powers. The Raman threshold is about 100 W in this experiment. For P0 < 100 W, the spectrum exhibits a multipeak structure, typical of SPM (see Section 4.1). However, the pump spectrum broadens and becomes highly asymmetric for P0 > 100 W. In fact, the two spectra corresponding to peak powers beyond the Raman threshold show features that are qualitatively similar to those of Fig. 8.12 (left column). The spectral asymmetry is due to the combined effects of XPM and pump depletion. Another phenomenon that can lead to new qualitative features in the pump spectrum is the XPM-induced modulation instability. It has been discussed in Section 7.2 for the case in which two pulses at different wavelengths are launched at the fiber input. However, the same phenomenon should occur even if the second pulse is internally

Stimulated Raman scattering

Figure 8.14 Experimental spectra of pump pulses at different input peak powers after 140-ps pump pulses have propagated through a 150-m-long fiber. The Raman threshold is reached near 100 W. (After Ref. [119]; ©1987 IEE.)

generated through SRS. Similar to the case of modulation instability occurring in the anomalous-GVD region (see Section 5.1), XPM-induced modulation instability manifests through the appearance of spectral side bands in the pulse spectra. Fig. 8.15 shows the observed spectra of the pump and Raman pulses in an experiment in which 25-ps pulses at 532 mm were propagated through a 3-m-long fiber [130]. The fiber-core diameter was only 3 µm to rule out the possibility of multimode four-wave mixing (see Chapter 10). The central peak in the pump spectrum contains considerable internal structure (Fig. 8.14) that remained unresolved in this experiment. The two sidebands provide clear evidence of XPM-induced modulation instability. The position of sidebands changes with fiber length and with peak power of pump pulses. The Stokes spectrum also shows side bands, as expected from the theory of Section 7.2, although they are barely resolved because of XPM-induced broadening of the spectrum. The temporal measurements of ultrafast SRS show features that are similar to those shown in Fig. 8.11 [113–117]. In one experiment, 5-ps pulses from a dye laser operating at 615 nm were propagated through a 12-m-long fiber with a core diameter of 3.3 µm [114]. Fig. 8.16 shows the cross-correlation traces of the pump and Raman pulses at the fiber output. The Raman pulse arrives about 55 ps earlier than the pump pulse; this is consistent with the group-velocity mismatch at 620 nm. More importantly, the Raman pulse is asymmetric with a sharp leading edge and a long trailing edge, features qualitatively similar to those shown in Fig. 8.11. Similar results have been obtained in other

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Figure 8.15 Spectra of pump and Raman pulses showing spectral sidelobes as a result of XPMinduced modulation instability. (After Ref. [130]; ©1988 Springer.)

Figure 8.16 Cross-correlation traces of pump and Raman pulses at the output of a 12-m-long fiber. The intensity scale is arbitrary. (After Ref. [114]; ©1985 Elsevier.)

experiments where the pulse shapes are directly recorded using a streak camera [119] or a high-speed photodetector [132]. The effects of pulse walk-off on SRS were studied by varying the peak power of 35-ps pump pulses (at 532 nm) over a range of 140 to 210 W while fiber length was varied over a range of 20 to 100 m [115]. Temporal measurements of the pump and Raman pulses were made with a high-speed CdTe photodetector and a sampling oscilloscope. The results show that the Raman pulse is produced within the first three to four walk-off lengths. The peak appears after about two walk-off lengths into the fiber for 20% energy conversion and moves closer to the input for higher peak powers. These conclusions are in agreement with the numerical results shown in Fig. 8.11. So far, only first-order SRS has been considered. When the input peak power of pump pulses exceeds the Raman threshold considerably, the Raman pulse may become intense enough to act as a pump to generate the second-order Stokes pulse. Such a cascaded SRS was seen in a 615-nm experiment using 5-ps pump pulses with a peak power of 1.5 kW [114]. In the near-infrared region multiple orders of Stokes can be generated using 1.06-µm pump pulses. The efficiency of the SRS process can be improved considerably by using silica fibers whose core has been doped with P2 O5 because of a relatively large Raman gain associated with P2 O5 glasses [133–135]. From a practical standpoint ultrafast SRS limits the performance of fiber-grating compressors [136]; the peak power of input pulses must be kept below the Raman threshold to ensure optimum performance. In this case SRS not only acts as a loss

Stimulated Raman scattering

Figure 8.17 Temporal and spectral output of a Raman laser in the case of resonant (dashed curve) and single-pass (full curve) operation. (After Ref. [117]; ©1987 Taylor & Francis.)

mechanism but also limits the quality of pulse compression by distorting the linear nature of the frequency chirp resulting from the XPM interaction between pump and Raman pulses [137]. A spectral-filtering technique had been used to improve the quality of compressed pulses even in the presence of SRS [132]. In this approach a portion of the pulse spectrum is selected by using an asymmetric spectral window so that the filtered pulse has nearly linear chirp across its entire width. Good-quality compressed pulses can be obtained in the strong-SRS regime but only at the expense of substantial energy loss [122].

8.3.6 Synchronously pumped Raman lasers The preceding section has focused on single-pass SRS. By placing the fiber inside a cavity (see Fig. 8.6), a single-pass SRS configuration can be turned into a Raman laser. Such lasers were discussed in Section 8.2.2 in the case of CW or quasi-CW operation (T0 ∼1 ns). This section considers Raman lasers pumped synchronously to emit short optical pulses with widths 1.3 µm). The fourth and fifth Stokes bands form a broad spectral band encompassing the range 1.3–1.5 µm that contains about half of the input energy. Autocorrelation traces of output pulses in the spectral region near 1.35, 1.4, 1.45, and 1.5 µm showed

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that the energy in the pedestal decreased as the wavelength increased. In fact, output pulses near 1.5 µm were nearly pedestal free. Even though Raman soliton lasers are capable of generating femtosecond pulses that are useful for many applications, they suffer from a noise problem that limits their usefulness. Measurements of intensity noise for a synchronously pumped Raman soliton laser indicated that the noise was more than 50 dB above the shot-noise level [167]. Pulse-to-pulse timing jitter was also found to be quite large, exceeding 5 ps at 1.6-W pump power. Such large noise levels can be understood if one considers the impact of the Raman-induced frequency shift (see Section 5.5) on the performance of such lasers. For synchronous pumping to be effective, the round-trip time of the Raman soliton inside the laser cavity must be an integer multiple of the spacing between pump pulses. However, the Raman-induced frequency shift changes the group velocity and slows down the pulse in such an unpredictable manner that synchronization is quite difficult to achieve in practice. As a result, Raman soliton lasers generate pulses in a way similar to single-pass Raman amplifiers and suffer from the noise problems associated with such amplifiers. The performance of Raman soliton lasers can be significantly improved if the Raman-induced frequency shift can be eliminated. It turns out that such frequency shifts can be suppressed with a proper choice of pump power and laser wavelength. In a 1992 experiment, a Raman soliton laser was synchronously pumped by using 200-ps pump pulses from a 1.32-µm Nd:YAG laser. This laser was tunable over a wavelength range of 1.37–1.44 µm [168]. The Raman-induced frequency shift was suppressed in the wavelength range of 1.41–1.44 µm for which the Raman-gain spectrum in Fig. 8.2 had a positive slope. Measurements of noise indicated significant reduction in both intensity noise and timing jitter [169]. Physically, Raman-gain dispersion is used to cancel the effects of the last term in Eq. (2.3.43) that is responsible for the Raman-induced frequency shift. It is difficult to operate a soliton laser at wavelengths below 1.3 µm because most fibers exhibit normal GVD for λ < 1.3 µm. A grating pair is often employed to solve this problem, but its use makes the device bulky. A compact, all-fiber, laser can be built if a suitable fiber exists that can provide anomalous dispersion at the Raman-laser wavelength. Such fibers have become available in recent years in the form of the so-called microstructured or holey fibers. In a 2003 experiment [170], a 23-m-long holey fiber was employed within the cavity of an all-fiber Raman laser to generate 2-ps output pulses at the 1.14-µm wavelength. The laser was pumped synchronously with 17-ps pulses from a Yb-doped fiber laser. The Raman gain was provided by 15 m of standard silica fiber with 3% Ge doping. In the wavelength region near 1.6 µm, one can employ dispersion-shifted fibers that permit the control of cavity dispersion. Such a fiber was used in a 2004 experiment to realize a compact, synchronously pumped, Raman laser that was also tunable

Stimulated Raman scattering

from 1620 to 1660 nm [171]. The 2.1-km-long fiber loop inside a ring cavity had its zero-dispersion wavelength at 1571 nm with a dispersion slope of about 0.1 ps3 /km. This laser also employed as a pump a gain-switched DFB semiconductor laser emitting 110-ps pulses at a 54.5 MHz repetition rate that were amplified to an average power level of up to 200 mW. Such a Raman laser produced pulses shorter than 0.5 ps over its entire 40-nm tuning range.

8.4.3 Soliton-effect pulse compression In some sense, Raman solitons formed in Raman amplifiers or lasers take advantage of the pulse compression occurring for higher-order solitons (see Section 5.2). Generally speaking, Raman amplification can be used for simultaneous amplification and compression of picosecond optical pulses by taking advantage of the anomalous GVD in optical fibers. In a 1991 experiment 5.8-ps pulses, obtained from a gain-switched semiconductor laser and amplified using an erbium-doped fiber amplifier to an energy level appropriate for forming a fundamental soliton, were compressed to 3.6-ps through Raman amplification in a 23-km-long dispersion-shifted fiber [172]. The physical mechanism behind simultaneous amplification and compression can be understood from Eq. (5.2.3) for the soliton order N. If a fundamental soliton is amplified adiabatically, the condition N = 1 can be maintained during amplification, provided the soliton width changes as P0−1/2 as its peak power P0 increases. In many practical situations, the input signal pulse is not intense enough to sustain a soliton inside the fiber during Raman amplification. It is possible to realize pulse compression even for such weak pulses through the XPM effects that invariably occur during SRS [173]. In essence, the pump and signal pulses are injected simultaneously into the fiber. The signal pulse extracts energy from the pump pulse through SRS and is amplified. At the same time, it interacts with the pump pulse through XPM that imposes a nearly linear frequency chirp on it and compresses it in the presence of anomalous GVD. As discussed in Section 7.5 in the context of XPM-induced pulse compression, the effectiveness of such a scheme depends critically on the relative groupvelocity mismatch between the pump and signal pulses. A simple way to minimize the group-velocity mismatch and maximize the XPM-induced frequency chirp is to choose the zero-dispersion wavelength of the fiber in the middle of the pump and signal wavelengths. Numerical simulations based on Eqs. (8.3.1) and (8.3.2) show that compression factors as large as 15 can be realized while the signal-pulse energy is amplified a millionfold [173]. Fig. 8.23 shows (A) the compression factor and (B) the amplification factor as a function of the propagation distance (ξ = z/LD ) for several values of the parameter N, related to the peak power P0 of pump pulses through N = (γp P0 LD )1/2 , where LD is defined in Eq. (8.3.5). The pump and signal pulses are assumed to be Gaussian, have the same width, and propagate at the same speed. Pulse compression is maximum for

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Figure 8.23 (A) Compression factor and (B) amplification factor as a function of normalized distance for several values of N in the case of Gaussian-shape pump and signal pulses. (After Ref. [173]; ©1993 OSA.)

an optimum fiber length, a feature similar to that of compressors based on higher-order solitons. This behavior is easily understood by noting that GVD reduces the XPMinduced chirp to nearly zero at the point of maximum compression (see Section 3.2). The main point to note from Fig. 8.23 is that weak input pulses can be amplified by 50–60 dB while getting compressed simultaneously by a factor of 10 or more. The quality of compressed pulses is also quite good with no pedestal and little ringing. The qualitative features of pulse compression remain nearly the same when pulse widths or group velocities of the pump and signal pulses are not the same, making this technique quite attractive from a practical standpoint. Simultaneous amplification and compression of picosecond optical pulses were indeed observed in a 1996 experiment [174].

8.5. Polarization effects In the scalar approach used so far, it has been implicitly assumed that both pump and signal are copolarized and maintain their state of polarization (SOP) inside the fiber. However, unless a polarization-maintaining fiber is used for making Raman amplifiers, birefringence of most fibers changes the SOP of both fields in a random fashion and leads to polarization-mode dispersion (PMD), a phenomenon discussed in Section 7.7.1. Here we develop a vector theory of SRS and use it to study the polarization effects that have attracted attention in recent years [175–182].

8.5.1 Vector theory of Raman amplification The starting point is Eq. (2.1.10) for the third-order nonlinear polarization induced inside the fiber material (silica glass) in response to an optical field E(r, t). In the scalar (3) is relevant for the case, this equation leads to Eq. (2.3.32) as only the component χxxxx nonlinear response. In the vector case, the situation is somewhat complicated. Following

Stimulated Raman scattering

Appendix B, we include both the Kerr and Raman contributions given in Eqs. (B.8) and (B.9), while also retaining the vector nature of the electric field E(r, t). In the case of SRS, we write the total electric field E and induced polarization PNL in the form E(t) = Re[Ep exp(−iωp t) + Es exp(−iωs t)],

(8.5.1)

PNL (t) = Re[Pp exp(−iωp t) + Ps exp(−iωs t)].

(8.5.2)

After some algebra, we can write Pj with j = p, s as 30 (3)

χ c0 (Ej · Ej )E∗j + c1 (E∗j · Ej )Ej 4 xxxx  + c2 (E∗m · Em )Ej + c3 (Em · Ej )E∗m + c4 (E∗m · Ej )Em ,

Pj =

(8.5.3)

where j = m and c0 = 13 (1 − fR ) + 12 fb . The remaining coefficients are defined as c1 = 23 (1 − fR ) + fa + 12 fb , c3 = 23 (1 − fR ) + 12 fb + fa h˜ a (),

c2 = 23 (1 − fR ) + fa + 12 fb h˜ b (),

(8.5.4)

c4 = 23 (1 − fR ) + 12 fb + 12 fb h˜ b (),

(8.5.5)

with  = ωj − ωk . In arriving at Eq. (8.5.3), we made use of the relation h˜ a (0) = h˜ b (0) = 1. We can now introduce the Jones vectors |Ap and |As as in Section 7.6.1. To keep the following analysis relatively simple, we assume that both the pump and Stokes are in the form of CW or quasi-CW waves and neglect the dispersive effects. With this simplification, we obtain the following set of two coupled vector equations [177]:

d|Ap αp i + |Ap + ωp bl · σ |Ap = iγp c1 Ap |Ap + c0 |A∗p A∗p | dz 2 2  + c2 As |As + c3 |As As | + c4 |A∗s A∗s | |Ap ,

i d|As αs + |As + ωs bl · σ |As = iγs c1 As |As + c0 |A∗s A∗s | dz 2 2  + c2 Ap |Ap + c3 |Ap Ap | + c4 |A∗p A∗p | |As ,

(8.5.6)

(8.5.7)

where the effects of linear birefringence are included through the birefringence vector bl . The coefficients c2 , c3 , and c4 contain the Fourier transforms, h˜ a () and h˜ b () of the two Raman response functions ha (t) and hb (t). Their imaginary parts can be used to define the corresponding Raman gains, ga and gb for the Stokes field as [4] gj = γs fj Im[h˜ j ()];

(j = a, b),

(8.5.8)

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where  = ωp − ωs is the Raman shift. For silica fibers, gb /ga is much smaller than 1. In the case of pump equation (8.5.6), we replace  with −; this sign change makes gj negative in Eq. (8.5.8). This is expected since the gain for Stokes results in a power loss for the pump. The real parts of h˜ a () and h˜ b () lead to small changes in the refractive index. Such changes invariably occur and are governed by the Kramers–Kronig relation. It is useful to introduce two dimensionless parameters δa and δb which are defined as δj = (fj /c1 )Re[h˜ j () − 1];

(j = a, b).

(8.5.9)

Eqs. (8.5.6) and (8.5.7) appear quite complicated in the Jones-matrix formalism. They can be simplified considerably if we write them in the Stokes space using the Pauli spin vector σ . After introducing the Stokes vectors for the pump and Stokes fields as P = Ap |σ |Ap ,

S = As |σ |As ,

(8.5.10)

and making use of the identities in Eqs. (7.6.12)–(7.6.14), we obtain the following two vector equations that govern the dynamics of P and S on the Poincaré sphere: ωp dP + αp P = − [(ga + 3gb )Ps P + (ga + gb )Pp S − 2gb Pp S3 ] + (ωp bl + W p ) × P, dz 2ωs (8.5.11) 1 dS + αs S = [(ga + 3gb )Pp S + (ga + gb )Ps P − 2gb Ps P3 ] + (ωs bl + W s ) × S, dz 2 (8.5.12)

where Pp = |P| is the pump power, Ps = |S| is the Stokes power, and 2γp [P3 + 2(1 + δb )S3 − (2 + δa + δb )S] , 3 2γs Ws = [S3 + 2(1 + δb )P3 − (2 + δa + δb )P] . 3

Wp =

(8.5.13) (8.5.14)

The vectors W p and W s account for the nonlinear polarization rotation on the Poincaré sphere through the SPM and XPM effects. Eqs. (8.5.11) and (8.5.12) simplify considerably when both the pump and signal are linearly polarized, as P and S then lie in the equatorial plane of the Poincaré sphere with P3 = S3 = 0. In the absence of birefringence (bl = 0), P and S maintain their initial SOPs with propagation. When the pump and signal are copolarized, the two gain terms add in phase, and Raman gain is maximum with a value g = ga + 2gb . In contrast, when the two are orthogonally polarized, the two gain terms add out of phase, and the Raman gain becomes minimum with the value g⊥ = gb . These two gains correspond to the solid and dashed curves of Fig. 8.2, respectively, which can be used to deduce

Stimulated Raman scattering

the depolarization ratio g⊥ /g . This ratio is about 1/3 near /(2π) = 1 THz but drops to below 0.05 close to the gain peak at  = R [13]. If we use the value g⊥ /g = 1/3 together with g /g⊥ = 2 + ga /gb , we deduce that gb = ga for small values of . However, close to the Raman gain peak, the contribution of gb becomes relatively small. In the case of circular polarization, the vectors P and S point toward the north or south pole on the Poincaré sphere. It is easy to deduce from Eqs. (8.5.11) and (8.5.12) that g = ga + gb and g⊥ = 2gb under such conditions. The depolarization ratio g /g⊥ for circular polarization is thus different compared to the case of linear polarization. In particular, it is close to 1 for small values of  for which gb ≈ ga . We now solve Eqs. (8.5.11) and (8.5.12) in the presence of fluctuating residual birefringence. Such fluctuations occur on a length scale (∼10 m) much smaller than a typical nonlinear length (>1 km). As a result, the tips of P and S move on the Poincaré sphere randomly at high rates proportional to ωp and ωs , respectively. However, the SRS process depends only on the relative orientation of P and S. To simplify the following analysis, we adopt a rotating frame in which P remains stationary and focus on the changes in the relative angle between P and S. Eqs. (8.5.15) and (8.5.16) then become [177] ωp dP + αp P = − [(ga + 3gb )Ps P + (ga + gb /3)Pp S] − γ¯p S × P, dz 2ωs 1 dS + αs S = [(ga + 3gb )Pp S + (ga + gb /3)Ps P] − (b + γ¯s P) × S, dz 2

(8.5.15) (8.5.16)

where b is related to bl through a rotation on the Poincaré sphere and γ¯j = (2γj /9)(4 + 3δa + δb ),

 = ωp − ωs .

(8.5.17)

To simplify Eqs. (8.5.15) and (8.5.16) further, we neglect the pump depletion as well as signal-induced XPM assuming that Ps Pp , a condition often valid in practice. The pump equation (8.5.15) then contains only the loss term and can be easily integrated to yield P(z) = pˆ Pin exp(−αp z) ≡ pˆ Pp (z),

(8.5.18)

where pˆ is the SOP of the pump at the input end. The term γ¯s P × S in Eq. (8.5.16) represents the pump-induced nonlinear polarization rotation of the signal being amplified. Such rotations of the Stokes vector do not affect the Raman gain. We can eliminate this term with a simple transformation to obtain [177] dS 1 + αs S = [(ga + 3gb )Pp S + (ga + gb /3)Ps P] −  b × S, dz 2

(8.5.19)

where b is related to b by a deterministic rotation that does not affect its statistical properties. It is modeled as a three-dimensional Gaussian process whose first-order and

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second-order moments are given in Eq. (7.7.6) in terms of the PMD parameter Dp of the fiber. The stochastic differential equation (8.5.19) should be treated on physical ground in the Stratonovich sense [183]. Eq. (8.5.19) can be solved by noting that the terms that do not change the direction of S can be removed with the transformation 

z

S = s exp 0





1 (ga + 3gb )Pp (z) − αs dz . 2

(8.5.20)

The new Stokes vector s then satisfies ds 1 = (ga + gb /3)Pp (z)s0 (z)ˆp −  b × s, dz 2

(8.5.21)

where s0 = |s| is the magnitude of vector s. The equation governing evolution of s0 along the fiber length is obtained by using the relation s20 = s · s and is given by ds0 s ds = · = (ga + gb /3)Pp (z)ˆp · s. dz s0 dz

(8.5.22)

Eq. (8.5.21) is derived assuming that both the pump and signal travel in the same direction (the so-called forward pumping scheme). In the case of backward pumping, the derivative dP/dz in Eq. (8.5.11) becomes negative. Following the preceding procedure, the Stokes equation can still be written in the form of Eq. (8.5.21) provided  is redefined as  = −(ωp + ωs ). Such a large value of  enhances the contribution of the birefringence term in Eq. (8.5.21). It will be seen later that backward pumping improves the performance of Raman amplifiers because of this feature.

8.5.2 PMD effects on Raman amplification The residual birefringence of fibers, responsible for PMD, affects the performance of Raman amplifiers considerably because of its statistical nature. Eq. (8.5.21) can be used to calculate the power Ps of the amplified signal as well its SOP at any distance within the amplifier. In practice, b(z) fluctuates along the fiber in a dynamic fashion as it depends on environmental factors such as stress and temperature. As a result, the amplified signal Ps (L ) at the output of an amplifier also fluctuates. Such fluctuations affect the performance of any Raman amplifier. In this section we calculate the average and the variance of the amplified signal fluctuating because of PMD. The signal gain is defined as G(L ) = Ps (L )/Ps (0). Its average value, Gav , and variance of signal power fluctuations, σs2 , can be calculated using Gav = Ps (L )/Ps (0),

σs2 = Ps2 (L )/[Ps (L )]2 − 1.

(8.5.23)

To find the average signal power Ps (L ) at the end of a Raman amplifier of length L, we calculate s0 by taking an average over b in Eq. (8.5.22). However, this average depends

Stimulated Raman scattering

on ρ = s0 cos θ , where θ represents an angle in the Stokes space between the pump and signal SOPs. Using a well-known technique [183], we obtain the following two coupled but deterministic equations [177]: ds0 1 = (ga + gb /3)Pp (z)ρ, dz 2 dρ 1 = (ga + gb /3)Pp (z)s0 − ηρ, dz 2

(8.5.24) (8.5.25)

where η = 1/Ld = Dp2 2 /3. The diffusion length Ld is a measure of the distance after which the SOPs of the two optical fields, separated in frequency by , become decorrelated. Eqs. (8.5.24) and (8.5.25) are two linear first-order differential equations that can be easily integrated. When PMD effects are large and Ld L, ρ reduces to zero over a short fiber section. The average gain in this case is given by Gav = exp[ 12 (ga + gb /3)Pin Leff − αs L ].

(8.5.26)

Compared to the ideal copolarized case (no birefringence) for which g = ga + 2gb , PMD reduces the Raman gain by a factor of about 2 [51]. This is expected on physical grounds. However, it should be stressed that the amplification factor GA is reduced by a large factor because of its exponential dependence on ga and gb . The variance of signal fluctuations requires the second-order moment Ps2 (L ) of the amplified signal. Following a similar averaging procedure [177], Eq. (8.5.21) leads to the following set of three linear equations: ds20 = (ga + gb /3)Pp (z)ρ1 , dz 1 d ρ1 = −ηρ2 + (ga + gb /3)Pp (z)[s20 + ρ2 ], dz 2 d ρ2 = −3ηρ2 + ηs20 + (ga + gb /3)Pp (z)ρ2 , dz

(8.5.27) (8.5.28) (8.5.29)

where ρ1 = s20 cos θ and ρ2 = s20 cos2 θ . These equations show that signal fluctuations have their origin in fluctuations of the relative angle θ between the Stokes vectors associated with the pump and signal. To illustrate the impact of PMD on the performance of Raman amplifiers, we consider a 10-km-long Raman amplifier pumped with 1 W of power using a single 1.45-µm laser and use ga = 0.6 W−1 /km with gb /ga = 0.012 near the signal wavelength 1.55 µm. Fiber losses are taken to be 0.273 dB/km and 0.2 dB/km at the pump and signal wavelengths, respectively. Fig. 8.24 shows how Gav and σs change with the PMD parameter Dp when the input signal is copolarized (solid curves) or orthogonally polarized (dashed

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Figure 8.24 (A) Average gain and (B) standard deviation of signal fluctuations at the output of a Raman amplifier as a function of PMD parameter in the cases of forward and backward pumping. The solid and dashed curves correspond to the cases of copolarized and orthogonally polarized signals, respectively. (After Ref. [177]; ©2003 OSA.)

curves) with respect to the pump. The curves are shown for both the forward and backward pumping schemes. When Dp is zero, the two beams maintain their SOP, and the copolarized signal experiences a maximum gain of 17.6 dB. In contrast, an orthogonally polarized signal exhibits a 1.7-dB loss, irrespective of the pumping configuration. The loss is not exactly 2 dB because of a finite value of gb . As the PMD parameter increases, the gain difference between the copolarized and orthogonally polarized cases decreases and disappears eventually. The level of signal fluctuations in Fig. 8.24 increases quickly with the PMD parameter, reaches a peak value, and then decreases slowly toward zero with a further increase in Dp because of an averaging effect produced by the PMD. The location of the peak depends on the pumping scheme as well as on the initial polarization of the pump; the noise level near this peak exceeds 40%. If a fiber with low PMD is used, the noise level can exceed 70% under some operating conditions. The behavior for backward pumping is similar to those for forward pumping but the peak shifts to smaller Dp values. This shift is due to a much larger value of || = ωp + ωs . Since the diffusion length scales as ||−2 , it is smaller by about a factor of (ωp + ωs )/(ωp − ωs ) ≈ 30 in the backward-pumping case. A smaller diffusion length results in rapid averaging over birefringence fluctuations and reduces the noise level. In practice, backward pumping is preferred because it produces less signal degradation. We briefly mention Raman-induced polarization pulling. It was found in 2009 by solving Eq. (8.5.19) numerically that the SOP of the signal is pulled toward the pump’s SOP inside a Raman amplifier when their frequencies differ by about 13 THz, and an experiment confirmed this behavior [181]. This can be understood by noting that

Stimulated Raman scattering

gb ga in Eq. (8.5.19), and the signal is amplified most when it is copolarized with the pump. In contrast when the frequency difference is small (∼1 THz) and gb is not negligible, signal’s SOP is pulled toward circular polarization, irrespective of the input SOP of the pump [182].

Problems 8.1 What is meant by Raman scattering? Explain its origin. What is the difference between spontaneous and stimulated Raman scattering? 8.2 Solve Eqs. (8.1.2) and (8.1.3) neglecting pump deletion. Calculate the Stokes power at the output of a 1-km-long fiber when 1-µW power is injected together with an intense pump beam. Assume gR Ip (0) = 2 km−1 and αp = αs = 0.2 dB/km. 8.3 Perform the integration in Eq. (8.1.8) using the method of steepest descent and derive Eq. (8.1.9). 8.4 Use Eq. (8.1.9) to derive the Raman-threshold condition given in Eq. (8.1.13). 8.5 Solve Eqs. (8.1.2) and (8.1.3) analytically after assuming αp = αs . 8.6 Calculate the threshold pump power of a 1.55-µm Raman laser whose cavity includes a 1-km-long fiber with 40-µm2 effective core area. Use αp = 0.3 dB/km and total cavity losses of 6 dB. Use the Raman gain from Fig. 8.2. 8.7 Explain the technique of time-dispersion tuning used commonly for synchronously pumped Raman lasers. Estimate the tuning range for the laser of Problem 8.6. 8.8 Solve Eqs. (8.3.1) and (8.3.2) analytically after setting β2p = β2s = 0. 8.9 Use the results of the preceding problem to plot the output pulse shapes for Raman amplification in a 1-km-long fiber. Assume λp = 1.06 µm, λs = 1.12 µm, γp = 10 W−1 /km, gR = 1 × 10−3 m/W, d = 5 ps/m, and Aeff = 40 µm2 . Input pump and Stokes pulses are Gaussian with the same 100-ps width (FWHM) and with 1 kW and 10 mW peak powers, respectively. 8.10 Solve Eqs. (8.3.16) and (8.3.17) numerically using the split-step Fourier method and reproduce the results shown in Figs. 8.11 and 8.12. 8.11 Solve Eqs. (8.3.16) and (8.3.17) in the case of anomalous dispersion and reproduce the results shown in Fig. 8.19. Explain why the Raman pulse forms a soliton when the walk-off length and the dispersion length have comparable magnitudes. 8.12 Design an experiment for amplifying 50-ps (FWHM) Gaussian pulses through SRS by at least 30 dB such that they are also compressed by a factor of 10. Use numerical simulations to verify your design. 8.13 Derive Eq. (8.5.3) after taking into account the properties of the χ (3) tensor discussed in Appendix B. 8.14 Derive the set of two coupled vector equations given as Eqs. (8.5.6) and (8.5.7) starting from the nonlinear polarization given in Eq. (8.5.3).

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8.15 Convert Eqs. (8.5.6) and (8.5.7) to the Stokes space and prove that the Stokes vectors for the pump and signal indeed satisfy Eqs. (8.5.11) and (8.5.12).

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[87] D.V. Gapontsev, S.V. Chernikov, J.R. Taylor, Opt. Commun. 166 (1999) 85. [88] H. Masuda, S. Kawai, K. Suzuki, K. Aida, IEEE Photonics Technol. Lett. 10 (1998) 516. [89] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, E. Rabarijaona, IEEE Photonics Technol. Lett. 11 (1999) 530. [90] H. Masuda, S. Kawai, IEEE Photonics Technol. Lett. 11 (1999) 647. [91] H. Suzuki, J. Kani, H. Masuda, N. Takachio, K. Iwatsuki, Y. Tada, M. Sumida, IEEE Photonics Technol. Lett. 12 (2000) 903. [92] S. Namiki, Y. Emori, IEEE J. Sel. Top. Quantum Electron. 7 (2001) 3. [93] M.N. Islam (Ed.), Raman Amplifiers for Telecommunications, Springer, 2004. [94] J. Bromage, J. Lightwave Technol. 22 (2004) 79. [95] D.F. Grosz, A. Agarwal, S. Banerjee, D.N. Maywar, A.P. Küng, J. Lightwave Technol. 22 (2004) 423. [96] C. Headley, G.P. Agrawal (Eds.), Raman Amplification in Fiber Optical Communication Systems, Academic Press, 2005. [97] Y. Feng, L.R. Taylor, D. Bonaccini Calia, Opt. Express 16 (2008) 10927. [98] L.R. Taylor, Y. Feng, D.B. Calia, Opt. Express 18 (2010) 8540. [99] I. Dajani, C. Vergien, C. Robin, B. Ward, Opt. Express 21 (2013) 12038. [100] A.R. Chraplyvy, Electron. Lett. 20 (1984) 58. [101] N.A. Olsson, J. Hegarty, R.A. Logan, et al., Electron. Lett. 21 (1985) 105. [102] A.R. Chraplyvy, J. Lightwave Technol. 8 (1990) 1548. [103] D.N. Christodoulides, R.B. Jander, IEEE Photonics Technol. Lett. 8 (1996) 1722. [104] J. Wang, X. Sun, M. Zhang, IEEE Photonics Technol. Lett. 10 (1998) 540. [105] M.E. Marhic, F.S. Yang, L.G. Kazovsky, J. Opt. Soc. Am. B 15 (1998) 957. [106] S. Bigo, S. Gauchard, A. Bertaina, J.P. Hamaide, IEEE Photonics Technol. Lett. 11 (1999) 671. [107] K.-P. Ho, J. Lightwave Technol. 18 (2000) 915. [108] X. Zhou, M. Birk, J. Lightwave Technol. 21 (2003) 2194. [109] T. Yamamoto, S. Norimatsu, J. Lightwave Technol. 21 (2003) 2229. [110] V.N. Lugovoi, Sov. Phys. JETP 44 (1976) 683. [111] J.I. Gersten, R.R. Alfano, M. Belic, Phys. Rev. A 21 (1980) 1222. [112] E.M. Dianov, A.Y. Karasik, P.V. Mamyshev, et al., JETP Lett. 39 (1984) 691. [113] E.M. Dianov, A.Y. Karasik, P.V. Mamyshev, A.M. Prokhorov, V.N. Serkin, Sov. Phys. JETP 62 (1985) 448. [114] B. Valk, W. Hodel, H.P. Weber, Opt. Commun. 54 (1985) 363. [115] R.H. Stolen, A.M. Johnson, IEEE J. Quantum Electron. QE-22 (1986) 2154. [116] P.M.W. French, A.S.L. Gomes, A.S. Gouveia-Neto, J.R. Taylor, IEEE J. Quantum Electron. 22 (1986) 2230. [117] K. Smith, P.N. Kean, D.W. Crust, W. Sibbett, J. Mod. Opt. 34 (1987) 1227. [118] D. Schadt, B. Jaskorzynska, J. Opt. Soc. Am. B 4 (1987) 856. [119] P.N. Kean, K. Smith, W. Sibbett, IEE Proc. 134 (1987), Pt. J. [120] J.T. Manassah, Appl. Opt. 26 (1987) 3747; J.T. Manassah, O. Cockings, Appl. Opt. 26 (1987) 3749. [121] A.S.L. Gomes, V.L. da Silva, J.R. Taylor, J. Opt. Soc. Am. B 5 (1988) 373. [122] J. Hermann, J. Mondry, J. Mod. Opt. 35 (1988) 1919. [123] R. Osborne, J. Opt. Soc. Am. B 6 (1989) 1726. [124] D.N. Cristodoulides, R.I. Joseph, IEEE J. Quantum Electron. 25 (1989) 273. [125] Y.B. Band, J.R. Ackerhalt, D.F. Heller, IEEE J. Quantum Electron. 26 (1990) 1259. [126] J.E. Sharping, Y. Okawachi, A.L. Gaeta, Opt. Express 13 (2005) 6092. [127] G. Fanjoux, T. Sylvestre, Opt. Lett. 33 (2008) 2506. [128] G. Fanjoux, T. Sylvestre, Opt. Lett. 34 (2009) 3824. [129] R.R. Alfano, P.L. Baldeck, F. Raccah, P.P. Ho, Appl. Opt. 26 (1987) 3492.

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[130] P.L. Baldeck, R.R. Alfano, G.P. Agrawal, in: T. Yajima, et al. (Eds.), Ultrafast Phenomena VI, Springer, 1988, p. 53. [131] M. Kuckartz, R. Schultz, A. Harde, Opt. Quantum Electron. 19 (1987) 237, J. Opt. Soc. Am. B 5 (1988) 1353. [132] A.M. Weiner, J.P. Heritage, R.H. Stolen, J. Opt. Soc. Am. B 5 (1988) 364. [133] K. Suzuki, K. Noguchi, N. Uesugi, Opt. Lett. 11 (1986) 656, J. Lightwave Technol. 6 (1988) 94. [134] A.S.L. Gomes, V.L. da Silva, J.R. Taylor, B.J. Ainslie, S.P. Craig, Opt. Commun. 64 (1987) 373. [135] E.M. Dianov, L.A. Bufetov, M.M. Bubnov, M.V. Grekov, S.A. Vasiliev, O.I. Medvedkov, Opt. Lett. 25 (2000) 402. [136] T. Nakashima, M. Nakazawa, K. Nishi, H. Kubota, Opt. Lett. 12 (1987) 404. [137] A.S.L. Gomes, A.S. Gouveia-Neto, J.R. Taylor, Opt. Quantum Electron. 20 (1988) 95. [138] E.M. Dianov, P.V. Mamyshev, A.M. Prokhorov, D.G. Fursa, JETP Lett. 45 (1987) 599, JETP Lett. 46 (1987) 482. [139] K. Tamura, M. Nakazawa, Opt. Lett. 21 (1996) 68. [140] M.E. Fermann, V.I. Kruglov, B.C. Thomsen, J.M. Dudley, J.D. Harvey, Phys. Rev. Lett. 26 (2000) 6010. [141] V.I. Kruglov, A.C. Peacock, J.D. Harvey, J.M. Dudley, J. Opt. Soc. Am. B 19 (2002) 461. [142] J. Limpert, T. Schreiber, T. Clausnitzer, et al., Opt. Express 10 (2002) 628. [143] A.C. Peacock, N.G.R. Broderick, T.M. Monro, Opt. Commun. 218 (2003) 167. [144] C. Finot, G. Millot, C. Billet, J.M. Dudley, Opt. Express 11 (2003) 1547. [145] C. Finot, G. Millot, S. Pitois, C. Billet, J.M. Dudley, IEEE J. Sel. Top. Quantum Electron. 10 (2004) 1211. [146] G.Q. Chang, A. Galvanauskas, H.G. Winful, T.B. Norris, Opt. Lett. 29 (2004) 2647. [147] V.A. Vysloukh, V.N. Serkin, JETP Lett. 38 (1983) 199. [148] E.M. Dianov, A.Y. Karasik, P.V. Mamyshev, A.M. Prokhorov, JETP Lett. 41 (1985) 294. [149] E.M. Dianov, A.M. Prokhorov, V.N. Serkin, Opt. Lett. 11 (1986) 168. [150] F.M. Mitschke, L.F. Mollenauer, Opt. Lett. 11 (1986) 659. [151] V.N. Serkin, Sov. Tech. Phys. Lett. 13 (1987) 366. [152] P. Beaud, W. Hodel, B. Zysset, H.P. Weber, IEEE J. Quantum Electron. 23 (1987) 1938. [153] A.S. Gouveia-Neto, A.S.L. Gomes, J.R. Taylor, Opt. Lett. 12 (1987) 1035, IEEE J. Quantum Electron. 24 (1988) 332. [154] K.J. Blow, N.J. Doran, D. Wood, J. Opt. Soc. Am. B 5 (1988) 381. [155] A. Höök, D. Anderson, M. Lisak, Opt. Lett. 13 (1988) 1114, J. Opt. Soc. Am. B 6 (1989) 1851. [156] E.A. Golovchenko, E.M. Dianov, P.V. Mamyshev, A.M. Prokhorov, D.G. Fursa, J. Opt. Soc. Am. B 7 (1990) 172. [157] A.S. Gouveia-Neto, J. Lightwave Technol. 10 (1992) 1536. [158] K. Drühl, R.G. Wenzel, J.L. Carlsten, Phys. Rev. Lett. 51 (1983) 1171. [159] C.R. Menyuk, Phys. Rev. Lett. 62 (1989) 2937. [160] J.C. Englund, C.M. Bowden, Phys. Rev. Lett. 57 (1986) 2661, Phys. Rev. A 46 (1992) 578. [161] D.N. Christodoulides, Opt. Commun. 86 (1991) 431. [162] M.N. Islam, L.F. Mollenauer, R.H. Stolen, J.R. Simpson, H.T. Shang, Opt. Lett. 12 (1987) 814. [163] J.D. Kafka, T. Baer, Opt. Lett. 12 (1987) 181. [164] A.S. Gouveia-Neto, A.S.L. Gomes, J.R. Taylor, Electron. Lett. 23 (1987) 537, Opt. Quantum Electron. 20 (1988) 165. [165] H.A. Haus, M. Nakazawa, J. Opt. Soc. Am. B 4 (1987) 652. [166] V.L. da Silva, A.S.L. Gomes, J.R. Taylor, Opt. Commun. 66 (1988) 231. [167] U. Keller, K.D. Li, M. Rodwell, D.M. Bloom, IEEE J. Quantum Electron. 25 (1989) 280. [168] M. Ding, K. Kikuchi, IEEE Photonics Technol. Lett. 4 (1992) 927, IEEE Photonics Technol. Lett. 4 (1992) 1109.

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CHAPTER 9

Stimulated Brillouin scattering Stimulated Brillouin scattering (SBS) is a nonlinear process that can occur in optical fibers at input power levels much lower than those needed for stimulated Raman scattering (SRS). It manifests through the generation of a backward-propagating Stokes wave that carries most of the input power, once the Brillouin threshold is reached. For this reason, SBS limits the channel power in optical communication systems. At the same time, it can be useful for making fiber-based Brillouin lasers and amplifiers. This chapter is devoted to the SBS phenomenon in optical fibers. Section 9.1 presents the basic concepts behind SBS with emphasis on the spectrum of Brillouin gain. Section 9.2 focuses on the Brillouin threshold in the case of a continuous-wave (CW) pump and the techniques used to control it. Fiber-Brillouin amplifiers and their properties are discussed in Section 9.3. The dynamic aspects of SBS are the focus of Section 9.4, where we discuss SBS for a pulsed pump together with phenomena such as SBS-induced modulation instability and optical chaos. Section 9.5 is devoted to Brillouin lasers operating continuously or pulsed.

9.1. Basic concepts The nonlinear phenomenon of SBS, first observed in 1964, has been studied extensively [1–10]. SBS is similar to SRS inasmuch as it manifests through the generation of a Stokes wave, whose frequency is downshifted from that of the incident light by an amount set by the nonlinear medium. However, several major differences exist between the two. For example, the Stokes wave propagates backward when SBS occurs in a single-mode optical fiber, in contrast to SRS that can occur in both directions. The Stokes shift (∼10 GHz) is smaller by three orders of magnitude for SBS compared with that of SRS. The threshold pump power for SBS depends on the spectral width associated with the pump wave. It can be as low as 2 mW for a CW pump, or when the pumping is in the form of relatively wide pulses (width >1 µs). Also, SBS nearly ceases to occur for short pump pulses (width 1 kW) as well as Q-switched pulses with energies exceeding 1 µJ [54–58]. SBS is often the limiting factor and its control is essential in such devices. As SBS threshold scales linearly with fiber length, a short fiber with high Yb content is employed. A considerable increase in the SBS threshold occurs for large-core fibers because the effective mode area Aeff in Eq. (9.2.4) scales quadratically with the core diameter. This is the approach adopted for modern Yb-doped fiber lasers. In a 2004

Stimulated Brillouin scattering

experiment [55], 1.36 kW of CW power was produced directly from such a laser by employing a 40-µm-core fiber with Aeff ≈ 900 µm2 . Even larger powers have been realized since then. A variety of schemes are used for SBS suppression in such high-power lasers and amplifiers. In addition to using a relatively large core, fiber structure is modified to decrease the overlap between the optical and acoustic modes either though different core dopants [59] or through air holes in the cladding [60]. Such modifications reduce the overlap factor fA in Eq. (9.1.5) below 1 and can increase the SBS threshold by as much as a factor of 20. The SBS threshold predicted by Eq. (9.2.4) is only approximate for large-core fibers because its derivation assumes a single-mode fiber. A fiber with core diameter of >20 µm may support hundreds of modes. Not only is the input pump power distributed among many fiber modes, but each pump mode can also contribute to multiple Stokes modes [61–63]. In particular, the threshold pump power increases from the prediction of Eq. (9.2.4) by a factor that depends on the numerical aperture of the fiber, in addition to the core diameter [61]. An increase by a factor of 2 is expected for fibers with a numerical aperture of 0.2 at wavelengths near 1.06 µm. See Section 14.4.3 for SBS in multimode fibers.

9.2.4 Experimental results In the 1972 demonstration of SBS in optical fibers, a xenon laser operating at 535.5 nm was used as a source of relatively wide (width ∼1 µs) pump pulses [6]. Because of large fiber losses (about 1300 dB/km), only small sections of fibers (L = 5–20 m) were used in the experiment. The measured Brillouin threshold was 2.3 W for a 5.8-m-long fiber and reduced to below 1 W for L = 20 m, in good agreement with Eq. (9.2.4) if we use the estimated value Aeff = 13.5 µm2 for the effective mode area. The Brillouin shift of νB = 32.2 GHz also agreed with Eq. (9.1.3). Fig. 9.3 shows the oscilloscope traces of incident and transmitted pump pulses, as well as the corresponding Stokes pulse, for SBS occurring in a 5.8-m-long fiber. The oscillatory structure is due to relaxation oscillations whose origin is discussed in Section 9.4.4. The oscillation period of 60 ns corresponds to the round-trip time within the fiber. The Stokes pulse is narrower than the pump pulse because SBS transfers energy only from the central part of the pump pulse where power is large enough to exceed the Brillouin threshold. As a result, peak power of the Stokes pulse can exceed the input peak power of the pump pulse. Indeed, SBS can damage the fiber permanently because of such an increase in the Stokes intensity [6]. In most of the early experiments, the SBS threshold was relatively high (>100 mW) because of extremely large fiber losses. As mentioned earlier, it can become as low as 1 mW if long lengths of low-loss fiber are employed. A threshold of 30 mW was measured when a ring dye laser, operating in a single longitudinal mode at 0.71 µm, was used to couple CW light into a 4-km-long fiber having a loss of about 4 dB/km [64].

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Figure 9.3 Oscilloscope traces of (A) input and transmitted pump pulses and (B) the Stokes pulse for SBS occurring inside a 5.8-m-long fiber. (After Ref. [6]; ©1972 AIP.)

Figure 9.4 Schematic of the experimental setup used to observe SBS in optical fibers. (After Ref. [65]; ©1982 IEE.)

In a later experiment, the threshold power was reduced to only 5 mW [65] by using a Nd:YAG laser operating at 1.32 µm. Fig. 9.4 shows a schematic of this experiment. The 1.6-MHz spectral width of the CW pump was considerably smaller than the Brillouingain bandwidth. The experiment employed a 13.6-km-long fiber with losses of only 0.41 dB/km, resulting in an effective length of 7.66 km. The isolator served to block the Stokes light from entering the laser. The transmitted and reflected powers were measured as a function of input power launched into the fiber. Fig. 9.5 shows the experimental data. At low input powers, the reflected signal is due to 4% reflection at the air–fiber interface. Brillouin threshold is reached at an input power of about 5 mW as manifested through a substantial increase in the reflected power through SBS. At the same time, the transmitted power decreases because of pump depletion, and reaches a saturation level of about 2 mW for input powers in excess of 10 mW. The SBS conversion efficiency is nearly 65% at that level. In a 1987 experiment, SBS was observed by using a semiconductor laser, emitting light near 1.3-µm in a single longitudinal mode (bandwidth ∼10 MHz) through the

Stimulated Brillouin scattering

Figure 9.5 Transmitted (forward) and reflected (backward) powers as a function of the input power launched into a 13.6-km-long single-mode fiber. (After Ref. [65]; ©1982 IEE.)

distributed-feedback mechanism [66]. The CW pump light was coupled into a 30-kmlong fiber with a loss of 0.46 dB/km, resulting in an effective fiber length of about 9 km. The Brillouin threshold was reached at a pump power of 9 mW. To check if the bandwidth of the pump laser affected the SBS threshold, the measurements were also performed using a Nd:YAG laser having a bandwidth of only 20 kHz. The results were virtually identical with those obtained with the semiconductor laser, indicating that the Brillouin-gain bandwidth νB was considerably larger than 15 MHz. As mentioned in Section 9.1.2, νB is considerably enhanced for optical fibers from its value in bulk silica (≈22 MHz at 1.3 µm) because of fiber inhomogeneities. The threshold power is smaller (∼1 mW) near 1.55 µm because of relatively low losses of optical fibers (0.2 dB/km) at that wavelength. In most experiments on SBS, it is essential to use an optical isolator between the laser and the fiber to prevent the Stokes signal from reentering into the fiber after reflection at the laser-cavity mirror. In the absence of an isolator, a significant portion of the Stokes power can be fed back into the fiber. In one experiment, about 30% of the Stokes power was reflected back and appeared in the forward direction [67]. It was observed that several orders of Stokes and anti-Stokes lines were generated in the spectrum as a result of the feedback. Fig. 9.6 shows the output spectra in the forward and backward directions for SBS occurring in a 53-m-long fiber. The 34-GHz spacing between adjacent spectral lines corresponds exactly to the Brillouin shift at λp = 514.5 nm. The anti-Stokes components are generated as a result of four-wave mixing between the copropagating pump and Stokes waves (see Section 10.1). The higher-order Stokes lines are generated when power in the lower-order Stokes wave becomes large enough to satisfy the threshold condition given in Eq. (9.2.4). Even when optical isolators are used

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Figure 9.6 Output spectra in the forward (upper trace) and backward (lower trace) directions showing multiple orders of Stokes and anti-Stokes waves generated without isolation between laser and fiber. (After Ref. [67]; ©1980 Elsevier.)

to prevent external feedback, Rayleigh backscattering, occurring within the fiber, can provide feedback to the SBS process. In a 1998 experiment, this internal feedback process in a high-loss fiber (length 300 m) was strong enough to produce lasing when the fiber was pumped using 1.06-µm CW radiation [68].

9.3. Brillouin fiber amplifiers Similar to the case of SRS, the SBS-produced gain in an optical fiber can be used to amplify a weak signal whose frequency is shifted from the pump frequency by an amount equal to the Brillouin shift. Of course, the pump and injected signal must propagate in opposite directions in the case of single-mode fibers if SBS were to transfer pump power to the signal. Such amplifiers were first studied during the 1980s [69–80] and are useful for sensing and other applications.

9.3.1 Gain saturation When the pump and signal waves counterpropagate inside an optical fiber, a large part of the pump power can be transferred to the Stokes wave when their frequencies differ by the Brillouin shift (about 10 GHz). Initially, the signal power increases exponentially, as indicated in Eq. (9.2.3). However, this growth is reduced when the Brillouin gain begins to saturate. To account for pump depletion, it is necessary to solve Eqs. (9.2.1) and (9.2.2). Their general solution is somewhat complicated [81]. However, if fiber losses are neglected by setting α = 0, we can make use of the fact that d(Ip − Is )/dz = 0. If we use Ip − Is = C, where C is a constant, Eq. (9.2.2) can be integrated along the fiber length to yield 



C + Is (z) Is (z) = exp(gB Cz). Is (0) C + Is (0)

(9.3.1)

Stimulated Brillouin scattering

Figure 9.7 (A) Evolution of pump and Stokes powers (normalized to the input pump power) along fiber length for bin = 0.001 (solid lines) and 0.01 (dashed lines). (B) Saturated gain as a function of Stokes output for several values of GA .

Using C = Ip (0) − Is (0), the Stokes intensity Is (z) is given by [4] Is (z) =

b0 (1 − b0 ) Ip (0), G(z) − b0

(9.3.2)

where G(z) = exp[(1 − b0 )g0 z] with b0 = Is (0)/Ip (0),

g0 = gB Ip (0).

(9.3.3)

The parameter b0 is a measure of the SBS efficiency as it shows what fraction of the input pump power is converted to the Stokes power. The quantity g0 is the small-signal gain associated with the SBS process. Eq. (9.3.2) shows how the Stokes intensity varies in a Brillouin amplifier along the fiber length when the input signal is launched at z = L and the pump is incident at z = 0. Fig. 9.7 shows this variation for two input signals such that bin = Is (L )/Ip (0) = 0.001 and 0.01. The value of g0 L = 10 corresponds to unsaturated amplifier gain of e10 or 43 dB. Because of pump depletion, the net gain is considerably smaller. Nonetheless, about 50 and 70% of the pump power is transferred to the Stokes for bin = 0.001 and 0.01, respectively. Note also that most of the power transfer occurs within the first 20% of the fiber length. The saturation characteristics of Brillouin amplifiers can be obtained from Eq. (9.3.2) if we define the saturated gain using Gs = Is (0)/Is (L ) = b0 /bin ,

(9.3.4)

and introduce the unsaturated gain as GA = exp(g0 L ). Fig. 9.7 shows gain saturation by plotting Gs /GA as a function of GA bin for several values of GA . It should be compared

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Figure 9.8 Schematic illustration of a Brillouin amplifier; ECL, ISO, and PD stand for external-cavity laser, isolator, and photodetector, respectively. Solid and dashed arrows show the path of pump and probe lasers. (After Ref. [69]; ©1986 AIP.)

with Fig. 8.9 where the saturation characteristics of Raman amplifiers are shown. The saturated gain is reduced by a factor of two (by 3 dB) when GA bin ≈ 0.5 for GA in the range of 20 to 30 dB. This condition is satisfied when the amplified signal power becomes about 50% of the input pump power. As typical pump powers are ∼1 mW, the saturation power of Brillouin amplifiers is also ∼1 mW.

9.3.2 Amplifier design and applications A semiconductor laser can be used to pump Brillouin amplifiers provided it operates in a single-longitudinal mode whose spectral width is considerably less than the Brillouingain bandwidth. Distributed-feedback or external-cavity semiconductor lasers [82] are most appropriate for pumping Brillouin amplifiers. In a 1986 experiment, two externalcavity semiconductor lasers, with line widths below 0.1 MHz, were used as pump and probe lasers [69]. Both lasers operated continuously and were tunable in the spectral region near 1.5 µm. Fig. 9.8 shows the experimental setup schematically. Radiation from the pump laser was coupled into a 37.5-km-long fiber through a 3-dB coupler. The probe laser provided a weak input signal (∼ 10 µW) at the other end of the fiber. Its wavelength was tuned in the vicinity of the Brillouin shift (νB = 11.3 GHz) to maximize the Brillouin gain. The measured amplification factor increased exponentially with the pump power. This is expected from Eq. (9.3.4). If gain saturation is neglected, the amplification factor can be written as GA = Is (0)/Is (L ) = exp(gB P0 Leff /Aeff − α L ).

(9.3.5)

The amplifier gain was 16 dB (GA = 40) at a pump power of only 3.7 mW because of a long fiber length used in the experiment. An exponential increase in the signal power with increasing pump powers occurs only if the amplified signal remains below the saturation level. The saturated gain Gs is reduced by 3 dB when GA (Pin /P0 ) ≈ 0.5,

(9.3.6)

Stimulated Brillouin scattering

for GA in the range 20–30 dB, where Pin is the incident power of the signal being amplified. As discussed in Section 9.3.1, the saturation power of Brillouin amplifiers is ∼1 mW. In spite of gain saturation, Brillouin amplifiers are capable of providing 30 dB gain at a pump power under 10 mW. However, because νB < 100 MHz, the bandwidth of such amplifiers is also less than 100 MHz, in sharp contrast with Raman amplifiers whose bandwidth exceeds 5 THz. In fact, the difference between the signal and pump frequencies should be matched to the Brillouin shift νB (about 11 GHz in the 1.55-µm region) with an accuracy to better than 10 MHz. For this reason, Brillouin amplifiers are not suitable for amplifying signals in fiber-optic communication systems. Brillouin amplifiers may be useful in practice for applications requiring selective amplification [70–72]. One such application consists of amplifying the carrier of a modulated signal selectively, while leaving its modulation sidebands unamplified [83]. The underlying principle is similar to that of homodyne detection, except that the amplified carrier acts as a reference signal. This feature eliminates the need of a local oscillator that must be phase locked to the transmitter—a difficult task in general. In a demonstration of this scheme, the carrier was amplified by 30 dB more than the modulation sidebands even at a modulation frequency as low as 80 MHz [70]. With a proper design, sensitivity improvements of up to 15 dB or more are possible at bit rates in excess of 100 Mb/s. The limiting factor is the nonlinear phase shift induced by the pump (a kind of crossphase modulation), if the difference between the pump and carrier frequencies does not match the Brillouin shift exactly. The calculations show [71] that deviations from the Brillouin shift should be within 100 kHz for a phase stability of 0.1 rad. Nonlinear phase shifts can also lead to undesirable amplitude modulation of a frequency-modulated signal [74]. Another application of narrowband Brillouin amplifiers consists of using them as a tunable narrowband optical filter for channel selection in a densely packed multichannel communication system [72]. If channel spacing exceeds but the bit rate is smaller than the bandwidth νB , the pump laser can be tuned to amplify a particular channel selectively. This scheme was demonstrated in 1986 using a tunable color-center laser as a pump [72]. Two 45-Mb/s channels were transmitted through a 10-km-long fiber. Each channel could be amplified by 20 to 25 dB by using 14 mW of pump power when pump frequency was tuned in the vicinity of the Brillouin shift associated with each channel. More importantly, each channel could be detected with a low bit-error rate ( 100 ns. When pulse widths are below 10 ns, it becomes necessary to consider the dynamics of acoustic modes participating in the SBS process. This section focuses on such time-dependent effects.

9.4.1 Coupled amplitude equations The coupled intensity equations (9.2.1) and (9.2.2) are valid only under steady-state conditions. To include the transient effects, we need to solve the wave equation (2.3.1)

Stimulated Brillouin scattering

with the following material equation [10]:

∂ 2ρ

2 ∂ρ 2 2

−  ∇ − vA ∇ ρ = −0 γe ∇ 2 (E · E), A ∂ t2 ∂t

(9.4.1)

where ρ = ρ − ρ0 is a change in the local density from its average value ρ0 , A is the damping coefficient, and γe = ρ0 (∂/∂ρ)ρ=ρ0 is the electrostrictive constant introduced earlier in Section 9.1. The nonlinear polarization PNL appearing in Eq. (2.3.1) also has an additional term that depends on ρ as .

PNL = 0 [χ (3) .. EEE + (γe /ρ0 )ρ E].

(9.4.2)

Its origin lies in changes in material’s dielectric constant in response to acoustically induced density variations. To simplify the following analysis, we assume that all fields remain linearly polarized along the x axis and introduce the slowly varying fields, Ap and As , as E(r, t) = xˆ Re[Fp (x, y)Ap (z, t) exp(ikp z − iωp t) + Fs (x, y)As (z, t) exp(−iks z − iωs t)], (9.4.3) where Fj (x, y) is the mode profile for the pump (j = p) or Stokes wave (j = s). The density ρ is written in a similar fashion as ρ (r, t) = Re[FA (x, y)Q(z, t) exp(ikA z − it)],

(9.4.4)

where  = ωp −ωs , kA = va , and FA (x, y) is the spatial distribution of the acoustic mode with the amplitude Q(z, t). If several acoustic modes participate in the SBS process [31], a sum over all modes should be used in Eq. (9.4.4). In the following we consider a single acoustic mode responsible for the dominant peak in the Brillouin-gain spectrum. By using Eqs. (9.4.1)–(9.4.4) and Eq. (2.3.1) with the slowly varying envelope approximation, we obtain the following set of three coupled amplitude equations: ∂ Ap 1 ∂ Ap α + = − Ap + iγ (|Ap |2 + 2|As |2 )Ap + iκ1 As Q, ∂z vg ∂ t 2 1 ∂ As ∂ As α − + = − As + iγ (|As |2 + 2|Ap |2 )As + iκ1 Ap Q∗ , ∂z vg ∂ t 2 ∂Q ∂Q + vA = −[ 12 B + i(B − )]Q + iκ2 Ap A∗s , ∂t ∂z

(9.4.5) (9.4.6) (9.4.7)

where B = k2A A is the acoustic damping rate and Ap is normalized such that |Ap |2 represents optical power. The two coupling coefficients are defined as κ1 =

ωp γe Fp2 FA 

2np c ρ0 Fp2 

,

κ2 =

ωp γe Fp2 FA 

2c 2 vA FA2 Aeff

(9.4.8)

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where the angle brackets denote integration over the entire x–y domain and we used Fs ≈ Fp , ωs ≈ ωp , and ns ≈ np in view of a relatively small Brillouin shift. Eqs. (9.4.5) through (9.4.7) govern the SBS dynamics under quite general conditions [95–104]. They include the nonlinear phenomena of SPM and XPM, but the effects of GVD are ignored. This is justified by noting that pulse widths typically exceed 1 ns, for which the dispersion length is so large that GVD plays little role in the SBS process. In view of a small frequency difference between the pump and Stokes waves (about 10 GHz), we assumed that γ and α are nearly the same for both waves. For pump pulses of widths T0  TB = B−1 , Eqs. (9.4.5) through (9.4.7) can be simplified considerably because the acoustic amplitude Q decays so rapidly to its steadystate value that we can neglect both derivatives in Eq. (9.4.7). If we introduce the optical power as Pj = |Aj |2 where j = p or s, the SBS process is governed by the following two simple equations: ∂ Pp 1 ∂ Pp gB () + =− Pp Ps − α Pp , ∂z vg ∂ t Aeff 1 ∂ Ps gB () ∂ Ps − + = Pp Ps − α Ps , ∂z vg ∂ t Aeff

(9.4.9) (9.4.10)

where gB () is the SBS gain given in Eq. (9.1.4). Its peak value gB (B ) takes the form in Eq. (9.1.5) if we define the acoustic overlap factor as [31] fA =

( Fp2 FA )2

Fp2  FA2 



Fp2 (x, y)FA (x, y) dx dy

2

 =  2 . ( Fp (x, y) dx dy)( FA2 (x, y) dx dy)

(9.4.11)

Thus, the SBS gain is reduced by a fraction fA whose numerical value depends on the extent of overlap between the optical and acoustic modes supported by the fiber. When the spatial profiles associated with the optical and acoustic modes occupy a comparable area within the fiber, the numerical value of fA is close to 1. However, it can be reduced considerably if the fiber is designed to reduce the overlap between the optical and acoustic modes participating in the SBS process [35]. In a 2009 study, aluminum doping was used to reduce this overlap, resulting in a 4.3 dB increase in the pump power at which the SBS threshold was reached [59].

9.4.2 SBS with Q-switched pulses Eqs. (9.4.5)–(9.4.7) can be used to study SBS in the transient regime applicable for pump pulses shorter than 100 ns. From a practical standpoint, two cases are of interest depending on the repetition rate of pump pulses. Both cases are discussed here. In the case relevant for optical communication systems, the repetition rate of pump pulses is >1 GHz, while the width of each pulse is 5 THz). In the case of smaller frequency spacings ( 0 for λ1 < λ0 , kM is positive in the visible or near-infrared region. Phase matching for λ1 < λ0 can be realized if we make kW negative by propagating different waves in different modes of a multimode fiber, as discussed in Section 14.4.1. In single-mode fibers, the waveguide contribution kW in Eq. (10.3.1) is very small compared with the material contribution kM for identically polarized waves, except near the ZDWL λ0 where the two become comparable. The three techniques for approximate phase matching consist of: (1) reducing kM and kNL by using small frequency shifts and low pump powers; (2) operating near the ZDWL so that kW nearly cancels kM + kNL ; and (3) working in the anomalous GVD regime so that kM is negative and can be canceled by kNL + kW . The fourth technique makes use of modal birefringence to realize phase matching by polarizing different waves differently with respect to a principal axis of a polarization-maintaining fiber. All of these techniques are discussed in this section.

Four-wave mixing

10.3.2 Nearly phase-matched four-wave mixing The gain spectrum shown in Fig. 10.1 indicates that significant FWM can occur even if phase matching is not perfect to yield κ = 0 in Eq. (10.3.1). The amount of tolerable wave-vector mismatch depends on how long the fiber is compared with the coherence length Lcoh . Assuming that the contribution kM dominates in Eq. (10.3.1), the coherence length can be related to the frequency shift s by using Lcoh = 2π/|κ| with Eq. (10.3.6) and is given by Lcoh =

2π 2π . = |kM | |β2 |2s

(10.3.7)

In the visible region, β2 ∼ 50 ps2 /km, resulting in Lcoh ∼ 1 km for frequency shifts νs = s /2π ∼ 100 GHz. Such large coherence lengths indicate that significant FWM can occur when the fiber length satisfies the condition L < Lcoh . In an early experiment, three CW waves with a frequency separation in the range 1–10 GHz were propagated through a 150-m-long fiber, whose 4-µm core diameter ensured single-mode operation near the argon-ion laser wavelength of 514.5 nm [6]. The FWM generated nine new frequencies such that ω4 = ωi + ωj − ωk , where i, j, k equal 1, 2, or 3 with j = k. The experiment also showed that FWM can lead to spectral broadening whose magnitude increases with an increase in the incident power. The 3.9-GHz linewidth of the CW input from a multimode argon laser increased to 15.8 GHz at an input power of 1.63 W after passing through the fiber. The spectral components within the incident light generate new frequency components through FWM as the light propagates through the fiber. In fact, SPM-induced spectral broadening discussed in Section 4.1 can be interpreted in terms of such a FWM process [39]. From a practical standpoint FWM can lead to crosstalk in multichannel (WDM) communication systems, where the channel spacing is typically in the range of 10 to 100 GHz. This issue attracted considerable attention during the 1990s with the advent of WDM systems [40–46]. In an early experiment [19], three CW waves with a frequency separation ∼10 GHz were propagated through a 3.5-km-long fiber and the amount of power generated in the nine frequency components was measured by varying the frequency separation and the input power levels. Fig. 10.2 shows measured variations for two frequency components f332 and f231 using the notation fijk = fi + fj − fk ,

fj = ωj /(2π).

(10.3.8)

In part (a), f3 − f1 = 11 GHz, f2 − f1 = 17.2 GHz, P1 = 0.43 mW, P2 = 0.14 mW, and P3 is varied from 0.15 to 0.60 mW. In part (b), f3 − f2 is varied from 10 to 25 GHz for P3 = 0.55 mW, while keeping all other parameters the same. The generated power P4 varies with P3 linearly for the frequency component f231 but quadratically for the frequency component f332 . This is expected from the theory of

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Figure 10.2 Variation of FWM-generated power at the output of a 3.5-km-long fiber with (A) input power P3 and (B) frequency separation. (After Ref. [19]; ©1987 IEEE.)

Figure 10.3 Phase-matching diagrams near the ZDWL for three values of the pump wavelength λ1 . Dotted, dashed, and solid lines show respectively kM , kW , and their sum. (After Ref. [9]; ©1981 OSA.)

Section 10.2 by noting that f231 results from nondegenerate pump waves but the pump waves are degenerate in frequency for f332 . More power is generated in the frequency component f231 because f231 and f321 are degenerate, and the measured power is the sum of powers generated through two FWM processes. Finally, P4 decreases with increasing frequency separation because of a larger phase mismatch. A noteworthy feature of Fig. 10.2 is that up to 0.5 nW of power is generated for input powers 0 provided β4 is negative. This situation can occur in dispersion-flattened fibers [49]. In practice, it can be realized in tapered or microstructured fibers [50–53] (see Chapter 11). In a fiber exhibiting normal GVD with β4 < 0, Eq. (10.3.9) leads to the following expression for s : 2s

=

6 |β4 |



 2 β2 + 2|β4 |γ P0 /3 + β2 .

(10.3.12)

Fig. 10.5 shows how the frequency shift, νs = s /(2π), varies with β2 and |β4 | for γ P0 = 10 km−1 . In the limit |β4 |γ P0  β22 , this shift is given by the simple expression s = (12β2 /|β4 |)1/2 . For typical values of β2 and β4 , the frequency shift can exceed 25 THz, indicating that FWM can amplify a signal that is more than 200 nm away from

Four-wave mixing

Figure 10.5 Frequency shift between the pump and signal waves plotted as a function of |β4 | for several values of β2 when the FWM-process is phase-matched by SPM in a dispersion-flattened fiber.

Figure 10.6 (A) Dispersion parameter D as a function of wavelength for a birefringent fiber for light polarized along the slow (solid line) and fast (dashed line) axes. (B) Signal and idler wavelengths as a function of λp at low power (dashed curve) and at 100-W power level. Dotted lines mark the predicted wavelengths at λp = 647 nm. (After Ref. [51]; ©2003 OSA.)

the pump wavelength. If both β2 and β4 are positive for a fiber, the signal and idler frequencies are determined by the sixth and higher-order dispersive terms in Eq. (10.3.10). It is important to stress that dispersion parameters change with the pump frequency, and one must use the actual dispersion relation β(ω) of a fiber to find them. Fig. 10.6(A) shows how the parameter D (related to β2 ) changes with wavelength for a slightly birefringent photonic crystal fiber [51]. These measurements were used to deduce β2 and β4 , and then to determine the signal and idler wavelengths from. Eq. (14.14). These are plotted in Fig. 10.6(B) as a function of pump wavelength λp for an input power level

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of 100 W (solid curves). The signal and idler wavelengths are different for light polarized along the slow and fast axes because of birefringence (dotted lines). The dashed curves show the low-power case. Several features are noteworthy in Fig. 10.6(B). When the pump wavelength λp exceeds the ZDWL λ0 of the fiber, pump travels in the anomalous-GVD regime, and the signal and idler wavelengths are relatively close to the pump wavelength as dictated by Eq. (10.3.9). In contrast, they differ by several hundred nanometers in the normal-GVD regime (β2 < 0), where β4 is required for phase matching. This type of highly nondegenerate FWM was observed in 2003 by launching 70-ps pump pulses at 647 nm, with peak powers of up to 1 kW, inside a 1-m-long fiber placed inside the cavity of a parametric oscillator [51].

10.3.5 Phase matching in birefringent fibers An important phase-matching technique in single-mode fibers takes advantage of the modal birefringence, resulting from different effective mode indices for optical waves propagating with orthogonal polarizations. The index difference δ n = nx − ny ,

(10.3.13)

where nx and ny represent changes in the refractive indices (from the material value) for the optical fields polarized along the slow and fast axes of the fiber, respectively. A complete description of the parametric gain in birefringent fibers should generalize the formalism of Section 10.2 by following an approach similar to that used in Section 6.1. Such a vector theory of FWM is discussed in Section 10.5. However, this theory is not needed for the following discussion of phase matching. If we assume that each of the four waves is polarized either along the slow axis or along the fast axis of a polarization-maintaining fiber, we can use the scalar theory of Section 10.2. The parametric gain is then given by Eq. (10.2.19) with only minor changes in the definitions of the parameters γ and κ . In particular, γ is reduced by a factor of 3 whenever the pump and signal are polarized orthogonally. The wave-vector mismatch κ still has three contributions as in Eq. (10.3.1). However, the waveguide contribution kW is now dominated by δ n. The nonlinear contribution kNL is also different from that given by Eq. (10.3.5). In the following discussion kNL is assumed to be negligible compared with kM and kW . As before, phase matching occurs when kM and kW cancel each other. Both of them can be positive or negative. For λ1 < λ0 , a range that covers the visible region, kM is positive because β2 is positive in Eq. (10.3.6). The waveguide contribution kW can be made negative if the pump is polarized along the slow axis, while the signal and idler are polarized along the fast axis. This can be seen from Eq. (10.3.4). As

Four-wave mixing

Table 10.1 Phase-matched FWM processes in birefringent fibers.a Process A1 A2 A3 A4 Frequency shift s Condition I s f s f δ n/(|β2 |c ) β2 > 0 II s f f s δ n/(|β2 |c ) β2 < 0 III s s f f (4π δ n/|β2 |λ1 )1/2 β2 > 0 f f s s (4π δ n/|β2 |λ1 )1/2 β2 < 0 IV a

The symbols s and f denote the direction of polarization along the slow and fast axes of the birefringent fiber, respectively.

n3 = n4 = ny and n1 = n2 = nx , the waveguide contribution becomes kW = [ny (ω3 + ω4 ) − 2nx ω1 ]/c = −2ω1 (δ n)/c ,

(10.3.14)

where Eq. (10.3.13) was used together with ω3 + ω4 = 2ω1 . From Eqs. (10.3.6) and (10.3.14), kM and kW compensate each other for a frequency shift s given by [8]  1/2 s = 4πδ n/(β2 λ1 ) ,

(10.3.15)

where λ1 = 2π c /ω1 . At a pump wavelength λ1 = 0.532 µm, β2 ≈ 60 ps2 /km. If we use a typical value δ n = 1 × 10−5 for the fiber birefringence, the frequency shift νs is ∼10 THz. In a 1981 experiment on FWM in birefringent fibers, the frequency shift was in the range of 10 to 30 THz [8]. Furthermore, the measured values of νs agreed well with those estimated from Eq. (10.3.15). Eq. (10.3.15) for the frequency shift is derived for a specific choice of field polarizations, namely, that the pump fields A1 and A2 are polarized along the slow axis while A3 and A4 are polarized along the fast axis. Several other combinations can be used for phase matching depending on whether β2 is positive or negative. The corresponding frequency shifts s are obtained by evaluating kW from Eq. (10.3.4) with nj (j = 1 to 4) replaced by nx or ny , depending on the wave polarization, and by using Eq. (10.3.6). Table 10.1 lists the four phase-matching processes that can occur in birefringent fibers together with the corresponding frequency shifts [25]. The frequency shifts of the first two processes are smaller by more than one order of magnitude compared with the other two processes. All frequency shifts in Table 10.1 are approximate because frequency dependence of δ n has been ignored; its inclusion can reduce them by about 10%. Several other phase-matched processes have been identified [14] but are not generally observed in a silica fiber because of its predominantly isotropic nature. From a practical standpoint, the four processes shown in Table 10.1 can be divided into two categories. The first two correspond to the case in which pump power is divided between the slow and fast modes. In contrast, the pump field is polarized along a principal axis for the remaining two processes. In the first category, the parametric gain is maximum when the pump power is divided equally by choosing θ = 45◦ , where

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Figure 10.7 Output spectra showing Stokes and anti-Stokes bands generated in a 20-m-long birefringent fiber when 15-ps pump pulses with a peak power ∼1 kW are incident at the fiber. Pump is polarized at 44◦ from the slow axis. (After Ref. [15]; ©1984 OSA.)

θ is the polarization angle measured from the slow axis. Even then, different processes compete with each other because the parametric gain is nearly the same for all of them. In one experiment, FWM occurring as a result of Process I was observed by using 15-ps pump pulses from a mode-locked dye laser operating at 585.3 nm [15]. Because of a relatively small group-velocity mismatch among the four waves in this case, Process I became dominant compared with the others. Fig. 10.7 shows the spectrum observed at the output of a 20-m-long fiber for an input peak power ∼1 kW and a pump-polarization angle θ = 44◦ . The Stokes and anti-Stokes bands located near ±4 THz are due to FWM phase-matched by Process I. As expected, the Stokes band is polarized along the slow axis, while the anti-Stokes band is polarized along the fast axis. Asymmetric broadening of the pump and Stokes bands results from the combined effects of SPM and XPM (see Section 7.4). Selective enhancement of the Stokes band is due to the Raman gain. The peak near 13 THz is also due to SRS. It is polarized along the slow axis because the pump component along that axis is slightly more intense for θ = 44◦ . An increase in θ by 2◦ flips the polarization of the Raman peak along the fast axis. The small peak near 10 THz results from a nondegenerate FWM process in which both the pump and the Stokes bands act as pump waves (ω1 = ω2 ) and the Raman band provides a weak signal for the parametric process to occur. Phase matching can occur only if the Raman band is polarized along the slow axis. Indeed, the peak near 10 THz disappeared when θ was increased beyond 45◦ to flip the polarization of the Raman band. The use of birefringence for phase matching in single-mode fibers has an added advantage in that the frequency shift νs can be tuned over a considerable range (∼4 THz).

Four-wave mixing

Such a tuning is possible because birefringence can be changed through external factors such as stress and temperature. In one experiment, the fiber was pressed with a flat plate to apply the stress [12]. The frequency shift νs could be tuned over 4 THz for a stress of 0.3 kg/cm. In a similar experiment, stress was applied by wrapping the fiber around a cylindrical rod [13]. The frequency shift νs was tuned over 3 THz by changing the rod diameter. Tuning is also possible by varying temperature as the built-in stress in birefringent fibers is temperature dependent. A tuning range of 2.4 THz was demonstrated by heating the fiber up to 700◦ C [16]. In general, FWM provides a convenient way of measuring the net birefringence of a fiber because the frequency shift depends on δ n [17]. The frequency shift associated with the FWM process depends on the pump power through the nonlinear contribution kNL in Eq. (10.3.1). This contribution has been neglected in obtaining Eq. (10.3.15) and other expressions of s in Table 10.1, but can be included in a straightforward manner. In general s decreases with increasing pump power. In one experiment, s decreased with pump power at a rate of 1.4% W−1 [18]. The nonlinear contribution kNL can also be used to satisfy the phasematching condition. This feature is related to modulation instability in birefringent fibers (see Section 6.4).

10.4. Parametric amplification Similar to the case of the Raman gain, the parametric gain in optical fibers can be used for making an optical amplifier. Such FWM-based devices are known as fiber-optic parametric amplifiers (FOPAs), or parametric oscillators if a FOPA is placed within an optical cavity providing feedback periodically. Although FOPAs were studied during the 1980s [54–56], they attracted considerably more attention after 1995 because of their potential applications in fiber-optic communication systems [57–63]. In this section we first review the early work and then focus on the single- and dual-pump configurations used for FOPAs.

10.4.1 Review of early work Phase-matching techniques discussed in Section 10.3 were used for making FOPAs soon after low-loss silica fibers became available. Most experiments until 2000 employed a single pump. In the original 1974 experiment [4], phase-matching was achieved by using different modes of a multimode fiber (see Section 14.4.1). The peak power of 532-nm pump pulses used in this experiment was ∼100 W, whereas the CW signal with a power ∼10 mW was tunable near 600 nm. The amplifier gain was quite small because of a short fiber length (9 cm) used in the experiment. In a 1980 experiment [7], phase matching was realized using a pump at 1.319 µm, a wavelength close to the ZDWL of the fiber employed (see Fig. 10.3). The peak power of the pump pulses was varied in

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Figure 10.8 Measured gain Gs of a parametric amplifier as a function of pump power for three values of the input signal power. (After Ref. [7]; ©1980 IEE.)

the range of 30 to 70 W. A CW beam at the 1.338-µm wavelength acted as the signal and propagated with the pump pulses in a 30-m-long fiber. The power of the amplified signal was measured to determine the amplification factor. Fig. 10.8 shows the signal gain Gs as a function of pump power P0 for three values of the input signal power P3 . Departure from the exponential increase in Gs with P0 is due to gain saturation occurring as a result of pump depletion. Note also that Gs reduces considerably as P3 increases from 0.26 to 6.2 mW. For the 0.26-mW input signal, the amplifier gain was as large as 46 dB for P0 = 70 W. Such large values indicated the potential of FWM for constructing FOPAs, provided phase matching could be realized. This requirement puts stringent limits on the range of frequency shift s between the pump and signal waves for which amplification can occur. The use of fiber birefringence for phase matching is attractive as birefringence can be adjusted to match s by applying external stress or by bending the fiber. FOPAs with such schemes were demonstrated during the 1980s. In one experiment, a 1.292-µm signal from a semiconductor laser was amplified by 38 dB as s was tuned by applying external stress to the fiber [54]. In another experiment, a 1.57-µm signal from a distributed feedback (DFB) semiconductor laser was amplified by 37 dB using a 1.319-µm pump [55]. The resurgence of interest in FOPAs can be traced back to the work on modulation instability [49] and phase conjugation [64] carried out during the 1990s. It was realized that a broadband FOPA can be designed by choosing the pump wavelength λ1 close to the ZDWL λ0 of the optical fiber. In one implementation of this idea [57], a DFB semiconductor laser operating near λp ≈ 1.54 µm was used for pump-

Four-wave mixing

ing, while a tunable external-cavity semiconductor laser provided the signal. The length of the dispersion-shifted fiber (with λ0 = 1.5393) used for parametric amplification was 200 m. The FOPA bandwidth changed considerably as pump wavelength was varied in the vicinity of λ0 and was the largest when pump wavelength was detuned such that λp − λ0 = 0.8 nm. Further research revealed that the use of dispersion management, dual-wavelength pumping, and highly nonlinear fibers (see Chapter 11) can provide >40-dB signal gain over a relatively large bandwidth [58–63]. Because of the fast nonlinear response of optical fibers, such FOPAs are useful for a variety of signal processing applications [61].

10.4.2 Gain spectrum and its bandwidth The most important property of any optical amplifier—and FOPAs are no exception—is the bandwidth over which the amplifier can provide a relatively uniform gain. Typically, FOPAs are pumped using one or two CW lasers acting as pumps. In the case of dual pumps, the FWM process is governed by Eqs. (10.2.1) through (10.2.4), and a complete description of parametric amplification requires a numerical solution. Considerable physical insight is gained by employing the approximate analytic solution given in Eqs. (10.2.17) and (10.2.18) and obtained under the assumption that the pumps are not depleted much. The constants a3 , b3 , a4 , and b4 in these equations are determined from the boundary conditions. If we assume that both signal and idler waves are launched at z = 0, we find that the constants a3 and b3 satisfy a3 + b3 = B3 (0),

g(a3 − b3 ) = (iκ/2)(a3 + b3 ) + 2iγ A1 (0)A2 (0)B4∗ (0).

(10.4.1)

Solving these equations, we obtain a3 = 12 (1 + iκ/2g)B3 (0) + iC0 B4∗ (0),

b3 = 12 (1 − iκ/2g)B3 (0) − iC0 B4∗ (0), (10.4.2)

where C0 = (γ /g)A1 (0)A2 (0). A similar method can be used to find the constants a4 and b4 . Using these values in Eqs. (10.2.17) and (10.2.18), the signal and idler fields at a distance z are given by B3 (z) = {B3 (0)[cosh(gz) + (iκ/2g) sinh(gz)] + iC0 B4∗ (0) sinh(gz)}e−iκ z/2 , B4∗ (z) = {B4∗ (0)[cosh(gz) − (iκ/2g) sinh(gz)] − iC0 B3 (0) sinh(gz)}eiκ z/2 .

(10.4.3) (10.4.4)

The preceding general solution simplifies considerably when only the signal is launched at z = 0 (together with the pumps), a practical situation for most FOPAs. Setting B4∗ (0) = 0 in Eq. (10.4.3), the signal power P3 = |B3 |2 grows with z as [11] P3 (z) = P3 (0)[1 + (1 + κ 2 /4g2 ) sinh2 (gz)],

(10.4.5)

where the parametric gain g is given in Eq. (10.2.19). The idler power P4 = |B4 |2 can be found from Eq. (10.4.4) in the same way. It can also be obtained by noting from

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Eqs. (10.2.13) and (10.2.14) that d(P3 − P4 )/dz = 0, or P4 (z) = P3 (z) − P3 (0). Using this relation, we obtain P4 (z) = P3 (0)(1 + κ 2 /4g2 ) sinh2 (gz).

(10.4.6)

Eq. (10.4.6) shows that the idler wave is generated almost immediately after the input signal is launched into the fiber. Its power increases as z2 initially, but both the signal and idler grow exponentially after a distance such that gz > 1. As the idler is amplified together with the signal all along the fiber, it can build up to nearly the same level as the signal at the FOPA output. In practical terms, the same FWM process can be used to amplify a weak signal and to generate simultaneously a new wave at the idler frequency. The idler wave mimics all features of the input signal except that its phase is reversed (or conjugated). Among other things, such phase conjugation can be used for dispersion compensation and wavelength conversion in a WDM system [47]. The amplification factor for the signal is obtained from Eq. (10.4.5) and can be written using Eq. (10.2.19) as Gs = P3 (L )/P3 (0) = 1 + (γ P0 r /g)2 sinh2 (gL ).

(10.4.7)

The parameter r given in Eq. (10.2.20) equals 1 when a single pump beam is used for parametric amplification. The parametric gain g in Eq. (10.2.19) varies with the signal frequency ω3 because of its dependence on the phase mismatch κ ≡ k + γ (P1 + P2 ). Here, k is the linear phase mismatch given in Eq. (10.1.8). It depends not only on ω3 but also on the pump frequencies and the dispersive properties of the fiber used to make the FOPA. Thus, the bandwidth of a FOPA can be controlled by tailoring the fiber dispersion and choosing the pump frequencies suitably. The gain expression (10.4.7) should be compared with Eq. (8.2.4) obtained for a Raman amplifier. The main difference is that the parametric gain depends on the phase mismatch κ and can become quite small if phase matching is not achieved. In the limit κ γ P0 r, Eqs. (10.2.19) and (10.4.7) yield Gs ≈ 1 + (γ P0 rL )2

sin2 (κ L /2) . (κ L /2)2

(10.4.8)

The signal gain is relatively small and increases with pump power as P02 if phase mismatch is relatively large. On the other hand, if phase matching is perfect (κ = 0) and gL 1, the amplifier gain increases exponentially with P0 as Gs ≈ 14 exp(2γ P0 rL ).

(10.4.9)

The amplifier bandwidth A is determined from Eq. (10.4.7). It is usually defined as the range of ω3 over which Gs (ω3 ) exceeds 50% of its peak value and depends on

Four-wave mixing

many factors such as fiber length L, pump power P0 , and the pumping configuration used for the FOPA. The gain peak occurs at a signal frequency for which κ = 0 and the phase-matching condition is perfectly satisfied. In the case of a single pump, κ equals k + 2γ P0 with k ≈ β2 2s , where s = |ω1 − ω3 | is the detuning given in Eq. (10.3.9). Thus, the gain peak occurs at s = (2γ P0 /|β2 |)1/2 . In the case of long fibers, the gain bandwidth A is set by the fiber length itself and corresponds to a maximum phase mismatch κm = 2π/L because Gs is reduced roughly by a factor of 2 for this value of κ . Using κm = β2 (s + A )2 + 2γ P0 with A  s , we obtain [11] A =

π π = (2γ P0 |β2 |)−1/2 . |β2 |s L L

(10.4.10)

For relatively short fibers, the nonlinear effects themselves limit the FOPA bandwidth. As seen from Fig. 10.1, the parametric gain vanishes when κm = 2γ P0 . Using this value, the FOPA bandwidth is approximated given by A ≈

γ P0 ≡ |β2 |s



γ P0 2|β2 |

1/2 .

(10.4.11)

As a rough estimate, the bandwidth is only 160 GHz for a FOPA designed using a dispersion shifted fiber such that γ = 2 W−1 /km and β2 = −1 ps2 /km at the pump wavelength, and the FOPA is pumped with 1 W of CW power. Eq. (10.4.11) shows that at a given pump power, it can only be increased considerably by reducing |β2 | and increasing γ . This is the approach adopted in modern FOPAs that are designed using highly nonlinear fibers with γ > 10 W−1 /km and choosing the pump wavelength close to the ZDWL of the fiber so that |β2 | is reduced. However, as β2 is reduced below 0.1 ps2 /km, one must consider the impact of higher-order dispersive effects. It turns out that the FOPA bandwidth can be increased to beyond 5 THz, for both singleand dual-pump FOPAs, by suitably optimizing the pump wavelengths. In the following discussion, we consider these two pump configurations separately.

10.4.3 Single-pump configuration In this configuration, the pump wavelength is chosen close to the ZDWL of the fiber to minimize the wave-vector mismatch governed by κ . In the limit β2 approaches zero, higher-order dispersion terms become important in the expansion of β(ω). If we expand k in Eq. (10.1.9) in a Taylor series around the pump frequency ω1 , we obtain [57] k(ω3 ) = 2

∞  β2m (ω3 − ω1 )2m , ( 2m )! m=1

(10.4.12)

where the dispersion parameters are evaluated at the pump wavelength. This equation is a generalization of Eq. (10.3.6) and shows that only even-order dispersion parameters

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contribute to the linear phase mismatch. Clearly, k is dominated by β2 when signal wavelength is close to the pump wavelength, but higher-order dispersion parameters become important when the signal deviates far from it. The FOPA bandwidth depends on the spectral range over which k is negative but large enough to balance the nonlinear phase mismatch of 2γ P0 . In practice, the bandwidth can be enhanced by choosing the pump wavelength such that β2 is slightly negative but β4 is positive. The quantities β2 and β4 can be related to the dispersion parameters, calculated at the zero-dispersion frequency ω0 of the fiber and defined as βm0 = (dm β/dωm )ω=ω0 , by expanding β(ω) in a Taylor series around ω0 . If we keep terms up to fourth-order in this expansion, we obtain β2 ≈ β30 (ω1 − ω0 ) + 12 β40 (ω1 − ω0 )2 ,

β4 ≈ β40 .

(10.4.13)

Depending on the values of the fiber parameters β30 and β40 , we can choose the pump frequency ω1 such that β2 and β4 have opposite signs. More specifically, as both β30 and β40 are positive for most silica fibers, one should choose ω1 < ω0 . The FOPA bandwidth can exceed 5 THz under such operating conditions [57]. As an example, consider a FOPA designed using a 2.5-km-long fiber with its ZDWL at 1550 nm such that β30 = 0.1 ps3 /km and β40 = 10−4 ps4 /km. The FOPA is pumped with 0.5 W of power, resulting in a nonlinear length of 1 km when γ = 2 W−1 /km. Fig. 10.9(A) shows the gain spectra for several values of pump detuning λp = λ1 − λ0 . The dotted curve shows the case λp = 0 for which pump wavelength coincides with the ZDWL exactly. The peak gain in this case is about 8 dB, and the gain bandwidth is limited to 40 nm. When the pump is detuned by −0.1 nm such that it experiences normal GVD, gain bandwidth is reduced to below 20 nm. In contrast, both the peak gain and the bandwidth are enhanced when the pump is detuned toward the anomalousGVD side. In this region, the gain spectrum is sensitive to the exact value of λp . The best situation from a practical standpoint occurs for λp = 0.106 nm because the gain is nearly constant over a wide spectral region. An interesting feature of Fig. 10.9(A) is that the maximum gain occurs when the signal is detuned relatively far from the pump wavelength. This feature is related to the nonlinear contribution to total phase mismatch κ . The linear part of phase mismatch is negative for λp > 0 and its magnitude depends on the signal wavelength. In a certain range of signal wavelengths, it is fully compensated by the nonlinear phase mismatch to yield κ = 0. When phase matching is perfect, FOPA gain increases exponentially with the fiber length, as indicated in Eq. (10.4.9), resulting in an off-center gain peak in Fig. 10.9(A). From a practical standpoint, one wants to maximize both the peak gain and the gain bandwidth at a given pump power P0 . Since the peak gain in Eq. (10.4.9) scales exponentially with γ P0 L, it can be increased by increasing the fiber length L. However, from Eq. (10.4.10) gain bandwidth scales inversely with L when L LNL . The only

Four-wave mixing

Figure 10.9 (A) Gain spectra of a single-pump FOPA for several values of pump detuning λp from the ZDWL. (B) Gain spectra for three different fiber lengths assuming γ P0 L = 6 in each case. The pump wavelength is optimized in each case, and only half of the gain spectrum is shown because of its symmetric nature.

solution is to use a fiber as short as possible. Of course, shortening of fiber length must be accompanied with a corresponding increase in the value of γ P0 to maintain the same amount of gain. This behavior is illustrated in Fig. 10.9(B) where the gain bandwidth increases considerably when large values of γ P0 are combined with shorter fiber lengths such that γ P0 L = 6 remains fixed. The solid curve obtained for a 250-m-long FOPA exhibits a 30-nm-wide region on each side of the pump wavelength over which the gain is nearly flat. The nonlinear parameter γ can be increased by reducing the effective mode area. Such highly nonlinear fibers (see Chapter 11) with γ > 10 W−1 /km are used to make modern FOPAs. A 200-nm gain bandwidth was realized as early as 2001 by using a 20-m-long fiber with γ = 18 W−1 /km [65]. The required pump power (∼10 W) was large enough that the signal was also amplified by SRS when its wavelength exceeded the pump wavelength. Another scheme for enhancing the bandwidth of FOPAs manages fiber dispersion along the fiber either through periodic dispersion compensation [59] or by using multiple fiber sections with different dispersive properties [66]. Fig. 10.10 shows the gain spectra at several pump-power levels for such a single-pump FOPA [67]. At a pump power of 31.8 dBm (about 1.5 W) at 1563 nm, the FOPA provided a peak gain of 39 dB. The 500-m-long FOPA was designed by cascading three fiber sections with ZDWLs at 1556.8, 1560.3, 1561.2 nm. Each fiber section had γ close to 11 W−1 /km and a dispersion slope around 0.03 ps/(nm2 /km). Such devices can be used to amplify multiple WDM channels simultaneously [68]. As discussed earlier, FOPAs can act as wavelength converters because the idler wave generated internally contains the same information as the signal launched at the input end but at a different wavelength [69]. As early as 1998, peak conversion efficiency of

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Figure 10.10 Measured signal gain as a function of signal wavelength at several pump powers. Solid curves show the theoretical prediction. (After Ref. [67]; ©2001 IEEE.)

28 dB was realized over a 40-nm bandwidth using a pulsed pump source [58]. More recently, transparent wavelength conversion over a 24-nm bandwidth (the entire pump tuning range) was realized using a single-pump FOPA made with just 115 m of a highly nonlinear fiber [70]. The performance of FOPAs is affected by several factors that must be considered during the design of such devices [71–80]. For example, although FOPAs benefit from an ultrafast nonlinear response of silica, they also suffer from it because fluctuations in the pump power are transferred to the signal and idler fields almost instantaneously. Indeed, noise in FOPAs is often dominated by the transfer of pump-power fluctuations to the amplified signal [71–74]. Moreover, as the pump beam is typically amplified using one or two erbium-doped fiber amplifiers (EDFAs) before it is launched inside the FOPA, spontaneous emission added by EDFAs can degrade the FOPA performance considerably. In practice, an optical filter is used to block such noise. Another source of noise is spontaneous Raman scattering that occurs whenever the signal or idler falls within the Raman-gain bandwidth [78]. In spite of these factors, noise figures of 4.2 dB for an FOPA with 27.2-dB gain [75] and of 3.7 dB with a 17-dB gain [76] were realized in two 2002 experiments. Such noise levels are close to the fundamental quantum limit of 3 dB for an ideal amplifier. Another factor that affects FOPAs is the onset of stimulated Brillouin scattering (SBS). As was seen in Section 9.2.2, the SBS threshold is around 5 mW for long fibers (>10 km) and increases to near 50 mW for fiber lengths of 1 km or so. Since FOPAs require pump power levels approaching 1 W, a suitable technique is needed that raises the threshold of SBS and suppresses it over the FOPA length. The techniques used in practice (i) control the temperature distribution along the fiber length [70] or (ii) modulate the pump phase either at several fixed frequencies [67], or over a broad frequency range using a pseudorandom bit pattern [77]. The pump-phase modulation technique suppresses SBS by broadening the pump spectrum. In practice, the FOPA gain is affected to some extent when pump phase is modulated because the phase-mismatch parameter

Four-wave mixing

κ depends on the phase of the pump [81]. Moreover, dispersive effects within the fiber

convert pump-phase modulation into amplitude modulation of the pump. As a result, the SNR of both the signal and the idler is reduced because of undesirable power fluctuations [82]. Pump-phase modulation also leads to broadening of the idler spectrum, making it twice as broad as the pump spectrum. Such broadening of the idler is of concern when FOPAs are used as wavelength converters. As seen later, it can be avoided in dual-pump FOPAs. An important issue associated with single-pump FOPAs is that their gain spectrum is far from being uniform over its entire bandwidth (see Fig. 10.9). The reason is that it is hard to maintain the phase-matching condition over a wide bandwidth in a single-pump FOPA. Since the linear contribution k → 0 when signal wavelength approaches the pump wavelength, κ ≈ 2γ P0 in the pump vicinity. This value of κ is quite large and results in only a linear growth of the signal (G = 1 + γ P0 L). As a result, the gain spectrum exhibits a dip in the vicinity of the pump wavelength. Although amplification over a range as wide as 200 nm is possible [65], the gain spectrum remains highly nonuniform, and the usable bandwidth is limited to a much smaller region of the gain spectrum. This problem can be solved to some extent by manipulating fiber dispersion [66]. Theoretically, a fairly flat gain spectrum is possible by using several fiber sections of suitable lengths and properly selecting dispersive properties of these fiber sections [83]. However, such a scheme is difficult to implement in practice because dispersive properties of fibers are rarely known with sufficient precision. A practical solution is to make use of the dual-pump configuration discussed next.

10.4.4 Dual-pump configuration Dual-pump FOPAs use the nondegenerate FWM process and employ two pumps at different wavelengths [84–89]. With a proper choice of the pump wavelengths, they can provide a relatively uniform gain over a wider bandwidth than that is possible with a single pump. The parametric gain in the dual-pump case is given by Eq. (10.2.19). Using r from Eq. (10.2.20), it can be written as g(ω3 ) = [4γ 2 P1 P2 − κ 2 (ω3 )/4]1/2 ,

(10.4.14)

where κ = k + γ (P1 + P2 ) and P1 and P2 are the input pump powers, assumed to remain undepleted. The amplification factor is related to g as indicated in Eq. (10.4.7) and is given by Gs = P3 (L )/P3 (0) = 1 + (2γ /g)2 P1 P2 sinh2 (gL ).

(10.4.15)

Similar to the single-pump case, the gain spectrum is affected by the frequency dependence of the linear phase mismatch k. If we introduce ωc = (ω1 + ω2 )/2 together

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with ωd = (ω1 − ω2 )/2, and expand k around ωc , we obtain [86]: k(ω3 ) = 2

∞ c    β2m (ω3 − ωc )2m − ωd2m , (2m)! m=1

(10.4.16)

where the superscript c indicates that dispersion parameters are evaluated at the frequency ωc . This equation differs from Eq. (10.4.12) by the ωd term, which contributes only when two pumps are used. The main advantage of dual-pump FOPAs over singlepump FOPAs is that the ωd term can be used to control the phase mismatch. By properly choosing the pump wavelengths, it is possible to use this term for compensating the nonlinear phase mismatch γ (P1 + P2 ) such that the total phase mismatch κ is maintained close to zero over a wide spectral range. The most commonly used configuration for dual-pump FOPAs employs a relatively large wavelength difference between the two pumps. At the same time, the center frequency ωc is set close to the zero-dispersion frequency ω0 of the fiber so that the linear phase mismatch in Eq. (10.4.16) is constant over a broad range of ω3 . To achieve a fairly wide phase-matching range, the two pump wavelengths should be located on the opposite sides of the ZDWL in a symmetrical fashion [87]. With this arrangement, κ can be reduced to nearly zero over a wide wavelength range, resulting in a gain spectrum that is nearly flat over this entire range. The preceding discussion assumes that only the nondegenerate FWM process, ω1 + ω2 → ω3 + ω4 , contributes to the parametric gain. In reality, the situation is much more complicated for dual-pump FOPAs because the degenerate FWM process associated with each pump occurs simultaneously. In fact, it turns out that the combination of degenerate and nondegenerate FWM processes can create up to eight other idler fields, in addition to the one at the frequency ω4 . Only four among these idlers, say, at frequencies ω5 , ω6 , ω7 , and ω8 , are important for describing the gain spectrum of a dual-pump FOPA. These four idlers are generated through the following FWM processes: ω1 + ω1 → ω3 + ω5 ,

ω2 + ω2 → ω3 + ω6 ,

ω1 + ω3 → ω2 + ω7 ,

ω2 + ω3 → ω1 + ω8 .

(10.4.17) (10.4.18)

The last two FWM processes are due to Bragg scattering [87] from a dynamic index grating created by beating of the signal with one of the pumps. They have attracted attention because they can transfer signal power to an idler wave, whose wavelength can be far from the signal, without adding additional noise [90–92]. In addition to the preceding processes, the energy conservation required for FWM is also satisfied by several other combinations of the eight frequencies involved. A complete description of dual-pump FOPAs becomes quite complicated [87] if one were to include all underlying FWM processes. Fortunately, the phase-matching conditions associated with these processes are quite different. When the two pumps are

Four-wave mixing

Figure 10.11 Gain spectrum of a dual-pump FOPA when all five idlers are included (solid curve). The dashed spectrum is obtained when only a single idler generated through the dominant nondegenerate FWM process is included in the model.

located symmetrically far from the ZDWL of the fiber, the FWM processes indicated in Eqs. (10.4.17) and (10.4.18) can only occur when the signal is in the vicinity of one of the pumps. For this reason, they leave unaffected the central flat part of the FOPA gain spectrum resulting from the process ω1 + ω2 → ω3 + ω4 . Fig. 10.11 compares the FOPA gain spectrum calculated numerically using all five idlers (solid curve) with that obtained using this sole nondegenerate FWM process. Other FWM processes only affect the edges of gain spectrum and reduce the gain bandwidth by 10 to 20%. The FOPA parameters used in this calculation were L = 0.5 km, γ = 10 W−1 /km, P1 = P2 = 0.5 W, β30 = 0.1 ps3 /km, β40 = 10−4 ps4 /km, λ1 = 1502.6 nm, λ2 = 1600.6 nm, and λ0 = 1550 nm. Dual-pump FOPAs provide several degrees of freedom that make it possible to realize a flat gain spectrum using just a single piece of fiber. A 2002 experiment employed a 2.5-km-long highly nonlinear fiber having its ZDWL at 1585 nm with γ = 10 W−1 /km [89]. It also used two pumps with their wavelengths at 1569 and 1599.8 nm. Such a FOPA exhibited a gain of 20 dB over 20-nm bandwidth when pump powers were 220 and 107 mW at these two wavelengths, respectively. The gain could be increased to 40 dB by increasing pump powers to 380 and 178 mW. It was necessary to launch more power at the shorter wavelength because of the Raman-induced power transfer between the two pumps within the same fiber used to amplify the signal through FWM. A bandwidth of close to 40 nm was realized for a dual-pump FOPA in a 2003 experiment [93]. Fig. 10.12 shows the experimental data together with a theoretical fit. In this experiment, two pumps were launched with powers of 600 mW at 1559 nm and 200 mW at 1610 nm. The 1-km-long highly nonlinear fiber had its ZDWL at 1583.5 nm with β30 = 0.055 ps3 /km, β40 = 2.35 × 10−4 ps4 /km, and γ = 17 W−1 /km. The theoretical

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Figure 10.12 Measured (diamonds) and calculated (solid) gain spectra as a function of signal wavelength for a dual-pump FOPA. (After Ref. [93]; ©2003 IEE.)

model included the effects of SRS to account for the Raman-induced power transfer among the pump, signal, and idler fields. The dual-pump configuration offers several other advantages over the single-pump one, in addition to providing a relatively flat gain over a wide bandwidth. As the pump wavelengths are located at the two edges of the gain spectrum, pump blocking is not necessary when signal wavelength lies in the central flat portion of the gain spectrum. Also, the required power for each pump to realize a certain value of gain is reduced by 50% compared with the single-pump case when both pumps are launched with equal powers. In practice, the shorter-wavelength pump requires more power to offset the impact of Raman-induced power transfer between the two pumps, but its power still remains well below the level of single-pump FOPAs. An additional advantage of dual-pump FOPAs is related to their use as wavelength converters. Although phase modulation of the pumps is still required to suppress SBS, spectral broadening of the idler is no longer a problem because the phases of two pumps can be manipulated such that a specific idler used for wavelength conversion is not broadened. For example, if the idler frequency ω4 ≡ ω1 + ω2 − ω3 is used for wavelength conversion, the two pumps should be modulated out of phase so that the sum φ1 (t) + φ2 (t) = 0 at all times [94]. In the case of binary phase modulation, the two bit streams differ in their phases by π to satisfy this condition [95]. In contrast, if the idler at the frequency ω7 or ω8 is used [see Eq. (10.4.18)], the two pumps should be modulated in phase [96]. Another advantage of the dual-pump configuration is that the state of polarization (SOP) of the two pumps can be controlled to mitigate the dependence of the parametric gain on the signal SOP. This problem affects all FOPAs because the FWM process itself is strongly polarization-dependent. In the case of single-pump FOPAs, a polarizationdiversity loop is sometimes employed. In the case of dual-pump FOPAs, the problem

Four-wave mixing

can be solved simply by using orthogonally polarized pumps. The polarization effects are considered in more detail in Section 10.5. Dual-pump FOPAs also suffer from several shortcomings. As already mentioned, wavelength difference between the two pumps often falls within the bandwidth of the Raman-gain spectrum. If the two pumps cannot maintain equal power levels along the fiber because of Raman-induced power transfer, the FWM efficiency is reduced, even though the total pump power remains constant. In practice, the power of the highfrequency pump is chosen to be larger than that of the low-frequency pump at the input end of the fiber. Although Raman-induced power transfer reduces the FOPA gain, it does not affect the shape of the gain spectrum because phase matching depends on the total power of the two pumps, which is conserved as long as the two pumps are not depleted significantly. As in single-pump FOPAs, modulation of pump phases, required for SBS suppression, reduces the signal SNR even for dual-pump FOPAs, if fiber dispersion converts phase modulation into amplitude modulation of the pump [82]. If the two pumps are amplified before launching them into the FOPA, noise added during their amplification also lowers the signal SNR [97]. A major limitation of all FOPAs results from the fact that a realistic fiber is far from having a perfect cylindrical core. In practice, the core shape and size may vary along the fiber length in a random fashion during the manufacturing process. Such imperfections produce random variations in the ZDWL of the fiber along its length. Since the phase-matching condition depends on this wavelength, and the parametric gain is extremely sensitive to dispersion parameters of the fiber, even small changes in the ZDWL ( 0.1 ps/ km, a central dip occurs, as seen in Fig. 10.17(A). The reason behind the dip formation is as follows. When the signal frequency is close to one of the pumps, that pump provides the

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Figure 10.17 (A) Average gain spectrum for a dual-pump FOPA for three values of Dp ; (B) average √ gain spectrum for three initially linear SOPs of the signal when Dp = 0.1 ps/ km. In both cases, the isotropic case is also shown. (After Ref. [108]; ©2004 IEEE.)

dominant contribution. However, as signal frequency moves toward the center, neither of the two pumps remains oriented parallel to the signal, and the gain is reduced. An important question is how PMD affects the independence of the FOPA gain on the signal SOP when the two pumps are orthogonally polarized (either linearly or circularly). As noted earlier, optical fields with different frequencies change their SOPs at different rates in the presence of PMD. As a result of this frequency dependence, the two pumps will not remain orthogonally polarized and would produce gain that depends on the signal SOP to some extent. Indeed, such a behavior has been observed for a dual-pump FOPA [61]. Fig. 10.17(B) shows the average gain spectra for three √ different signal SOPs for a specific value of Dp = 0.1 ps/ km. The two pumps are launched with SOPs that are linear and orthogonal to each other. The signal is also linearly polarized with its SOP inclined at an angle of 0, 45, or 90◦ from that of the shorter-wavelength pump. The gain spectrum expected in the absence of PMD is also shown as a dotted curve. Two features of Fig. 10.17(B) are noteworthy. First, at signal wavelengths close to the pump, gain can change by as much as 12 dB as signal SOP is varied; variations are much smaller in the central region. The reason for this behavior can be understood as follows. If the signal has its wavelength close to that of a pump, their relative orientation does not change much along the fiber. As a result, it experiences the highest or smallest gain depending on whether it is initially polarized parallel or orthogonal to the that pump. Second, the PMD effects enhance the gain considerably compared with the case of an isotropic fiber with no birefringence. The reason is understood by recalling from Fig. 10.16 that the FWM efficiency is the smallest when the pumps are orthogonally polarized. However, because of the PMD, pump SOPs do not remain orthogonal along

Four-wave mixing

the fiber, and may even become parallel occasionally; this results in a higher gain at all signal wavelengths. PMD effects can be made negligible for FOPAs designed with a relatively short length (∼100 m) of low-PMD fibers and pumped at wavelengths spaced apart by 0 and β4 < 0 were satisfied at the pump wavelength. As discussed in Section 10.3.4, such large frequency shifts are possible only in that situation.

10.6.2 Tunable fiber-optic parametric oscillators Perhaps, the simplest application of the parametric gain is to use it for making a laser by placing the fiber inside an optical cavity and pumping it with a pump laser. Since no signal beam is launched, the signal and idler waves are created initially from noise through spontaneous FWM at frequencies dictated by the phase-matching condition; the two are subsequently amplified through stimulated FWM. As a result, the laser emits the signal and idler beams simultaneously at frequencies that are located symmetrically on opposite sides of the pump frequency. Such lasers are called parametric oscillators; the term four-photon laser is also sometimes used. They were made soon after FWM was realized inside an optical fiber. As early as 1980, a parametric oscillator pumped at 1.06 µm exhibited a conversion efficiency of 25% [127]. In a 1987 experiment, a parametric oscillator emitted 1.15-µm pulses of about 65-ps duration when it was pumped with 100-ps pulses obtained from a Nd:YAG laser [128]. Synchronous pumping was realized by adjusting the cavity length such that each laser pulse overlapped with the next pump pulse after one round trip. A new kind of parametric oscillator, sometimes called the modulation-instability laser, can be made by pumping a fiber in the anomalous-GVD region so that FWM is phase-matched through the SPM effect. Such a laser was first made in 1988 by pumping a fiber-ring cavity (length ∼100 m) synchronously with a mode-locked, color-center

Four-wave mixing

laser (pulse width ∼10 ps) operating in the 1.5-µm wavelength region [129]. For a 250-m ring, the laser reached threshold at a peak pump power of 13.5 W. The laser generated the signal and idler bands with a frequency shift of about 2 THz, a value in agreement with the theory of Section 5.1. The use of vector modulation instability (see Section 6.4) makes it possible to operate such a parametric oscillator in the visible region [130]. A modulation-instability laser is different from a conventional parametric oscillator in the sense that it can convert a CW pump into a train of short optical pulses. This objective was realized in 1999 by pumping a 115-m-long ring cavity with a CW laser. The laser reached threshold at a pump power of about 80 mW and emitted a pulse train at the 58-GHz repetition rate when pumped beyond its threshold [131]. The spectrum exhibited multiple peaks, separated by 58 GHz, that were generated through a cascaded FWM process. A parametric oscillator, tunable over a 40-nm range centered at the pump wavelength of 1539 nm, was realized in a 1999 experiment [132]. This laser employed a nonlinear Sagnac interferometer (with a loop length of 105 m) as a FOPA that was pumped by using 7.7-ps mode-locked pulses from a color-center laser operating at 1539 nm. Such an interferometer separates pump light from the signal and idler waves while amplifying both of them. In effect, it acts as a mirror of a Fabry–Perot cavity with internal gain. A grating at the other end of the cavity separates the idler and signal waves so that the Fabry–Perot cavity is resonant for the signal only. This grating is also used for tuning laser wavelength. The laser emitted 1.7-ps pulses at the 100-MHz repetition rate of the pump laser. After 2001, the use of highly nonlinear fibers became common for making parametric oscillators [133–141]. Such devices can operate continuously with a narrow line width, or they can be forced to emit a train of short pulses by pumping them with a suitable pulsed source. The use of FWM is essential for realizing fiber-optic parametric oscillators that emit femtosecond pulses and are tunable over a wide wavelength range. A tuning range of more than 200 nm was realized in 2005 by employing a ring cavity containing a 65-cm-long photonic crystal fiber [137]. The cavity was pumped synchronously using 1.3-ps pulses from a ytterbium fiber laser, and pulses shorter than 0.5 ps were obtained at the signal and idler wavelengths. Fig. 10.19(A) shows how these two wavelengths change when the pump wavelength is varied over a relatively narrow range of 25 nm. These curves are obtained using the phase-matching condition in Eq. (14.14), retaining only two terms in the sum, and employing the specific values of β2 and β4 for the PCF used in the experiment. The parametric oscillator is, in principle, tunable over 300 nm by changing the pump wavelength over a 20-nm range. In practice, the walk-off effects, resulting from the group-velocity mismatch, limited the tuning range to around 200 nm. The dashed curve in Fig. 10.19(B) shows that the walk off between the pump and laser pulses exceeds 1 ps for a 150-nm detuning. The extent of temporal walk off between the pump and laser pulses can be reduced either by using shorter fibers or by employing longer pump pulses, and both help in

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Figure 10.19 (A) Phase matching curves as a function of pump wavelength for a PCF having its ZDWL at 1038 nm. Tuning curve of the pump laser is also shown. (B) Walk-off delay (dashed curve) as a function of pump wavelength and a typical output spectrum showing the signal and idler bands on opposite sides of the pump. (After Ref. [137]; ©2005 OSA.)

increasing the tuning range of fiber-optic parametric oscillators. In a 2007 experiment, tuning range was enhanced considerably (covering one octave) when length of the microstructured fiber was reduced to 3 cm [138]. In contrast, another 2007 experiment employed a 40-m-long fiber but avoided the walk-off problem by using much longer 4-ns pump pulses [139]. Tuning over a very wide range was realized in 2008 by using a 1.3-m-long PCF with a butt-coupled mirror at both ends, thus forming a Fabry– Perot cavity [140]. The PCF exhibited normal GVD at wavelengths λ < 725 nm. The pump pulses were 8 ps wide but their wavelength was tunable from 705 to 725 nm. Fig. 10.20(A) shows that the laser frequency can be shifted from 23 to 164 THz on both sides of the pump frequency for the PCF used in the experiment. As a result, the signal (Stokes) and idler (anti-Stokes) wavelengths could be tuned over a broad range by simply changing the pump wavelength from 705 to 725 nm. The relative walk off for the three pulse pairs is shown in Fig. 10.20(B). As it exceeds the width of pump pulses for frequency shifts >60 THz, the output power is reduced considerably for large frequency shifts. The use of 100-ps pulses from a pump operating near 1550 nm has produced lasers whose wavelength can be tuned over a wide range extending from 1320 to 1820 nm [141]. Recent work on tunable parametric oscillators has followed several distinct directions [142–149]. Continuous-wave (CW) pumping was used in 2009 to realize a CW laser tunable over 240 nm [142]. The FWM process was affected considerably by the Raman gain in this experiment. In the other extreme limit, wavelength-tunable pulses shorter than 100 fs were produced in a 2010 experiment by pumping with 400-fs pump pulses at wavelengths near 1030 nm [143]. The length of the fibers used in this experiment varied from 2 to 6 cm. The duration of idler pulses was only 48 fs for a 2.7-cm-long fiber. In a 2011 experiment, the objective was to obtain high output powers from a

Four-wave mixing

Figure 10.20 (A) Phase matching curve as a function of pump wavelength for a 1.3-m-long PCF fiber having its ZDWL at 725 nm; inset shows β2 (λ) for this PCF. (B) Temporal walk off for three pulse pairs as a function of frequency shift. (After Ref. [140]; ©2008 OSA.)

tunable parametric oscillator. Indeed, the output average power exceeded 1.9 W over the entire tuning range of 1350–1790 nm when the 26-ns pump pulses were tuned over 16 nm in the vicinity of 1550 nm [144]. In a different approach, the intracavity fiber was heated up to 500◦ C to tune the oscillator [145]. Parametric oscillators can also be actively mode-locked using a phase modulator inside the laser’s cavity. In a 2013 study, such a device produced 9 to 10-ps pulses at a repetition rate of 10 GHz that were tunable over 30 nm range near 1550 nm [146]. Similar results were obtained in a 2016 study [147]. However, passive mode locking must be employed for shorter pulses. The use of a semiconductor saturable-absorber mirror for this purpose was proposed in a 2018 study [149]. Numerical simulations indicated that pulse widths of 1 to 2 ps could be obtained with this approach. Femtosecond pulses can be realized by pumping the FWM fiber with short pump pulses. In a 2017 experiment, 39-fs-wide idler pulses (in the wavelength region near 1330) were obtained from a parametric oscillator containing a 2-cm-long fiber that was pumped with 300-fs pump pulses at 1027 nm [148]. With a suitable design, pulses as short as 26 fs were possible when linearly chirped pulses emitted by the oscillator were compressed externally.

10.6.3 Ultrafast signal processing The ultrafast nature of the FWM process in optical fibers stems from the electronic nature of fiber nonlinearity. As a result, rapid variations in the input power of the signal or the pumps are transferred to the FOPA output almost instantaneously. It is this feature that converts FOPAs into signal-processing devices capable of responding at picosecond timescales. Moreover, all FOPAs generate one or more idler waves that mimic the launched signal but have a different wavelength. From a practical perspective,

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Figure 10.21 Measured output powers in the signal and three idlers as a function of input signal power for a dual-pump FOPA. Temporal profiles are shown for input signal (A), output signal (B), and two idlers. (After Ref. [153]; ©2003 IEEE.)

these idlers represent a replica of the signal (except for a phase reversal) and thus can serve as a wavelength-converted signal. When FOPAs are used as a wavelength converter in optical communication systems, one or two CW pump beams are injected together with the signal such that the idler wave is created at the desired wavelength [150]. As the idler is generated through FWM only when the pumps and signal are present simultaneously, it appears in the form of a pulse train consisting of the same sequence of “1” and “0” bits as the signal. In effect, FWM transfers the signal data to the idler at a new wavelength with perfect fidelity. It can even improve the signal quality by reducing the noise level [151–153]. The reason is related to the nonlinear power-transfer characteristics of FOPAs. Fig. 10.21(A) shows the measured output powers at the FOPA output as the input power of the signal is varied over a 30-dB range [153]. The temporal profile of the input signal (A) is compared with the output signal (B) and two idlers (C and D). The idler (D) exhibits much less noise for both 1 and 0 bits and can serve as a regenerated signal. Clearly, FWM may prove useful for all-optical regeneration of channels in WDM systems Another application employs FWM for demultiplexing a time-division multiplexed bit stream [154]. In such a bit stream, bits from different channels are packed such that the bits belonging to a specific channel are separated by N bits, when N channels are multiplexed together in the time domain. A specific channel can be demultiplexed if the pump is in the form of an optical pulse train at the bit rate of an individual channel (referred to as an optical clock). The idler wave is generated only when the pump and signal overlap in the time domain and is thus a replica of the bit pattern associated with a single channel. The FWM process was used for time-domain demultiplexing as

Four-wave mixing

early as 1991 [155]. In a 1996 experiment, 10-Gb/s channels were demultiplexed from a 500-Gb/s TDM signal using clock pulses of 1-ps duration [156]. This scheme can also amplify the demultiplexed channel inside the same fiber [157]. The same scheme can be used for all-optical sampling of signal pulses [158]. The basic idea is to employ pump pulses shorter than the signal pulse so that the idler power provides a sample of the signal in the time window associated with the pump pulse. By 2005, subpicosecond temporal resolution over more than 60-nm bandwidth had been realized with the use of FWM [159]. An interesting application uses FWM to convert a CW signal into a pulse train at high repetition rates through pump modulation [60]. The pump is modulated sinusoidally at the desired repetition rate before it is launched into the fiber where FWM occurs. Because of the exponential dependence of the signal gain on the pump power, the signal is amplified mostly in the central part of each modulation cycle and is thus in the form of a pulse train. By 2005, a 40-GHz train of short pulses (width about 2 ps) generated with this technique was used for information transmission at bit rates of up to 160 Gb/s [160]. An important feature of FWM is that a FOPA can be used as an optical gate that opens for a short duration controlled by the pump. This feature has been used for optical switching at bit rates as high as 40 Gb/s [161]. The basic idea is similar to that used for time-domain demultiplexing. In the case of a dual-pump FOPA, one pump remains CW but the other is turned on only for time intervals during which the optical gate should remain open. Because the idler wave is generated when both pumps overlap with the signal, it contains temporal slices of the signal whose duration is dictated by the second pump. As all idlers (as well as the signal) at the FOPA output contain only these temporal slices, a single pump can be used for multicasting the selected signal information at multiple wavelengths. This approach was used in a 2005 experiment to select either individual bits or a packet of bits (packet switching) from a 40-Gb/s signal [161]. With the advent of coherent receivers, information is often coded by modulating the phase of an optical carrier [154]. FWM can be used for all-optical signal processing even for such systems [162,163].

10.6.4 Quantum correlation and noise squeezing The simultaneous generation of the signal and idler photons during the FWM process indicates that each photon pair is correlated in the quantum sense. This correlation has found a number of applications, including reduction of quantum noise through a phenomenon called squeezing [164–166]. Squeezing refers to the process of generating the special states of an electromagnetic field for which noise fluctuations in some frequency range are reduced below the quantum-noise level. An accurate description of squeezing in optical fibers requires a quantum-mechanical approach in which the signal and idler amplitudes B3 and B4 are replaced by corresponding annihilation operators.

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Figure 10.22 Noise spectrum under minimum-noise conditions. Horizontal line shows the quantumnoise level. Reduced noise around 45 and 55 MHz is a manifestation of squeezing in optical fibers occurring as a result of FWM. (After Ref. [168]; ©1986 APS.)

From a physical standpoint, squeezing can be understood as deamplification of signal and idler waves for certain values of the relative phase between the two [56]. Spontaneous emission at the signal and idler frequencies generates photons with random phases. FWM increases or decreases the number of specific signal-idler photon pairs depending on their relative phases. A phase-sensitive (homodyne or heterodyne) detection scheme shows noise reduced below the quantum-noise level when the phase of the local oscillator is adjusted to match the relative phase corresponding to the photon pair whose number was reduced as a result of FWM. The observation of squeezing in optical fibers is hindered by the competing processes such as spontaneous and stimulated Brillouin scattering. A particularly important noise process turned out to be Brillouin scattering caused by guided acoustic waves [167]. If noise generated by this phenomenon exceeds the reduction expected from FWM, squeezing is washed out. Several techniques have been developed to reduce the impact of this noise source [164]. A simple method consists of immersing the fiber in a liquid-helium bath. Indeed, a 12.5% reduction in the quantum-noise level was observed in a 1986 experiment in which a 647-nm CW pump beam was propagated through a 114-m-long fiber [168]. Fig. 10.22 shows the observed noise spectrum when the local oscillator phase is set to record the minimum noise (two large peaks are due to guided acoustic waves). Squeezing occurs in the spectral bands located around 45 and 55 MHz. During the 1990s several other types of squeezing was realized [164]. After 2000, much attention focused on optical sources capable of emitting single photons or entangled photon pairs for applications such as quantum cryptography and quantum computing [169]. The nonlinear process of FWM occurring inside optical fibers provides a simple way to generate quantum-correlated photon pairs within a single spatial mode [170–176]. In this application, only a pump beam is launched into the fiber, and the spontaneous FWM process is employed to generate the correlated photon pairs

Four-wave mixing

Figure 10.23 Output spectrum observed when a highly nonlinear fiber is pumped with Q-switched pulses at 1047 nm. (After Ref. [172]; ©2005 OSA.)

at different frequencies satisfying the FWM condition ω3 + ω4 = 2ω1 , where ω1 is the pump frequency. If it is desirable to have a source that emits correlated photon pairs at the same frequency, a dual-pump configuration is used in which the nondegenerate FWM process produces the signal and idler photons at the same frequency such that ω3 = ω4 = 12 (ω1 + ω2 ). It was observed in several experiments that the quality of the photon-pair source is severely deteriorated by spontaneous Raman scattering that accompanies the spontaneous FWM process inevitably and cannot be avoided in practice [170–174]. In the single-pump configuration, Raman scattering is weak if the pump-signal detuning is kept relatively small ( 10 W−1 /km were developed; they are collectively referred to as highly nonlinear fibers (HNLFs). This chapter deals with the properties of such fibers. The techniques used to measure the nonlinear parameter are described first in Section 11.1. The following four sections then focus on the four kinds of HNLFs that have been developed to enhance the nonlinear effects. In each case, dispersive properties of the fibers are also described because they play an important role whenever HNLFs are used for practical applications. It will be seen in Chapters 12 and 13 that the combination of unusual dispersive properties and a high value of γ makes HNLFs useful for a variety of novel nonlinear effects. Section 11.6 shows how the design of some HNLFs modifies the effective value of the nonlinear parameter in the case of narrow-core fibers.

11.1. Nonlinear parameter The nonlinear parameter γ , defined in Eq. (2.3.30), can be written as γ = 2π n2 /(λAeff ), where λ is the wavelength of input signal and Aeff is the effective mode area given in Eq. (2.3.31). This area depends on the fiber design, and it can be reduced with a proper design to enhance γ . On the other hand, the nonlinear-index coefficient n2 is a material parameter related to the third-order susceptibility, as indicated in Eq. (2.3.13). This parameter is fixed for each glass material. Thus, the only practical approach for enhancing γ for silica-based optical fibers is to reduce the effective mode area Aeff . The use of non-silica glasses provides an alternative approach to making the HNLFs. Before focusing on the design of such fibers, it is important to discuss the techniques used to determine n2 experimentally. Accurate measurements of both γ and Aeff are necessary for this purpose.

11.1.1 Units and values of n2 Before proceeding, it is important to clarify the units used to express the numerical values of the Kerr coefficient n2 [1]. The nonlinear part of the refractive index in Eq. (2.3.12) is expressed as δ nNL = n¯ 2 |E|2 . In the standard metric system, the electric Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00018-X

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field E has units of V/m. As δ nNL is dimensionless, the units of n¯ 2 are m2 /V2 . In practice, it is more convenient to write the nonlinear index in the form δ nNL = n2 I, where I is the intensity of the optical field related to E as I = 12 0 cn|E|2 .

(11.1.1)

Here 0 is the vacuum permittivity (0 = 8.8542 × 10−12 F/m), c is the velocity of light in vacuum (c = 2.998 × 108 m/s), and n is the effective modal refractive index (n ≈ 1.45 for silica fibers). The parameter n2 is related to n¯ 2 as indicated in Eq. (2.3.30): n2 = 2n¯ 2 /(0 nc ).

(11.1.2)

As n2 I is a dimensionless quantity, n2 has units of m2 /W. Extensive measurements of n2 were first carried out during the 1970s for several bulk glasses [2–6]. In the case of fused silica glass, the value n2 = 2.73 × 10−20 m2 /W was measured (with an accuracy of ±10%) at a wavelength of 1.06 µm [4]. A compilation of the measurements made at wavelengths ranging from 248 to 1550 nm indicates that the nonlinear index coefficient n2 for fused silica decreases at longer wavelengths [7]. However, n2 varies slowly with wavelength in the spectral region extending from 800 to 1600 nm; its value decreases only by about 5% over this wavelength range. The earliest measurements of n2 for silica fibers [8] were carried out in 1978 using SPM-induced spectral broadening of 90-ps optical pulses, obtained from an argon-ion laser operating near 515 nm. The estimated value of 3.2 × 10−20 m2 /W from this experiment was used almost exclusively in many studies of the nonlinear phenomena in optical fibers, even though the measured value of γ depends on several factors. The wavelength region near 1550 nm became relevant for nonlinear fiber optics with the advent of fiber-optic communication systems. The measured value of 3.2 × 10−20 m2 /W at 515 nm should not be used near 1550 nm because n2 is reduced by at least 10% because of its frequency dependence. The growing importance of the nonlinear effects in optical fibers led to more accurate measurements of γ during the 1990s, especially because fiber manufacturers are often required to specify the numerical value of γ for their fibers [9]. Several different experimental techniques were developed to measure n2 and applied to different types of fibers [10–19]. These techniques make use of one of the nonlinear effects discussed in earlier chapters. In fact, all of the three major nonlinear effects—SPM, XPM, and FWM—have been used for this purpose. All techniques measure γ and deduce n2 from it. Table 11.1 summarizes the results obtained in several experiments in the wavelength region near 1550 nm using a standard fiber, a dispersion-shifted fiber (DSF), or a dispersion-compensating fiber (DCF). As seen there, the measured values vary in the range of 2.2 to 3.9×10−20 m2 /W. Such large variations result partly from estimating the effective mode area Aeff from the mode-field diameter and partly from a wide range of

Highly nonlinear fibers

Table 11.1 Measured values of n2 for different fibers. Method λ Fiber type Measured n2 used (μm) (10−20 m2 /W)

SPM

XPM

FWM

1.319 1.319 1.548 1.550 1.550 1.550 1.550 1.550 1.550 1.550 1.550 1.548 1.548 1.555 1.553

silica core DSF DSF DSF standard DSF DCF silica core standard DSF DCF standard standard DSF DSF

2.36 2.62 2.31 2.50 2.20 2.32 2.57 2.48 2.63 2.98 3.95 2.73 2.23 2.25 2.35

Experimental conditions

110-ps pulses [12] 110-ps pulses [12] 34-ps pulses [13] 5-ps pulses [16] 50-GHz modulation [17] 50-GHz modulation [17] 50-GHz modulation [17] 7.4-MHz modulation [14] 7.4-MHz modulation [14] 7.4-MHz modulation [14] 7.4-MHz modulation [14] 10-MHz modulation [18] 2.3-GHz modulation [18] two CW lasers [11] 10-ns pulses [15]

pulse widths employed. In the remainder of this section, we discuss several measurement techniques and indicate why they may yield different values of n2 even at the same wavelength.

11.1.2 SPM-based techniques The SPM technique makes use of the broadening of the pulse spectrum (see Section 4.1) and was first used in 1978 [8]. As seen from Eq. (4.1.16), this technique actually measures the maximum value of the nonlinear phase shift, φmax , a dimensionless quantity that is related to γ linearly through Eq. (4.1.6). Once γ has been determined, n2 is estimated from it using the relation n2 = λAeff γ /(2π). The accuracy of measurements depends on how well one can characterize input pulses because the SPM-induced spectral broadening is quite sensitive to the shape of optical pulses used in the experiment. In spite of the uncertainties involved, the SPM technique is often used in practice [19]. In a set of measurements performed in 1994, mode-locked pulses of 110-ps duration were obtained from a Nd:YAG laser operating at 1.319 µm [12]. Pulses were broadened spectrally inside the test fiber, and their spectrum measured using a scanning Fabry–Perot interferometer. The input power was adjusted such that the measured spectrum corresponded to one of the shapes shown in Fig. 4.2 so that φmax was a multiple of π/2. The effective mode area was calculated from the measured refractiveindex profiles of the fiber and used to deduce the value of n2 . For a silica-core fiber (no dopants inside the core), the measured value of n2 was 2.36 × 10−20 m2 /W with

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Figure 11.1 Experimental setup for measuring n2 using SPM-induced spectral broadening. (After Ref. [19]; ©1998 IEEE.)

an uncertainty of about 5%. The measured n2 values were larger for DSFs (average value 2.62 × 10−20 m2 /W) because of the contribution of dopants. As discussed in Section 6.6.3, all measured values were reduced by a factor of 8/9 (compared with the bulk silica glass) because the fiber used did not maintain the linear state of polarization of launched pulses during their transmission. The same technique was used in 1998 to measure n2 of relatively long dispersionshifted fibers near 1.55 µm [19]. Fig. 11.1 shows the experimental setup schematically. A mode-locked fiber laser provided 51.7 ps pulses that were first amplified and filtered and then launched into a 20-km-long test fiber. It was necessary to account for changes in the width and peak power of pulses along the fiber because of its long length. The dispersive and modal characteristics of the test fiber were quantified through separate measurements to ensure the accuracy. The NLS equation was then solved to fit the measured spectra and to deduce the values of γ and n2 . The measured value, n2 = 2.45 × 10−20 m2 /W, is smaller in the 1.55-µm wavelength region compared with its values measured near 1.3-µm. A reduction by about 2% is expected from the frequency dependence of n2 . The remaining change may be related to different amounts of dopants or measurement errors. In place of deducing the SPM-induced phase shift from spectral broadening, a related technique deduces it from spectral changes occurring when light from two lasers operating at slightly different wavelengths is transmitted through the fiber. Such an approach does not require short optical pulses, and measurements can be performed with CW lasers. In a 1996 experiment [17], two CW semiconductor lasers (both DFB type) were used and their wavelength difference (0.3–0.5 nm) was stabilized by controlling the laser temperature. The optical signal entering the fiber oscillates sinusoidally at the beat frequency (∼ 50 GHz) and its amplitude is given by Ein (t) = Re[A1 exp(−iω1 t) + A2 exp(−iω2 t)] = Re[A1 cos( ωt) exp(−iωav t)], (11.1.3)

Highly nonlinear fibers

where ω = ω1 − ω2 is the beat frequency, ωav = 12 (ω1 + ω2 ) is the average frequency, and the two fields are assumed to have the same power (|A1 | = |A2 |). When such a signal propagates inside the fiber, the SPM-induced phase shift is also time-dependent. Similar to the discussion in Section 4.1.1, if we neglect the effects of fiber dispersion, total optical field at the fiber output is given by Eout (t) = Re{A1 cos( ωt) exp(−iωav t) exp[iφmax cos2 ( ωt)]},

(11.1.4)

where φmax = 2γ Pav Leff and Pav is the average power of the launched signal. It is easy to see by taking the Fourier transform of Eq. (11.1.4) that the optical spectrum at the fiber output exhibits peaks at multiples of the beat frequency because of the SPM-induced phase shift. The ratio of the peak powers depends only on φmax and can be used to deduce the value of n2 . In particular, this power ratio for the central and first sideband is given by [17] P0 J02 (φmax /2) + J12 (φmax /2) = . P1 J12 (φmax /2) + J22 (φmax /2)

(11.1.5)

This equation provides φmax from a simple measurement of the power ratio, which can be used to determine γ and n2 . For standard telecommunication fibers, the value of n2 was found to be 2.2 × 10−20 m2 /W. This technique was also used to measure n2 for several DSFs and DCFs with different amounts of dopants (see Table 11.1). The main limitation of the technique is that it cannot be used for long fibers for which dispersive effects become important. In fact, the fiber length and laser powers should be properly optimized for a given fiber to ensure accurate results [20]. The SPM-induced phase shift can also be measured with an interferometric technique. In one experiment, the test fiber was placed inside a fiber loop that acted as a Sagnac interferometer [16]. Mode-locked pulses of ≈5-ps width were launched into the loop such that pulses acquired a large SPM-induced phase shift in one direction (say, clockwise). In the other direction, a 99:1 coupler was used to reduce the peak power so that the pulses acquired mostly a linear phase shift. An autocorrelator at the loop output was used to deduce the nonlinear phase shift and to obtain n2 from it. A CW laser can also be used with a Sagnac interferometer [21]. Its use simplifies the technique and avoids the uncertainties caused by the dispersive effects. Fig. 11.2 shows the experimental setup schematically. The interferometer is unbalanced by using a fiber coupler that launches unequal amounts of laser power in the counterpropagating direction, and the transmitted power is measured for low and high values of the input power. The transmissivity of such a Sagnac interferometer changes at high power levels because of the SPM-induced phase shift and thus provides a way to measure it. The measured value of n2 was 3.1 × 10−20 m2 /W at 1064 nm for a fiber whose core was doped with 20% (by mol) of germania.

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Figure 11.2 Experimental setup for measuring n2 using a Sagnac interferometer and a CW laser. (After Ref. [21]; ©1998 OSA.)

A self-aligned Mach–Zehnder interferometer (MZI) has also been used for measuring n2 [22]. In this scheme, pulses pass through the MZI twice after being reflected by a Faraday mirror. The path lengths in the two arms of MZI are different enough that the two pulses are separated by more than their widths at the MZI output and thus do not interfere. They pass through the fiber twice and accumulate the SPM-induced nonlinear phase shift. After a complete round trip, a single pulse has three different delays as it may pass twice through the long arm, twice through the short arm, or once through long and short arms. In the last case, the power at the detector depends on the phase relationship between the interfering signals and can be used to measure the nonlinear phase shift. Such an interferometer is called self-aligned because the path lengths of the two interfering signals are automatically matched.

11.1.3 XPM-based technique The XPM-induced phase shift was used as early as 1987 to measure n2 [10]. A 1995 experiment employed a pump-probe configuration in which the pump and probe signals were obtained from CW sources [14]. Fig. 11.3 shows the experimental setup schematically. When the pump light was modulated at a low frequency (< 10 MHz), the probe spectrum developed FM sidebands because of the XPM-induced phase shift. The theory behind the formation of the FM sidebands is similar to that discussed earlier in the SPM case when two CW lasers were used. The main difference is that one of the lasers acts as the pump and its intensity is changed in a periodic fashion using a modulator. As a result, the phase shift induced on the probe through XPM beam becomes time dependent. If we again neglect the dispersive effects, the probe field at the fiber output can be written as E2 (t) = Re{A2 exp(−iω2 t) exp[ibφmax cos2 (2π fm t)]},

(11.1.6)

where φmax = 2γ P0 Leff , fm is the modulation frequency, P0 is the peak power of the pump beam, and the parameter b is defined in Eq. (6.2.11). If we calculate the probe

Highly nonlinear fibers

Figure 11.3 Pump-probe configuration for measuring n2 using a technique based on the XPMinduced phase shift. (After Ref. [14]; ©1995 OSA.)

spectrum by taking the Fourier transform of Eq. (11.1.6), the spectrum is found to exhibit peaks at multiples of fm because of the XPM-induced phase shift. As before, the power ratio of two neighboring peaks depends on φmax and can be used to deduce the value of n2 . As discussed in Section 6.2.2, the XPM-induced phase shift depends on the relative states of polarizations associated with the pump and probe beams through the parameter b. If the state of polarization is not maintained inside the test fiber for the two beams, one must make sure that the measured value is not affected by such polarization effects. In practice, pump light is depolarized before it enters the fiber [14]. Under such conditions, the relative polarization between the pump and probe varies randomly, and the measured value of nonlinear phase shift corresponds to b = 2/3. In the experimental setup shown in Fig. 11.3, two CW beams differing in their wavelengths by about 10 nm acted as the probe and the pump. The pump intensity was modulated at a relatively low frequency of 7.36 MHz. XPM inside the fiber converted pump-intensity modulations into phase modulations for the probe. A self-delayed heterodyne technique was used to measure the nonlinear phase shift as a function of pump power. A Sagnac interferometer can also be used to measure the XPM-induced phase shift and to deduce n2 from it [23]. The values of n2 obtained with the XPM technique for several kinds of fibers are listed in Table 11.1. They are consistently larger than those obtained with the SPMbased short-pulse technique. The explanation for this discrepancy is related to the electrostrictive contribution to n2 that occurs for pulse widths >1 ns (or modulation frequencies 1000 W−1 /km at a wavelength near 1.55 µm when their core diameter was reduced to below 2 µm. Fig. 11.23 shows the cross section of a fiber whose narrow core (diameter 2.1 µm) is almost completely surrounded with large air holes. The effective mode area of this fiber was estimated to be only 3.1 µm2 The nonlinear parameter γ was estimated for three fibers with different core sizes from the measurements of the SPM-induced nonlinear phase shifts as a function of the input power. Fig. 11.23 shows the measured values together with the prediction of a model in which the core is taken to be completely surrounded with air. The γ values range from 460 to 1100 W−1 /km, the largest value occurring for a fiber with the core diameter of 2.1 µm. The theoretical model predicts that values of γ close to 2000 W−1 /km are possible with a further reduction in the core size. Bismuth-oxide fibers have been used in a variety of applications relevant for telecommunication systems. Because of their relatively high losses (∼1 dB/m), fiber length is

Highly nonlinear fibers

typically 2 µm, this ratio exceeds 0.25. As a result, γ1 term contributes significantly to the nonlinear evolution at such wavelengths. Numerical simulations reveal that it can reduce considerably the Ramaninduced frequency shift, among other things [153]. The main point to note is that the frequency dependence of the mode profile may become important in practice, depending on the fiber design.

Highly nonlinear fibers

Figure 11.25 (A) calculated values of γ0 and γ1 as a function of wavelength for a PCF designed with a dispersion profile shown in part (B), where the ratio γ1 /γ0 is also plotted. (After Ref. [153]; ©2010 APS.)

Clearly, the theory of optical fibers with core diameters 1 µm or less (often called silica nanowires) is quite involved. Eq. (11.6.8) ignores polarization effects by assuming that the input pulse maintains its linear polarization. A full vectorial theory of silica nanowires has also been developed for including the polarization effects. If we ignore the frequency dependence of mode profiles, e(x, y, ω) with its value at ω0 , and follow Section 6.1, we obtain the following coupled-mode equations [151]: ∂ Ax ∂ Ax iβ2 ∂ 2 Ax + β1x + ∂z ∂t 2 ∂ t2 2  = iγ11 |Ax | Ax + iγ12 |Ay |2 Ax + iγ12 A2y A∗x exp(−2i β z),

(11.6.11)

∂ 2A

∂ Ay ∂ Ay iβ2 y + β1y + 2 ∂z ∂t 2 ∂t  = iγ22 |Ay |2 Ay + iγ12 |Ax |2 Ay + iγ12 A2x A∗y exp(2i β z),

(11.6.12)

where β is related to the linear birefringence of the fiber. These equations are similar to Eqs. (6.1.11) and (6.1.12) in structure with one major difference. They contain six different nonlinear parameters of the form γij (defined in Ref. [151]) that depend on the fiber design. In particular, the SPM coefficients γ11 and γ22 are not the same for the two polarization components. Also, the factors of 2/3 and 1/3 appearing in Eqs. (6.1.11) and (6.1.12) are modified for narrow-core fibers. The implications of these changes have been discussed in Ref. [152]. In particular, they may lead to nonlinear self-flipping of the state of polarization of the optical field in such fibers.

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Problems 11.1 The refractive index of an optical fiber is specified as n = n0 + n2 I, where n0 = 1.45, n2 = 2.7 × 10−20 m2 /W and I is the optical intensity. If it is written in terms of the electric field as n = n0 + nE2 |E|2 , find the magnitude of nE2 . 11.2 Light from two CW lasers differing in their wavelengths by a small amount λ is transmitted through a fiber of length L. Derive an expression for the optical spectrum at the fiber output after including the effects of SPM-induced phase shift. Assume equal input powers for the two lasers and neglect fiber loss and dispersion for simplicity. 11.3 Explain how the optical spectrum obtained in the preceding problem can be used to estimate the nonlinear parameter γ . Also, derive the expression for the power ratio given in Eq. (11.1.5). 11.4 Discuss how the nonlinear phenomenon of XPM can be used to deduce γ for an optical fiber. Why is it important to depolarize the pump light when a pump–probe configuration is employed? 11.5 The measured values of γ are found to be smaller by about 20% when pulses shorter than 1 ns are employed. Explain the physical process responsible for this reduction. What would happen if femtosecond pulses are employed for measuring γ ? 11.6 Explain why large values of γ cannot be realized by reducing the core diameter of a fiber designed with silica cladding to below 1 µm? Using Aeff = π w 2 with √ √ w = a/ ln V , prove that the maximum value of γ occurs when the V = e, where V parameter of the fiber is defined as in Eq. (11.2.1). 11.7 Solve the eigenvalue equation (2.2.8) given in Section 2.2.1 for a tapered fiber with air cladding using n1 = 1.45, n2 = 1, and a = 1 µm and plot β(ω) in the wavelength range of 0.4 to 2 µm. Also plot second and third-order dispersion parameters, β2 and β3 , over this range and find the zero-dispersion wavelength of this fiber. 11.8 Repeat the preceding problem for tapered fibers with diameters 0, the frequency shift of the dispersive wave is positive. As a result, the NSR is emitted at a higher frequency (or a shorter wavelength) than that of the soliton. This was the case for the numerical results shown in Fig. 12.1. Indeed, the frequency of NSR in this figure agrees well with the prediction of Eq. (12.1.7). The situation changes if both β2 and β3 are negative; the NSR is then emitted at wavelengths longer than that of the soliton. It is important to stress that the fourth- and higher-order dispersion terms can modify the predictions of Eq. (12.1.7) if they are not negligible. As a second example, consider the special case in which β3 can be set to zero in Eq. (12.1.6) so that the soliton is perturbed by the fourth-order dispersion alone. Assuming δβ1 = 0 and neglecting the contribution of the γ term, we predict that two dispersive waves will be generated simultaneously, shifted from the soliton frequency by ± ≈ ±(−12β2 /β4 )1/2 .

(12.1.8)

For solitons propagating in the region of a fiber such that β2 < 0 and β4 > 0, both solutions are real and have opposite signs, indicating that the NSR appears on both sides of the soliton’s spectrum. However, if β4 0 is negative for this fiber, both solutions become imaginary, and no dispersive waves will be produced. One can verify the preceding predictions by solving the generalized NLS equation numerically. If we use the normalized form of this equation given in Eq. (5.5.8) and retain only the dispersive perturbation by setting s = τR = 0, it takes the form i

∂ u 1 ∂ 2u ∂ 3u ∂ 4u 2 + | u | u = i δ − δ , + 3 4 ∂ξ 2 ∂τ 2 ∂τ 3 ∂τ 3

(12.1.9)

where the last term representing fourth-order dispersion has been added, and the two normalized dispersion parameters are defined as δ3 =

β3 , 6|β2 |T0

δ4 =

β4 . 24|β2 |T02

(12.1.10)

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Figure 12.4 (A) Temporal and (B) spectral evolutions of a third-order soliton (N = 3) over a distance z = 1.5LD when it is perturbed by the fourth-order dispersion alone (δ4 = 0.0005). A decibel scale over 70 dB range is used such that a darker shade represents a more intense region.

Assuming δβ1 = 0, the polynomial in Eq. (12.1.6) can be written in the following normalized form [16]: δ4 f 4 + δ3 f 2 − 12 f 2 = 12 (2N − 1)2 ,

(12.1.11)

where f = T0 and Ps in Eq. (12.1.7) was replaced with P1 in Eq. (12.1.2), assuming that dominant contribution to dispersive waves comes from the shortest soliton formed after the fission process. Fig. 12.4 shows the temporal and spectral evolutions of a third-order soliton (N = 3) over a distance of 3LD /2 by solving Eq. (12.1.9) numerically with δ3 = 0 and δ4 = 0.0005. This figure should be compared with Fig. 12.1 obtained by solving the same equation with δ3 = 0.01 and δ4 = 0. Similar to the case of TOD, fourth-order dispersion also leads to the soliton fission at a distance near ξ = 0.3. However, in contrast with the TOD case, we see a sudden emergence of two new spectral peaks after the fission on both the blue and red sides of the original pulse spectrum. The frequencies of these peaks agree with the prediction of Eq. (12.1.10). Indeed, if we neglect the last term and set δ3 = 0 in this equation, the solution is given by f = ±(2δ4 )−1/2 , resulting in f /2π = ±5.03, in good agreement with the results in Fig. 12.4(B). Temporal evolution in Fig. 12.4(A) shows that two dispersive waves are generated simultaneously, one propagating faster and the other traveling slower than the input pulse. We also observe

Novel nonlinear phenomena

Figure 12.5 NSR peak power (relative to input peak power) as a function of δ3 when a second-order soliton undergoes fission. (After Ref. [14]; ©2009 American Physical Society.)

the formation of temporal and spectral fringes at distances z > 0.5LD . These can be understood to result from an interference among the three fundamental solitons and two dispersive wave propagating at different speeds with different phase shifts. The preceding discussion clearly shows that the generation of dispersive waves during soliton fission is sensitive to minute details of the dispersion relation β(ω) of the fiber, and it may be necessary in some situations to include dispersion terms even higher than the fourth order. A numerical approach with multiple higher-order dispersion terms shows that all odd-order terms generate a single NSR peak on the blue or the red side of the carrier frequency of the pulse, depending on the sign of the corresponding dispersion parameter [15]. In contrast, even-order dispersion terms with positive signs always create two NSR peaks on opposite sides of the carrier frequency. In general, the frequencies as well as the amplitudes of the NSR peaks depend not only on the magnitudes of various dispersion parameters of the fiber but also on higher-order nonlinear effects discussed in the following section. An important question is how much pulse energy is transferred to dispersive waves during soliton fission. Although several approximate analytic approaches have been used for this purpose [6–11], an accurate answer requires numerical solutions of the generalized NLS equation [14]. Fig. 12.5 shows peak power of the NSR spectral band (relative to input peak power) as a function of δ3 when a second-order soliton undergoes fission. The gray curve shows the case when Eq. (12.1.9) is solved with δ4 = 0 and predicts that the NSR peak power can exceed 1% of the input peak power for δ3 > 0.02. Since higher-order nonlinear effects can also modify the fission process, the black curve is obtained by solving Eq. (2.3.36). One sees clearly that, for any value of δ3 , the inclusion of intrapulse Raman scattering lowers the amplitude of the NSR peak. Next section provides more details on the impact of intrapulse Raman scattering on femtosecond pulses.

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12.2. Intrapulse Raman scattering As discussed in Section 5.5.4, the spectrum of an ultrashort pulse shifts toward longer wavelengths because of intrapulse Raman scattering. Such a shift was first noticed around 1985 in the context of solitons and was called the soliton self-frequency shift [17–19]. The term Raman-induced frequency shift (RIFS) was used in Section 4.4.3 because a spectral shift can occur even in the normal-GVD region of an optical fiber where solitons do not form [20]. With the advent of highly nonlinear fibers, it was discovered that the RIFS of solitons can become quite large in such fibers, and that it can be used for providing a wavelength-tunable source of femtosecond pulses. Later experiments revealed that the RIFS can also become smaller under certain conditions. This section focuses on such novel aspects of intrapulse Raman scattering.

12.2.1 Enhanced RIFS through soliton fission In a 1986 experiment, 560-fs optical pulses were propagated as fundamental solitons inside a 0.4-km-long fiber and their spectrum was found to shift by up to 8 THz [18]. It was discovered soon after that the fission of higher-order solitons generates a much larger RIFS [4]. This is clearly evident in Fig. 12.2(A) where the shortest soliton created after the fission exhibits a shift of more than 20 THz after 1-km of propagation inside a fiber. A soliton whose wavelength shifts continuously during propagation is sometimes referred to as the Raman soliton. Although the properties of Raman solitons attracted some attention during the 1990s [21–26], it was only after 1999 that the RIFS was observed and studied in microstructured and other narrow-core fibers [27–37]. The generalized NLS equation (5.5.8) of Section 5.5 can be used for studying RIFS in microstructured fibers. Neglecting the self-steepening term (s = 0) and including the effects of fourth-order dispersion, it takes the form i

∂ u 1 ∂ 2u ∂ 3u ∂ 4u ∂|u|2 + |u|2 u = iδ3 3 − δ4 3 + τR u + , 2 ∂ξ 2 ∂τ ∂τ ∂τ ∂τ

(12.2.1)

where τR = TR /T0 is a dimensionless parameter with TR ≈3 fs, as defined in Eq. (2.3.42). As an example, Fig. 12.6 shows the temporal and spectral evolutions of a third-order soliton (N = 3) over a distance of 1.5LD by solving Eq. (12.2.1) numerically with δ3 = 0.01, δ4 = 0, and τR = 0.04. This figure should be compared with Fig. 12.1, obtained by solving the same equation with τR = 0 so that the Raman effects are absent. A comparison of two figures shows how intrapulse Raman scattering affects the pulse evolution. A dispersive wave is still generated on the blue side of the input spectrum but its amplitude is much smaller for τR = 0.04, in agreement with the results in Fig. 12.5. More importantly, we observe in Fig. 12.6(A) that the shortest soliton created after the fission process separates from the main part of the pulse and its trajectory bends continuously toward the right side. Physically, this can happen if it travels slower and slower as

Novel nonlinear phenomena

Figure 12.6 (A) Temporal and (B) spectral evolutions of a third-order soliton (N = 3) over a distance of 1.5LD when δ3 = 0.01, δ4 = 0, and τR = 0.04. A decibel scale over 70 dB range is used such that a darker shade represents a more intense region.

it propagates down the fiber. This deacceleration of the soliton is produced by the RIFS that continuously shifts the spectrum of the shortest soliton toward the red side. Such a spectral shift is indeed seen in Fig. 12.6(B). Notice that the spectral bandwidth oscillates along the fiber’s length just after the fission of the third-order soliton occurs near ξ = 0.3. As before, temporal and spectral fringes in Fig. 12.6 result from interference of multiple optical fields. The important issue from a practical standpoint is how the RIFS depends on the input parameters associated with the optical pulse and the fiber used. The theory of Section 5.5.4 can be employed to study how RIFS scales with these parameters. In general, one should employ Eq. (5.5.17) to take into account fiber losses and changes in the soliton’s width. In the case of negligible fiber losses and a nearly constant soliton width, we can employ Eq. (5.5.18). If we introduce νR = p /(2π), the RIFS grows linearly along the fiber at a rate 4TR |β2 | 4TR (γ P0 )2 d νR =− =− , 4 dz 15π |β2 | 15π T0

(12.2.2)

where we used Eq. (12.1.1). This equation shows that the RIFS produces a spectral shift toward lower frequencies (a red shift) and it scales with the width as T0−4 , or quadratically with the product γ P0 . However, it should be used with caution. The bent trajectory of

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Figure 12.7 (A) Raman gain spectrum (solid curve) and its linear approximation (gray line). (B) Rate of RIFS (d νR /dz) experienced by a fundamental soliton as a function of the pulse bandwidth in the two cases. (After Ref. [37]; ©2010 OSA.)

the Raman soliton in Fig. 12.6(A) shows clearly that the soliton’s width does not remain constant because |β2 | itself changes as the spectrum shifts toward the red side. It follows from Eq. (12.2.2) that RIFS can be made large by propagating shorter pulses with higher peak powers inside highly nonlinear fibers. Using the definition of −2 the nonlinear length, LNL = (γ P0 )−1 , the RIFS is found to scale as LNL . As an example, 2 if we use a highly nonlinear fiber with β2 = −30 ps /km and γ = 100 W−1 /km, LNL = 10 cm for 100-fs (FWHM) input pulses with P0 = 100 W, and the pulse spectrum shifts inside the fiber at a rate of about 1 THz/m. Under such conditions, it will shift by 50 THz over a 50-m-long fiber, provided that N = 1 can be maintained over this distance. Eq. (12.2.2) does not apply to the propagation of high-order solitons. However, it can be used to estimate RIFS of fundamental solitons after the fission process provided we replace T0 and P0 with the values given in Eq. (12.1.2). Even then, it provides only a rough estimate of the RIFS because its derivation is based on the approximate form of the generalized NLS equation given in Eq. (2.3.43). Moreover, its use requires a numerical value of the parameter TR . As indicated in Eq. (2.3.42), this parameter depends on the slope of the Raman-gain spectrum at origin. The dashed line in Fig. 12.7 shows that such a linear approximation deviates considerably from the actual shape of the Raman-gain spectrum, especially for short pulses with a relatively wide spectrum [37]. As a result, it overestimates the rate of RIFS in silica fibers for pulses shorter than 200 fs or so, and the actual Raman-gain spectrum should be used for predicting the RIFS. An alternative is to use the modified form of the Raman response function given in Eq. (2.3.40). Its predictions agree well with the results obtained using the actual Raman-gain spectrum [33]. The use of Eq. (2.3.39) should be avoided because it overestimates considerably the RIFS of pulses shorter than 100 fs. Relatively large RIFS were realized in a 2001 experiment in which 200-fs pulses at an input wavelength of 1300 nm were launched into a 15-cm-long fiber, tapered

Novel nonlinear phenomena

Figure 12.8 Left Panel: Measured (A) spectrum and (B) autocorrelation trace of the Raman soliton at the output of a tapered fiber. Right Panel: Numerically simulated (A) evolution of pulse spectrum and (B) output spectrum. Dashed curves show the input spectrum. (After Ref. [30]; ©2001 OSA.)

down to a 3-µm core diameter [30]. Fig. 12.8 shows the experimentally measured output spectrum and the autocorrelation trace, together with numerical predictions of the generalized NLS equation (2.3.43). The parameters used for simulations corresponded to the experiment and were LNL = 0.6 cm, LD = 20 cm, and TR = 3 fs. The effects of third-order dispersion were included using LD = 25 m. In this experiment, a spectral shift of 45 THz occurs over a 15-cm length, resulting in a rate of 3 THz/cm, a value much larger than that expected from Eq. (12.2.2) or Fig. 12.7. This discrepancy can be resolved by noting that the soliton order N exceeded 1 under the experimental conditions, and the input pulse propagates initially as a higher-order soliton inside the fiber. The qualitative features of Fig. 12.8 can be understood from the soliton-fission scenario discussed in Section 12.1. Recall that soliton fission breaks a higher-order soliton into multiple fundamental solitons whose widths and peak powers are related to N as indicated in Eq. (12.1.2). When N = 2.1, two fundamental solitons created through the fission process have widths of T0 /3.2 and T0 /1.2. The narrower soliton exhibits a much larger RIFS compared with the broader one because of the T0−4 dependence seen in Eq. (12.2.2). For the N = 2.1 example, the enhancement factor for the narrower soliton

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is about 105. Fig. 12.8 agrees with this scenario as the Raman soliton, whose spectrum has been shifted by 250 nm, has a width of 62 fs, about 3.3 times smaller than the input pulse width of 205 fs. Numerical simulations reveal that fission occurs within 2 cm of propagation inside the tapered fiber. The spectrum of the ultrashort soliton shifts rapidly toward the red side, as dictated by Eq. (12.2.2). After 15 cm, the spectrum has shifted to near 1550 nm. This predicted behavior agrees with the experimental data qualitatively.

12.2.2 Cross-correlation technique Fig. 12.8 shows evidence of soliton fission through the pulse spectrum observed at the fiber output together with a separate autocorrelation measurement of the pulse width in the time domain. As seen in Fig. 12.6, the shortest soliton separates from other solitons because it moves slower and slower as its spectrum shifts toward longer wavelengths. To observe the temporal and spectral features simultaneously, an extension of the frequency-resolved optical gating technique, discussed in Section 3.3.3, is used in practice. It is referred to as the X-FROG technique [38] and it consists of performing cross-correlation of the fiber’s output with a narrow reference pulse (with an adjustable relative delay) inside a nonlinear crystal through second-harmonic or sum-frequency generation. More precisely, one records a series of optical spectra at the crystal output for different relative delays. Mathematically, the cross-correlation is given by  

S(τ, ω) = 

∞ −∞

2 

A(L , t)Aref (t − τ )eiωt dt ,

(12.2.3)

where A(L , t) is the amplitude at the end of a fiber of length L and Aref (t − τ ) is the amplitude of the reference pulse delayed by an adjustable amount τ . In practice, rather than employing another source of femtosecond pulses, the same laser is used to provide both the input and reference pulses by splitting the laser power into two parts. In a different approach, the output spectrum was filtered to extract the Raman soliton, which was then used as a reference pulse for performing cross-correlation [39]. In this experiment, 110-fs pulses from a fiber laser operating at 1556 nm were launched into a polarization-maintaining fiber exhibiting a dispersion of only −0.1 ps2 /km at the laser wavelength. The output was divided into two beams, one of which was passed through a low-pass filter (with its cutoff wavelength at 1600 nm) and a delay line before reaching the KTP crystal. The spectrum of the sum-frequency signal was recorded using a monochromator for a range of delay times. Fig. 12.9 shows the X-FROG traces and optical spectra of output pulses for fiber lengths of 10 m and 180 m, when 110-fs pulses at a 48-MHz repetition rate are launched with 24-mW average power. In the case of a 10-m-long fiber (left panel), the Raman soliton is shifted to near 1650 nm and is delayed by about 3 ps, compared with the residual part of the input pulse located near 1550 nm. This delay is due to a change in

Novel nonlinear phenomena

Figure 12.9 X-FROG traces (top row) and optical spectra (bottom row) for fiber lengths of 10 m (left column) and 180 m (right column). Dashed curves show the delay time as a function of wavelength. (After Ref. [39]; ©2001 OSA.)

the group velocity that accompanies any spectral shift. The most interesting feature is the appearance of NSR near 1440 nm in the normal-GVD region of the fiber. In the case of a 180-m-long fiber (right panel), the Raman soliton has shifted even more, and its spectrum is located close to 1700 nm. It is also delayed by about 200 ps from the residual input pulse (not shown in Fig. 12.9). The radiation in the normal-GVD region now appears near 1370 nm and is spread temporally over a 4-ps-wide window. The group velocity of the NSR may not necessarily coincide with that of the Raman soliton. However, Fig. 12.9 reveals that the two overlap temporally for both the short and long fibers. This overlap is related to the phenomenon of soliton trapping, discussed earlier in Section 6.5.2 in the context of a birefringent fiber. As soon as the NSR and Raman soliton, propagating at different speeds, approach each other, they interact through XPM. This interaction modifies the spectrum of the NSR such that the two begin to move at the same speed. In other words, the Raman soliton traps the NSR and

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Figure 12.10 Output spectra for several input average powers when 70-fs pulses were launched into a 60-cm-long photonic crystal fiber. (After Ref. [31]; ©2002 IEEE.)

drags it along. Such trapping was observed in a 2002 experiment in which the NSR and the soliton were first separated using low- and high-pass optical filters, and then launched into a 170-m-long highly nonlinear fiber [40]. As the soliton spectrum shifted through the RIFS, the NSR spectrum also shifted with it such that they moved at the same speed. This trapping mechanism has also been used for ultrafast optical switching.

12.2.3 Wavelength tuning through RIFS An important application of RIFS consists of tuning the wavelength of a model-locked laser by sending its output pulses through a highly nonlinear fiber. The wavelength tuning occurs because the RIFS in Eq. (12.2.2) depends on the peak power P0 of input pulses, in addition to the fiber’s length and its parameters β2 and γ . In addition, β2 depends on the laser’s wavelength and can be varied if the pulse source itself is tunable. Thus, the RIFS can be used for tuning the wavelength of a model-locked laser over a wide range toward the red side by controlling the operating wavelength and the average power of the laser. A 60-cm-long photonic crystal fiber (PCF) having its zero-dispersion wavelength (ZDWL) at 690 nm was used in a 2002 experiment [31]. Fig. 12.10 shows the experimentally measured output spectra at several average power levels when 70-fs pulses with a 48-MHz repetition rate at a wavelength of 782 nm were launched into this fiber. After the input power exceeded a value large enough to ensure N > 1, the fission process created a Raman whose wavelength shifted continuously through the RIFS inside the fiber. The total shift at the output of fiber became larger as the input power was increased. The spectra at different powers had a nearly “sech” shape with a width of about 18 nm and contained as much as 70% of input energy in the form of a Raman soliton. Note the appearance of a second Raman soliton when input power was 3.6 mW. This

Novel nonlinear phenomena

is expected when the soliton order exceeds 2 for the input pulse. The wavelength shift is close to 120 nm (15% of input wavelength) at the 3.6-mW level. Although the tuning range was initially limited to near 100 nm [27–29], the RIFS technique was soon employed to produce widely tunable sources of femtosecond pulses [41–49]. As early as 2001, a 220-m-long polarization-maintaining fiber (core diameter 5.8 µm) was used to tune the wavelength of 110-fs pulses from 1560 to 2030 nm [41]. Clearly, much shorter lengths would be needed if microstructured fibers with narrower core diameters were used for shifting the pulse’s wavelength. Moreover, because the ZDWL of such fibers shifts toward shorter wavelengths, they can be used with lasers operating at wavelengths near or below 1 µm. In a 2002 experiment, a 4.7-m-long holey fiber with a core diameter of 3 ps/m. For θ > 66◦ , the Raman soliton forms only along the fast axis because the input power along the slow axis becomes too small to form a soliton. An interesting feature of such a dual-wavelength pulse pair was discovered in a 2002 experiment in which a fiber with reduced birefringence was employed [51]. It was

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Figure 12.14 (A) X-FROG spectrogram of trapped pulses at the output of a 140-m-long fiber. (B) Measured wavelengths of slow (circles) and fast (squares) polarization components for three different fiber lengths. Filled circles show power of the fast component at the central wavelength. (After Ref. [51]; ©2002 OSA.)

found that the more-intense pulse can trap the other one, if the two Raman solitons overlap temporally when they are formed and interact through XPM. After this trapping occurs, the pair moves together at a common group velocity. Fig. 12.14(A) shows the X-FROG spectrogram of trapped pulses at the output of a 140-m-long fiber exhibiting a birefringence of 3 × 10−4 . In this experiment, 110-fs pulses at a 48-MHz repetition rate were launched at a wavelength near 1550 nm and were polarized such that a larger part of pulse energy was in the mode polarized along the slow axis of the fiber. As seen in Fig. 12.14(A), the spectrum of the Raman soliton for this mode is centered near 1684 nm. The other mode should have shifted less because of its lower power, but its spectrum was actually located near 1704 nm. Such a large spectral shift was attributed to trapping of the lower-energy pulse by the soliton. Fig. 12.14(B) shows the measured wavelengths of the two components for three fiber lengths. Clearly, once trapping occurs, the two pulses move together and their wavelengths are shifted by the same amount. An unexpected feature of Fig. 12.14(B) is that the energy of the trapped pulse polarized along the fast axis increases exponentially along the fiber. This is due to amplification of the trapped pulse by the Raman soliton polarized along the slow axis through the Raman gain. The soliton acts as a pump and amplifies the trapped pulse located 20 nm away on the Stokes side. Even though the Raman gain is not very large for an orthogonally polarized pump, numerical simulations based on two coupled NLS equations show that it can still amplify the trapped pulse enough that the peak power of the amplified pulse becomes comparable to that of the soliton [51]. Such a trapping mechanism can be used for ultrafast optical switching by selectively dragging one pulse from a pulse train [52]. The preceding approach can be used to create more than two Raman solitons. The basic idea consists of creating multiple pulses with different peak powers from a single

Novel nonlinear phenomena

pump pulse. For example, if the pulse is first passed through a high-birefringence fiber of short length and large core diameter so that nonlinear effects are negligible inside it, the pulse will split into two orthogonally polarized pulses with a temporal spacing set by the birefringence of this fiber. If these two pulses are then launched into a highly nonlinear, birefringent fiber whose principal axes are oriented at an angle to the first fiber, each pulse would create two Raman solitons, resulting in the formation of four solitons at four different wavelengths. Time-domain multiplexing or polarization-mode dispersion inside a fiber can be used to make multiple copies of one pulse with different amplitudes, each of which then creates a Raman soliton at a different wavelength in the same highly nonlinear fiber [53].

12.2.5 Suppression of Raman-induced frequency shifts One may ask whether the RIFS of solitons can be reduced, or even suppressed, under some conditions. The question attracted attention soon after this phenomenon was discovered, and several techniques have been proposed in response [55–65]. Bandwidthlimited amplification of solitons for RIFS suppression was suggested in 1988 [55] and verified in a 1989 experiment [56]. Other possibilities include the use of cross-phase modulation [57], frequency-dependent gain [58], or an optical resonator [59]. More recently, the unusual properties of PCFs [60] and photonic bandgap fibers [65] have been utilized for this purpose. In this section we focus on the use of PCFs. As seen in Fig. 12.3, the GVD parameter β2 of any fiber has a parabolic shape when fourth-order dispersion is included. Depending on their design, β2 in some microstructured fibers vanishes at two wavelengths, one lying in the visible region and the other located in the infrared region. The dispersion slope, governed by the TOD parameter β3 , is positive near the first zero but becomes negative near the second one. This change in the sign of β3 changes the frequency associated with the dispersive wave generated during soliton fission. As seen from Eq. (12.1.7), the frequency shift of the NSR becomes negative for solitons propagating in the anomalous-GVD region near the second ZDWL, where both β2 and β3 take negative values. As a result, soliton loses its energy to NSR at a wavelength that lies in the infrared region beyond the second ZDWL. It turns out that spectral recoil from this NSR suppresses the RIFS as the soliton approaches the second ZDWL [61]. Fig. 12.15(A) shows the spectral evolution of 200-fs pulses, launched with a peak power of 230 W at a wavelength of 860 nm, into a photonic crystal fiber exhibiting β2 = 0 at two wavelengths near 600 and 1300 nm [60]. Under these experimental conditions, the launched pulse formed a fourth-order soliton (N = 4) that created 4 fundamental solitons after its fission. The shortest soliton had a width of only about 28 fs and shifted rapidly toward the red side through the RIFS. The spectra of other solitons shifted much less because of their larger widths. The shortest soliton approached the second β2 = 0 frequency (shown by a vertical dashed line) at a distance of about 1.25 m.

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Figure 12.15 (A) Experimental and (B) numerical spectra plotted as a function of propagation distance when 200-fs pulses are launched with 230-W peak power into a PCF exhibiting two ZDWLs. A darker shade represents a more intense region. A vertical dashed line marks the location of second frequency where β2 = 0. (After Ref. [60]; ©2003 AAAS.)

As seen in Fig. 12.15, its spectrum suddenly stopped shifting, a clear indication that the RIFS was suppressed beyond this point. At the same time, a new spectral peak appeared on the red side of the second ZDWL. This peak represents the dispersive wave to which the soliton transfers a part of its energy. Its frequency is smaller (and wavelength is larger) than that of the Raman soliton because of a change in the sign of β3 . The generalized NLS equation (2.3.36) was used to simulate pulse propagation numerically under the experimental conditions [61], and the results are shown in Fig. 12.15(B). The agreement with the experimental data is excellent, although few quantitative differences are also apparent. The main reason for these differences is related to the use of the approximate form of the Raman response function hR (t) given in Eq. (2.3.39). As discussed in Section 2.3.2, this form approximates the Raman-gain spectrum with a single Lorentzian profile. The use of hR (t) based on the actual gain spectrum (see Fig. 2.2) should provide a better agreement. The question is why the emission of dispersive waves near the second ZDWL suppresses the RIFS in Fig. 12.15. To answer it, we need to understand how the energy lost to the NSR affects the soliton itself [61]. The requirement of energy conservation dictates that a soliton would become wider as it loses energy to NSR, as N = 1 must be maintained. This broadening is relatively small in practice if solitons lose energy slowly. Much more important is the requirement of momentum conservation. It dictates that, as NSR is emitted in the normal-GVD region, the soliton should “recoil” into the anomalous-GVD region [11]. This spectral recoil mechanism is responsible for the suppression of RIFS in Fig. 12.15. Near the first ZDWL, spectral recoil is negligible because the RIFS forces soliton spectrum to shift away from this wavelength, and the loss of energy decreases as β2 increases. However, the exact opposite occurs near the second ZDWL, where the RIFS moves the soliton closer to it, and the rate of energy

Novel nonlinear phenomena

Figure 12.16 Measured spectra as a function of average input power when 110-fs pulse at wavelengths (A) 1400, (B) 1510, and (C) 1550 nm were launched into a PCF with its second ZDWL near 1510 nm (horizontal dashed line). A darker shading represents higher power. Right panel shows the spectra at the 9-mW power level. (After Ref. [64]; ©2005 Elsevier.)

loss to NSR increases drastically. The resulting spectral recoil reduces the RIFS, and a steady state is reached in which spectral recoil is just large enough to nearly cancel the RIFS. The X-FROG technique, as well as numerical simulations, confirm the preceding physical scenario [62]. The moment method has also been used to study momentum conservation near the second ZDWL [63]. All results confirm that the unusual soliton dynamics near the second ZDWL, in combination with the NSR produced by higher-order dispersive effects, is responsible for the suppression of RIFS. The pulse-propagation scenario seen in Fig. 12.15 depends on the input wavelength of pulses because it sets the initial values of the dispersion parameters. In one study, 110-fs pulses were launched at wavelengths close to the second ZDWL (around 1510 nm) of a 1-m-long PCF [64]. Fig. 12.16 shows the measured spectra at the fiber output as a function of the average input power (at a 80-MHz repetition rate) for three input wavelengths of 1400, 1510, and 1550 nm. In case (A), a Raman soliton is generated through fission, but its spectrum stops shifting beyond 1470 nm because of the RIFS suppression induced by the NSR appearing in the normal-GVD regime lying beyond 1510 nm. In case (B), a part of the pulse lies in the normal-GVD regime, and it disperses without forming a soliton. The remaining part propagates in the anomalousGVD regime, and its spectrum shifts toward shorter wavelengths with increasing power.

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Figure 12.17 Measured spectra for fiber lengths ranging from 0.4 to 5 m as when 270-fs pulses are launched into a photonic bandgap fiber. The gray shading marks a bandgap edge at 1535 nm. (After Ref. [65]; ©2010 OSA.)

At a certain value of the input power, soliton effects begin to dominate, and the spectrum shifts through the RIFS toward longer wavelengths, resulting in the comma-like shape (marked by an arrow). At even higher powers, the RIFS is arrested by the spectralrecoil mechanism. In case (C), the pulse propagates in the normal-GVD regime and mostly experiences SPM-induced spectral broadening. At high powers, the spectrum broadens enough that a part of the pulse energy lies in the anomalous-GVD regime and begins to form a soliton. As discussed in Section 11.4, some microstructured fibers exhibit a photonic bandgap (PBG) such that light can propagate inside the core only if its wavelength falls within this bandgap region. Although a hollow core is often used, PBG fibers can be designed with a solid core. In a 2010 experiment, 270-fs pulses at 1200 nm were launched into such a fiber to study the RIFS phenomenon [65]. Fig. 12.17 shows the measured output spectra for fiber lengths ranging from 0.4 to 5 m. The fission of high-order solitons occurs within first 10 cm of fiber, and the spectrum of two shortest solitons, marked as S1 and S2 , shifts rapidly toward longer wavelengths. However, the spectrum of S1 approaches the PBG edge after 2.5 m of propagation and it stops shifting beyond that distance, with its central wavelength locked at 1510 nm. Clearly, RIFS is nearly suppressed close to a PBG edge. Numerical results based on the generalized NLS equations confirm these results. The physical explanation of this type of RIFS reduction is quite different from that used for Fig. 12.15, where the soliton approached the ZDWL of a photonic crystal fiber. In the case of a PBG fiber, the parameters β2 and γ change considerably as the PBG edge is reached. Moreover, changes occur such that β2 increases and γ decreases at the same time. As a result, the requirement that the parameter N in Eq. (12.1.1) remain close to 1 can only be satisfied if soliton width increases substantially. Since the RIFS rate in Eq. (12.2.2) scales as Ts−4 , where Ts is the local soliton width, this rate is reduced drastically as the soliton approaches the PBG edge. Indeed, the spectral width of the

Novel nonlinear phenomena

soliton was measured [65] for the data in Fig. 12.17, and it was found to contract by a factor of almost 8. This spectral contraction translates into a temporal broadening factor of 8, and a reduction in the RIFS rate by a factor of more than 4000. Two other mechanisms can suppress RIFS inside optical fibers. First, the XPM interaction of a Raman soliton with another pulse may produce a blue shift that can reduce, or even cancel the RIFS [66]. Such blue shifts can also occur during temporal reflection of the Raman soliton [67]. Second, the frequency dependence of the nonlinear parameter in fibers doped with silver nanoparticles can suppress RIFS under appropriate conditions [68].

12.2.6 Soliton dynamics near a zero-dispersion wavelength It is clear from Fig. 12.15 that the evolution of a soliton in the anomalous-GVD region near a ZDWL of the fiber is affected strongly by the signs of the third- and fourth-order dispersion parameters, δ3 and δ4 , appearing in Eq. (12.2.1). The moment method has been applied to this equation to study the soliton evolution and the behavior of RIFS near a ZDWL [13]. When |δ3 | =  0, one finds that RIFS is reduced for δ3 > 0 because of an increase in |β2 | and a corresponding increase in the soliton width. In contrast, when δ3 < 0, soliton width decreases, and RIFS rate increases until the soliton wavelength approaches the ZDWL, where the RIFS is nearly suppressed. As discussed earlier, the use of Eq. (12.2.1) becomes questionable for studying RIFS because it includes the Raman effects approximately through a single parameter τR . An accurate description requires the use of generalized NLS equation (2.3.36). In its normalized form, this equation can be written as i

∞  ∂u 1 ∂ 2u ∂ nu 2 + | u | u + in δn n + 2 ∂ξ 2 ∂τ ∂t n=3     ∞ ∂ = i 1 + is u(ξ, τ ) R(τ  )|u(ξ, τ − τ  )|2 dτ  , ∂τ 0

(12.2.4)

where s = γ1 /γ ≈ (ω0 T0 )−1 and fiber losses have been neglected. This equation includes dispersion to all orders but the third- and fourth-order parameters, δ3 and δ4 , play the most critical roles near a ZDWL. It turns out that even relatively small values of δ4 affect the fission process considerably. As an example, Fig. 12.18 shows the spectral and temporal evolution of a thirdorder soliton (N = 3) in four cases: Top and bottom rows correspond to δ3 = ±0.02, while δ4 = ±0.0002 for the left and right columns. Thus, the four cases cover four possible sign combinations of β3 and β4 . It is remarkable how much the evolution of a high-order soliton is affected by small changes in the dispersion characteristics when solitons propagate near a ZDWL of the fiber.

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Figure 12.18 Temporal and spectral evolution of a third-order soliton over a distance 3LD on a 50-dB intensity scale when (A) δ3 = 0.02 with δ4 = 0.0002, (B) δ3 = 0.02 with δ4 = −0.0002, (C) δ3 = −0.02 with δ4 = 0.0002, and (D) δ3 = −0.02 with δ4 = −0.0002.

Consider first the case β3 > 0 shown in the top row. When β4 is positive but relatively small, the fission generates a dispersive wave on the blue side (as expected for β4 = 0) and the shortest soliton is separated from others as its speed is reduced because of a continuously increasing RIFS. These features are expected from the earlier discussion in this section. However, if the sign of β4 is negative, as seen in part (B), the dispersive wave is totally suppressed. At the same time, the RIFS of the shortest soliton is reduced. Now consider the case β3 < 0 shown in the bottom row of Fig. 12.18. Two things occur when β4 is positive and relatively small. First, the dispersive wave, as expected from Eq. (12.1.7), is generated on the red side of the input spectrum. Second, RIFS is nearly suppressed, and this suppression is accompanied with a large increase in the amount of NSR in the normal-GVD region lying beyond the ZDWL of the fiber. As discussed earlier, RIFS suppression is related to spectral recoil induced by the NSR pressure. In

Novel nonlinear phenomena

Figure 12.19 Nonlinear propagation of a 50-fs pulse over 15 m for two nearly identical holey fibers. Top row shows D(λ), middle row shows spectral evolution, and bottom row compares the input and output pulse spectra. Soliton spectral tunneling occurs in case (A) but not in case (B). (After Ref. [71]; ©2008 IEEE.)

the final case shown in part (D), both β3 and β4 are negative. We now see the emergence of two dispersive waves, one on the blue and one on the red side together with some RIFS. Note also considerable spectral changes resulting from optical interference. A new phenomenon can occur when both β3 and β4 are negative for a fiber [13]. In this case, the frequency dependence of β2 exhibits an inverted parabola shown schematically in Fig. 12.4. If this parabola crosses the frequency axis, two ZDWLs exist such that the spectral region between them exhibits normal dispersion and is surrounded on both sides with regions exhibiting anomalous dispersion. When a soliton, undergoing RIFS, approaches one of the ZDWLs, it may tunnel across the entire normal-GVD band and transfer most of its energy to a new soliton in the second anomalous-GVD region. This phenomenon, called soliton spectral tunneling, was first predicted in 1993 [69]. With the advent of microstructured fibers, it attracted attention again in the context of a PCF with sub-wavelength air holes [70]. A more systematic numerical study has revealed that soliton spectral tunneling may be difficult to observe experimentally because of an extreme sensitivity of this phenomenon to the cladding-design parameters such as the diameter and spacing of air holes [71]. As an example, Fig. 12.19 shows numerical simulations for two holey fibers that are identical in all respects (core diameter 1.8 µm) except for a difference of 1% in the air-hole diameter. As seen there, this minor difference changes fiber dispersion such that two ZDWLs exist for one fiber but not for the other. As a result, soliton spectral tunneling occurs in case (A), but not in case (B). However, the spectral evolution of a 50-fs pulse over 15 m appears similar for both fibers, with nearly identical spectra at the

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Figure 12.20 (A) Experimental spectra over 45 cm of a PCF for a 110-fs pulse launched with 8.5-kW peak power. (B) Output spectra for two PCFs of 1 m (black) and 15 m (gray) lengths. (After Ref. [73]; ©2007 OSA.)

fiber output. Since no normal-GVD region exists for the situation in part (B), it may be difficult in practice to pin-point the spectral-tunneling phenomenon.

12.2.7 Multipeak Raman solitons The results shown in Fig. 12.18 correspond to an input pulse with a peak power P0 such that the pulse propagates initially as a third-order soliton N = 3. It was noticed in a 2007 experiment that pulses launched with larger peak powers exhibited a new phenomenon in which two Raman solitons created after the fission process come close to each other and formed a bound pair under certain conditions. Such bound states were first discovered in 1996 [72] and have been studied further after their experimental observation [73–75]. Fig. 12.20 shows the experimental spectra recorded over distances of up to 45 cm when a 110-fs pulse was launched into a PCF with 8.5-kW peak power (N ≈ 12). One sees clearly that two rightmost Raman solitons form a bound pair and their spectra shift together. Part B shows that this bound state continues to hold over distances as long as 15 m even though peak amplitudes decrease considerably because of internal losses. Numerical simulations based on Eq. (12.2.4) reveal that the formation of such a bound pair is very sensitive to the precise value of the input peak power. To understand the underlying physics, it is best to focus on a much simpler theoretical model given in Eq. (5.5.19) and reproduced here: i

∂ u 1 ∂ 2u ∂|u|2 2 + | u | u = τ u + . R ∂ξ 2 ∂τ 2 ∂τ

(12.2.5)

This equation includes only the second-order dispersion, ignoring higher-order dispersion completely, but the Raman effect is included through a dimensionless parameter

Novel nonlinear phenomena

τR . In spite of these limitations, it is useful to study the propagation of individual fun-

damental solitons created after the fission of a higher-order soliton. As mentioned in Section 5.5.4, Eq. (12.2.5) was solved approximately in 1990 using the following transformation [76]: u(ξ, τ ) = f (η) exp[iξ(q + bτ − b2 ξ 2 /3)],

(12.2.6)

where η = τ − bξ 2 /2 is a new temporal variable in a reference frame moving with the soliton. The parameter b depends on q and τR as b = 32τR q2 /15. Using the preceding form of u(ξ, τ ) in Eq. (12.2.5), f (η) is found to satisfy the following second-order differential equation: 1 d2 f d |f |2 2 = ( q + b η) f − | f | f + τ f . R 2 dη2 dη

(12.2.7)

Solutions of this equation with the appropriate boundary conditions determine the eigenvalue q together with the pulse shape f (η). An approximate analytic solution in the form of a single Raman soliton with q = 12 was first obtained in Ref. [76] and its functional form is given in Eq. (5.5.20). It was discovered numerically in 1996 that Eq. (12.2.7) also has solutions in the form of two Raman solitons of different amplitudes that form a pair and move as one unit [72]. More recently, it has been found that multiple Raman solitons of differing amplitudes can also form a multipeak bound state [75]. Fig. 12.21 shows the temporal profiles of four solutions of Eq. (12.2.5) obtained numerically using τR = 0.1. The single-peak solution in part (A) corresponds to a Raman soliton whose spectrum as well as temporal position shifts continually with ξ (see Fig. 12.18). The two-peak solution in part (B) corresponds to a soliton pair that has been observed in experiments (see Fig. 12.20). The most noteworthy feature of Fig. 12.21 is that amplitudes of multiple peaks decrease with a constant ratio r. The origin of this feature is related to the term bη in the eigenvalue equation (12.2.7). If we ignore the last two nonlinear terms, this equation represents effectively the motion of a unit-mass particle of energy q in a gravity-like potential U (η) = bη, the parameter b playing the role of gravity [77]. This potential is shown by a dashed in line in Fig. 12.21 and is the reason why peak amplitudes should decrease with a constant ratio r that depends on the value of q. Another consequence of this potential is that all solutions in Fig. 12.21 exhibit an oscillatory structure on the leading edge of the pulse (not seen on a linear scale because of its relatively low amplitude). The reason is easily understood from Eq. (12.2.7), whose solution is in the form of an Airy function near the leading edge of the pulse where the last two nonlinear terms are negligible [72]. An important issue is related to the stability of the multipeak intensity profiles seen in Fig. 12.21. Stability can be tested by solving Eq. (12.2.5) numerically over long distances with the input corresponding to a specific multipeak intensity profile. The results are

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Figure 12.21 Temporal profiles of four solutions obtained numerically using τR = 0.1. The dashed line shows the gravity-like potential that enforces a constant ratio r between successive peaks. (After Ref. [75]; ©2010 OSA.)

Figure 12.22 Temporal evolution of multipeak Raman solitons over hundreds of dispersion lengths. (After Ref. [75]; ©2010 OSA.)

shown in Fig. 12.22 for the two-, three-, and four-peak structures of Fig. 12.21. As seen there, even though they are not absolutely stable, a multipeak Raman soliton can propagate over hundreds of dispersion lengths before the bound state is destroyed.

12.3. Frequency combs and cavity solitons Frequency combs are optical sources with a broad spectrum containing equally spaced, discrete spectral lines. They are useful in view of their applications in frequency metrology, among other things [78]. An example of frequency combs is provided by mode-

Novel nonlinear phenomena

locked lasers emitting a train of ultrashort pulses with a broad spectrum. FWM inside a micro-ring resonator made of a nonlinear material can also be exploited to realize them [79]. Such combs are called Kerr frequency combs. This section focuses on the use of nonlinear effects inside optical fibers for making frequency combs.

12.3.1 CW-pumped ring cavities Optical fibers can be used to make a ring cavity with the help of a fiber coupler. Indeed, such ring cavities are routinely used for making fiber lasers that emit short pulses, referred to as dissipative solitons because their formation requires a gain mechanism to balance the cavity loss [80]. The gain is often provided using a suitably pumped doped fiber. However, modulation instability can also be used to provide the needed gain. Solitons forming though this mechanism inside a ring cavity make use of the Kerr nonlinearity, and their spectrum is in the form of a Kerr frequency comb. Such solitons are known as temporal cavity solitons. Historically, modulation instability inside a fiber cavity was studied as early as 1988 [81]. It attracted considerable attention during the 1990s for generating a train of ultrashort pulses [82–86]. It was found that the instability can occur even in the normal-GVD region of the fiber because of the feedback provided by cavity mirrors [82]. Moreover, a relatively small feedback occurring at the fiber–air interface (about 4%) is enough for this to occur [83]. Numerical and analytical results showed that such a laser can generate ultrashort pulse trains through CW pumping [84]. In a 2001 experiment, a pulse train indeed formed when a cavity containing 115-m-long fiber was pumped with a 1555-nm CW beam [86]. The threshold pump power was close to 90 mW for this modulation-instability laser. The pulses emitted by such a laser are related to cavity solitons because they require both the Kerr nonlinearity and anomalous GVD. In a 2010 experiment, cavity solitons were used as bits for all-optical storage [87]. Since then, such solitons have attracted considerable attention [88–94]. The NLS equation (5.1.1) can be used to understand the formation of cavity solitons, but it must be supplemented to account for the external CW pumping using [95]

√

A(m+1) (0, T ) = 1 − ρ A(m) (L , T )eiφ0 + ρ Ap ,

(12.3.1)

where A(m) (z, T ) is the intracavity field at the location z during the mth round trip inside a fiber cavity of length L, φ0 is the linear phase accumulated over one round trip with respect to the pump, ρ is the fraction of power leaving the cavity at z = L, √ and ρ Ap is the pump field entering the cavity. Eq. (5.1.1) together with Eq. (12.3.1) must be solved over thousands of round trips before a steady state is realized. Such an approach is quite time-consuming in practice. In a high-Q cavity with relatively low losses, A(z, T ) does not change much during a single round trip, as L is typically much smaller than both LD and LNL . Under such

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conditions, one can introduce a slow time scale ts = αc (t/tR ), measured in steps of the round-trip time tR , and use an averaging approach to obtain [96] ∂U ∂ 2U + sgn(β2 )i 2 = −(1 + i )U + i|U |2 U + S, ∂ ts ∂τ

(12.3.2)

where αc = (α L + ρ)/2 represents the total cavity loss, = (ω − ωc )tR /αc is a normalized detuning from the cavity resonance ωc closest to the pump frequency, and the normalized variables are defined as



τ=

2αc T, β2 L

U=

γL , αc



S=

γ Lρ Ap . αc3

(12.3.3)

Eq. (12.3.2) is known as the mean-field Lugiato–Lefever equation [97]. It can be solved numerically much faster than the NLS equation because ts is effectively the round-trip number. Before using numerical simulations, let us consider the steady-state solution of Eq. (12.3.2) when a CW beam is injected into the cavity. Setting both time-derivative terms to zero in this equation, the intracavity power Y = |U |2 satisfies the relation [(Y − )2 + 1]Y = |S|2 .

(12.3.4)

This is a cubic equation in Y , indicating that up to three solutions may exist for specific parameter values, among which at most two are stable against CW perturbations [97]. Thus, a nonlinear ring cavity exhibits optical bistability with two stable branches over a certain range of the pumping conditions. However, when time-dependent perturbations are considered, the modulation instability is found to destabilize portions of the upper branch [98]. In this region, the CW solution is converted into a pulse train that, depending on the input parameters, may or may not evolve into a train of ultrashort cavity solitons with a comb-like spectrum. This instability can occur even in the case of normal GVD (β2 > 0). It can also be interpreted through a FWM process in which pump energy is transferred to multiple neighboring cavity modes.

12.3.2 Nonlinear dynamics of ring cavities Fig. 12.23 show the results obtained by solving Eq. (12.3.2) numerically using |S|2 = 10 for the CW pump power [96]. The tilted peak in part (A) results from three solutions occurring for > 4. The lower branch starting near = 4 is always stable The dotted middle branch is always unstable. The dashed portion of the upper branch is unstable owing to modulation instability starting at MI = 1 − |S|2 − 1 = −2. In the shaded region marked MI, a pulse train is created through modulation instability but the spectrum is not wide and contains only a few cavity modes. In contrast, in the shaded region

Novel nonlinear phenomena

Figure 12.23 (A) Stability diagram showing intracavity peak power versus detuning for a CW pump. Dashed and dotted lines indicate unstable solutions. Shaded areas represent modulation-instability (MI) and cavity-soliton (CS) regions. Two crosses mark the parameter values for which the temporal and spectral profiles are shown in parts (B)–(E). (After Ref. [96]; ©2013 OSA.)

marked CS, the pulse train consists of cavity solitons whose spectrum spans hundreds of cavity modes. These results show that the combination of dispersive and nonlinear effects in fiber-ring cavities leads to complicated dynamics. Indeed, a careful adjustment of the pump power |S|2 and the detuning is required in practice for observing cavity solitons. Often, the laser frequency is tuned over a cavity resonance to reach the upper branch. In some experiments, short pulses at another wavelength were launched together with the CW beam to produce an XPM-induced phase shift that helped in exciting the cavity solitons [89]. In a 2013 experiment, the rich nonlinear dynamics predicted by Eq. (12.3.2) was observed using a 380-m-long ring cavity [88]. The largest values of the two parameters in this equation that could be realized in the experiment were |S|2 = 8.5 and = 4.1. As seen in Fig. 12.23, the stable region of cavity solitons is not reachable under such conditions. Nevertheless, the recoded temporal traces showed a periodic MI region that became chaotic when was close to 4. A clear evidence of cavity solitons was seen in a 2014 experiment in which cavity length was close to 110 m [89]. Two types of fibers were employed to control the average GVD of the cavity to values as low as −0.23 ps2 /km. As a result, the TOD played a significant role, and the spectrum showed the presence of a dispersive wave emitted by the perturbed cavity soliton. Even though the total spectral width of the frequency comb was less than 3 THz in this experiment, it contained millions of comb lines separated by about 1 MHz. The width of the cavity soliton was estimated to be 650 fs through numerical simulations. Further research has revealed interesting dynamics of cavity solitons and has confirmed that they differ from a regular pulse train created through modulation instability [90,91,93]. In a 2015 experiment, it was found that individual cavity solitons may ap-

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Figure 12.24 (A) Stability diagram showing intracavity peak power versus detuning for |S|2 = 30. Dashed and dotted lines indicate unstable solutions. Evolution of cavity solitons (numerical) for (B)

= 8, (C) = 7.45, and (D) = 6. (After Ref. [92]; ©2016 OSA.)

pear or disappear spontaneously from the train of pulses produced by a 100-m-long fiber cavity [90]. Because of a relatively long round-trip time (tR = 0.5 µs), it was possible to record temporal evolution over many round trips. The results show that cavity solitons behave independently and do not always form a perfectly periodic pulse train. In a later experiment, phase modulation of the pump beam itself was used to create or destroy individual cavity solitons [91]. Spatiotemporal instabilities of temporal cavity solitons were observed in a 2016 experiment [92] by focusing on the dashed part of the CS line in Fig. 12.23. In this experiment, pump power was large enough to realize |S|2 = 30. As the detuning parameter was varied from 6 to 8, different dynamic regions of cavity solitons were observed. In some regions, solitons decayed toward the CW state, while in others they produced a chaotic pattern with increasing ts . Fig. 12.24 shows the results of numerical simulations based on Eq. (12.3.2). In the range = 6–8, cavity solitons exhibit different dynamical features. In particular, they produce chaos when is close to 6. In the stable region (the solid part of the CS line in Fig. 12.24), the output is in the form of a pulse train with a frequency-comb like spectrum. However, as individual cavity solitons can be controlled externally, they can be useful for optical storage. This was first demonstrated in 2015 by showing that individual solitons could be written or

Novel nonlinear phenomena

Figure 12.25 Manipulation of a 3-bit pattern of cavity solitons through intensity modulation at different times indicated on the y axis. The crosses mark the time when addressing pulses are applied to create or erase a cavity soliton. (After Ref. [93]; ©2018 OSA.)

erased by just changing the phase of the external pump field [91]. In a 2018 experiment, cavity solitons were addressed with a single control pulse producing intensity modulation [93]. Fig. 12.25 shows changes in the 3-bit pattern of cavity solitons created through intensity modulation at specific times. These results clearly show the potential of cavity solitons for making all-optical buffers.

12.3.3 Frequency combs without a cavity A practical problem with the use of fiber cavities is that the frequency separation between the comb lines is relatively small (1%. It turns out that Eqs. (12.4.4) and (12.4.5) can be solved analytically even when the pump is allowed to deplete. The solution in the form of elliptic functions was first obtained in 1962 [128]. The periodic nature of the elliptic functions implies that the power flows back to the pump after the SHG power attains its maximum value. The analysis also predicts the existence of a parametric mixing instability induced by SPM and XPM [126]. The instability manifests as doubling of the spatial period associated with the frequency-conversion process. As pump depletion is negligible in most experimental situations, such effects are difficult to observe in optical fibers. The derivation of Eq. (12.4.8) assumes that the χ (2) grating is created coherently all along the fiber. This would be the case if the pump used during the preparation process were a CW beam of narrow spectral width. In practice, mode-locked pulses of duration ∼100 ps are used. The use of such short pulses affects grating formation in two ways. First, a group-velocity mismatch between the pump and second-harmonic pulses leads to their separation within a few walk-off lengths LW . If we use T0 ≈ 80 ps and |d12 | ≈ 80 ps/m in Eq. (1.2.14), the values appropriate for 1.06-µm experiments, LW ≈ 1 m. Thus, the χ (2) grating stops to form within a distance ∼1 m for pump pulses of 100 ps duration. Second, SPM-induced spectral broadening reduces the coherence length Lcoh over which the χ (2) grating can generate the second harmonic coherently. It turns out that Lcoh sets the ultimate limit because Lcoh < LW under typical experimental conditions. This can be seen by noting that each pump frequency creates its own grating with a slightly different period 2π/ kp , where kp is given by Eq. (12.4.2). Mathematically, Eq. (12.4.1) for Pdc should be integrated over the pump-spectral range to include the contribution of each grating. Assuming Gaussian spectra for both pump and second-harmonic waves, the effective dc polarization becomes [116] eff Pdc = Pdc exp[−(z/Lcoh )2 ],

Lcoh = 2/|d12 δωp |,

(12.4.9)

where d12 is defined in Eq. (1.2.13) and δωp is the spectral half-width (at 1/e point). For |d12 | = 80 ps/m and 10-GHz spectral width (FWHM), Lcoh ≈ 60 cm. In most experiments performed with 1.06-µm pump pulses, the spectral width at the fiber input is ∼10 GHz. However, SPM broadens the pump spectrum as the pump pulse travels down the fiber [see Eqs. (4.1.6) and (4.1.11)]. This broadening reduces the coherence length considerably, and Lcoh ∼ 10 cm is expected. Eq. (12.4.8) should be modified if Lcoh < L. In a simple approximation [109], L is replaced by Lcoh in eff Eq. (12.4.8). This amounts to assuming that Pdc = Pdc for z ≤ Lcoh and zero for z > Lcoh . One can improve over this approximation using Eq. (12.4.9). Its use requires that γSH in Eq. (12.4.7) be multiplied by the exponential factor exp[−(z/Lcoh )2 ]. If Lcoh  L, Eq. (12.4.7) can be integrated with the result [116] 2 P2 (κ) = (π/4)|γSH Pp Lcoh |2 exp(− 12 κ 2 Lcoh ).

(12.4.10)

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This expression is also approximate as it is based on Eqs. (12.4.4) and (12.4.5) that are valid only under quasi-CW conditions. For short pump pulses, the GVD effects can be included in Eqs. (12.4.4) and (12.4.5) by replacing the spatial derivatives by a sum of partial derivatives as indicated in Eq. (10.2.23).

12.5. Third-harmonic generation In contrast with the SHG, third harmonic of incident light should be created readily inside silica fibers, if the corresponding phase-matching condition is satisfied, because it originates from the same third-order nonlinear process that leads to FWM and other nonlinear effects. Indeed, third-harmonic generation (THG) has been observed in several experiments, although efficiency is typically low because phase matching is difficult to realize in standard single-mode fibers [163–169]. In a 1993 experiment, the doping level of germania was found to affect the THG process [165]. The presence of other dopants such as erbium or nitrogen also helps, and efficiencies as high as 2 × 10−4 were realized by 1997.

12.5.1 THG in highly nonlinear fibers It turns out that phase matching is much easier to accomplish in highly nonlinear fibers. In a 2000 experiment the THG light appeared in a higher-order mode of a 50-cm-long microstructured fiber when 1-kW pulses at 1064 nm were launched into it [150]. A PCF was used for THG in a 2001 study [170]. Since then, this nonlinear process has attracted considerable attention [171–179,181,182]. To understand why THG is easier to observe in microstructured fibers, consider first why it is difficult to realize phase matching in standard silica fibers. For a quasi-CW pump launched at the frequency ω, the phase-matching condition takes the form

β ≡ β(3ω) − 3β(ω) = (3ω/c )[¯n(3ω) − n¯ (ω)] = 0,

(12.5.1)

where β(ω) ≡ n¯ (ω)ω/c is the propagation constant and n¯ (ω) is the effective mode index. The values of n¯ at ω and 3ω differ enough for the fundamental mode that the THG process cannot be phase-matched in single-mode fibers. However, if the fiber supports multiple modes at the third-harmonic wavelength, n¯ (3ω) can match n¯ (ω) for a third harmonic propagating in a specific higher-order mode. However, this can only occur if the difference, n¯ (3ω) − n¯ (ω), is less than the core–cladding index difference, a quantity that rarely exceeds 0.01 for most standard fibers. As seen in Fig. 1.4, the refractive index of pure silica increases by more than 0.01 at the third-harmonic wavelength, falling in the range of 300 to 500 nm in most experiments. Since no bound modes can have its effective index less than that of the cladding, phase matching cannot be realized for fibers in which the core–cladding index difference is 0.01 or less. If this index difference is enhanced by increasing the doping level

Novel nonlinear phenomena

Figure 12.29 Measured optical spectra near (A) third harmonic and (B) pump wavelengths when 170-fs pulses were launched into a 95-cm-long fiber. (After Ref. [170]; ©2001 OSA.)

of germania within the core, THG can become phase-matched [165,178]. THG can also be realized through the Cherenkov-type phase matching in which third harmonic propagates in a leaky mode of the fiber [166]. However, a microstructured fiber with air holes in its cladding provides the best environment for THG because the effective core–cladding index difference can exceed 0.1, especially in fibers with large air holes. Indeed, it was observed easily in a 2000 experiment performed with a 50-cm-long microstructured fiber [150]. As expected, the third harmonic appeared in a higher-order mode of the fiber. In the case of ultrashort pulses, the THG process is affected by other nonlinear effects occurring simultaneously within the same fiber. Fig. 12.29(A) shows the spectrum of third harmonic at the output of a 95-cm-long PCF when 170-fs pulses at 1550 nm (with 45 mW average power at a 80-MHz repetition rate) were launched into it [170]. The 517-nm peak is expected as this wavelength corresponds to third harmonic of the 1550-nm pump. The origin of the second broad peak near 571 nm becomes clear from the output pump spectrum shown in Fig. 12.29(B), where the 1700-nm peak corresponds to the Raman soliton formed through the RIFS of the pump pulse. The 571-nm peak is generated by this Raman soliton. It is broader than the 517-nm peak because the spectrum of Raman soliton shifts continuously inside the fiber. The observed farfield pattern (with six bright lobes around a central dark spot) indicated clearly that the third harmonic was generated in a higher-order mode for which the phase-matching condition was satisfied. In general, the THG spectrum produced by femtosecond pump pulses is asymmetric and exhibits considerable structure [173]. The formation of Raman solitons with continuously shifting spectra helps considerably in generating third harmonic in a higher-order guided mode of the same fiber [175].

12.5.2 Effects of group-velocity mismatch In the case of short pump pulses with a broad spectrum, phase matching of the THG process is more complicated because the effects of group-velocity mismatch cannot

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be ignored. Physically speaking, the generated THG pulse width is comparable to the pump pulse, but the two move at different speeds through the fiber. The THG pulse can grow coherently only if it overlaps with the pump pulse. In fact, the spectrum of third harmonic may not be located exactly at 3ωp for a pump pulse whose spectrum is centered at ωp because of the effects of group-velocity mismatch. To understand this feature, we should expand β(ω) appearing in Eq. (12.5.1) in a Taylor series around ωp . A similar expansion should be carried out for β(3ω). If we retain terms only up to first order in this expansion, we find that [173]

β = β0 + β1 ,

(12.5.2)

where = 3(ω − ωp ) is the frequency shift of third harmonic and

β0 = β(3ωp ) − 3β(ωp ),

−1 −1

β1 = vgh − vgp .

(12.5.3)

The quantities vgj and β2j represent the group velocity and its dispersion at the frequency ωj with j = p for the pump and j = h for the third harmonic. Clearly, the condition β = 0 is satisfied when third harmonic is shifted from 3ωp by an amount = −( β0 / β1 ). This shift depends on the group-velocity mismatch and can be positive or negative depending on the fiber mode for which phase matching occurs. Eq. (12.5.2) does not include the contribution of SPM and XPM. This nonlinear contribution to the phase mismatch also affects the exact value of the frequency shift . It is possible for the THG spectrum to exhibit several distinct peaks, if the phasematching condition in Eq. (12.5.2) is satisfied for several different values of such that each peak corresponds to THG in a different fiber mode [173]. This feature was observed in a 2006 experiment in which 60-fs pulses at a wavelength of 1240 nm were launched into a 8-cm-long microstructured fiber with 4-µm core diameter [174]. Fig. 12.30 shows the spectrum of third harmonic at the fiber output for pulses with 0.5 nJ energy. The peak located near 420 nm is shifted from the third harmonic of 1240 nm at 413.3 nm by more than 6 nm. Moreover, several other peaks occur that are shifted by as much as 60 nm from this wavelength. A theoretical model predicts that three higher-order modes, EH14 , TE04 , and EH52 , provide phase matching for spectral peaks located near 370, 420, and 440 nm, respectively. The simple theory developed earlier for SHG can be extended for the case of THG, with only minor changes, when pump pulses are relatively broad. However, one must include the dispersive effects in the case of short pump pulses by changing dA/dz as indicated in Eq. (10.2.23). The resulting set of equations can be written as ∂ Ap 1 ∂ Ap iβ2p ∂ 2 Ap i ∗ + + = iγp (|Ap |2 + 2|Ah |2 )Ap + γTH Ah A∗p2 ei β0 z , (12.5.4) ∂z vgp ∂ t 2 ∂ t2 3 ∂ Ah 1 ∂ Ah iβ2h ∂ 2 Ah + + = iγh (|Ah |2 + 2|Ap |2 )Ah + iγTH A3p e−i β0 z , ∂z vgh ∂ t 2 ∂ t2

(12.5.5)

Novel nonlinear phenomena

Figure 12.30 Measured third-harmonic spectrum when 60-fs pulses at 1240 nm with 0.5 nJ energy were launched into a 8-cm-long fiber. (After Ref. [174]; ©2006 American Physical Society.)

where γj and β2j are the nonlinear and GVD parameters with j = p and h for the pump and its third harmonic, respectively. The THG growth is governed by γTH . The preceding set of two equations should be solved numerically in general. An analytic solution is possible if we neglect GVD, pump depletion, and the SPM and XPM terms. If we set T = t − z/vgp and work in a reference frame moving with the pump pulse, Eq. (12.5.5) is reduced to ∂ Ah ∂ Ah + β1 = iγTH A3p (T )e−i β0 z , ∂z ∂T

(12.5.6)

where Ap (T ) is the amplitude of pump pulse at the input end at z = 0. This equation can easily be solved in the frequency domain to obtain the following expression for the third-harmonic power [173]: |Ah (z, )|2 = |γTH |2    × 

sin2 [( β0 + β1 )z/2] ( β0 + β1 )2

∞ −∞

2 

Ap ( − 1 )Ap ( 1 − 2 )Ap ( 2 ) d 1 d 2  .

(12.5.7)

Clearly, the maximum growth of third harmonic occurs for a value = −( β0 / β1 ), the same value predicted by Eq. (12.5.2). The solution (12.5.7) is useful for calculating the spectrum of THG pulses for a given pump-pulse spectrum.

12.5.3 Effects of fiber birefringence As we have seen earlier in the context of FWM, the birefringence of an optical fiber affects the phase-matching condition. In essence, birefringence doubles the number of modes supported by a fiber by forcing the two orthogonal polarized components of

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Figure 12.31 Measured THG spectra when 170-fs pulses at 1550 nm are launched along the (A) slow or (B) fast axis of a 20-cm-long microstructured fiber. (After Ref. [171]; ©2003 OSA.)

each mode to have slightly different mode indices. As a result, the third-harmonic mode for which the phase-matching condition in Eq. (12.5.2) is satisfied may be different depending on whether the pump is polarized along the slow or fast axis. This behavior was observed in a 2003 experiment in which 170-fs pulses at 1550 nm were launched into a 20-cm-long fiber with a slightly elliptical core [171]. Fig. 12.31 shows the spectrum of third harmonic at the fiber output as a function of average input power when pump pulse propagates either along the slow or along the fast axis. The THG process creates a spectral band near 514 nm in both cases. However, another spectral band centered at 534 nm is also created when the average power exceeds 16 mW and the pump pulse is launched along the fast axis. At high enough input powers, only this second band appears. Of course, both spectral bands are present simultaneously when both polarization modes are excited by pump pulses. Similar to the case shown in Fig. 12.29, the 534-nm band is due to THG from the Raman soliton. A detailed study of the birefringence was carried out in 2003 using birefringent microstructured fibers with core diameters ranging from 1.5 to 4 µm [172]. An optical parametric oscillator, providing 100-fs pulses with a 82-MHz repetition rate at wavelengths tunable from 1450 to 1650 nm, was employed as a pump source, and the average input power was varied in the range of 5–80 mW. For a 6-cm-long fiber with 2.5 µm core diameter, the THG spectra exhibited peaks near 510 and 550 nm, but the peak wavelengths shifted by 5 nm or so when pump pulses were polarized along the two principal axes. Each peak appeared as a doublet whose relative heights varied with the average power launched into the fiber. As in Fig. 12.31, the longer-wavelength THG peak results from the Raman soliton that separates from the main pump pulse after a short distance into the fiber as a result of a large RIFS. The observed far-field patterns associated with the THG peaks were complex with eight lobes around a central dark spot and corresponded to a fiber mode whose order was relatively high (>20). A theo-

Novel nonlinear phenomena

retical model predicted the observed patterns quite well. For a fiber with 1.5-µm core diameter, the mode order for which THG was phase-matched was as high as 48. Several experiments have used tapered fibers for THG in which a narrow silica core is exposed to air that acts as a cladding. The large core-cladding difference makes it easy to satisfy the phase-matching condition for THG in a higher-order mode of such fibers. In a 2012 study, the THG efficiency was relatively small even for tapered fibers with a core diameter of less than 2.5 µm [179]. Much higher efficiencies were predicted by 2017 through the technique of quasi-phase matching that would allow THG in the fundamental mode of the fiber [181]. Non-silica fibers have also been used for THG [176]. In a 2015 experiment, a 2-cm-long tellurite microstructured fiber with a 4.5-µm core diameter was used [180]. Tunable THG was realized by launching 200 fs pump pulses whose wavelength was tunable from 1700 to 3000 nm, but the THG efficiency was relatively low (< 0.1%).

Problems 12.1 A short pulse with 100 W peak power propagates as a fundamental soliton inside a 10-m-long highly nonlinear fiber with γ = 100 W−1 /km and β2 = −10 ps2 /km. Calculate the pulse width (FWHM) and the dispersion and nonlinear lengths. 12.2 A fourth-order soliton with 150-fs width (FWHM) undergoes fission inside a 10-m-long fiber with γ = 100 W−1 /km and β2 = −10 ps2 /km. Calculate the widths, peak powers, and Raman-induced spectral shifts for the four fundamental solitons created. 12.3 What is meant by the Cherenkov radiation in the context of optical fibers? Under what conditions is this radiation emitted by a soliton? Derive an expression for the radiation frequency assuming that solitons are perturbed only by the third-order dispersion. 12.4 Use Eq. (12.1.11) and plot the frequencies of the two dispersive waves for a third-order soliton as a function of δ4 for δ3 = −0.01, 0, and 0.01. Estimate these frequencies when a 100-fs (FWHM) input pulse is launched and δ4 = 0.001 for the fiber. 12.5 Extend the MATLAB code given in Appendix B for solving Eq. (12.1.9) and reproduce the plots given in Figs. 12.1 and 12.4. 12.6 Extend the MATLAB code for the preceding problem to include the Raman term in Eq. (12.2.1) and reproduce the plot given in Fig. 12.6. 12.7 Explain why the spectrum of an ultrashort pulse propagating as a soliton shifts toward longer wavelengths. Do you expect this shift to occur if the pulse were to propagate in the normal-GVD region of the fiber? Discuss the logic behind your answer.

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12.8 Explain the FROG technique and sketch a typical experimental setup used for it. How would you extend this technique to make use of cross-correlation in place of autocorrelation? 12.9 Discuss why the Raman-induced frequency shift of a soliton is suppressed in a microstructured fiber exhibiting two zero-dispersion wavelengths when the soliton spectrum approaches the second wavelength. 12.10 Derive the phase-matching condition given in Eq. (14.14). Consider the terms up to fourth-order dispersion and calculate the frequency shift in THz when β2 = 5 ps2 /km, β4 = −1 × 10−4 ps4 /km, and LNL = 1 cm. 12.11 Discuss how a microstructured fiber can be used inside the cavity of a fiber-optic parametric oscillator to extend the tuning range of such lasers. What is the main factor that limits the tuning range when such a laser is pumped with short optical pulses? 12.12 Explain how thermal poling can be used to generate second harmonic inside in an optical fiber. Describe the physical mechanism involved in detail. 12.13 Why is it difficult to realize phase matching for generating third harmonic inside standard optical fibers with a core-cladding index difference of 0.01 or less? How is this problem solved in microstructured fibers?

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J.K. Ranka, R.S. Windeler, A.J. Stentz, Opt. Lett. 25 (2000) 796. L. Ji, P. Lu, W. Chen, et al., J. Opt. Soc. Am. B 25 (2008) 513. K. Suizu, K. Kawase, Opt. Lett. 32 (2007) 2990. V. Tombelaine, C. Lesvigne, P. Leproux, L. Grossard, V. Couderc, J.-L. Auguste, J.-M. Blondy, Opt. Express 13 (2005) 7399. S. Richard, J. Opt. Soc. Am. B 27 (2010) 1504. V. Couderc, A. Tonello, C. Buy-Lesvigne, P. Leproux, L. Grossard, Opt. Lett. 35 (2010) 145. M.A. Eftekhar, Z. Sanjabi-Eznaveh, J.E. Antonio-Lopez, F.W. Wise, D.N. Christodoulides, R. Amezcua-Correa, Opt. Lett. 42 (2017) 3478. V. Grubsky, J. Feinberg, Opt. Commun. 274 (2007) 447. G. Brambilla, F. Xu, P. Horak, et al., Adv. Opt. Photonics 1 (2009) 107. J. Lægsgaard, J. Opt. Soc. Am. B 27 (2010) 1317. F. Xu, Z.-X. Wu, Y.-Q. Lu, Prog. Quantum Electron. 55 (2017) 35. J.-M. Ménard, P.S.J. Russell, Opt. Lett. 40 (2015) 3679. J.-M. Ménard, F. Köttig, P.S.J. Russell, Opt. Lett. 41 (2016) 3795. J.M. Gabriagues, Opt. Lett. 8 (1983) 183. U. Österberg, Electron. Lett. 26 (1990) 103. D.L. Nicácio, E.A. Gouveia, N.M. Borges, A.S. Gouveia-Neto, Appl. Phys. Lett. 62 (1993) 2179. J. Thøgersen, J. Mark, Opt. Commun. 110 (1994) 435. J.M. Hickmann, E.A. Gouveia, A.S. Gouveia-Neto, D.C. Dini, S. Celaschi, Opt. Lett. 20 (1995) 1692. M.T. de Araujo, M.V.D. Vermelho, J.M. Hickmann, A.S. Gouveia-Neto, J. Appl. Phys. 80 (1996) 4196. I.A. Bufetov, M.V. Grekov, K.M. Golant, E.M. Dianov, R.R. Khrapko, Opt. Lett. 22 (1997) 1394. F.G. Omenetto, A.J. Taylor, M.D. Moores, J. Arriaga, J.C. Knight, W.J. Wadsworth, P.S.J. Russel, Opt. Lett. 26 (2001) 1158. F.G. Omenetto, A. Efimov, A.J. Taylor, J.C. Knight, W.J. Wadsworth, P.S.J. Russell, Opt. Express 11 (2003) 61. A. Efimov, A.J. Taylor, F.G. Omenetto, J.C. Knight, W.J. Wadsworth, P.S.J. Russell, Opt. Express 11 (2003) 910. A.M. Zheltikov, J. Opt. Soc. Am. B 22 (2005) 2263, Phys. Rev. A 72 (2005) 043812. A.A. Ivanov, D. Lorenc, I. Bugar, et al., Phys. Rev. E 73 (2006) 016610. E.E. Serebryannikov, A.B. Fedotov, A.M. Zheltikov, J. Opt. Soc. Am. B 23 (2006) 1975. G. Qin, M. Liao, C. Chaudhari, X. Yan, C. Kito, T. Suzuki, Y. Ohishi, Opt. Lett. 35 (2010) 58. K. Tarnowski, B. Kibler, C. Finot, W. Urbanczyk, IEEE J. Quantum Electron. 47 (2011) 622. K. Bencheikh, S. Richard, G. Mélin, G. Krabshuis, F. Gooijer, J.A. Levenson, Opt. Lett. 37 (2012) 289. T. Lee, Y. Jung, C.A. Codemard, M. Ding, N.G.R. Broderick, G. Brambilla, Opt. Express 20 (2012) 8503. T. Cheng, D. Deng, X. Xue, M. Matsumoto, H. Tezuka, T. Suzuki, Y. Ohishi, Appl. Opt. 54 (2015) 1326. X. Jiang, T. Lee, J. He, M.I.M.A. Khudus, G. Brambilla, Opt. Express 25 (2017) 22626. X. Jiang, D. Zhang, T. Lee, G. Brambilla, Opt. Lett. 43 (2018) 2728.

CHAPTER 13

Supercontinuum generation As we saw in Chapter 12, when short optical pulses propagate through a highly nonlinear fiber, their temporal and spectral features are affected not only by a multitude of the nonlinear effects but also by the dispersive properties of this fiber. As most nonlinear processes are capable of generating new frequencies within the pulse spectrum, for sufficiently intense pulses, the pulse spectrum can become so broad that it extends over a frequency range exceeding 100 THz. Such extreme spectral broadening is referred to as supercontinuum generation, a phenomenon first observed around 1970 in solid and gaseous nonlinear media [1–3]. In the context of optical fibers, a supercontinuum was first observed in 1976 using 10-ns pulses from a dye laser [4]. Although this topic attracted some attention during the decades of 1980s and 1990s, it was only after 2000, with the emergence of microstructured and photonic crystal fibers (PCFs), that the use of optical fibers for supercontinuum generation became common [5–7]. As physical mechanisms responsible for creating a supercontinuum depend on the pulse width, we discuss in the first two sections the cases of picosecond and femtosecond pulses separately. Section 13.3 focuses on the physical understanding of supercontinuum generation when femtosecond input pulses are used to excite a relatively high-order soliton. Section 13.4 is devoted to the situation of continuous-wave (CW) pumping. The focus of Section 13.5 is on the polarization effects, and the coherence properties of a supercontinuum are discussed in Section 13.6. The techniques used to extend the supercontinuum into the ultraviolet (UV) and mid-infrared (mid-IR) regions are described in Section 13.7. In the last section we discuss the optical analog of a rogue wave and its relation to a supercontinuum.

13.1. Pumping with picosecond pulses Supercontinuum generation in optical fibers was first observed in 1976 by launching Q-switched pulses (width ∼10 ns) from a dye laser into a 20-m-long fiber [4]. The output spectrum spread over 180 nm when the peak power of input pulses exceeded 1 kW. In a 1987 experiment, 25-ps pulses were launched into a 15-m-long fiber supporting four modes at the input wavelength of 532 nm [8]. The output spectrum extended over 50 nm because of the combined effects of SPM, XPM, SRS, and FWM (see Fig. 14.13). Similar features were expected to occur in the case a single-mode fiber [9]. Indeed, when a 1-km-long single-mode fiber was used in a 1987 experiment [10] to launch 830-fs pulses with 530-W peak power, a 200-nm-wide spectrum was observed Nonlinear Fiber Optics https://doi.org/10.1016/B978-0-12-817042-7.00020-8

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at the end of the fiber (see Fig. 12.2). Similar results were obtained later with longer pulses [11,12]. Starting in 1993, supercontinuum generation in single-mode fibers was used as a practical tool for generating pulse trains at multiple wavelengths by launching a single picosecond pulse train at a wavelength near 1.55 µm [13]. Such a device acts as a useful source for WDM systems. In a 1994 experiment [14], 6-ps pulses at a 6.3-GHz repetition rate with a peak power of 3.8 W were propagated in the anomalous-GVD region of a 4.9-km-long fiber (|β2 | < 0.1 ps2 /km). The generated supercontinuum was wide enough (>40 nm) that it could be used to produce 40 WDM channels by filtering fiber’s output with an optical filter exhibiting periodic transmission peaks. The resulting 6.3-GHz pulse trains at different wavelengths contained nearly chirp-free pulses whose widths varied in the range of 5 to 12 ps. By 1995, this technique produced a 200-nmwide supercontinuum, resulting in a 200-channel WDM source [15]. The same technique was used to produce even shorter pulses by enlarging the bandwidth of each transmission peak of an optical filter. In fact, pulse widths could be tuned in the range of 0.37 to 11.3 ps when an array-waveguide grating with a variable bandwidth was used [16]. A supercontinuum source was used in 1997 to demonstrate data transmission at a bit rate of 1.4 Tb/s using seven WDM channels with 600-GHz spacing [17]. In this experiment, time-division multiplexing was used to operate each WDM channel at 200 Gb/s. Individual 10-Gb/s channels were demultiplexed from the 200-Gb/s bit stream through FWM inside a separate fiber designed for this purpose.

13.1.1 Nonlinear mechanisms One should ask what nonlinear mechanisms are responsible for broadening the spectrum of a picosecond pulse. As we saw in Section 4.1.3, SPM can produce considerable spectral broadening at a fiber’s output. The spectral broadening factor in Eq. (4.1.11) is related to the maximum SPM-induced phase shift, φmax = γ P0 Leff , where P0 is the peak power of the pulse and Leff is the effective fiber length. For typical values of the experimental parameters, this factor is ∼10. As the input spectral width is ∼1 nm for picosecond pulses, SPM alone cannot produce a supercontinuum extending over 100 nm or more. Another nonlinear mechanism that can generate new wavelengths is SRS. If the input peak power P0 is large enough, SRS creates a Stokes band on the long-wavelength side, shifted by about 13 THz from the center of the pulse spectrum. Even if the peak power is not large enough to reach the Raman threshold, the Raman gain can amplify the pulse spectrum on the long-wavelength side as soon as SPM broadens it by 5 nm or more. Clearly, SRS would affect any supercontinuum by enhancing it selectively on the long-wavelength side, and thus making it asymmetric. However, SRS cannot generate any frequency components on the short-wavelength side.

Supercontinuum generation

FWM is the nonlinear process that can create sidebands on both sides of the pulse spectrum, provided a phase-matching condition is satisfied. Indeed, FWM is often behind a supercontinuum generated using optical fibers. FWM is also the reason why dispersive properties of the fiber play a critical role in the formation of a supercontinuum. Because of a large spectral bandwidth associated with any supercontinuum, the GVD parameter β2 cannot be treated as being constant over the entire bandwidth, and its wavelength dependence should be included in any theoretical modeling through the higher-order dispersion parameters. Moreover, the process of supercontinuum generation may be improved, if GVD is allowed to change along the fiber’s length. As early as 1997, numerical simulations showed that the uniformity or flatness of the supercontinuum improved considerably if β2 increased along the fiber such that an optical pulse experienced anomalous GVD near the front end of the fiber and normal GVD close to the output end [18]. By 1998, a 280-nm-wide supercontinuum could be generated by using a special kind of fiber in which dispersion not only varied along the fiber’s length but was also remained relatively flat over a 200-nm bandwidth in the wavelength region near 1.55 µm [19]. Dispersion flattening turned out to be quite important for supercontinuum generation. In a 1998 experiment, a supercontinuum extending over a 325-nm bandwidth (at 20-dB points) could be generated when 3.8-ps pulses were propagated in the normal-GVD region of a dispersion-flattened fiber [20]. At first sight, it appears surprising that supercontinuum can be produced in a fiber exhibiting normal GVD (β2 > 0) along its entire length. However, recall from Section 10.3 that, even though β3 is nearly zero for a dispersion-flattened fiber, the fourth-order dispersion governed by β4 plays a critical role. As seen in Fig. 10.5, FWM can be phase-matched even when β2 > 0 provided β4 < 0, and it generates signal and idler bands that are shifted by >10 THz. It is this FWM process that can produce a wideband supercontinuum in the normal-GVD region of an optical fiber. The 280-nm-wide supercontinuum was generated when 0.5-ps-wide pulses (obtained from a mode-locked fiber laser) were propagated with a small positive value of β2 . Fig. 13.1 shows the output spectra at several power levels when 2.2-ps-wide pulses were launched into a 1.7-kmlong fiber with β2 = 0.1 ps2 /km at 1569 nm [21]. Input spectrum is also shown as a dashed curve for comparison. The nearly symmetric nature of the spectra indicates that the Raman gain played a relatively minor role in this experiment. The combination of SPM, XPM, and FWM is responsible for most of spectral broadening seen in Fig. 13.1. In a 1999 experiment such a supercontinuum was used to produce 10-GHz pulse trains at 20 different wavelengths, with nearly the same pulse width in each channel [22]. For applications as a multichannel WDM source, the supercontinuum does not have to extend over a very wide range. What is really important is the flatness of the supercontinuum over its entire bandwidth. A dispersion-flattened fiber was used in a 2001 experiment to produce 150 channels spaced apart by 25 GHz [23]. The total bandwidth of all channels occupied only a 30-nm bandwidth. Of course, the number of channels

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Figure 13.1 Supercontinuum spectra measured at average power levels of (A) 45 mW, (B) 140 mW, and (C) 210 mW. Dashed curve shows the spectrum of input pulses. (After Ref. [21]; ©1998 IEEE.)

can be increased to beyond 1000 if the supercontinuum bandwidth were to increase beyond 200 nm. Indeed, it was possible to create a WDM source with 4200 channels, spaced apart by only 5 GHz, when the bandwidth was close to 200 nm [24]. By 2003, such a WDM source provided 50-GHz-spaced channels over the spectral range extending from 1425 to 1675 nm, thus covering the three main telecommunication bands [25].

13.1.2 Experimental progress after 2000 Until the year 2000, most experiments employed long fibers (length ∼1 km) for generating a supercontinuum. However, shorter fibers with lengths 1700 nm with a spectral density of more than 1 mW/nm. In a 2006 experiment, the PCF was tapered such that its ZDWL decreased continuously along its length [30]. Fig. 13.4(A) shows the measured spectra for several taper lengths when the PCF was pumped with 600-ps pulses from a Nd:YAG microchip laser with peak powers of about 15 kW. Fig. 13.4(B) shows similar spectra when 4-ps pulses from a Yb-doped fiber laser with 50 kW peak power were sent through several tapers of different lengths. In both cases a high-brightness supercontinuum is obtained for PCF

Supercontinuum generation

Figure 13.5 Supercontinuum generated in a 75-cm-long microstructured fiber when 100-fs pules with 0.8 nJ energy are launched close to the ZDWL near 767 nm. The dashed curve shows for comparison the spectrum of input pulses. (After Ref. [33]; ©2000 OSA.)

lengths of 1 m or more. A 2-m-long PCF was used in a 2010 experiment to obtain an ultrawide supercontinuum (expending from 0.4 to 2.3 µm) with a 39 W output power and spectral density of more than 30 mW/nm [32].

13.2. Pumping with femtosecond pulses The use of femtosecond pulses became practical with the advent of narrow-core microstructured fibers whose ZDWL was close to 800 nm, the wavelength region around which tunable Ti:sapphire lasers operate. Starting in 2000, femtosecond pulses from such mode-locked lasers were used for supercontinuum generation. In a 2000 experiment, 100-fs pulses at 770 nm were launched into a 75-cm-long section of a microstructured fiber exhibiting anomalous GVD near this wavelength [33]. Fig. 13.5 shows the spectrum observed at the fiber’s output when the peak power of input pulses was about 7 kW. Even for such a short fiber, not only the supercontinuum was extremely broad, extending from 400 to 1600 nm, it was also relatively flat (on a logarithmic power scale) over the entire bandwidth. Similar features were observed in another 2000 experiment in which a 9-cm-long tapered fiber with 2-µm core diameter was employed [34]. Fig. 13.6 shows the output spectra of 100-fs pulses observed at average power levels ranging from 60 to 380 mW. It is important to emphasize that the optical power in Figs. 13.1 to 13.6 is plotted on a decibel scale; the spectra would appear much more nonuniform on a linear scale. How should one characterize the spectral width of such a supercontinuum because the concept of a FWHM cannot be applied? A common approach is to employ the 20-dB spectral width, defined as the frequency or wavelength range over which the power level exceeds 1% of the peak value. The 20-dB spectral width exceeds 1000 nm in Fig. 13.5 for the supercontinuum extending over two octaves.

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Figure 13.6 Optical spectra at the output of a 9-cm-long tapered fiber with a 2-μm waist. The average power was 380, 210, and 60 mW (from top to bottom) for the three curves. The input spectrum is also shown for comparison. (After Ref. [34] ©2000 OSA.)

It is evident from the superbroad and nearly symmetric nature of the spectra in Figs. 13.5 and 13.6 that a different physical mechanism is at play in the case of femtosecond pulses. For the high peak power levels and anomalous GVD used in the experiments, input pulses can form a relatively high-order soliton inside the fiber. The order of a soliton is governed by the parameter N, defined in Eq. (12.1.1) as N 2 = LD /LNL , where LD = T02 /|β2 | is the dispersion length and LNL = 1/(γ P0 ) is the nonlinear length. As discussed in Sections 12.1 and 12.2, high-order solitons are perturbed considerably by third-order dispersion and intrapulse Raman scattering and they breakup into multiple, much narrower, fundamental solitons through soliton fission. It turns out that if N is relatively large (more than 10), the phenomenon of soliton fission can produce a supercontinuum [35] similar to that seen in Figs. 13.5 and 13.6. We discuss such details in the following section and focus here on the experimental results. The 2000 observation of supercontinua generated using short PCFs led to an explosion of research activity during 2002 in the nascent field [36–46]. The choice of the pump wavelength relative to the ZDWL of the fiber was found to be a critical factor for supercontinuum generation [42]. When the input wavelength fell in the normal-GVD region of the fiber, spectral broadening was reduced considerably because higher-order solitons could not form in this situation. Input wavelengths longer than 1000 nm were used in several experiments. A Yb-doped fiber laser producing parabolic-shape pulses was used in one 2002 experiment [43]. In another 2002 experiment [44], femtosecond pulses at 1260 nm with 750 pJ of energy were launched close to the second ZDWL of a tapered fiber. The resulting supercontinuum extended from 1000 to 1700 nm. The initial width of input pulses also plays an important role in the supercontinuum formation. Pulses shorter than 20 fs were launched in a 2002 experiment inside cobweb-type PCFs with core diameters ranging from 1 to 4 µm [45]. Since the PCF core was not perfectly circular, the fiber exhibited some birefringence. The spectral widths of observed supercontinua varied considerably with the polarization angle of incident light with respect to the slow axis of the fiber. The output spectra were also found to depend on the extent of frequency chirp on input pulses. The input phase for

Supercontinuum generation

Figure 13.7 Observed spectra for several input chirp levels at the output of a 4.1-cm-long PCF with a 2.5-μm core. The average input power was 400 mW. (After Ref. [45] ©2002 OSA.)

a linearly chirped pulse varies with time as φ(t) = φ0 + ct2 . Fig. 13.7 shows the output spectra of 18-fs pulses observed at the end of a 4.1-cm-long fiber for several values of c. The supercontinuum is the widest for unchirped pulses (c = 0) with a distinct peak near 500 nm. In another experiment, 200-fs pulses were launched into a 6-m-long PCF (its ZDWL at 806 nm) at a wavelength of 850 nm [46]. The observed spectra exhibited multiple distinct peaks on the long-wavelength side that corresponded to Raman solitons formed after the fission of a high-order soliton. Pulses shorter than 10 fs pulses were used in a 2003 experiment [47]. Most of these experiments made use of a bulky Ti:sapphire laser as a source of femtosecond pulses. An all-fiber version was realized in 2003 by employing 200-fs pulses from a mode-locked erbium fiber laser [48]. This experiment also used a relatively long silica fiber (up to 600 m length) with a γ of only 10 W−1 /km. The observed supercontinuum was octave spanning as it extended from 1100 to 2200 nm. A much flatter supercontinuum was realized in 2004 by employing a hybrid fiber made of three sections with different dispersive and nonlinear properties [49]. First section (a 17-cmlong polarization-maintaining fiber) was followed with a 4.5-cm-long second section of dispersion-shifted fiber that was sliced to a 1-m-long third section of a fiber exhibiting normal GVD at the input wavelength. In this experiment, 100-fs pulses at 1560 nm (from a mode-locked erbium fiber laser) were employed. Fig. 13.8 shows the output spectra at the end of each fiber section together with the input spectrum. The final spectrum extends from 1200 to 2100 nm. A single 5-m-long highly nonlinear fiber

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Figure 13.8 Input spectrum (A) and observed spectra at the end of (B) first, (C) second, and (D) third fiber section. The peak power of 100-fs input pulses was 3.7 kW. Corresponding numerical simulated spectra are also shown. (After Ref. [49] ©2004 OSA.)

was used in a 2005 experiment to produce a supercontinuum extending from 1000 to 2500 nm [50]. Starting in 2003, nonsilica fibers were used for supercontinuum generation, and it was discovered that they can produce spectra even wider than those generated by silica fibers [51–61]. A 30-cm-long fluoride fiber was pumped in the 2003 experiment with 60-fs pulses at 1560 nm with 210 pJ energy. The resulting supercontinuum extended from 350 nm to >1750 nm [52]; spectral measurements were limited to 1750 nm because of the spectrum analyzer employed used in the experiment. This limit was overcome by 2006 using an infrared HgCdTe detector that could go up to 3000 nm, and it was found that the spectrum extended beyond this limit even for a fiber that was only 0.57 cm long [54]. In this experiment, 110-fs pulses at a wavelength of 1550 nm were launched into a 2.6-µm-core PCF made of lead-silicate glass (Schott SF6), for which n2 is 10 times larger than that of silica [54]. The experiment was repeated for average launched powers in the range of 1 to 70 mW. Fig. 13.9 shows the recorded spectra over this entire range of input powers as a surface plot. The fiber length is only 0.57 cm in the image (A) and increases to 70 cm for the image (B). In the case of a long fiber, one can clearly see the formation of multiple Raman solitons with different RIFS after the fission process that occurs for Pav > 1 mW; the number of Raman solitons at the output end, of course, depends on the launched power. Fig. 13.9(B) shows that the wavelength of the shortest soliton exceeds 1800 nm for Pav > 30 mW. This wavelength reached 3000 nm for Pav > 70 mW, as verified using a HgCdTe detector.

Supercontinuum generation

Figure 13.9 Recorded spectra as a function of average power at the output of a microstructured fiber when 110-fs pulses are propagated over a fiber length of (A) 0.57 and (B) 70 cm. Arrows indicate solitons created after the fission. (After Ref. [54]; ©2006 OSA.)

The optical spectra in Fig. 13.9(A) were observed for a lead-silicate PCF that was only 0.57 cm long. The fiber length in this case is so much shorter than the dispersion length (about 40 cm) that the soliton fission does not occur. A nearly symmetric nature of the optical spectra indicates that the supercontinuum extending from 350 to 1800 nm is formed mainly through the SPM process. Such a wide spectrum forms because of a relatively large value of γ for the non-silica fiber used. As discussed later, a supercontinuum generated through SPM is smoother and much more spectrally coherent, compared with the one formed through soliton fission. Tellurite-glass-based PCFs have also been used for supercontinuum generation. In a 2008 experiment [56], 100-fs pulses with 1.9 nJ energy at a 1550-nm wavelength were transmitted through a 0.8-cm-long PCF. The output spectrum extended from 790 to 4870 nm when measured at the points 20 dB below the peak spectral power. In another 2008 experiment [57], a 9-cm-long microstructured tellurite fiber with a large effective mode area of 3000 µm2 was pumped with 120-fs pulses. Even in this case the observed supercontinuum extended from 900 to 2500 nm. A 6-cm-long hexagonal-core fiber was pumped in a 2009 experiment using 400-fs pulses at 1557 nm [61]. At pulse energies close to 0.6 nJ, the observed supercontinuum extended from visible region to beyond 2.4 µm. Chalcogenide fibers have attracted attention because of their low losses in the infrared region and a relatively large n2 parameter (400 times larger than silica). Moreover, their effective mode area can be reduced to below 1 µm2 by reducing the core diameter. As a result, the nonlinear parameter γ was close to 100 W−1 /m (80,000 times larger than that of silica fibers) in a 2008 experiment [58]. The 3-cm-long fiber was pumped at 1550 nm using 250-fs pulses. Fig. 13.10 shows the output spectra observed for several values of peak powers. Notice that these power levels were much smaller than those used in Fig. 13.38 for a fluoride fiber. Nevertheless, the observed spectrum

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Figure 13.10 Supercontinuum observed at the output of a 3-cm-long chalcogenide fiber for peak powers of 250-fs pulses. The effective mode area of this fiber was only 0.5 μm2 because of a nanoscale core. (After Ref. [58] ©2008 OSA.)

extended from 1150 to 1650 nm, a range large enough for practical applications. The shape of observed spectra was consistent with the soliton-fission scenario. Starting in 2010, chalcogenide fibers were used to extend the supercontinuum into the mid-IR region (see Section 13.7).

13.3. Temporal and spectral evolution of pulses In this section we consider in more detail the temporal and spectral evolution of femtosecond pulses and focus on the physical phenomena that create the supercontinuum at the fiber output. A numerical approach is necessary for this purpose. We present first the model based on the generalized NLS equation and then discuss the role of soliton fission, dispersive waves, and Raman solitons. As we saw in Section 12.1, fission of a Nth-order soliton produces N fundamental solitons whose widths and peak powers follow Eq. (12.1.2). All of these solitons are shorter than the original input pulse, the shortest one being narrower by a factor of 2N − 1. For 100-fs or shorter input pulses propagating with N > 5, the shortest soliton after fission would have a width below 10 fs and its spectrum would be relatively wide (>40 THz). As discussed in Section 12.2, such short pulses are affected considerably by intrapulse Raman scattering and their spectra shift rapidly toward longer and longer wavelengths as they propagate inside the fiber. For a large value of N, many new spectral peaks corresponding to multiple solitons of different widths would form on the long-wavelength side of the original pulse spectrum. At the same time, solitons created through the fission process and perturbed by third- and higher-order dispersion, would emit nonsolitonic radiation (NSR) in the form of dispersive waves, whose wavelengths fall on the short-wavelength side of the input spectrum for β3 > 0. The only remaining

Supercontinuum generation

question is why various peaks merge together to form a broadband supercontinuum. As we see later in this section, the nonlinear phenomena of XPM and FWM provide an answer to this question.

13.3.1 Numerical modeling of supercontinuum The supercontinuum generation process can be studied by solving the generalized NLS equation of Section 2.3.2. As it is important to include the dispersive effects and intrapulse Raman scattering as accurately as possible, one should employ Eq. (2.3.36) for modeling supercontinuum generation. Because of its importance, we rewrite it here as   M  ∂A 1 ∂ βm ∂ m A + A+ im−1 α0 + iα1 ∂z 2 ∂t m! ∂ tm m=2     ∞ ∂   2  = i γ 0 + i γ1 A(z, t) R(t )|A(z, t − t )| dt , ∂t 0

(13.3.1)

where α0 and γ0 are the loss and nonlinear parameters at the central frequency ω0 of the input pulse and M represents the order up to which dispersive effects are included. The m = 1 term has been removed from Eq. (13.3.1), assuming that t represents time in a reference frame moving at the group velocity of the input pulse. The α1 terms should be included if fiber losses vary considerably over the bandwidth of a supercontinuum. Both loss terms are often negligible for short fibers. The approximation γ1 ≈ γ0 /ω0 is often employed in practice [see Eq. (2.3.37)] but its use is not always justified [62]. Eq. (13.3.1) has proven quite successful in modeling most features of supercontinua observed experimentally by launching ultrashort optical pulses into highly nonlinear fibers [38–42]. The split-step Fourier method discussed in Section 2.4.1 is used often in practice for solving this equation. The choice of M in the sum appearing in Eq. (13.3.1) is not always obvious. Typically, M = 6 is used for numerical simulations, although values of M as large as 12 are sometimes employed. In fact, dispersion to all orders can be included numerically if we recall that the dispersive step in the split-step Fourier method is carried out in the spectral domain of the pulse, after neglecting all nonlinear terms. In the Fourier domain, all dispersion terms in Eq. (13.3.1) can be replaced with ˜ where −iD(ω)A, D(ω) = β(ω) − β(ω0 ) − β1 (ω0 )(ω − ω0 ),

(13.3.2)

˜ (z, ω) is the Fourier transform of A(z, t). This approach requires a knowledge and A of β(ω) over the whole frequency range covered by a supercontinuum. As discussed in Section 11.3, in the case of a fiber whose core and cladding have constant refractive indices, one can solve the eigenvalue equation (2.2.8) numerically to obtain the propagation constant β(ω). This approach is not appropriate for a microstructured fiber

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designed with air holes within its cladding. As mentioned in Section 11.4, several other numerical techniques can provide β(ω) for such fibers. As a relevant example of supercontinuum generation in highly nonlinear fibers, a fiber with a 2.5-µm-diameter silica core, surrounded by air acting as a cladding, was used in one numerical study [38]. Frequency dependence of the propagation constant β(ω) was found by solving Eq. (2.2.8) numerically, and dispersion parameters were calculated from it through a Taylor-series expansion. The ZDWL for this fiber is near 800 nm. If we assume that 150-fs pulses from a laser operating at 850 nm are used for supercontinuum generation, dispersion parameters at this wavelength are: β2 = −12.76 ps2 /km, β3 = 8.119 × 10−2 ps3 /km, β4 = −1.321 × 10−4 ps4 /km, β5 = 3.032 × 10−7 ps5 /km, β6 = −4.196 × 10−10 ps6 /km, and β7 = 2.570 × 10−13 ps7 /km. The higherorder terms have a negligible influence on the following results. The nonlinear effects are included through the terms appearing on the right side of Eq. (13.3.1). The nonlinear parameter γ0 for the 2.5-µm-waist fiber can be estimated from Fig. 11.9 and is found to be about 100 W−1 /km. The nonlinear response function R(t) has the form R(t) = (1 − fR )δ(t) + fR hR (t) given in Eq. (2.3.33) with fR = 0.18. The Raman response function hR (t) is sometimes approximated by Eq. (2.3.39). However, the numerical accuracy is improved if the oscillatory trace seen in Fig. 2.2 is employed because it takes into account the actual shape of the Raman gain spectrum. Any implementation of the split-step Fourier method for solving Eq. (13.3.1) requires a careful consideration of the step size z along the fiber length and the time resolution t used for the temporal window. As discussed in Section 2.4, the size of the spectral window is set by ( t)−1 . Since this window must be wide enough to capture the entire supercontinuum, t should be 2 fs or less if the supercontinuum extends over 300 THz. At the same time, the temporal window should be wide enough that all parts of the pulse remain confined to it over the entire fiber length. Typically, the temporal window should extend over 20 ps to allow for the Raman-induced changes in the group velocity of the pulse. For this reason, it is often necessary to employ 10,000 points or more for the FFT algorithm. The step size z should be a small fraction of the shortest nonlinear length. It can become 1000 steps/cm along the fiber length. Clearly, numerical simulations for modeling supercontinuum formation in optical fibers can be time-consuming. However, continuing advances in the computing technology have made it possible to perform such simulations on a personal computer. In an alternative approach, Eq. (13.3.1) is solved in the frequency domain. By taking the Fourier transform of A(t), this equation can be written in the form [6]    ∞ ˜ ∂A 1 ˜ = iD(ω)A ˜ + iγ (ω)FT A(z, t) + α(ω)A R(t )|A(z, t − t )|2 dt , (13.3.3) ∂z 2 0

Supercontinuum generation

Figure 13.11 (A) Spectral and (B) temporal evolution of a 150-fs pulse launched with 10-kW peak power into a 10-cm-long tapered fiber with 2.5-μm waist. (After Ref. [38]; ©2002 IEEE.)

where FT represents the Fourier-transform operation. The resulting set of ordinary differential equations can be solved with any finite-difference scheme. In particular, we can use the RK4IP scheme discussed in Section 2.4.2 because Eq. (13.3.3) is in the form of Eq. (2.4.14). This scheme allows for an adaptive step size in the z direction, provides control over numerical accuracy, and can be faster than the split-step Fourier method in some cases. It also allows us to include the full frequency dependence of the loss, dispersion, and nonlinear parameters. Note that the convolution in Eq. (13.3.3) becomes a product in the frequency domain. Fig. 13.11 shows the (A) spectral and (B) temporal evolution of a 150-fs (FWHM) “sech” pulse launched into a 10-cm-long tapered fiber with 10-kW peak power. If we use T0 = 85 fs and P0 = 10 kW for the pulse parameters, the dispersion and nonlinear lengths are found to be about 57 cm and 1 mm, respectively, and the soliton order N is about 24 under such conditions. As seen in Fig. 13.11(B), the pulse undergoes an initial compression phase, a common feature of any high-order soliton, while its spectrum broadens because of SPM. However, after 2 cm, the soliton undergoes the fission process, and the pulse breaks up into multiple fundamental solitons [35]. The spectrum changes qualitatively after that and acquires features at a distance of z = 4 cm that are typical of a supercontinuum. A clue to the sudden spectral change is provided by the pulse shape at z = 4 cm. It exhibits a narrow spike that has separated from the main pulse. As discussed in Section 12.2, this separation is due to a change in the group velocity of the shortest fundamental soliton, occurring when spectrum of this soliton shifts toward longer wavelengths because of intrapulse Raman scattering. As the spectrum of a soliton shifts,

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its group velocity is reduced (because of GVD), and it separates from the main pulse. Other solitons also experience such a Raman-induced frequency shift (RIFS), but the magnitude is smaller for them because of their larger widths and smaller peak powers. They also eventually separate from the main pulse, as seen in Fig. 13.11(B) at a distance z = 10 cm. Thus, the spectral broadening beyond 900 nm is due to a combination of SPM and RIFS. In the supercontinuum seen in Fig. 13.11, the spectral components are also generated on the short-wavelength (or anti-Stokes) side of the input wavelength. As discussed in Section 12.1, Cherenkov radiation or NSR is responsible for them [63]. Because of the third- and higher-order dispersive effects, each Raman soliton loses some of its energy in the form of a dispersive wave. The wavelength of this NSR is governed by the phasematching condition given in Eq. (12.1.5). For a fiber with β3 > 0, the phase-matching condition is satisfied for a wavelength that lies below 800 nm. Since this wavelength is different for different solitons, multiple spectral peaks form in Fig. 13.11(A) in the visible region.

13.3.2 Role of cross-phase modulation Numerical results in Fig. 13.11 show that multiple spectral peaks merge together to form a broadband supercontinuum. It turns out that the nonlinear phenomena of XPM and FWM are responsible for this merging. To understand their role, we need to realize that, even though the Raman solitons and dispersive waves have widely separated optical spectra, they may still overlap in the time domain. Any temporal overlapping of a Raman soliton with a dispersive wave results in their mutual interaction through XPM. As we have seen in Section 7.4, such a nonlinear interaction can create new spectral components and broaden the spectrum of dispersive waves in an asymmetric fashion. For this reason, the XPM effects play an important role during supercontinuum formation inside highly nonlinear fibers [67–74]. A simple way to visualize temporal overlapping of Raman solitons and dispersive waves is to produce a spectrogram through the X-FROG technique discussed in Section 12.2.2. Fig. 13.12 shows the temporal and spectral evolutions of a N = 6 soliton over two dispersion lengths together with the spectrogram at z = 2LD . A normalized form of Eq. (13.3.1), given in Eq. (12.2.4), was used for these numerical results. Moreover, only the third-order dispersion term was retained so that a single parameter (δ3 = 0.05) was required. The parabolic dashed curve in part (C) shows changes as a function of frequency in the derivative dβ/dω, which is inversely related to the group velocity. Fig. 13.12 provides considerable physical insight into the process of supercontinuum generation inside optical fibers. Parts (A) and (B) show clearly how two Raman solitons separate from the main pulse as their spectra shift toward the Stokes side. They also show dispersive waves that spread in time domain but their spectrum lies on the anti-Stokes

Supercontinuum generation

Figure 13.12 (A) Spectral and (B) temporal evolutions of a pulse launched as a sixth-order soliton over a distance 0 to 2LD . (C) Corresponding spectrogram at z = 2LD . The parabolic dashed curve shows changes in group velocity with frequency.

side. The new feature seen in the spectrogram in part (C) is that the two Raman solitons (dark spots on the left side) overlap temporally with their dispersive waves and thus interact with them through XPM and FWM. Indeed, one can notice broadening of the spectrum associated with the dispersive wave that overlaps with the shortest Raman soliton, the one whose spectrum shifts the most and is thus delayed the most in the time domain because of a continuing reduction in its speed. The XPM-induced broadening of spectral peaks within a supercontinuum was clearly observed in a 2002 experiment [42], where the peak power of input pulses was varied over a wide range to change the soliton order associated with the input pulse from N < 1 to N > 6. Fig. 13.13 shows the spectra observed at the output of a 5-m-long microstructured fiber when 100-fs pulses were launched (at 804 nm) and the average launched power was varied from 0.05 to 55 mW. The fiber had its ZDWL at 650 nm such that β2 = −57.5 ps2 /km at 804 nm, resulting in a dispersion length of 6 cm or so. At power levels of 1 mW or less, the output spectrum exhibits some SPM-induced broadening, but it neither shifts spectrally nor produces any NSR, as the nonlinear effects play a relatively minor role. By the time the average power is >4 mW, the soliton order N exceeds 1, and one or more Raman solitons are created through the fission process, and distinct spectral peaks appear on the Stokes side whose number depends on the launched power; three of them are present when Pav = 16 mW. At the same time, each Raman soliton produces an NSR peak in the visible region that corresponds to a different dispersive wave that is phase-matched to that specific soliton. A supercontinuum begins to form for Pav > 40 mW, a power level at which N exceeds 5 for the input pulse, as various peaks merge together because of XPM-induced coupling among solitons and dispersive waves. Numerical simulations confirm that NSR peaks

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Figure 13.13 (A) Recorded spectra at the output of a 5-m-long fiber for 8 average power levels of 100-fs input pulses. (After Ref. [42]; ©2002 OSA.)

in the visible region are not generated if third- and higher-order dispersive effects are not included in the calculation. The X-FROG technique has also been used in several experiments on supercontinuum generation [64–67]. In one approach, cross-correlation of a spectrally filtered output pulse with a reference pulse (with an adjustable delay) is recorded as a function of filter wavelength through two-photon absorption inside a silicon avalanche photodiode [66]. Mathematically, the intensity cross-correlation is given by 

S(τ, ωc ) =

∞ −∞

If (t, ωc )Iref (t − τ ) dt,

(13.3.4)

where Iref (t − τ ) is the intensity of a reference pulse delayed by τ and If (t, ωc ) is the intensity of the fiber’s output, filtered using an optical filter centered at the frequency ωc . Fig. 13.14 shows an example of such a spectrogram, created by launching a 100-fs pulse into a 2-m-long dispersion-shifted fiber with 2.5-kW peak power [66]. The fiber’s parameters at the input wavelength of 1.56 µm were β2 = −0.25 ps2 /km and γ = 21 W−1 /km. The right panel of Fig. 13.14 shows the measured spectrum and the temporal profile of the output pulse deduced from the X-FROG measurements. The group delay as a function of wavelength and the chirp across the pulse can also be deduced from the FROG data. As a check on the accuracy of the reconstruction process, spectrogram retrieved from the measured spectral data and the deduced amplitude and phase profiles (using an iterative phase-retrieval algorithm) is also shown for comparison. Several features are noteworthy in the spectrogram displayed in Fig. 13.14. Two solitons with their spectra shifted through RIFS toward the Stokes side are clearly visible.

Supercontinuum generation

Figure 13.14 Left: Experimental (A) and reconstructed (B) spectrograms at the end of a 2-m-long fiber for 100-fs input pulses with 2.5 kW peak power. Right: (A) Measured spectrum and calculated group delay; (B) reconstructed intensity and chirp profiles. (After Ref. [66]; ©2003 OSA.)

The spectral components on the anti-Stokes side spread much more temporally because dispersive waves associated with them propagate in the normal-GVD region of the fiber. They represent relatively low-power pulses that are broadened because of GVD. In fact, the nearly parabolic shape of the spectrogram results from a quadratic phase of the form 12 β2 ω2 z that is added by GVD during propagation inside the fiber. Note also that blue-shifted dispersive waves travel slower in the normal-GVD region, compared with the central part of the pulse. As a result, they slow down and begin to overlap with the red-shifted solitons in the time domain. Once this happens, the two interact nonlinearly through XPM and their spectra begin to change. Moreover, solitons can trap dispersive waves, as seen in Fig. 13.14. This phenomenon is similar to the soliton trapping discussed in Section 6.5.2 in the context of highly birefringent fibers and is discussed further in the following subsection.

13.3.3 XPM-induced trapping The XPM interaction between a Raman soliton and a dispersive wave can lead to trapping of the dispersive wave [75–79]. If the dispersive wave travels in the normal-GVD region, the group-velocity mismatch between this wave and the soliton can be relatively small, depending on their respective wavelengths. If the two overlap temporally, it is

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Figure 13.15 (A) Experimental spectra at several power levels at the output of a 1.5-m-long fiber. (B) Magnified view of the spectrum for Pav = 118 mW. Phase mismatch of the dispersive wave is also shown. (After Ref. [69]; ©2004 OSA.)

possible for a Raman soliton to trap a nearby dispersive wave such that the two move together. In a 2004 study of the XPM effects on supercontinuum generation, the XPM was found to create another spectral peak whose wavelength was shifted toward the blue side, and the magnitude of shift depended on the input power launched into the fiber [69]. Fig. 13.15(A) shows the observed spectra at the output of a 1.5-m-long microstructured fiber for several values of the average power when 27-fs pulses at 790 nm with a 1-GHz repetition rate were launched into it. This fiber exhibited two ZDWLs at 700 and 1400 nm. The Raman soliton and NSR are visible for Pav = 53 mW, and their wavelengths shift in the opposite directions with increasing power. The peak near 1600 nm in the spectrum for Pav = 118 mW also represents a dispersive wave. It is created when the Raman soliton approaches the second ZDWL of the fiber, and its spectrum stops shifting because of the RIFS suppression phenomenon discussed in Section 12.2. Fig. 13.15(B) shows the spectrum for Pav = 118 mW on a magnified vertical scale, together with the phase mismatch as a function of wavelength between the shortest soliton and its dispersive wave [69]. The phase matching occurs for 515 nm, and the peak located there corresponds to the dispersive wave. The peak on the blue side of this peak, shifted by more than 60 nm, is due to XPM. Numerical simulations based on the generalized NLS equation reproduce well the observed spectral features qualitatively. They also help in understanding how the second peak is created. The spectrograms at different distances within the fiber reveal that the soliton slows down as its spectrum shifts toward the red and begins to overlap with the dispersive wave; their interaction

Supercontinuum generation

Figure 13.16 Spectrograms for two PCFs with air-hole spacings of 1.2 μm (left) and 1.3 μm (right) when 13-ps pulses with 15-kW peak power are launched. Two horizontal lines indicate the ZDWLs of each fiber. (After Ref. [70]; ©2005 OSA.)

through XPM creates the blue-shifted peak. However, the two eventually separate from each other because of a mismatch in their group velocities, a feature that indicates that soliton trapping did not occur in this experiment. Note that the second dispersive wave created near 1600 nm travels faster than the soliton. The XPM-induced spectral peak was not created in this case because the dispersive wave separated rapidly from the soliton. Since the exact wavelength of the dispersive waves created through NSR depends on the dispersive properties of the fiber employed, the group-velocity mismatch between the soliton and the dispersive wave generated by it depends on multiple parameters associated with the fiber and the pulse. Soliton trapping can only occur when this mismatch is not too large. It was found in a 2005 numerical study that XPM-coupled soliton pairs, similar to those discussed in Section 7.3, can form under certain conditions in fibers exhibiting two ZDWLs [70]. Fig. 13.16 shows the spectrograms for two photonic crystal fibers, with the only difference that the spacing of 1-µm-diameter air holes changes from 1.2 to 1.3 µm. In the 1.2-µm case shown on left in Fig. 13.16, the second ZDWL of the fiber occurs near 1330 nm. When 13-ps pulses with 15-kW peak power are launched into such a fiber at a wavelength of 804 nm, the Raman soliton emits NSR first in the visible region, and then in the infrared region lying beyond 1330 nm, as its RIFS is suppressed close to the second ZDWL. The infrared radiation is emitted in large amounts such that its group velocity is close to that of the soliton. As a result, the two form a soliton pair at a distance of z = 35 cm through the XPM-induced coupling between them [70]. This pair moves as one unit for z > 35 cm. The temporal width of the dispersive wave did not change much even at z = 60 cm as this wave was propagating in the anomalous-GVD

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region of the fiber. This phenomenon was not observed for the second fiber with a 1.3-µm hole spacing because its ZDWL lies close to 1500 nm. We can find the condition under which the XPM-induced pairing is possible by noting that the parameter β1 ≡ 1/vg for a dispersive wave at the frequency ωd can be written as β1 (ωd ) ≈ β1 (ωs ) + β2 (ωs ) + 12 β3 (ωs )2 ,

(13.3.5)

where  = ωd − ωs and ωs is the frequency at which soliton spectrum is centered, and we have retained terms only up to third order. Clearly, β1 (ωd ) = β1 (ωs ) when  = −2β2 /β3 . If this condition is satisfied for a fiber, the dispersive wave can be trapped by the soliton, and the two can move together as a pair through the XPM coupling between them. The XPM effects can also be initiated by launching the main intense pulse into a highly nonlinear fiber together with another pulse at a different wavelength [71–75]. This configuration allows more control, as the second pulse can be delayed or advanced such that the two overlap after some distance inside the fiber. It also permits one to choose the wavelength, width, and energy of the second pulse. In practice, the second pulse is chosen at the second-harmonic wavelength of the main pulse for experimental convenience. The main advantage of using two input pulses is that XPM interaction between the two leads to considerable spectral broadening in the visible region of the generated supercontinuum. The use of XPM for blue-light generation was proposed in a numerical study in which 27-fs pulses at 900 nm with 1.7-kW peak power were propagated inside a microstructured fiber with 200-fs pulses at 450 nm with only 5.6-W peak power [73]. The ZDWL of the fiber was set at 650 nm so that the two pulses propagate with nearly the same group velocity (resulting in enhanced XPM interaction). The width of the second pulse as well as its relative delay was found to affect the visible spectrum. Fig. 13.17 shows the predicted spectra in the blue region at the output of a 80-cm-long fiber as a function of (A) initial delay of 200-fs pulses and (B) width of the second-harmonic pulse with zero delay. In a 2005 experiment, 30-fs pulses at a wavelength of 1028 nm were launched into a 5-m-long microstructured fiber with and without weak second-harmonic pulses [75]. Their relative delay was adjusted using a delay line. The fiber had two ZDWLs located at 770 and 1600 nm. Fig. 13.18 compares the output spectra as a function of launched power when pump pulses were propagated (A) alone and (B) with the second-harmonic pulses. When only pump pulses are launched, the supercontinuum is generated through the formation of Raman solitons that shift toward longer wavelengths until their RIFS is suppressed near the second ZDWL at 1600 nm. The NSR emitted by them near 1700 nm is clearly visible. However, the supercontinuum does not extend in the spectral region below 900 nm. When second-harmonic pulses at 514 nm are also launched, the preceding scenario does not change because these pulses are too weak to affect the main pump pulse. However, the XPM-induced phase shift produced by pump pulses

Supercontinuum generation

Figure 13.17 Predicted spectra when 27-ps pulses with 1.7 kW peak power propagate inside a microstructured fiber together with second-harmonic pulses at 450 nm with only 5.6 W peak power: (A) spectra for different initial delay of 200-fs pulses; (B) spectra for different pulse widths launched with no initial delay. (After Ref. [73]; ©2005 OSA.)

Figure 13.18 Measured spectra as a function of launched power at the output of a 5-m-long fiber when 30-fs input pulses were propagated (A) without and (B) with weak second-harmonic pulses. The black rectangle indicates the region covered in part (A) on the left. (After Ref. [75]; ©2005 OSA.)

chirps the green pulses such that the resulting supercontinuum extends into the visible region and covers a wide range from 350 to 1700 nm. In another experiment, when the power ratio between the pump and second-harmonic pulses was adjusted, it was possible to produce a smooth and nearly flat supercontinuum that covered the entire visible region [71]. Clearly, XPM can be used as a tool to tailor the properties of the generated supercontinuum.

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13.3.4 Role of four-wave mixing As discussed earlier, fission of a high-order soliton creates multiple Raman solitons and multiple dispersive waves with distinct spectral peaks that cover a wide wavelength range. Clearly, each Raman soliton can act as a pump and interact with a dispersive wave through FWM to create an idler wave, provided an appropriate phase-matching condition is nearly satisfied. Such a FWM process has been observed experimentally and its theory has also been developed [80–84]. It turns out that two distinct FWM mechanisms exist in the case of supercontinuum generation, and the corresponding phase-matching conditions are given by [83] β(ω3 ) + β(ω4 ) = βs (ω3 ) + βs (ω4 ),

β(ω3 ) − β(ω4 ) = βs (ω3 ) − βs (ω4 ), (13.3.6)

where βs (ω) and β(ω) are the propagations constants associated with a soliton and a dispersive wave, respectively. As seen in Eq. (12.1.5), the dispersion relation for a soliton has the form βs (ω) = β(ωs ) + β1 (ω − ωs ) + 12 γ Ps , where Ps is soliton’s peak power. In contrast, the dispersion relation β(ω) of dispersive waves does not depend on power but contains terms that vary with frequency quadratically and cubically. The idler frequencies ω4 corresponds to the solutions of Eq. (13.3.6). The first phase-matching condition corresponds to traditional FWM in which two pump photons taken from the soliton are converted into the signal and idler photons. In this case, the idler frequency is far from the original dispersive wave as it lies on the opposite side of the input spectrum. In the case of a positive third-order dispersion, such a FWM process creates new frequency components on the Stokes side of the soliton frequency, and it plays an important role in expending the supercontinuum into the infrared side of the input wavelength. The second phase-matching condition corresponds to a Bragg-scattering-type FWM process in which beating of a soliton and a dispersive wave creates a moving index grating that produces an idler wave. In this situation, the idler frequency lies close to the original dispersive wave. In the case of a positive third-order dispersion, such a FWM process creates new frequency components on the anti-Stokes side of the dispersive wave and helps to extend the supercontinuum toward the blue side of the input wavelength [82]. Several experiments have shown that both FWM processes take place inside optical fibers. In a 2005 experiment, a CW wave was launched together with an optical pulse propagating as a soliton [81], and it was observed that new spectral components were created as predicted by Eq. (13.3.6). In a 2006 experiment, 200-fs pulses at 800 nm were launched with 6.2-kW peak power inside a 2-m-long PCF with its ZDWL at 820 nm, and the evolution of the output spectrum along its length was observed [83]. Numerical simulations based on Eq. (13.3.1) agreed with the experimental spectra showing the generation of FWM-induced spectral peaks in the visible region.

Supercontinuum generation

Figure 13.19 X-FROG spectrograms calculated numerically at six locations within a PCF when 200-fs pulses at 800 nm are launched with 6.2-kW peak power. A dashed vertical line marks the ZDWL of the PCF used. (After Ref. [83]; ©2006 OSA.)

Fig. 13.19 shows six X-FROG spectrograms calculated numerically at six locations within a 2 m long fiber. The spectrograms reveal graphically how a supercontinuum is produced when femtosecond pulses propagate inside a fiber. Only SPM-induced spectral broadening occurs at a distance of 3 cm, and the entire spectrum still lies in the normal-GVD region of the fiber. Spectrum broadens with further propagation, and one of the spectral lobes lies in the anomalous-GVD region at a distance of 9 cm. Its energy forms a high-order soliton that undergoes fission. One can see the formation of a Raman soliton at z = 20 cm, together with its dispersive wave near 700 nm. With further propagation, the spectra of Raman solitons shift toward the red side and FWM sets in. At z = 50 cm, we see a clear evidence of FWM band near 650 nm. At a distance of 1.3 m and 2 m, several FWM bands can be seen together with the trapping of dispersive waves by various Raman solitons.

13.4. CW or quasi-CW pumping The use of picosecond or shorter pulses is not essential for supercontinuum generation. As mentioned earlier, 10-ns pulses were used in the original 1976 experiment [4]. In a 2003 experiment, 42-ns pulses from a Q-switched Nd:YAG laser were launched into a 2-m-long microstructured fiber to produce a wide supercontinuum at 10-kW power levels [85]. In fact, even CW lasers can produce a supercontinuum at sufficiently high

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power levels. CW lasers were used for this purpose as early as 2003 and, by 2010, such supercontinuum sources were being used for a variety of applications [85–98]. In this section we first discuss the nonlinear mechanisms behind CW supercontinuum generation and then review the experimental progress. It should be stressed that the following discussion also applies to quasi-CW pumping in which optical pulses of widths >1 ns are employed.

13.4.1 Nonlinear mechanisms It should come as no surprise that the phenomenon of modulation instability is behind the CW supercontinuum generation [99–103]. As was seen in Section 5.1, when a CW beam propagates in the anomalous-GVD region of an optical fiber, modulation instability can create two sidebands at specific frequencies ω0 ± max , where ω0 is the frequency of the CW beam. The frequency shift max depends on the input power P0 in Eq. (5.1.11), in addition to the fiber’s parameters: max =



2γ P0 /|β2 |.

(13.4.1)

In the case of spontaneous modulation instability, the growth of these sidebands is initiated by intensity fluctuations within the CW beam. Recall from Section 10.3 that modulation instability can be viewed as a FWM process in which the input CW beam acts as the pump and the noise at the two sidebands frequencies seeds the signal and the idler waves. The amplitudes of the two sidebands grow initially with distance z as exp(2z/LNL ), where LNL = 1/(γ P0 ) is the nonlinear length. This growth manifests in the time domain as sinusoidal oscillations of the form P (z, t) = P0 + pm (z) cos[max t + ψ(z)] with the period Tm = 2π/ max . The exponential growth continues as long as the fraction of the power in the two sidebands remains a small fraction of the total power (the so-called linear regime). Once the modulation-instability process enters the nonlinear regime, evolution of the optical field can only be studied by solving the NLS equation (5.1.1) numerically with noise added to the input CW beam. Fig. 13.20 shows temporal power variations predicted for a 20 W input power at the output of a 50-m-long fiber for several values of the β2 and γ parameters of the fiber. The nonlinear length is 10 m for γ = 5 W−1 /km and reduces to 1 m when γ is increased by a factor of 10. Fig. 13.20(A) shows that the ratio L /LNL should exceed 10 or so before significant modulations appear on the pump beam. As the nonlinear length shortens, modulations become sharper and sharper and take the form of a train of short optical pulses of different widths that propagate as solitons in the anomalous-GVD regime of the fiber. We can estimate the average width of these solitons using a simple argument [103]. The energy of an optical soliton of peak power Ps is related to its width Ts as Es = 2Ps Ts (see Section 5.2.2). This energy must equal the energy P0 Tm of the CW beam in one

Supercontinuum generation

Figure 13.20 Temporal structures at the output of a 50 m long fiber for a CW input beam with 20 W pump power; (A) γ is varied for β2 = 12 ps2 /km; (B) β2 is varied for γ = 40 W−1 /km. The bars show the value of Tm . (After Ref. [103]; ©2010 Cambridge University Press.)

modulation period Tm . However, for a fundamental soliton, we must also have LD = LNL or Ps Ts2 = |β2 |/γ . Using max from Eq. (13.4.1), we find the simple relation 

Ts =

Tm π2

=

2|β2 | 1 = 2|β2 |LNL . π 2 γ P0 π

(13.4.2)

This equation shows, in agreement with Fig. 13.20, that smaller values of |β2 | and larger values of γ help in reducing the width of solitons formed through modulation instability. If modulation instability is induced by launching a weak signal at the sideband frequency, all solitons within the pulse train would have the same width. However, when modulation instability is seeded by noise and is of spontaneous kind, pulse widths fluctuate in a way seen in Fig. 13.20. The formation of a supercontinuum proceeds as follows. First, modulation instability converts the CW beam into a train of pulses of different widths and peak powers that propagate as fundamental solitons. Since the RIFS depends on the pulse width, different solitons shift their spectra by different amounts toward longer wavelengths. At the same time, blue-shifted radiation is generated in the form of dispersive waves because of perturbations of these solitons by third-order dispersion. As a soliton shifts its spectrum, it also slows down as long as it experiences anomalous GVD. As a result, solitons

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Figure 13.21 Temporal (bottom) and spectral (top) evolutions over 500 m of fiber when a CW beam is launched with 4 W of power. Dark-gray curves represent an average over 50 light-gray spectra. (After Ref. [101]; ©2005 OSA.)

collide (overlap temporally) with neighboring solitons and dispersive waves and interact with them though XPM and FWM. It turns out that such a collision can transfer energy to the slowing soliton, which reduces its width further (to maintain the condition N = 1) and slows down even more, and its spectrum shifts even further toward longer wavelengths. Multiple soliton collisions eventually produce a supercontinuum that is extended mostly toward the red side of the input wavelength. It is clear from the preceding description that the noisy nature of an input CW beam plays a critical role because it seeds the process of modulation instability. Even a CW laser beam is only partially coherent because of its finite spectral width resulting from intrinsic phase fluctuations. Any numerical modeling must include such fluctuations [99–101]. Moreover, as most measurements of optical spectra represents a long-time average of fiber’s output, one must perform an ensemble average of the numerically simulated spectra. Fig. 13.21 shows the temporal and spectral evolution over 500 m of a highly nonlinear fiber with γ = 10.6 W−1 /km, β2 = −0.17 ps2 /km, and β3 = 0.0393 ps3 /km at the 1486 nm wavelength of the CW beam launched with 4 W of power [101]. The dark-gray curves represent an ensemble average over 50 light-gray spectra. The NLS equation given in Eq. (13.3.1) was solved numerically to obtain these results. Modulation instability occurs within the first 100 m, and it leads to rapid temporal oscillations and creates two spectral sidebands shifted from the central peak by about 3.6 THz. After 300 m of propagation, we see a clear evidence of different solitons of different widths undergoing large red shifts through intrapulse Raman scattering. Two temporal spikes at z = 500 m (marked as a and b) correspond to two solitons whose

Supercontinuum generation

spectra have shifted the most. Multiple peaks on the blue side of the input wavelength correspond to dispersive waves.

13.4.2 Experimental results The two important ingredients for any recipe of CW supercontinuum are a high-power laser and a highly nonlinear fiber so that the product γ P0 L exceeds 30 or so, where P0 is the CW power launched into a fiber of length L. This condition can be satisfied for a 100-m-long fiber with γ = 100 W−1 /km at a pump-power level of a few watts. Such power levels are readily available from Yb-doped fiber lasers. In a 2003 experiment, a 100-m-long microstructured fiber was employed, and a Yb-fiber laser was used for its CW pumping at 1065 nm [86]. The resulting supercontinuum extended from 1050 to 1380 nm when 8.7 W of CW power was coupled into the fiber. In a 2004 experiment, three highly nonlinear fibers of lengths 0.5, 1, and 1.5 km were used for supercontinuum generation by launching a CW beam at 1486 nm [87]. The ZDWL of the fibers was 1800 nm when pump power was close to 4 W. The spectrum was highly asymmetric with much more power on the long-wavelength side. As discussed earlier, this asymmetry is due to intrapulse Raman scattering that selectively extends the spectrum toward the longwavelength side. A 20-m-long PCF with γ = 43 W−1 /km was used in a 2008 experiment [92]. It exhibited two ZDWLs located near 810 and 1730 nm. As a result, dispersion was relatively large (65 ps/km/nm) at the pump wavelength of 1070 nm but it decreased for longer wavelengths. Fig. 13.22 shows the observed supercontinuum at the fiber output when 44 W of CW power was launched into the fiber. The supercontinuum spans from 1050 to 1680 nm. More importantly, the output power is close to 29 W, and the spectral power density exceeds 50 mW/nm up to 1400 nm. These features are useful for applications of such a supercontinuum source for biomedical imaging. Numerical simulations reproduce most of the experimental features seen in Fig. 13.22. More importantly, they shed light on the formation and collisions of multiple soliton that plays an important role. Fig. 13.23 shows the X-FROG spectrogram computed numerically [92]. One sees clearly the formation of solitons (round objects) through modulation instability, together with their different spectral shifts and different speeds (leading to different delays). Collisions among these solitons are also apparent from their temporal overlap. Eventually the spectrum of short solitons approaches the ZDWL near 1730 nm, and it stops shifting because of the radiation pressure induced by the corresponding dispersive waves (cigar-like objects) emitted at wavelengths longer than 1730 nm. The interaction (collision) of solitons with these dispersive waves gener-

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Figure 13.22 Observed supercontinuum on (A) semi-log and (B) linear scales at the fiber output of a 20-m-long PCF when 44 W of CW power at 1070 nm was launched into the fiber. (After Ref. [92]; ©2008 OSA.)

Figure 13.23 X-FROG spectrogram calculated numerically under the conditions of the supercontinuum in Fig. 13.22. (After Ref. [92]; ©2008 OSA.)

ates new spectral components through FWM [97] that can be seen in Fig. 13.22 in the wavelength region near 1900 nm. The formation of CW supercontinuum in the visible region has also attracted attention [104–108]. This is not easy to do because the most practical source of CW radiation is a high-power Yb-fiber laser emitting light near 1060 nm. When such a laser is used with a suitable PCF having its ZDWL near 1000 nm, the observed supercontinuum rarely extends below 900 nm. In a 2006 experiment, a tapered PCF was employed for this purpose whose core diameter, and hence the ZDWL, decreased along its length [30]. A quasi-CW source (a Nd:YAG microchip laser) emitting nanosecond

Supercontinuum generation

Figure 13.24 Experimental spectra for a launched power of (A) 70 W in a uniform-core PCF and (B) 45 W in a tapered-core PCF; the inset shows the output dispersed by a prism. Photographs of the output beam (C) and fiber spool (D) are also shown. (After Ref. [107]; ©2009 OSA.)

pulses was used to produce the optical spectra shown in Fig. 13.4(A) for several fiber lengths. As seen there, the spectrum extended from 350 to 1750 nm with a high uniformity when fiber length exceeded 5 m. The extension of the supercontinuum into the visible region was possible because a monotonically decreasing |β2 (z)| allowed the FWM phase-matching condition to be satisfied for progressively shorter idler wavelengths. CW pumping near 1060 nm of a constant-diameter PCF can also create spectral components at wavelengths below 900 nm. In a 2008 experiment, a 100-m-long PCF with its ZDWL at 1068 nm was pumped with a 50-W Yb-fiber laser emitting light at 1070 nm, and wavelengths shorter than 900 nm were observed at the fiber output at high power levels [104]. FWM was again behind this extension of the supercontinuum, although visible wavelengths were not produced in this experiment. Much better results were obtained in another 2008 experiment by using two cascaded PCFs such that a 100-m-long section followed another 100-m-long tapered section in which |β2 | decreased with length [105]. When pumped with 12.5 W of power at 1064 nm, the output spectrum contained a broad peak in the green region together with a supercontinuum extending from 700 to 1300 nm. By 2009, the use of a PCF whose core was both tapered and doped with GeO2 created a CW supercontinuum that extended toward wavelengths as short as 450 nm [107]. Fig. 13.24 compares the spectra obtained with the uniform-core and tapered-core PCFs.

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In case (A) of a uniform-core PCF, the launched CW power at 1075 nm was 70 W, and the supercontinuum extended from 550 nm to >1750 nm with spectral power variations of less than 12 dB. In case (B), a uniform-core section of 50 m was followed with a 130-m-long PCF section whose outer diameter decreased from 135 to 85 µm. When pumped with 40 W of CW power, the supercontinuum extended from 470 to >1750 nm, and thus covered the entire visible region, as also evident from the photo of the output beam seen in part (C). When a prism was used to disperse the output light, a rainbow-like spectrum was observed, as seen in the inset of Fig. 13.24. These results clearly show that an ultrabroad spectrum covering both the visible and near-infrared regions can be produced by pumping a PCF at 1060 nm provided the PCF is designed suitably.

13.5. Polarization effects The numerical model based on Eq. (13.3.1) assumes that the state of polarization (SOP) of an input pulse is preserved during its propagation inside the fiber. As we have seen in Chapter 6, this assumption is often not valid in practice. In the case of a highly birefringent fiber, the SOP may remain preserved if the input pulse is launched with its polarization oriented along a principal axis of the fiber. However, if it is initially tilted at an angle, the input pulse will split into two orthogonally polarized parts that would begin to separate from each other because of their different speeds. Even in an ideal isotropic fiber, nonlinear polarization rotation, induced by the XPM-induced coupling between the two orthogonally polarized parts of a pulse can affect the process of soliton fission. The important question is how such a coupling affects the generation of a supercontinuum in highly nonlinear fibers [109–117]. Microstructured fibers, with a cladding containing multiple air holes, often do not have a circular core. The polarization dependence of a supercontinuum in such fibers has been observed in several experiments [109–111]. In a 2002 experiment, PCFs with a hexagonal symmetry of air holes in its cladding were employed [45]. The birefringence should be relatively weak in these fibers because of a nearly circular core. Nevertheless, it was found that the supercontinuum changed with the input SOP. Moreover, different parts of the supercontinuum exhibited different SOPs, indicating a nonlinear coupling between the two orthogonally polarized pulses within the fiber. Fig. 13.25 shows changes in the recorded supercontinuum spectra with ± 30◦ rotations in the SOP of a linearly polarized input pulse, realized by rotating the 4.1-cm-long fiber used. In this experiment, 18-fs input pulses from a Ti:sapphire laser operating at 790 nm were launched into the fiber. Fig. 13.25 shows that spectral details of a supercontinuum are quite sensitive to the input SOP of the pulse. In a 2008 experiment, a highly birefringent microstructured fiber with an elliptical core was employed [116]. Fig. 13.26 shows changes in the output spectra observed as

Supercontinuum generation

Figure 13.25 Supercontinuum spectra at the output of a photonic crystal fiber for three different orientations of linearly polarized 18-fs input pulses. (After Ref. [45]; ©2002 OSA.)

Figure 13.26 Supercontinuum spectra at the output of a highly birefringent fiber (micrograph on the right) for several different polarization angles. (After Ref. [116]; ©2008 American Physical Society.)

the angle θ (that the SOP direction of the input pulse makes from the slow axis) was varied, together with a micrograph of fiber on the right. This experiment launched 15-fs pulses with 300 mW average power at a wavelength of 800 nm into a 12-cm-long fiber with an effective mode area of about 2 µm2 . The ZDWL of the fiber was 683 and 740 nm for light polarized along the fast and slow axes of the fiber, respectively. When light is polarized along these two axes (θ = 0 or 90◦ ), only one of the orthogonally polarized modes is excited. Differences observed in the spectra in these two cases are

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due to different dispersive properties of the fiber experienced by the pulse because of a large birefringence. For values of θ between 0 and 90◦ in Fig. 13.26, both modes are excited because pulse energy splits along the slow and fast axes. The two orthogonally polarized pulses travel with different speeds because of birefringence and separate from each other quickly (the walk-off length is 0 are 4-fold degenerate in the weakly guiding approximation. Fibers with V > 10 support a large number of modes, scaling as V 2 /2. For example, a multimode fiber with a = 25 µm and  = 5 × 10−3 has V 15 near λ = 1.5 µm and supports 112 modes. Fig. 14.3 shows the spatial intensity patterns of four low-order modes for a fiber with V = 5. Only even versions are shown for the LP11 and LP21 modes; odd versions can be obtained by rotating the spatial patterns. Note also that the phase is not spatially uniform for modes with l = 0. As we saw in Chapter 2, evolution of optical pulses inside single-mode fibers depended on several frequency derivatives of β(ω). In the case of a multimode fiber, these derivatives become different for different LP modes. As the group velocity of a mode is inversely related to the first derivative of β [see Eq. (1.2.9)], non-degenerate modes

Multimode fibers

travel at different speeds inside a multimode fiber. One consequence of this feature is that, unless the entire pulse energy is coupled onto one specific mode of the fiber, a single pulse excites several LP modes simultaneously that exit the fiber at different times, resulting in the so-called differential group (or mode) delay (DGD). The group-velocity dispersion (GVD) parameter β2 , related to the second frequency derivative of βlm , is also expected to vary from mode to mode. It is possible to calculate β1 and β2 for various fiber modes [4] in terms of the first two derivatives of the b(V ) curves shown in Fig. 14.1. The same process can be used for the higher-order dispersion parameters, if they are needed.

14.1.2 Graded-index fibers The refractive index of a graded-index (GRIN) fiber decreases radially from the core center, taking the value nc at ρ = a for a core of radius a. Its radial variation in most GRIN fibers is nearly parabolic and has the form n2 (x, y) = n21 [1 − 2(ρ/a)2 ] (ρ ≤ a),

(14.1.7)

with n(x, y) = nc for ρ ≥ a. The parameter  = (n1 − nc )/n1 is the same used earlier for step-index fibers. The modes are still obtained by solving the Helmholtz equation ˜ + n2 (x, y)k2 E ˜ ∇ 2E 0 = 0.

(14.1.8)

Although a numerical approach is necessary in general, considerable insight is gained by solving Eq. (14.1.8) analytically with the assumption that the index profile in Eq. (14.1.7) applies for all values of ρ . This assumption is reasonable for low-order modes in fibers designed with a core radius a  λ (a = 25 µm typically). Eq. (14.1.8) can be solved using either the Cartesian or cylindrical coordinates. In the cylindrical coordinates, the method of separation of variables leads to the following form of the electric field for the x-polarized even modes [3]: Ex (x, y, z) = F (x, y)eiβ z = R(ρ) cos(lφ)eiβ z ,

(14.1.9)

where F (x, y) is the spatial distribution, l = 0, 1, . . . is an integer, and the radial function R(ρ) satisfies 



 l2 ρ2  d2 R 1 dR 2 2 2 + k n 1 − 2  − R = 0. − β + 0 1 dρ 2 ρ dρ a2 ρ2

(14.1.10)

This equation has analytic solutions in terms of the associated Laguerre polynomials: 

Rlm (ρ) = Nlm

ρ ρ0

l



Lml −1

   ρ2 ρ2 exp − 2 , ρ02 2ρ0

(14.1.11)

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√

where ρ0 = a/ V , and the parameter V is defined in Eq. (14.1.2). An important property of the modes for all fibers is that they are orthogonal to each other. If we use a single subscript p to denote mode profiles as Fp (x, y), the orthogonality of modes requires that



Fp∗ (x, y)Fq (x, y) dx dy = δpq ,

(14.1.12) 

where each mode is assumed to be normalized such that |Fp (x, y)|2 dx dy = 1 and all integrations extend from −∞ to +∞, unless specified otherwise. The constant Nlm can be found using this normalization. Similar to the case of step-index fibers, the integer m labels multiple solutions for each value of the integer l. Also, except for l = 0, the odd version of each mode with l = 0 exists with sin(lφ) in place of cos(lφ) in Eq. (14.1.9). The value of βlm for both sets of modes is given by [3] βlm (ω) = k0 n1 [1 − (l + 2m − 1)(4/V )]1/2 .

(14.1.13)

The main advantage of this analytic approach is that we have a closed-form expression for the effective index (n = β/k0 ) of each LPlm mode. It shows clearly that groups of modes for which the integer g = l + 2m − 1 has the same value are degenerate because n has the same value for the entire group. This degeneracy occurs in addition to the degeneracy discussed earlier for step-index fiber. Keeping this in mind, the g = 1 group has a single member with l = 0 and m = 1, and this LP01 mode is 2-fold degenerate. The next group with g = 2 also has a single member, the 4-fold degenerate LP11 mode. The next group with g = 3 has two members LP02 and LP21 and is 6-fold degenerate. The total group degeneracy increases rapidly as the integer g increases. If we label the modes with their group number g and use Eq. (14.1.13) in Eq. (14.1.6), we obtain an amazingly simple expression, b = 1 − 2g/V , which can be used to write the effective mode index in the form √

n ≈ n1 (1 − 2g/V ) = n1 − g 2/(k0 a).

(14.1.14)

Several differences between the GRIN and step-index fibers are now apparent. First, the curves in Fig. 14.2 for step-index fibers are replaced by the function b(V ) = 1 − 2g/V for GRIN fibers that is valid only for V > 2g. Fig. 14.4 shows the dispersion curves for mode groups up to g = 6. The values near V = 2g are not accurate since the boundary condition at the core-cladding interface has not been applied in this simple treatment. Second, the spatial mode profiles are different but they look similar in shape to those shown in Fig. 14.3 for step-index fibers. Third, the effective mode index n for GRIN √ fibers decreases by a constant value 2/(k0 a) from one mode group to the next. As we discuss later, this ladder-like structure of n leads to periodic self-imaging of optical beams

Multimode fibers

Figure 14.4 Normalized effective index b(V ) of several mode groups (indicated by the value of g) plotted as a function of the V parameter for GRIN fibers.

inside a GRIN fiber. Note also that the problem of finding modes of a GRIN fiber is analogous to finding the energy levels of a harmonic oscillator in quantum mechanics. Similar to the case of step-index fibers, Eq. (14.1.13) can be used to calculate the dispersion parameters for each mode by taking derivatives with respect to ω. For low-order modes for which 4g(/V )  1, the first derivative is found to be √ β1 = k0 n1g − g(a 2)−1 (d/dω),

(14.1.15)

where n1g is the group index of the core material. Physically, β1 represents the group delay and its inverse provides the group velocity. One is often interested in the DGD of a mode group with respect to the fundamental LP01 mode. Using this definition, DGD can be written as √ τg1 = (g − 1)(a 2)−1 (d/dω).

(14.1.16)

Typical values of DGD are ∼0.1 ps/m for g = 2 at wavelengths near 1.55 µm and they increase linearly for higher groups. The GVD parameter β2 for any group can also be found by calculating d2 β/dω2 . In commercial GRIN fibers, β2 for g = 1 vanishes at the zero-dispersion wavelength near 1.3 µm and is about −25 ps2 /km near 1.55 µm.

14.1.3 Multicore fibers Multicore fibers, as seen in Fig. 14.1, contain multiple cores within the same cladding. The cores are often designed to support a single mode. If the cores are relatively far apart, their individual modes overlap negligibly, and the multicore fiber behaves as a bundle of single-mode fibers. However, mode overlap is not negligible if the cores are closely spaced. One can find the modes of the whole structure by solving the Helmholtz

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equation (14.1.8) numerically such that n(x, y) = n1 in the core regions and becomes nc in the cladding regions. Such modes are called supermodes and their number equals the number of single-mode cores within a multicore fiber. An approximate analytic technique makes use of the coupled-mode theory [6], used extensively for fiber couplers that constitute an example of a two-core fiber [5]. In this approach, the supermodes result from a linear combination of the modes supported by isolated individual cores [7–11]. Focusing on the x-polarized supermodes of a N-core fiber, the electric field has the form Ex (x, y, z) =

N

Cm (z)Fm (x, y) eiβ z ,

(14.1.17)

m=1

where Fm (x, y) is the modal field of the mth core, β is the propagation constant of the supermode, and Cm (z) is the amplitude that evolves with z because of mode coupling. Substituting the preceding form in Eq. (14.1.8) and following a standard procedure [5], we obtain a set of N coupled-mode equations: dCm = i(βcm − β)Cm + iκmn Cn , dz n=m N

(14.1.18)

where m = 1, 2, . . . , N and the coupling coefficient κmn is defined as k2 κmn = 0 2β



[n2 (x, y) − n2cm (x, y)]Fm∗ (x, y)Fn (x, y) dx dy.

(14.1.19)

Here ncm (x, y) is the refractive-index distribution when only the mth core is considered by itself and βcm is the corresponding modal propagation constant. The modal fields  Fm (x, y) are assumed to be normalized such that |Fm (x, y)|2 dx dy = 1. Eq. (14.1.18) shows that Cm is not constant when light is launched into one core of a multicore fiber. Its solution for a two-core fiber reveals that the power is transferred to the second core in a periodic fashion, a feature that is used to design fiber couplers [5]. For a supermode, Cm should not vary with z. Setting dCm /dz = 0, we obtain a matrix equation, MC = 0, where C is a column vector containing C1 , C2 , . . . , CN as its elements and M is a N × N matrix. A nontrivial solution exists when the determinant of M vanishes. This condition is used to find the propagation constants β for all supermodes. In general, they depend on the shapes, sizes, and geometrical locations of the cores and require a numerical approach. An analytical solution can be obtained when the cores are identical, equally spaced, and arranged in a circle such that each core couples only with its two nearest neighbors. In this case, M becomes a tridiagonal symmetric matrix, yielding the following solution [11]: β (n) = βc + 2κ cos(2π(n − 1)/N ],

Cmn = N −1/2 exp[2iπ(m − 1)(n − 1)/N ], (14.1.20)

Multimode fibers

Figure 14.5 Spatial distributions of the six supermodes of a multicore fiber with V = 5. The phase information is included by color-coding the field amplitudes from −1 to 1. The core regions are indicated by black circles. (After Ref. [9]; ©2011 OSA).

where β (n) is the propagation constant of the nth supermode and βc is its value for each isolated core. This technique can be generalized to fibers containing multiple circular rings of cores [11]. As an example, Fig. 14.5 shows the spatial field distributions of the six supermodes of a fiber containing a single ring of six cores [9]. The phase information is included by color-coding field amplitudes in the range of −1 to 1. Different spatial patterns result from different phases among the neighboring cores. We can apply Eq. (14.1.20) to the specific case of a two-core fiber that has been studied extensively in the context of fiber couplers [5,12]. If the two cores are separated by a distance d, they lie on a circle of diameter d centered in the middle. Using N = √ 2 in Eq. (14.1.20), the two supermodes are the even and odd combinations, (F1 ± F2 )/ 2, with the propagation constants βc ± 2κ , respectively. When an optical field is launched into a specific core, both supermodes are excited. The different phase shifts acquired by them during propagation results in a periodic transfer of power from one core to another, a feature that can be used to make directional couplers. Eq. (14.1.20) can also be applied to a fiber with three equidistant cores. Using N = 3, the propagation constants in this case become β (1) = βc + 2κ,

β (2) = βc + κ,

β (2) = βc − κ.

(14.1.21)

The corresponding supermodes are linear combinations of the three spatial profiles Fm (x, y) with coefficients Cmn obtained from Eq. (14.1.20). It is important to note that

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Fm (x, y) is localized in the vicinity of the mth core, whereas supermodes extend over all cores.

14.1.4 Excitation of fiber modes As we have seen, a multimode fiber can support a large number of modes, depending on its core size. The important question from a practical standpoint is how many of these modes are exited when an optical beam is used to launch light into this fiber. In a typical experiment, the size, shape, and polarization of the input beam is controlled using lenses and other optical elements. Assuming that the beam is polarized along the x axis, the electric field at the input plane of the fiber can be written as Ex (x, y, t) = A0 (t)F0 (x, y), where A0 (t) is related to the temporal shape of each pulse and F0 (x, y) denotes spatial shape of the beam. To find which fiber modes are excited by this beam, we expand Ex in terms of all fiber modes as Ex (x, y, t) = A0 (t)

N

Cn Fn (x, y).

(14.1.22)

n=1

We can find the expansion coefficients by multiplying this equation by Fp∗ (x, y), integrating over the whole transverse plane, and using the mode-orthogonality relation in Eq. (14.1.12). The result is given by

Cp = 0

2π

0

∞

Fp∗ (ρ, φ)F0 (ρ, φ)ρ dρ dφ.

(14.1.23)

It shows that the extent of spatial overlap of each mode with the input beam determines how strongly that mode will be excited. The quantity |Cp |2 is called the coupling efficiency and its magnitude represents the fraction of total pulse energy carried by the pth mode. Consider the case of a Gaussian beam with F0 (ρ, φ) ∝ exp(−ρ 2 /w02 ) peaking precisely at the core center. Here, w0 is the spot size of the beam. Since this beam is radially symmetric (no φ dependence), it is easy to deduce that Cp vanishes all LPlm modes for which l = 0. As a result, such a Gaussian beam excites only the LP0m modes. Moreover, Cp is the largest for the fundamental LP01 mode because it has a central peak at ρ = 0 and overlaps most strongly with the input Gaussian beam. By careful adjusting the spot size w0 , it is possible to couple most of the pulse energy into this mode. On the other hand, many fiber modes can be excited simultaneously if we break the radial symmetry by shifting the intensity peak of the input beam from the core center. In this case pulse energy appears even in the modes with l = 0. For some applications, it is desirable to excite a single higher-order mode of the fiber. This is indeed possible, and many techniques have been developed for this purpose that make use of devices such as amplitude and phase masks, fiber gratings, spatial

Multimode fibers

light modulators, and photonic lanterns [13–19]. In particular, a spatial light modulator can be programmed to excite one or more specific modes of the fiber. Selective excitation of fiber modes has attracted attention because of the possibility of mode-division multiplexing for enhancing the capacity of optical communication system. Fiber modes for which l = 0 can be used to create a donut-shape spatial profile that carries orbital angular momentum (OAM) whose value depends on value of the integer l. This can happen in both the step-index and GRIN fibers and is a consequence of the factor cos lφ in Eq. (14.1.1). Recalling that each LPlm mode with l = 0 is four-fold degenerate, a superposition of these modes can lead to a spatial profile whose phase dependence is of the form R(ρ)e±ilφ such that R(ρ) vanishes at ρ = 0. The integer l is sometimes called the topological charge and represents the OAM of this beam [20,21]. In practice, it is not easy to maintain the OAM modes over significant fiber lengths because LPlm modes are not true modes of the fiber and represent linear combinations of several nearly degenerate HE-type modes discussed in Section 2.2. New kinds of fibers, whose high-index region is designed to be in the shape of a ring, have been fabricated for solving this problem [22]. In recent years, several techniques have been developed for generating and propagating OAM beams over considerable lengths of both the step-index and GRIN multimode fibers [23–26].

14.2. Nonlinear pulse propagation Although theory of nonlinear pulse propagation in multimode fibers attracted attention during the 1980s [27–29], it was only in 2008 that the coupled-mode equations were derived that included both the Kerr and the Raman nonlinearities for studying propagation of ultrashort pulses inside such fibers [30]. Since then, the same approach has been used in several other studies focused on different applications [31–36]. The single-mode NLS equation of Section 2.3 is extended in this section to multimode fibers.

14.2.1 Multimode propagation equations It was implicitly assumed in Section 2.3.1 that the input pulse was propagating inside a single-mode fiber with the spatial distribution F (x, y). In the case of a multimode fiber, unless special precautions are taken, pulse energy is likely to be distributed among several modes. In the coupled-mode approach, the Fourier transform of the electric field is expanded linearly in terms of all modes supported by the optical fiber and has the general form E˜ (r, ω) =



˜ m (z, ω) exp[iβm (ω)z], eˆ m Fm (x, y, ω)A

(14.2.1)

m

where Fm governs the shape of a specific mode, βm is the propagation constant of this mode, and eˆ m is its state of polarization. The sum should generally include both the

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guided and radiation modes of the fiber. The analysis is much simpler if the sum is limited to include only the guided modes of a multimode fiber. The starting point is Eq. (2.3.1). Taking its Fourier transform and using Eq. (2.1.13), it can be written in the form ˜ + n2 (r, ω) ∇ 2E

ω2 ˜ ω2 ˜ E = − PNL (r, ω), c2 0 c 2

(14.2.2)

where n is the linear part of the refractive index. We substitute Eq. (14.2.1) into the preceding equation and note that each fiber mode satisfies ∇T2 Fm + n2 (r, ω)(ω2 /c 2 )Fm = βm2 Fm ,

(14.2.3)

˜m where the subscript T denotes the transverse part of the ∇ 2 operator. Assuming that A varies slowly with z and neglecting its second derivative, we obtain M n=1

2iβn

˜n ∂A ω2 eˆ n Fn (x, y, ω)eiβn z = − 2 P˜ NL (r, ω). ∂z 0 c

(14.2.4)

Multiplying the preceding equation with Fm∗ , integrating over the transverse plane, and using Eq. (14.1.12), the mode amplitudes are found to satisfy ˜ m iωe−iβm z ∂A = ∂z 2 0 cnm



Fm∗ (x, y, ω)ˆe∗m · P˜ NL (r, ω) dx dy,

(14.2.5)



where βm = nm ω/c and modes are normalized such that Fm∗ Fm dx dy = 1. We need to convert this equation to the time domain, following the method outlined in Section 2.3.1. This is a difficult task without making some simplifications. First, we ignore linear coupling among the modes that can occur in most fibers because of fabrication imperfections. Second, the input pulse is assumed to remain linearly polarized with no excitation of the orthogonally polarized fiber modes. Third, we assume that spatial mode profiles do not change significantly over the pulse bandwidth and replace Fm (x, y, ω) with its value at the carrier frequency ω0 . Fourth, we expand βm (ω) in a Taylor series [see Eq. (2.3.23)] and replace nm (ω) with its value at ω0 . Noting that ω − ω0 is replaced with i ∂∂t in the time domain, we obtain (see Appendix C) ∞

ik βkm ∂ k Am ∂ Am iω0 −i(β0m z−ω0 t) −i = e k ∂z k! ∂ t 2 0 cnm k=1

 i ∂ × 1+ Fm∗ (x, y)xˆ · PNL (r, t) dx dy, (14.2.6) ω0 ∂ t

where βkm = (dk βm /dωk )ω=ω0 is the kth-order dispersion parameter for the mth mode of the fiber with the effective mode index nm .

Multimode fibers

At this point we need to specify the nonlinear polarization PNL (r, t). For ultrashort pulses, one must use the general form given in Eq. (2.3.32) so that both the Kerr and Raman contributions are included. We initially neglect the Raman contribution and use the form given in Eq. (2.3.6) together with electric field, E(r, t) = xˆ



Fm (x, y)Am (z, t) exp[i(β0m z − ω0 t)],

(14.2.7)

m

and retain only the terms oscillating at the carrier frequency ω0 . When the result is substituted into Eq. (14.2.6), we obtain ∂ Am ∂ Am iβ2m ∂ 2 Am + β1m + = i γ fmnpq An A∗p Aq eiβmnpq z , ∂z ∂t 2 ∂ t2 n p q

(14.2.8)

where the triple sum extends over the number of modes M supported by the fiber, βmnpq = β0n − β0m + β0q − β0p

(14.2.9)

is the phase mismatch, and we kept only the dominant first two dispersion terms on the left side. We also neglected the self-steepening term containing the time derivative on the right side of Eq. (14.2.6). The preceding equation extends the single-mode NLS equation to the multimode case and reduces to it when M = 1. The parameter γ is defined as in Eq. (2.3.30) using the effective core area Aeff of the fundamental mode. The right side of Eq. (14.2.8) includes all the nonlinear effects related to the Kerr nonlinearity (SPM, XPM, and FWM). The relative strength of various intermodal terms is governed by the dimensionless overlap factors defined as



fmnpq = Aeff

Fm∗ (x, y)Fn (x, y)Fp∗ (x, y)Fq (x, y) dx dy.

(14.2.10)

As f1111 = 1 by definition, fmnpq involving four modes represents strength of that nonlinear term relative to the SPM term of the fundamental mode. The triple sum on the right side of Eq. (14.2.8) takes into account intermodal nonlinear coupling among various fiber modes. The magnitude of this coupling depends on fmnpq , or on the overlap of the four modes participating in the corresponding FWM process. The form of Eq. (14.2.8) indicates that the SPM and XPM phenomena can be interpreted as degenerate FWM processes that are self-phase matched. For example, f1122 corresponds to an intermodal XPM process that is phase matched because β1122 = 0.

14.2.2 Few-mode fibers It is clear from Eq. (14.2.8) that nonlinear propagation in multimode fibers is much more complex compared to single-mode fibers. The degree of complexity depends on

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the number of modes excited by an input pulse, say M0 , which may be less than the total number M of modes supported by the fiber. Even when the linear mode coupling is ignored assuming an ideal fiber, one needs to solve M0 coupled NLS equations, each containing M03 nonlinear terms. Even when a single higher-order LP0m mode is excited using a device such as a spatial light modulator, some nonlinear terms in Eq. (14.2.8) couple to other modes of the fiber. However, these are not generally phase-matched, and the situation is similar to that of a single-mode fiber with an effective nonlinear parameter fmmmm γ . The case of few-mode fibers is interesting because it allows one to gain insight into the intermodal nonlinear effects by considering only two or three modes. The simplest case is that of a step-index fiber with 2.405 < V < 3.8 because such a fiber supports only the LP01 and LP11 modes (see Fig. 14.2). Since the LP11 mode is two-fold degenerate, let us denote its even and odd versions as modes 2 and 3, mode 1 being the fundamental mode. We ignore polarization degeneracy here, assuming that the optical fiber does not exhibit any birefringence. The first step is to find 81 overlap factors fmnpq resulting from 3 possible values of the 4 subscripts. Their magnitudes can be calculated using the mode profiles and carrying out the spatial integrals in Eq. (14.2.10). A numerical approach is necessary for step-index fibers, but approximate analytical expressions can be obtained for GRIN fibers [31]. Accurate numerical values has been provided in Ref. [37] for a specific GRIN fiber. It turns out that 60 out of 81 coefficients vanish. The remaining 21 elements take five distinct values. Three SPM coefficients are f1111 = 1 and f2222 = f3333 = 0.747 ≈ 3/4. The 12 intermodal XPM coefficients break into three groups: f1122 = f2211 = f1133 = f3311 = 0.496, f1212 = f2121 = f1313 = f3131 = 0.496, f2233 = f3311 = f2323 = f3232 = 0.249.

(14.2.11)

The remaining six coefficients represent intermodal FWM and have values f1221 = f2112 = f1331 = f3113 = 0.496,

f2332 = f3223 = 0.249.

(14.2.12)

These numerical values differ from 1/2 and 1/4 because the modes of the GRIN fiber were calculated more accurately by matching the boundary conditions at the core– cladding interface. As a simple example, we consider the two-mode case in which two input pulses at the same wavelength are injected into two specific LP modes of a few-mode fiber. Similar to the single-mode case, we use the reduced time T = t − β1p /z in a frame moving at the speed of the pth mode and obtain the following two coupled NLS equations:

   ∂ Ap iβ2p ∂ 2 Ap 2 2 2 ∗ 2iβ z + = i γ f | A | + 2f | A | A + f A A e , pppp p ppqq q p pqqp q p ∂z 2 ∂T 2

(14.2.13)

Multimode fibers

   ∂ Aq ∂ Aq iβ2q ∂ 2 Aq 2 2 2 ∗ −2iβ z + dqp + = i γ f | A | + 2f | A | A + f A A e , qqqq q qqpp p q qppq 1 2 ∂z ∂T 2 ∂T 2

(14.2.14) where β = β0q − β0p is the difference in the propagations constants of the two modes and dqp = β1q −β1p is the DGD parameter for them. These equations should be compared with those in Section 6.2 obtained for the two orthogonally polarized components of a single mode. Their forms are quite similar but the XPM and FWM coefficients of 2/3 and 1/3 are replaced with the nonlinear overlap factors involving the two spatial modes. If the two modes are not degenerate (β = 0) and the fiber length L is long enough that β L  1, the last term in Eqs. (14.2.13) and (14.2.14) does not contribute significantly because of its rapidly oscillating nature and can be neglected. In this specific case, these equations can be written as   ∂ Ap iβ2p ∂ 2 Ap 2 2 + = i γ f | A | + 2f | A | Ap , p q pp pq ∂z 2 ∂T 2

(14.2.15)

  ∂ Aq ∂ Aq iβ2q ∂ 2 Aq 2 2 + dqp + = i γ f | A | + 2f | A | Aq , qq q pq p ∂z ∂T 2 ∂T 2

(14.2.16)

where we used the contracted notation f pq ≡ fppqq . These equations should be compared to the coupled NLS equations in Section 7.2 for two pulses of different wavelengths propagating inside a single-mode fiber. Indeed, they reduce to them if all overlap factors are set to 1. The two-mode case provides physical insight into the relative importance of SPM, XPM, and FWM terms. If an input pulse excites more than two modes of the fiber, the number of nonlinear terms increases so rapidly that it is better to use the compact form given in Eq. (14.2.8).

14.2.3 Random linear mode coupling It is well known that a relatively long multimode fiber suffers from random linear coupling among its modes because of manufacturing variations in the shape and size of the fiber’s core along its length [33–35]. Such coupling also occurs if the fiber is bent or suffers from micro-bending. The extent of coupling between any two LP modes depends inversely on the difference in their propagation constants, δβ = βlm − βl m ; smaller is the difference, stronger becomes the coupling. For this reason, degenerate mode pairs (δβ = 0) are most strongly coupled in any multimode fiber. Random fluctuations in the fiber’s birefringence lead to coupling between the orthogonally polarized versions of the same LP mode because of their degenerate nature. Similarly, the even and odd versions of any LP mode with l > 0 become strongly coupled. Other mode pairs also exhibit some coupling depending on the magnitude of δβ for them. As fiber lengths can exceed hundreds or even thousands of kilometers for some telecommunication systems,

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the power launched into a specific fiber mode is distributed among many modes of a multimode fiber, resulting in the linear as well as nonlinear crosstalk. The linear mode coupling not only requires that the full set of 2M coupled NLS equations must be solved for a fiber supporting M spatial modes (two accounts for polarization degeneracy), but also that this set be solved hundreds of times with different seeds to include the random nature of this coupling. In this section we extend the coupled multimode equations to include random coupling among fiber modes by replacing n2 (x, y) in Eq. (14.2.2) with n2 (x, y) + δ (x, y, z), where δ  n2 is a perturbation that varies with z in a random fashion because of fluctuations in the core’s shape and size (or stress) along the fiber. Using the expansion in Eq. (14.2.1), the modes are still obtained from Eq. (14.2.3) but Eq. (14.2.4) is replaced with M 

2iβn

n=1

˜n ω2  ω2 ˜ ∂A + δ 2 eˆ n Fn (x, y, ω)eiβn z = − 2 P NL (r, ω). ∂z c 0 c

(14.2.17)

Multiplying the preceding equation with Fm∗ , integrating over the transverse plane, and using the orthogonal nature of various modes, the mode amplitudes are found to satisfy −iβm z ˜m ∂A ˜ n ei(βn −βm )z = iωe −i κmn A ∂z 2 0 cnm n



Fm∗ (x, y, ω)ˆe∗m · P˜ NL (r, ω) dx dy,

(14.2.18)

where the coupling coefficients are defined as iω κmn (z) = nn c



δ (x, y, z)Fm∗ (x, y, ω)Fn (x, y, ω) dx dy.

(14.2.19)

The next step is to convert Eq. (14.2.18) to the time domain. However, we can no longer ignore the polarization effects since orthogonally polarized modes are strongly coupled because of their degenerate nature. Thus, we must use the vector form of the nonlinear polarization given in Eq. (6.1.3) together with Eq. (6.1.4). A simple way to include the polarization effects is to employ Jones vectors for each mode by defining a column vector as Am = [Amx Amy ]T . The final result can then be written in a vector form as [34] ∂ Am ∂ Am iβ2m ∂ 2 Am + β1m + =i κmn An ei(β0n −β0m )z 2 ∂z ∂t 2 ∂t n iγ T ∗ iβmnpq z + fmnpq [2(AH , p An )Aq + (An Aq )Ap ]e

3

n

p

(14.2.20)

q

where the superscripts H and T denote the Hermitian and transpose operations on a Jones vector. This equation is appropriate for relatively wide (> 1 ps) pulses; higherorder dispersive and nonlinear terms must be added for shorter pulses. It can only be

Multimode fibers

solved numerically. However, such an approach requires considerable computational resources. Because of the random nature of the coupling coefficients κmn , the preceding stochastic set of multimode equations must be solved many times before the average behavior of the underlying physical process can be predicted. It is thus natural to ask under what conditions random coupling among fiber modes can be ignored. Linear mode coupling is a well-understood effect in the context of fiber couplers [5]. If we consider only two modes (m = 1, 2) excited by a weak CW beam, neglect the nonlinear terms and time derivatives in Eq. (14.2.20), and assume Am = Bm exp(−iδa z) with δa = (β01 − β02 )/2, we obtain the following set of two coupled linear equations: dB1 = iδa B1 + iκ12 B2 , dz

dB2 = −iδa B2 + iκ21 B1 . dz

(14.2.21)

It is relatively easy to solve these equations. Assuming B2 (0) = 0 initially, we obtain [5] B2 (z) = (iκ21 /κe )B1 (0) sin(κe z),

κe =

 κ12 κ21 + δa2 .

(14.2.22)

This equation shows that power is transferred to the second mode in a periodic fashion with maximum transfer occurring for the first time at a distance Lc = π/(2κe ) known as the coupling length. The fraction of power transferred, η = |κ21 /κe |2 , depends on δa and becomes 100% when the two modes have the same propagation constants (δa = 0). Clearly, η is small if two modes are weakly coupled because of a small value of κ21 . However, η also depends on δa and decreases as 1/δa2 with increasing δa . Noting that δa = (π/λ)δ n, where δ n is the difference in the effective mode indices of the two modes involved, one can show that η becomes small for δ n > 10−5 . This is the reason why mode coupling is the strongest for two degenerate modes (δ n = 0) and becomes negligible for δ n > 10−4 . As discussed in Section 14.1, the degeneracy of the LPlm modes of step-index fibers depends on the integer l; modes with l = 0 are two-fold degenerate and those with l > 0 are four-fold degenerate. This suggests that launching light selectively into an LP0m mode may help in avoiding mode coupling. As any LP0m mode is radially symmetric, its coupling to nearby modes with l = 0 is not very strong because of a relatively small overlap between them. Moreover δ n between two neighboring LP0m modes increases with m and can easily exceed 10−5 for m > 5. It was found in 2006 that light launched into the LP07 mode of a step-index fiber with 86-µm core did not suffer from mode coupling over a 12-m length [38]. Since then, this approach to avoiding mode coupling has been used extensively. In essence, even a multimode fiber acts as a single-mode fiber over lengths ∼10 m when light is launched into a high-order LP0m mode. As the effective mode area is relatively large for such fibers, the nonlinear parameter γ is reduced considerably. This feature is useful for making high-power fiber amplifiers and lasers [39].

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Random mode coupling cannot be avoided in mode-multiplexed communication systems in which multiple modes of fibers are intentionally excited to carry different data streams. Since fiber lengths can exceed hundreds of kilometers, all modes suffer from random coupling, and Eq. (14.2.20) must be used for modeling such systems. In this situation, an averaging procedure, similar to that used in Section 6.6 in the context of random birefringence in single-mode fibers, can be carried out for multimode fibers to obtain Manakov-like equations [33–35]. In the weak-coupling regime, one assumes that degenerate modes within each mode group are strongly coupled but no coupling occurs between the mode groups [34]. In the strong-coupling regime, all fiber modes are assumed to be equally coupled. The Manakov-like equations were first obtained in these two limiting cases. More recently, a more general approach was developed that provided such equations for all coupling regimes [40].

14.2.4 Graded-index fibers The coupled NLS equations given in Eq. (14.2.8) or (14.2.20) can be used for GRIN fibers by using the modal solution in Eq. (14.1.9) for calculating the nonlinear overlap factors fmnpq given in Eq. (14.2.10). However, a much simpler approach is available for a GRIN fiber whose core is wide enough that it supports a relatively large number of modes. When a pulsed Gaussian beam is launched into such a fiber, it excites many modes that propagate with different speeds and acquire different phase shifts. As a result, they interfere and create a spatial pattern in the transverse plane that changes considerably along the fiber length. Such a problem requires extensive numerical calculations if the multimode coupled NLS equations are used to solve it. However, the quadratic nature of the refractive index variations within the GRIN core simplifies the problem considerably. It turns out that this feature reduces the pulse-propagation problem to a single NLS equation that can be solved much faster numerically [41]. The basic idea is to solve Eq. (14.2.2) approximately with the variational method after including both the parabolic index profile given in Eq. (14.1.7) and the Kerr nonlinearity, but without paying attention to the fiber modes. As a further simplification, let us first consider a CW beam propagating along the z axis of a GRIN fiber, ignore its cladding completely, and assume that the CW beam remains confined to the fiber core because of the parabolic refractive index and Kerr self-focusing. We also neglect all polarization effects and assume that the CW beam remains polarized along the x axis. Using E˜ (r) = xB ˆ (r)eiβ0 z with β0 = n1 ω0 /c in Eq. (14.2.2) and assuming that B(r) is a slowly varying function of z, we obtain i

∂B 1 2 n2 ω0 2 ρ2 + ∇T B − β0  2 B + |B| B = 0, ∂ z 2β0 a c

(14.2.23)

where ∇T2 is the transverse part of the Laplacian operator. This equation was solved in 1992 with the variational method [42], assuming a Gaussian shape for the spatial

Multimode fibers

profile whose amplitude, width, and phase were allowed to change with z. The resulting solution can be written as B(r) = A0 F (r), where A0 is the amplitude at z = 0 and F (r) is given by F (r) =



(x2 + y2 ) w0 i φ( r ) . exp − + w (z) 2w2 (z)

(14.2.24)

Here φ(r) is a spatially varying phase whose functional form is known (but not needed) and w0 is the initial spot size at z = 0. The spatial width w evolves with z in a periodic fashion as w 2 (z) = w02 f (z),

f (z) = cos2 (π z/zp ) + Cg2 sin2 (π z/zp ).

(14.2.25)

The spatial period zp and parameter Cg in the preceding equation are defined as πa zp = √ , 2



Cg =

1 − pzp , πβ0 w02

(14.2.26)

where p is related to the initial amplitude A0 as p = n2 (β0 w0 A0 )2 /2n1 . The Kerr parameter n2 appears only through the dimensionless number p that is related to the beam collapse known to occur in any nonlinear Kerr medium [43]. The physical meaning of the CW solution in Eq. (14.2.24) is clear. It shows that a Gaussian CW beam, launched into a GRIN fiber at z = 0, maintains its Gaussian shape inside a GRIN fiber, but its amplitude and width (also phase) change in a periodic fashion such that it recovers its input shape periodically at distances z = mzp (m is a positive integer), a phenomenon known as self-imaging. At distances z = (m − 12 )zp , w (z) takes its minimum value Cg w0 , that is, Cg governs the extent of beam compression during each cycle. Typically zp < 1 mm for GRIN fibers. Using  = 0.01 with a = 25 µm, we find zp = 0.55 mm, a remarkably short distance. It is important to emphasize that self-imaging is a linear effect and occurs in GRIN fibers even in the absence of any nonlinearities. Eq. (14.2.25) shows that self-imaging is preserved even when the Kerr nonlinearity is included, as long as p < 1 so that Cg is a positive quantity. Indeed, p = 1 corresponds to the beam collapse induced by the Kerr nonlinearity [42]. In many cases of practical interest, p  1 at the power levels employed and we can replace 1 − p in Eq. (14.2.26) with 1. To apply the preceding CW solution to a pulsed Gaussian beam, we assume that the bandwidth of the pulse is narrow enough that the spatial profile F (r) of the beam does not vary with ω and write the solution of Eq. (14.2.2) at any frequency ω within the pulse bandwidth as ˜ (z, ω)F (r)eiβ(ω)z , E˜ (r, ω) = A

(14.2.27)

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where β(ω) = n1 (ω)ω/c is the propagation constant at that frequency. We can now follow the procedure used in Section 2.3 for single-mode fibers, while keeping in mind that F (r) is not the spatial profile of a specific mode but changes with z along the fiber length as indicated in Eqs. (14.2.24) and (14.2.25). After eliminating the transverse coordinates through a spatial integration and going back to the time domain, the final result can be written in the form [41] ∂A ∂ A iβ2 ∂ 2 A α + β1 + + A = iγ (z)|A|2 A, ∂z ∂t 2 ∂ t2 2

(14.2.28)

where the nonlinear parameter is defined as  γ (z) =

ω0 n2 , cAeff (z)

Aeff (z) =

2 ∞ 2 dx dy | F ( r )| −∞  ∞ . 4 −∞ |F (r)| dx dy

(14.2.29)

This is a remarkable result. It shows that the pulse evolution inside a GRIN fiber can be studied by solving a single NLS equation, even though multiple spatial modes may be propagating simultaneously inside the fiber. The oscillating spatial width of a Gaussian beam, resulting from the GRIN nature of the refractive index, gives rise to an effective nonlinear parameter γ (z) that is periodic in z. This is not surprising since the intensity at a given distance z depends on the beam width, becoming larger when the beam compresses and smaller as it spreads. One can also interpret the same effect as a spatially varying, effective mode area. The spatial integrals appearing in Eq. (14.2.29) can be performed analytically using the functional form of F (r) in Eq. (14.2.24). The result can be written in the form γ (z) = γ /f (z), where γ is defined as in Section 2.3, except that the initial value of Aeff at z = 0 is used to calculate it. As a final step, if we neglect fiber losses and use the reduced time T = t − β1 z, Eq. (14.2.28) takes the form ∂ A iβ2 ∂ 2 A + = iγ f −1 (z)|A|2 A, ∂z 2 ∂T 2

(14.2.30)

where f (z) is given in Eq. (14.2.25). This equation reduces to the standard NLS equation when Cg = 1, resulting in f (z) = 1. We use Eq. (14.2.30) in later sections to discuss the nonlinear phenomena in GRIN fibers.

14.3. Modulation instability and solitons The set of coupled NLS Equations given in Eq. (14.2.8) or in Eq. (14.2.20) can be used to study how the nonlinear effects such as SPM, XPM, and FWM affect the evolution of optical fields inside multimode fibers. The multiple nonlinear terms within the triple sum in these equations include all intermodal couplings induced by the Kerr nonlinearity. In this section we first discuss the phenomenon of modulation instability

Multimode fibers

and then investigate whether soliton-like propagation of optical pulses is feasible in multimode fibers.

14.3.1 Modulation instability We have seen in Section 5.1 how modulation instability, occurring in the anomalousGVD region of a single-mode fiber, produces temporal oscillations on a CW beam that are reshaped by the Kerr nonlinearity into a train of optical solitons. It is thus natural to wonder whether modulation instability occurs in multimode fibers [44–47] and whether temporal solitons can form in such fibers, in spite of different group velocities associated with different modes [48,49]. This section is devoted to answering such questions. Consider first the simplest case in which a CW beam excites only two modes of a few-mode fiber that do not couple linearly to other modes but are coupled nonlinearly. In this situation, Eqs. (14.2.13) and (14.2.14) govern the stability of the CW solution. If the last nonlinear term in these equations is negligible because of a relatively large phase-mismatch, then Eqs. (14.2.15) and (14.2.16) can be used to analyze the CW instability with the approach outlined in Section 7.2. The results show that modulation instability can occur in the normal-GVD region only when the XPM dominates over the SPM and requires the following condition on the overlap factors: 

2f pq > f pp f qq .

(14.3.1)

These parameters can be calculated for any multimode fiber and are different for different mode pairs. As a result, the spectral sidebands appear around the CW-beam wavelength only for specific mode pairs. In a 2017 study, performed using a 40-cmlong GRIN fiber of 11-µm core radius [47], spectral sidebands were observed for the mode pair involving the LP01 and LP02 modes because the condition (14.3.1) holds for this pair (f pp = 1, f qq = fpq = 1/2). The observed frequency shift was close to 10 THz at power levels of 50 kW for 400-ps input pulses used in the experiment. In contrast, the condition (14.3.1) was not satisfied for the (LP01 , LP21 ) mode pair. The general case of M-mode fibers has been analyzed including both polarization components of each mode [46]. The linear stability analysis of the CW solution was based on Eq. (14.2.20) but it neglected any linear coupling among the modes. The analysis involving 2M modes is cumbersome as it leads to a 4M × 4M matrix whose eigenvalues provide the gain spectrum of modulation instability. The numerical results reveal that, depending on the fiber parameters, up to 4M spectral sidebands can form in different spectral regions, covering a wide bandwidth on both sides of the CW pump. As an example, Fig. 14.6 shows the gain spectrum of the modulation instability occurring in a four-mode fiber with 12-µm core radius (V ≈ 5 at λ = 1.55 µm), assuming that the 4-kW pump power is distributed equally among the four x-polarized modes. The

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Figure 14.6 (A) Gain spectra of modulation instability for a specific four-mode fiber pumped with 4 kW power distributed equally among the four spatial modes. Parts (B) and (C) show magnified views covering different frequency ranges with labels marking different sidebands. (After Ref. [46]; ©2015 APS).

analysis included orthogonally polarized degenerate modes but ignored the odd versions of the LP11 and LP21 modes. The gain and frequency scales are normalized in Fig. 14.6 using the nonlinear length associated with the LP01 mode. All sidebands are shown in part (A), with two magnified views of the positive-side sidebands in parts (B) and (C). Both the peak gain and the bandwidth vary from one sideband to next. It is easy to conclude from this figure that the nature of modulation instability is quite complex in multimode fibers. The stability of a CW Gaussian beam inside a GRIN fiber has also been studied [50]. As discussed in Section 14.2.4, such fibers exhibit a self-imaging feature that results in periodic focusing of a Gaussian beam along the fiber’s length. Because of this feature, Eq. (14.2.30) can be used to find all features of modulation instability in GRIN fibers in an analytic form. A similar equation was encountered in Section 5.1.5 in the context of a fiber-optic communication link employing optical amplifiers periodically for compensating fiber losses. Since the peak power of optical pulses varies in a periodic fashion in such communication links, Eq. (14.2.30) with a different functional form of f (z) was used in 1993 to study modulation instability [51].

Multimode fibers

As discussed in Section 5.1.1, the stability of the CW solution is analyzed by perturbing the steady-state solution of Eq. (14.2.30) in the form



A = ( P0 + a) exp(iφNL ),

φNL = γ P0

z

f −1 (z) dz,

(14.3.2)

0

where a(z, t) is a small perturbation. Linearizing Eq. (14.2.30) in a, we obtain ∂a iβ2 ∂ 2 a iγ P0 =− + (a + a∗ ). ∂z 2 ∂ T 2 f (z)

(14.3.3)

We seek its solution in the form a = a1 e−iT + a2 eiT , where  is the modulation frequency and the amplitudes a1 and a2 satisfy: ∂ ak i i γ P0 = β2 2 ak + (ak + a∗3−k ), ∂z 2 f (z)

k = 1, 2.

(14.3.4)

Because of the periodic nature of f (z), the preceding coupled equations can be solved approximately by expanding f −1 (z) in a Fourier series as f −1 (z) =

∞

cm eimKz ,

m=−∞

cm =

1 zp



zp

f −1 (z)e−imKz dz,

(14.3.5)

0

where K = 2π/zp and zp is the self-imaging distance given in Eq. (14.2.26). Each term in this Fourier series can satisfy a phase-matching condition and results in one pair of sidebands located on opposite sides of the pump’s frequency. The frequencies at which the gain of each sideband pair becomes maximum are found to satisfy [51] 2m =

2π m 2c0 − , β2 zp β2 LNL

(14.3.6)

where m is an integer and the nonlinear length LNL = (γ P0 )−1 . The peak gain of the mth sideband depends on the Fourier coefficient cm and is given by gm = 2γ P0 |cm |. It is easy to show that c0 = 1/Cg . The other Fourier coefficients can also be computed numerically. The m = 0 sideband corresponds to the single-mode case discussed in Section 5.1. This sideband exists only if β2 < 0, i.e., GVD of the fiber must be anomalous at the pump wavelength. Eq. (14.3.6) shows that a CW Gaussian beam launched inside a GRIN fiber can become unstable to small perturbations, even when it propagates in the normal-GVD region of the fiber. The gain spectrum of modulation instability exhibits a rich structure with an infinite number of sideband pairs at frequencies that are not equally spaced (in contrast with the FWM sidebands). The peak gain of each sideband depends on the spatial pattern of the oscillating Gaussian beam through the function f (z). Since spatial

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variations play a crucial role, this instability is also known as a geometric parametric instability, or as a spatio-temporal instability [52–54]. The numerical values of the sideband frequencies can be estimated using typical values for commercial GRIN fibers. We saw earlier that the self-imaging period zp is 0.1 ps if we use β2 = −20 ps2 /km, a typical value near 1550 nm for silica fibers. As a result, q is a large number (q > 100) under typical experimental conditions. Physically, it represents the number of times the spatial beam width oscillates inside a GRIN fiber within one dispersion length. We saw in Chapter 5 that the dispersion length provides the scale over which solitons evolve [68]. Indeed, solitons cannot respond to beam width changes taking place on a scale of 1 mm or less when LD exceeds 10 cm. If we write the solution of Eq. (14.3.12) as U = U + u, where U represents average over one spatial period zp , the perturbations |u(ξ, τ )| induced by spatial beam-width variations remain small enough that they can be neglected (as long as q  1). In other words, the average dynamics of the soliton can be captured by solving the standard NLS equation i

∂ U 1 ∂ 2U 2 + N |U |2 U = 0, + ∂ξ 2 ∂τ 2

(14.3.14)

where N is the effective soliton order defined as 2

N = N 2 f −1 (ξ ) = N 2 c0 = N 2 /Cg ,

(14.3.15)

where we evaluated the integral in Eq. (14.3.5) to find c0 = 1/Cg . As discussed before, Cg represents the fraction by which the beam width shrinks at zp /2 before recovering its original width at z = zp . Clearly, Eq. (14.3.12) has a solution in the form of a fun damental soliton when we choose N = 1 or N = Cg . This solution exists for a wide range of input pulse widths as long as the peak power is adjusted to satisfy this soliton condition. Because of an oscillating beam width, the input peak P0 must be adjusted to make sure that N = 1 on average along the GRIN fiber. Fig. 14.9 compares the temporal and spectral profiles of a GRIN soliton (N = 1), after it has propagated a distance of 100LD inside a GRIN fiber using Cg2 = 0.2 and q = 100. These results were obtained by solving Eq. (14.3.12) numerically with the initial field U (0, τ ) = U0 sech(τ ), where U0 was chosen to ensure N = 1. On the logarithmic scale used here, perturbations induced by spatial oscillations are evident but their magnitude remains below the 50 dB level even after 100 dispersion lengths. Higherorder solitons for which both the spatial and temporal widths evolve periodically can also form inside GRIN fibers. As seen in Section 5.2.3 in the context of single-mode

Multimode fibers

Figure 14.9 Temporal and spectral profiles of a multimode soliton (N = 1) before (dashed line) and after it has propagated a distance of 100LD inside a GRIN fiber with Cg2 = 0.2 and q = 100.

fibers, the period of temporal oscillations is set by the dispersion length (soliton period z0 = π LD /2) and is much longer than the period zp of spatial-width oscillations. Numerical results indicate that higher-order solitons are less stable compared to the fundamental GRIN soliton as they remain intact only over distances ∼ 10LD [65]. They also undergo fission and breakup into multiple fundamental solitons if third-order dispersion is included in Eq. (14.3.12).

14.3.3 Solitons in specific fiber modes The single-mode solitons considered in Chapter 5 may also exist inside a multimode fiber when the input pulse is launched into a specific mode of the fiber that does not couple to any other mode linearly. An interesting question is whether two solitons (at the same or different wavelengths) can propagate stably inside a multimode fiber if they are initially launched into two distinct modes of a multimode fiber. This scenario first attracted attention during the 1980s and has remained of continued interest [70–73]. The simplest example of such a soliton pair is the vector soliton discussed in Section 6.5 in the context of birefringent single-mode fibers supporting two orthogonally polarized modes. As was discussed there, because of the XPM-induced coupling between the two polarization components, the two pulses shift their spectra in the opposite directions such that both of them propagate at a common speed, in spite of the DGD related to linear birefringence of the fiber. Another example is provided by the XPMpaired solitons of Section 7.3, where two pulses at different wavelengths were found to propagate inside a single-mode fiber at a common speed through the XPM-induced spectral shifts. These examples clearly suggest that two pulses launched into two different modes of a multimode fiber may also be able to overcome their modal DGD through the nonlinear intermodal coupling between them. To study the formation of

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Figure 14.10 (Evolution over 100 LD of two solitons launched into two lowest-order modes of a fiber with a DGD of 10 ps/km. (After Ref. [60]; ©2015 OSA.)

such coupled solitons, one must solve Eqs. (14.2.13) and (14.2.14) numerically. The general M-mode problem requires a solution of M coupled NLS equations, given in Eq. (14.2.8) and containing a large number of nonlinear intermodal terms. If orthogonally polarized modes are also involved together with linear coupling, the problem becomes quite complicated and requires the use of the set given in Eq. (14.2.20). Consider the simplest case of two pulses at the same central wavelength launched into two lowest-order modes of a step-index fiber. If both pulses are linearly polarized along the same direction and the fiber is short enough that linear mode coupling can be ignored, we can use Eqs. (14.2.13) and (14.2.14) to study whether they can propagate as two mutually supporting solitons. These equations differ from Eqs. (7.4.1) and (7.4.2) in two ways: the presence of a FWM-like term and different coefficients appearing in the XPM terms. If the FWM term can be neglected because of a relatively large phase-mismatch, we may expect each pulse to propagate as a soliton if the DGD is not too large and the peak powers are adjusted properly to satisfy the soliton condition N = 1. Numerical simulations have been used to verify this expectation [60]. Fig. 14.10 shows the evolution of two fundamental solitons in the LP01 and LPe11 modes, assuming a DGD of 10 ps/km between the two modes. The trajectory of the LP01 soliton (left side) would be vertical in the absence of second soliton. Because of nonlinear coupling, trajectories of both solitons tilt in such a way that they overlap temporally. This can only happen if the two solitons shift their spectra in the opposite directions and move

Multimode fibers

at a common speed. More specifically, mode 1 slows down while mode 2 accelerates just enough that the two solitons trap each other and propagate as one unit. However, this situation is not expected to remain stable over long distances. As seen in Fig. 14.10, each soliton sheds some energy in the form of dispersive waves. Moreover, energy is transferred continuously from the LP11 mode to the LP01 mode such that the former has lost more than half of its energy after 100 dispersion lengths. If the DGD between the two modes is larger, only a fraction of energy in mode 2 is trapped by the mode 1. For example, less than 10% of pulse energy is trapped when DGD exceeds 50 ps/km, and no trapping occurs for a DGD of 100 ps/km.

14.4. Intermodal nonlinear phenomena The existence of multiple modes within the same optical fiber adds a new dimension to the three nonlinear effects discussed in Chapters 8 to 10. In this section we consider four-wave mixing (FWM), stimulated Raman scattering (SRS), and stimulated Brillouin scattering (SBS) in multimode fibers. In each case, two or three pulses at different wavelengths propagate in different modes of the fiber, while interacting with each other through the intermodal nonlinear coupling induced by the Kerr nonlinearity of the fiber.

14.4.1 Intermodal FWM As we discussed in Section 10.3, the phase-matching condition in Eq. (10.3.1), kM + kW + kNL = 0,

(14.4.1)

plays a critical role in any FWM process. This condition shows that the impact of phase mismatch resulting from material dispersion (kM ) can be reduced through the waveguide (kW ) and nonlinear contributions (kNL ) to the phase-matching condition. It is much easier to satisfy Eq. (14.4.1) for multimode fibers because kW can be made negative when four waves participating in FWM are allowed to propagate in different fiber modes. Indeed, several early experiments during the 1970s employed multimode fibers for this purpose [74–80]. Although single-mode fibers were used mostly after the year 1980, FWM in multimode fibers has continued to attract attention [81–90]. The magnitude of kW depends on the choice of fiber modes in which four waves participating in the FWM process propagate. The eigenvalue equation (14.1.5) can be used for step-index fibers to calculate kW for the four LP modes participating in the FWM process. Fig. 14.11 shows the resulting |kW | as a function of the pump–idler frequency shift s for a few-mode fiber with a = 5 µm and V ≈ 6 at the pump wavelength of 532 nm. A dashed line shows the quadratic variation of kM with s using Eq. (10.3.6). Its intersection with the solid lines determines the frequency shift in a

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Figure 14.11 Phase-matching diagrams for (A) mixed-mode and (B) single-mode pump propagation. Solid and dashed lines show variations of |kW | and kM with frequency shift. Dotted lines illustrate the effect of increasing core radius by 10%. The four modes of the FWM process are indicated in the LPlm notation. (After Ref. [74]; ©1975 IEEE.)

Figure 14.12 Idler power as a function of wavelength generated by tuning the signal wavelength (upper scale). Far-field patterns corresponding to the two dominant peaks are shown as insets. (After Ref. [1]; ©1974 AIP.)

FWM experiment, assuming kNL is negligible. Two cases in Fig. 14.11 correspond to whether the pump power (A) is divided between two different fiber modes or (B) is launched only in the LP01 mode. As seen in part (B), frequency shifts are much larger in the latter case. The exact value of frequency shift is quite sensitive to various fiber parameters. The dotted lines show how much s changes with a 10% increase in the core radius of the fiber. In general, the phase-matching condition can be satisfied for several combinations of the fiber modes. In the 1974 demonstration of multimode FWM in silica fibers, pump pulses at 532 nm with peak powers ∼100 W were launched in a 9-cm-long fiber, together with a CW signal (power ∼10 mW) that was tunable from 565 to 640 nm [1]. FWM generated an idler wave in the blue region (ω4 = 2ω1 − ω3 ). Fig. 14.12 shows the observed idler spectrum obtained by varying ω3 . The five peaks correspond to different combinations of fiber modes for which phase matching was realized. The far-field patterns of the two dominant peaks clearly indicate that the idler was generated in different fiber modes

Multimode fibers

Figure 14.13 Observed spectra at the output of a 150-m-long multimode fiber as the peak power of 25-ps pump pulses is increased from (A) to (D). Supercontinuum formed (bottom trace) when peak intensity was increased to 1.5 GW/cm2 . (After Ref. [80]; ©1987 IEEE.)

for two different signal frequencies. In this experiment, the entire pump power was launched in the LP01 mode. As expected from Fig. 14.11, phase-matching occurred for relatively large frequency shifts (50 to 60 THz). In another experiment, the frequency shift was as large as 130 THz, a value that corresponds to 23% change in the pump frequency [78]. FWM with much smaller frequency shifts (νs < 10 THz) can occur when the pump power is divided between two different fiber modes (see Fig. 14.11). This configuration is also relatively insensitive to variations in the core diameter [74]. When s is close to 10 THz, the SRS process can interfere with the FWM process as this value is near the location of the Raman-gain peak (see Fig. 8.2). In a 1975 experiment in which 532-nm pump pulses with peak powers ∼100 W were sent through a multimode fiber, the Stokes band was always more intense than the anti-Stokes band because of its Raman amplification [74]. When the pump beam is in the form of a train of short optical pulses, the FWM process is affected not only by SRS but also by SPM, XPM, and GVD. In a 1987 experiment, 25-ps pump pulses were transmitted through a 15-m-long fiber, supporting four modes at the pump wavelength of 532 nm [80]. Fig. 14.13 shows the optical

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spectra observed at the fiber’s output as the pump’s peak intensity was increased from 0.1 to 1.5 GW/cm2 . As nothing else was launched in this experiment, the signal and idler waves buildup from noise through spontaneous FWM occurring at high pump powers. Indeed, only the pump spectrum is seen at the lowest pump power (trace A). Three pairs of FWM sidebands with frequency shifts in the range 1–8 THz appear spontaneously just above the FWM threshold of 0.5 GW/cm2 (trace B). As all sidebands have nearly the same amplitude as SRS does not play a significant role at this pump power. As pump power is increased further, the red-shifted lines become more intense than the blue-shifted lines as a result of Raman amplification (trace C). With a further increase in the pump power, the Stokes line closest to the Raman-gain peak becomes as intense as the pump itself, whereas the anti-Stokes lines are nearly depleted (trace D). At the same time, the pump and the dominant Stokes line exhibit spectral broadening and splitting that are characteristic of SPM and XPM (see Section 7.4). As the pump power is increased further, higher-order Stokes bands are generated through cascaded SRS. At a pump intensity of 1.5 GW/cm2 , the broadened multiple Stokes bands merge and form a supercontinuum extending from 530 to 580 nm (bottom trace). Note that solitons played no role here as GVD was normal (β2 > 0) in the entire spectral range. Supercontinuum generation is discussed further in Section 14.5. With the advent of photonic crystal fibers (PCFs) during the late 1990s (see Section 11.4), it was realized that such fibers could be used to realize FWM with large frequency shifts if they supported multiple modes. In a 2009 experiment, a spectral shift of 140 THz was realized by launching femtosecond pump pulses at 808 nm into the fundamental mode of a 20-cm-long PCF with a core diameter of 10-µm [81]. Both the signal and idler pulses appeared in higher-order modes of the fiber, and their spectra were located at wavelengths near 590 nm and 1250 nm. Moreover, intermodal FWM with such a large spectral shift (about 135 THz) was realized with the pump experiencing normal GVD inside the PCF. The spectral shift was even larger (175 THz) in a 2015 experiment in which the signal and idler wavelengths were close to 550 nm and 1590 nm, respectively, when 800-nm pulses were launched into a 28-cm-long PCF [83]. Also, the FWM efficiency was close to 20%. In a later experiment, cascaded intermodal FWM led to the generation of ultraviolet light at 376 nm when the signal at 538 nm became intense enough that it acted as a pump and created its own FWM sidebands in another mode of a multimode PCF [88]. A GRIN fiber was first used in 1981 for FWM [77]. The phase-matching condition (14.4.1) can be used to find the sideband frequencies in an analytical form for GRIN fibers using the relatively simple expression for the effective mode indices given in Eq. (14.1.14). In the degenerate case in which both pump photons come from the same mode group gp , the waveguide contribution is found to be [86] kW = −mg (π/zp ),

mg = gs + gi − 2gp ,

(14.4.2)

Multimode fibers

where gs and gi are the mode groups for the signal and idler waves, respectively. It follows that kW is negative if mg is a positive integer. The material contribution kM is given in Eq. (10.3.6). If we neglect the relatively small nonlinear contribution in Eq. (14.4.1), the frequency shift 2s satisfies a quadratic equation whose solution is given by [89] 2s

=

6 β4

 −β2 +

 β22

+ mg

πβ4

3zp

 .

(14.4.3)

When β4 is negligible, the shift from the pump frequency is given by a remarkably  simple expression: s = ± mg π/(β2 zp ) for the signal and idler sidebands. Here the integer mg > 0 is restricted to take only even values because of a constraint set by the conservation of angular momentum [86]. Even with this restriction, FWM can occur for several mode combinations when the pump propagates in the LP01 mode (gp = 1). For example, mg = 2 for the two (gs , gi ) combinations (1,3) and (3,1). Thus, one of the FWM sidebands must be generated in the LP02 mode belonging to the third group. The sideband frequencies in Eq. (14.3.7) found in the context of modulation instability coincide with the preceding s values because the two have the same origin (periodic self-imaging). However, the analysis of Section 14.3.1 assumed that the GRIN fiber supported a large number of modes and that the pump beam excited several of these modes simultaneously. The intermodal FWM requires that the pump propagate in a single mode or at most in two distinct modes. In a 2017 experiment, a short, 1-m-long, GRIN fiber was used to avoid random mode coupling. Its core diameter of 22 µm was chosen to ensure that it supported only a few mode groups [87]. The pump was at 1064 nm in the form of a 400-ps pulse train, and its spatial size was controlled to ensure that it excited mostly the LP01 mode. Fig. 14.14 shows the observed output spectra at four peak power levels ranging from 44 to 80 kW. As seen there, spontaneous intermodal FWM generated many sidebands, covering a wide spectral range from 400 to 1600 nm. The SRS also played a role at such high power levels. The four sidebands close to the pump result from non-degenerate FWM (pump in two different modes) and are affected by SRS. The most interesting feature of Fig. 14.14 is the narrow spectral line at 625 nm at the 44-kW pump level. This line is due to intermodal FWM, and its frequency shift of 181 THz corresponds to the choice mg = 1 in Eq. (14.4.2); its partner sideband lies in the mid-infrared region beyond 2.5 µm. As the pump power is increased, the 625-nm feature broadens, becomes more intense, and eventually generates its own set of FWM sidebands acting as a pump. Their frequencies can also be predicted from Eq. (14.4.2) by using gp = 1 because this wave was generated in the LP01 mode. Even though the 625-nm line can also be generated in the second mode group, the efficiency of that FWM process, governed by the extent of mode overlaps, is relatively small [86]. The beam profiles recorded at 433, 473, and 625 nm agree with this interpretation [87].

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Figure 14.14 Measured spectra at the end of a 1-m-long GRIN fiber at four peak power levels of 1064-nm input pulses. The 625-nm peak acts as a pump at high power levels and creates its own FWM sidebands that extend in the UV region. (After Ref. [87]; ©2017 OSA.)

In a 2018 experiment, a 1-m-long, GRIN fiber, whose 50-µm core supported a large number of modes, was used to observe intermodal FWM [89]. The spatial extent of the 1064-nm pump in the form of a 400-ps pulse train was controlled to ensure that its power was coupled mostly into the LP01 mode. The observed output spectra at peak powers ranging from 20 to 70 kW were qualitatively similar to those in Fig. 14.14, with multiple sidebands covering a spectral range from 400 to 2400 nm. The 625-nm peak seen in this figure moved to near 809 m because of a larger core size of the fiber. Since its idler sideband was located near 1555 nm, the FWM process could be seeded by injecting light at this wavelength from a CW diode laser. Seeding not only amplified the 1555-nm band (as expected) but also created new spectral peaks in the visible and infrared regions. In essence, the GRIN fiber was used as a parametric amplifier in the seeded mode of operation. Multimode parametric amplifiers require long fibers, and lengths ∼ 1 km were used in several studies [91–94]. A 5-km-long fiber supporting three spatial modes was used in a 2013 experiment to observe intermodal FWM [37]. The two pumps and the signal were launched into distinct fiber modes, and an idler wave was observed in the spectrum in two different pumping configurations. The FWM efficiency was relatively low (< 1%) in this experiment. The results could be explained using the coupled set in Eq. (14.2.20) that includes random linear mode coupling along the fiber length [92]. In a later experiment, the idler power was found to fluctuate considerably with the signal wavelength when two pumps at 1541 and 1542 nm wavelengths were launched into the LP11 mode [93]. A specific mode-coupling model based on fiber’s random birefringence was employed in another study to understand the process of parametric amplification [94]. The FWM efficiency depends on the relative magnitudes of the coherence length LC (related to random rotations of the fast and slow axes) and the beat length LB related to the magnitude of the birefringence. The so-called Manakov region

Multimode fibers

can be approached when LC is smaller than the shortest LB , resulting in better FWM efficiency. The idler power at the fiber output can exceed the injected signal power by up to a factor of 100 for fiber lengths > 1.5 km and pump power levels ∼ 0.2 W.

14.4.2 Intermodal SRS As discussed in Chapter 8, a Stokes pulse is generated though SRS when the pump pulse is intense enough to exceed the Raman threshold. Both pulses belong to the same mode in single-mode fibers, but they travel at different speeds because their spectra are shifted by 13.2 THz (as dictated by the Raman-gain spectrum). SRS in multimode fibers has also been studied extensively [95–108]. Such fibers offer the possibility that the Stokes pulse may appear in a mode different from that of the pump pulse but the two pulses propagate at nearly the same speeds inside the fiber. SRS in a higher-order mode of a step-index fiber (core diameter 100 µm, length 7.5 m) was observed as early as 1987 by launching 25-ps pump pulses into the fiber at the 532-nm wavelength such that pump energy was distributed among several modes of the fiber [97]. At energy levels below the SRS threshold (< 1 nJ), the output beam covered the entire fiber core and exhibited a speckle pattern resulting from multimode interference. However, at the energy level of 10 nJ, a narrow ring appeared within the speckle pattern. The 11-µm-diameter ring represented the spatial pattern of a higherorder mode in which the Stokes pulse was created through SRS at the Raman-shifted wavelength of 550 nm. This Raman process was relatively efficient because the ring contained about 50% of the input energy. It was argued that self-focusing induced by the Kerr nonlinearity played a role by creating a spatially varying index profile, but other explanations are also possible [100]. When an optical filter is used to block the pump, improved quality of the output beam realized through SRS is referred to as Ramanbeam cleanup [109]. Indeed, the use of GRIN fibers for this purpose has attracted considerable attention [105]. For sufficiently intense pump pulses, the Stokes pulse (red-shifted by 13.2 THz in silica fibers) may become intense enough that it acts as a pump and creates its own Stokes pulse that is red-shifted again by the same amount. This process can occur multiple times and is referred to as cascaded SRS; it results in the formation of new pulses at longer and longer wavelengths. In a 1988 experiment, 150-ps pump pulses at 1064 nm were injected into a 50-m-long GRIN fiber with a core diameter of 38 µm [98]. The output spectrum exhibited multiple spectral bands extending up to 1700 nm, each belonging to a different Stokes pulse and red-shifted by about 13.2 THz from the preceding band. The pulses whose wavelengths exceeded 1300 nm were narrower than the pump pulse because they experienced anomalous GVD and formed solitons. An investigation of cascaded SRS was carried out in 1992 by launching 3-ns pulses (at 585 nm) into a 30-m-long GRIN fiber with a core diameter of 50 µm [99]. A grating was employed to separate the pump and Stokes pulses and to record their spatial

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Figure 14.15 (A) Cascaded SRS peaks observed at the end of a 1-km-long GRIN fiber by launching 8-ns pulses at 523 nm with 22 kW peak power; (B) spectrum in the infrared region at higher pulse energies. Spectral power of the data in part (A) plotted on a log scale (right) as a function of the frequency shift. (After Ref. [107]; ©2013 AIP.)

Figure 14.16 Measured spatial profiles using a CCD-based beam profiler. Image (A) was obtained without any filter. The other 4 images were obtained using color filters centered at (B) 610 nm, (C) 700 nm, (D) 770 nm, and (E) 890 nm. (After Ref. [107]; ©2013 AIP.)

patterns at the fiber output. With careful adjustment of the launch conditions, four different Stokes pulses were found to propagate in different low-order modes of the fiber. Cascaded SRS over a much wider wavelength range was observed in a 2013 experiment by launching 8-ns pulses at 523 nm into a 1-km-long GRIN fiber with a 50-µm core diameter [107]. As seen in Fig. 14.15, the output covered a large wavelength range with many cascaded Stokes peaks that were separated by about 13 THz when plotted on a frequency scale. The spectrum also extended into the infrared region up to 1750 nm with a spectral dip around 1300 nm. The intermodal nature of SRS was verified through spatial profiles of different SRS peaks. The pattern (A) in Fig. 14.16 was obtained without any optical filter. Remaining four patterns were obtained by using color filters centered at wavelengths of the four specific peaks in Fig. 14.15; they correspond to different modes of the GRIN fiber used in the experiment. Two important questions are: (i) why a pump whose power may be distributed over many fiber modes ends up creating a Stokes pulse that propagates in a specific fiber mode, and (ii) how that mode is selected as the SRS process builds up from noise. In

Multimode fibers

general, one must consider amplification of noise in each fiber mode through SRS by all fiber modes in which pump power resides. In a simple approach, we can generalize Eq. (8.2.3) obtained for Raman amplification in single-mode fibers to the multimode case as [99]

Gm = exp gR (P0 /Aeff )L





f mn pn ,

(14.4.4)

n

where Gm is the amplification factor for the mth fiber mode, gR is the Raman gain coefficient at the pump wavelength, L is the fiber length, pn is the fraction of total pump power P0 in the nth mode, and f mn is the overlap factor obtained using Eq. (14.2.10):



f mn = fmmnn = Aeff

|Fm (x, y)|2 |Fn (x, y)|2 dx dy.

(14.4.5)

As before, Aeff is the effective mode area of the fundamental mode. The overlap factors depend on the fiber design and can be easily calculated for any fiber. Even though spontaneous Raman noise is amplified in all fiber modes, the mode for which Gm is the largest wins the race because of the exponential nature of the amplification process. Eq. (14.4.4) has been used to compare relative magnitudes of the modal Raman gain for different fibers [104]. It has also been used to explain why the Raman-beam cleanup occurs in GRIN fibers but not in step-index fibers [105]. When the pump is in the form of a short optical pulse, the effects of fiber dispersion begin to play an important role. As the group velocity in a fiber depends on the wavelength of a pulse, the Raman and Stokes pulses do not travel at the same speed even inside a single-mode fiber. As discussed in Section 8.3.3, energy transfer between the pump and Stokes pulses is reduced considerably because they separate from each other within a short distance when pump pulses are shorter than 10 ps. Indeed, no Stokes pulse is generated through SRS for femtosecond pump pulses propagating inside singlemode fibers. In its place, the spectrum of pump pulse undergoes a continuous red-shift through intrapulse Raman scattering (see Section 12.2). The situation is different in a multimode fiber because the speed of a pulse also depends on the mode in which it is traveling. As a result, a femtosecond pump pulse, traveling in one mode of the fiber, may undergo intermodal SRS and generate the Stokes pulse in another mode whose group velocity happens to nearly coincide with that of the pump, even if the modal Raman gain Gm is smaller for this mode. Indeed, this phenomenon was observed in an experiment [110] in which a 100-fs pump pulse was launched into the LP0,19 mode of a 12-m-long step-index fiber (core diameter 87 µm), and its energy was transferred to several lower-order LP0,m modes in a cascaded fashion such that m = 15 at the end of the fiber. The dynamical details of this process require multimode numerical simulations because one must consider many dispersive and nonlinear effects, including intrapulse Raman scattering that red-shifts the spectrum

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Figure 14.17 Numerically simulated spectra at the end of a 20-m-long step-index fiber for pulses of different widths, launched initially into the LP0,19 mode at the 1100-nm wavelength. Pulse width varies from 60 to 100 fs in steps of 5 fs. (Courtesy of Aku Antikainen)

of the pulse before it jumps to the next mode [111]. As an example, Fig. 14.17 shows the numerically simulated spectra in the same step-index fiber using pump-pulse widths in the range 60 to 100 fs. Input pulses at 1100 nm excite the LP0,19 mode and propagate a fundamental soliton inside the fiber. For the 100-fs pump pulse, three spectral peaks belong to three different fiber modes. They originate from an intermodal SRS process in which energy of a pulse is transferred to both the LP1,18 and LP0,18 modes depending on how closely their group velocities match with that of the LP0,19 mode. For pump pulses shorter than 80 fs, which red-shift faster through intrapulse Raman scattering, most of the energy is transferred directly to the LP0,18 mode. This behavior can be understood from Fig. 14.18 where the group index (ng = c /vg ) is shown as a function of wavelength for the three modes involved in this intermodal process. As we saw in Section 8.2.3, SRS can be used for Raman amplification if a Stokes pulse is launched into a single-mode fiber together with the pump. The use of multimode fibers for Raman amplification has attracted attention in recent years [112–116]. In the multimode case, pump and Stokes pulses may propagate in the same mode or may be launched into two different modes. Both cases can be studied by solving Eqs. (14.2.15) and (14.2.16), after modifying them to include the Raman gain. One way to do so is by making the Kerr nonlinearity n2 a complex quantity such that its imaginary part is related to the Raman susceptibility [101]. A more accurate approach should follow the analysis in Section 8.3 and modify Eqs. (8.3.1) and (8.3.2) with the

Multimode fibers

Figure 14.18 Effective group index as a function of wavelength for the three modes involved in Fig. 14.17; their shapes are shown in the insets.

appropriate overlap factors to obtain ∂ Ap iβ2p ∂ 2 Ap gp + = iγp [f pp |Ap |2 + (2 − fR )f ss |As |2 ]Ap − f ps |As |2 Ap , (14.4.6) 2 ∂z 2 ∂T 2 ∂ As ∂ As iβ2s ∂ 2 As g s −d + = iγs [f ss |As |2 + (2 − fR )f pp |Ap |2 ]As + f ps |Ap |2 As , (14.4.7) ∂z ∂T 2 ∂T 2 2

where T = t − z/vgp and d = β1p − β1s is the DGD between the modes p and s in which the pump and Stokes pulses are propagating. The parameter d accounts for the group-velocity mismatch between the pump and Stokes pulses. The Raman-gain coefficients are different for the two pulses because of their different carrier frequencies. The preceding equations can be used to study Raman amplification for any mode pairs by employing the corresponding overlap factors. They still need to be modified for femtosecond optical pulses by including higher-order dispersion terms and intrapulse Raman scattering through the Raman response function. An important conclusion can be drawn by comparing Eqs. (14.4.6) and (14.4.7) with Eqs. (8.3.1) and (8.3.2) of Section 8.3 for single-mode fibers. In particular, we can understand the detrimental role played by the DGD from Fig. 8.11, where only a fraction of pump energy is transferred to the Stokes pulse because of the walk-off effects that separate it from the pump pulse within a short distance. In the case of multimode fibers, the parameter d can be close to zero if the GVD of the fiber helps in canceling the DGD between the two fiber modes in which the pump and Stokes pulses are propagating (see Fig. 14.17). As a result, entire pump energy can be transferred to the Stokes pulse for this specific mode pair, resulting in an efficient amplification process.

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Eqs. (14.4.6) and (14.4.7) are not suitable for fibers long enough that linear mode coupling cannot be ignored. Theory of Raman amplification in multimode fibers with random mode coupling was developed in 2013 by considering the Raman gain for both co-polarized and cross-polarized pump and Stokes pulses [112]. It can be simplified considerably by recalling that fiber modes can be divided into multiple groups of nearly degenerate modes. One can then assume that modes within each group are strongly coupled but modes in two different groups do not couple linearly, and use it to obtain a set of equations describing how the pump and Stokes powers evolve in different mode groups. Such an approach has been used to optimize the gain of a multimode Raman amplifier [113]. Raman amplification is often used to compensate for fiber losses in fiber-optic communication systems [69]. In this case, the same fiber that is used to transmit information is also used for distributed Raman amplification by pumping it periodically, resulting in long amplifier lengths (60 to 100 km) over which random linear mode coupling cannot be ignored. In a 2016 experiment, this scheme was used to transmit data over 1050 km of a six-mode fiber by pumping it bidirectionally every 70 km [114]. The pump power was launched into the LP11 mode group (four-fold degenerate) because the same Raman gain could be realized in all six fiber modes in this configuration. The experiment showed that fiber losses could be fully compensated for all modes through Raman amplification inside a six-mode fiber.

14.4.3 Intermodal SBS As discussed in Chapter 9, a backward propagating Stokes wave is generated though SBS in single-mode fibers when the pump is intense enough to exceed the Brillouin threshold. Its frequency is down-shifted from that of the pump by about 11 GHz when pump wavelength is near 1550 nm. This Brillouin shift corresponds to the frequency of an acoustic wave that accompanies the SBS process. The gain spectrum of SBS can exhibit multiple narrow-bandwidth peaks that correspond to different acoustic modes of the fiber [117]. In the case of a multimode fiber, launched pump power is generally divided into multiple modes, all of which can participate in generating the Stokes through excitation of multiple acoustic modes of the fiber [118–128]. This affects the pump power level at which the SBS threshold is reached. A theoretical model was developed in 2001; it predicted that the SBS threshold increases in multimode fibers by a factor that depends on the numerical aperture, but not on the core diameter of the fiber [118]. In a non-modal approach, increase in the SBS threshold was attributed to inhomogeneous broadening of the Brillouin-gain spectrum. A simple model was able to fit the experimental dependence of the gain bandwidth on fiber’s numerical aperture [119]. This model was later extended to predict the SBS threshold of GRIN fibers [121]. Similar to the case of single-mode fibers, SBS threshold depends inversely on the width of pump pulses.

Multimode fibers

In a 2003 study the threshold power increased by a factor of eight as the pulse width decreased from 80 to 20 ns [120]. In a more accurate calculation of the SBS process, the analysis of Section 9.4 is extended to the case of multimode fibers [126–128]. In this approach, Eq. (9.4.1) is solved by using modal expansion for the pump and Stokes fields in the form E(r, t) = xˆ Re



Fk (x, y, ωp )Ak (z, t) exp[iβk (ωp )z − iωp t]

k

+





Fk (x, y, ωs )Bk (z, t) exp[−iβk (ωs )z − iωs t] ,

(14.4.8)

k

where the sum extends over all fiber modes with the propagation constants βk (ω). A similar expansion of the material density in terms of the acoustic modes of the fiber takes the form [127] kl 



N

ρ  (r, t) = Re

Wklu (x, y)Qklu (z, t) exp(ikA z − it) ,

(14.4.9)

k,l u=1

where Wklu (x, y) is the spatial profile of the acoustic mode with the amplitude Qklu , frequency  = ωp − ωs , and propagation constant kA . For each pair of the pump and Stokes modes, several acoustic modes may contribute. This is indicated by the sum over u whose upper limit Nkl denotes the number of acoustic modes involved in that specific pair. Clearly, such a modal analysis of SBS is cumbersome. However, if we assume CW pumping, it eventually leads to the coupled power equations in the form [127] kl d |Ak |2 = Gklu |Ak |2 |Bl |2 , dz l u=1

(14.4.10)

kl d|Bl |2 − = Gklu |Ak |2 |Bl |2 , dz k u=1

(14.4.11)

N

N

where the intermodal Brillouin gain Gklu , resulting from a specific acoustic mode, is given by Gklu () =

B ( /2)2 |f |2 gklu klu klu . ( − klu )2 + (klu /2)2

(14.4.12)

Here klu is the frequency and klu is the damping rate of the acoustic mode with the B , and f peak Brillouin gain gklu klu is the overlap factor defined similar to Eq. (14.2.10) and

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is given by



fklu = Aeff

∗ Fk (x, y, ωp )Fl∗ (x, y, ωs )Wklu (x, y) dx dy.

(14.4.13)

The physical meaning of Eqs. (14.4.10) and (14.4.11) is quite clear. They show that the pumps in all fiber modes contribute in building up the Stokes signal in any fiber mode, but the amount of SBS gain depends on the acoustic modes involved and the overlap factors fklu . If we neglect pump depletion, the net Brillouin gain for the Stokes propagating in a specific fiber mode is given by glB () =

Nkl

Gklu ()(Pk /P0 ),

(14.4.14)

k u=1

where Pk = |Ak |2 is the pump power in the kth mode and P0 is the total pump power. Since each Gklu has a Lorentzian spectral profile, glB is a superposition of several such profiles, with the fraction of power in that mode acting as a weighting factor. Even when the pump is propagating in only one mode, the Brillouin-gain spectrum may have more than one peak resulting from the sum over u in Eq. (14.4.14). When a pump beam is launched into the fiber, the Stokes signal builds up from noise and the overlap factors fklu play an important role in selecting the mode in which the Stokes wave propagates in the backward direction. It turns out that fklu is the largest when l in Eq. (14.4.14) corresponds to the LP01 mode of the fiber, irrespective of how the pump power is distributed among the fiber modes. As the Stokes power builds up exponentially from a noise seed, the mode with the largest gain wins the competition and carries the Stokes wave generated through SBS. It was noticed in several experiments that SBS occurring inside multimode fibers can be used to cleanup the spatial profile at the fiber’s output end [129,130]. SBS can also be used to combine several lowquality pump beams into one high-quality Stokes beam [131]. Indeed, SBS-induced beam cleanup and beam combining are important SBS applications. Another useful feature of SBS is that the Stokes beam is often phase-conjugated, i.e., its spatial phase is negative of that of the pump [132]. This feature has been known since 1978 for SBS occurring inside multimode waveguides [133]. The Brillouin gain given in Eq. (14.4.14) has been measured through amplification of an input Stokes signal inside fibers supporting a few modes. A 2013 experiment employed a 100-m-long fiber with an elliptic core that supported only the LP01 and LP11 modes [124]. The pump power from a 1550-nm laser was coupled selectively into one of these modes and the SBS gain spectra were recorded for all four possible mode combinations. A two-peak spectrum with 30-MHz spacing was recorded only when both the pump and Stokes were in the LPe11 mode. In a later experiment, a circular-core fiber (a = 5.09 µm,  = 1.15%) supporting four mode groups was used and gain spectra

Multimode fibers

Figure 14.19 Measured intermodal gain spectra for SBS occurring inside a step-index fiber supporting LP01 , LP11 , LP21 and LP01 mode groups. Thin lines show fitting three Lorentzian profiles used to fit the data. (After Ref. [125]; ©2013 OSA.)

were recorded for all mode combinations [125]. Fig. 14.19 shows the measured spectra in four cases. As seen there, spectra exhibited more than one peak and could be fitted by superimposing three Lorentzian profiles whose center frequencies varied from 10.25 to 10.5 GHz with bandwidths ranging from 33 to 39 MHz. Numerical estimates show that the Brillouin shift does not vary much for intermodal SBS processes involving Stokes generation into the LP01 mode from a pump in the LP0m mode, where m is an integer [123]. This is the reason why a Gaussian pump beam that couples its power into multiple LP0m modes ends up creating a single Stokes wave confined to the LP01 mode.

14.5. Spatio-temporal dynamics In this section we focus on the dynamic interplay between the spatial and temporal features when a pulsed optical beam is launched into a multimode fiber. As mentioned earlier, this problem requites solving Eq. (14.2.23), which couples the spatial and temporal evolution through the Kerr nonlinearity. This is so because the same Kerr coefficient n2 that leads to SPM of optical pulses in time also provides self-focusing in the transverse

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Figure 14.20 Spatial beam profiles from a 20-cm-long GRIN fiber at six energy levels of 80-fs input pulses; black bar is 13 μm wide. Two images on the right are for 0.9 ns pulses inside a 12-m-long GRIN fiber; black bar is 10 μm wide. (After Ref. [135] ©2016 OSA.)

spatial dimensions. Such spatio-temporal dynamics, studied for bulk Kerr media in the past [58], has also been observed in the context of multimode GRIN fibers [134–137].

14.5.1 Spatial beam cleanup As we saw in the preceding section, both SRS and SBS can be used to cleanup the spatial profile of a pump beam at the output of a multimode fiber. It is thus natural to ask whether the Kerr nonlinearity itself can be used for this purpose, resulting in Kerr-induced self-cleaning. The term ‘beam cleanup’ generally implies that the optical beam exiting at the output end of the fiber has a smooth spatial intensity profile with a single peak. A 20-cm-long GRIN fiber with a 62.5-µm core diameter was used in 2015 for studying Kerr-induced self-cleaning [135]. When 80-fs pulses at 1030 nm were launched into this fiber such that their energy coupled into multiple modes of the fiber, near-field images of the fiber’s output indicated that the spatial pattern changed considerably as pulse energy was increased. Fig. 14.20 shows six images for energies ranging from 0.44 to 46.6 nJ. At the highest energy level, the Kerr nonlinearity leads to a smooth intensity profile with a nearly circular spot. In the time-domain, the pulse at this energy level was only 34 fs wide with a broad spectrum. In another experiment, much longer pulses (width 0.9 ns) at 1064 nm were launched into a 12-m-long GRIN fiber [136]. The 40-µm-width of the input beam was wide enough to excite nearly 200 modes of this fiber. Two images taken at the fiber’s output end are shown on the right in Fig. 14.20. At the low peak power of 3.7 W, nonlinear effects are negligible, and the beam exhibits a speckle pattern. When the peak power of input pulses is increased to 1.2 kW, the pattern becomes smoother and narrower with a single central peak. These results show that the Kerr-induced beam cleanup occurs over a wide range of pulse widths. Also, soliton effects play little role in this process because the fiber’s GVD was normal for most modes at the wavelengths used in the two experiments. The beam’s cleanup indicates intermodal energy transfer. Numerical results

Multimode fibers

Figure 14.21 Output beam diameter as a function of the peak power of 60-ps input pulses launched into a 11-m-long fiber. The images on the top and bottom show spatial patterns for the pump and Stokes beams at few input power levels. (After Ref. [139] ©2018 OSA.)

also show a nonlinear transfer of energy from higher-order modes to the fundamental mode [60]. The use of a GRIN fiber is not essential for observing Kerr self-cleaning. A Ybdoped step-index fiber with the 50-µm core diameter was used in one experiment. It was found that the beam quality improved dramatically beyond a certain peak power of 0.5-ns input pulses [137]. This fiber acted as an amplifier when it was pumped suitably. The self-cleaning threshold was near 40 kW for the unpumped fiber (because of high losses), but it was reduced to below 0.5 W when the fiber was pumped to provide 23-dB gain. A multicore Yb-doped fiber was used in another experiment to show that the spatial shape of the amplified beam can be controlled through the nonlinear effects [138]. The competition between the Kerr- and Raman-induced beam cleaning has also been studied using a microstructured fiber whose 30-µm silica core was surrounded with a holey cladding [139]. The beam diameter at the output of the 11-m-long fiber was measured as a function of the peak power of 60-ps input pulses at 1064 nm. The results are shown in Fig. 14.21 together with the observed beam patterns at several power levels. The pump’s beam diameter shrinks initially due to Kerr self-cleaning, takes a minimum value at the 18-kW peak power where the SRS threshold is reached,

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and increases once a fraction of the pump’s energy is transferred to the Stokes beam. The diameter of the Stokes beam reaches a minimum value of about 10 µm at a peak power near 70 W and then begins to increase after this beam generates the second-order Stokes through a cascaded Raman process. The physical mechanism behind spatial beam cleanup occurring inside multimode fibers has two origins. First, self-focusing must play a role because it reduces beam size even in a bulk medium, and it leads to beam collapse if the input power is large enough. In most experiments power levels remained below this collapse but some self-focusing of the beam did occur. Second, different propagation constants of fiber modes lead to spatial beating. The resulting intensity modulations along the fiber length are converted into index modulations by the Kerr nonlinearity. This effect is the strongest in GRIN fibers, where equal spacing of modal propagation constants results in beam-width oscillations with a period of < 1 mm. The Kerr nonlinearity converts these oscillations into periodic variations of the refractive index that acts as a grating along the fiber length (period about 0.5 mm). Such a grating helps in transferring energy from higherorder modes to the fundamental mode. This is the reason why the threshold of Kerr self-cleaning is lower for GRIN fibers.

14.5.2 Supercontinuum generation Although two-mode fibers were considered earlier in the context of fiber’s birefringence [140–142], it was only after 2008 that supercontinuum generation in multimode fibers attracted attention [143–154]. As intrapulse Raman scattering plays an important role in this process (see Section 13.3), the set of multimode NLS equations in Eq. (14.2.20) must be generalized to include the delayed Raman response of optical fibers. For this purpose we extend the vector theory of Section 8.5.1 to the multimode case [30,112]. Only the nonlinear term in Eq. (14.2.20), containing the triple sum changes. ˆ it takes the form If we denote this term by N, Nˆ =

∞  iγ iβmnpq z    fmnpq e 2Aq (t) R(t )[AH p (t − t )An (t − t )]dt 3 n p q 0 + A∗p (t)



∞

0





R(t )[ATn (t − t )Aq (t − t )]e2iω0 t dt ,

(14.5.1)

where R(t) is the nonlinear response function (see Appendix B). The form of R(t) reduces to that in Eq. (2.3.36) if we ignore the cross-polarization Raman gain. One more change is often made in Eq. (14.2.20) before solving it numerically. The exponential term containing the phase-mismatch βmnpq in Eq. (14.5.1) oscillates rapidly along the fiber. It can be removed by defining Bm = Am exp[i(β0m − β0r )z],

T = t − β1r z,

(14.5.2)

Multimode fibers

where β0r and β1r serve as reference values. In practice, the fundamental mode of the fiber is used for setting these reference values. In terms of Bm , Eq. (14.2.20) takes the form ∞ ∂ Bm ∂ Bm ik βkm ∂ k Bm + (β1m − β1r ) − = i (β − β ) B + i κmn Bn ei(β0n −β0m )z 0m 0r m ∂z ∂T 2 ∂T k n k=2

∞  iγ    + fmnpq 2Bq (t) R(t )[BH p (t − t )Bn (t − t )]dt

3

n

p

q

+ B∗p (t)

0



∞

0





R(t )[BTn (t − t )Bq (t − t )]e2iω0 t dt ,

(14.5.3)

where we also included higher-order dispersion terms through a sum in the third term. This equation incorporates all important dispersive and nonlinear effects and is appropriate for studying supercontinuum generation in multimode fibers. If a fiber is short enough that linear mode coupling is not a major factor, one can set κmn = 0 in Eq. (14.5.3) to simplify it. The equation resulting from this simplification has been solved numerically to study the nonlinear intermodal effects in few-mode fibers. Typical simulations consider femtosecond pulses launched in the anomalous-GVD region of the fiber such that their energy is carried by the fundamental mode alone, or by the first two or three modes. When only the fundamental mode is excited, energy is transferred to the orthogonally polarized version of this mode but not to higher-order modes [146]. When more than one mode is exited initially, other LP0m modes may also be excited, but the fraction of energy transferred to them remains relatively small [143]. Experiments have shown that when a femtosecond pulse is launched into a fewmode optical fiber, some of the spectral bands within the supercontinuum propagate in higher-order modes of the fiber [144–147]. In all cases, the soliton effects play a dominant role, and supercontinuum generation evolves in a fashion similar to that in single-mode fibers. Starting in 2013, multimode GRIN fibers (core diameter 50 µm) were used in several experiments. The input beam diameter was kept large enough to excite multiple modes of the fiber simultaneously. In one experiment, 0.5-ps pulses at 1550 nm were launched into a 1-m-long GRIN fiber [134]. The output was in the form of a supercontinuum extending from violet to mid-infrared regions. Moreover, it exhibited multiple peaks in the visible region that were not equally spaced. The results are shown in Fig. 14.22 together with the numerical simulations [148]. The spectral peaks seen there have the same origin as those in Fig. 14.7. They originate from periodic spatial oscillations of the beam width inside GRIN fibers discussed in Section 14.2.4 and constitute an example of spatio-temporal coupling in such fibers [41]. Physically, narrow spectral peaks in Fig. 14.22 result from a Kerr-induced index grating (created by periodic oscillations of the peak intensity along the fiber) that helps

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Figure 14.22 Numerical (top) and experimental (bottom) spectra at the end of a GRIN fiber for 0.5-ps input pulses launched at 1550 nm. (After Ref. [148] ©2015 APS.)

in phase matching of dispersive waves (see Section 12.1.2). Their frequencies can be calculated after modifying the phase-matching condition given in Eq. (12.1.6) as β2

2

2 +

β3

6

3 +

β4

24

4 = δβ1  +

c0 2π m γ Ps + , 2 zp

(14.5.4)

where the last term represents the contribution of the Kerr-induced index grating. The integer m can take any positive or negative value. As before, c0 = 1/Cg with Cg given in Eq. (14.2.26). If we keep only the β2 term and assume δβ1 = 0, the frequency shift  can be calculated from 2 =

1/Cg 4π m + , β2 zp β2 LNL

(14.5.5)

compared where LNL = (γ Ps )−1 is the nonlinear length. In practice, zp is much smaller  to LNL , and the first term dominates. In this approximation, T0 = ± 4π |m|q, where q = LD /zp . The effects of higher-order dispersion can be included through Eq. (14.5.4). The results in Fig. 14.22 were obtained using femtosecond pulses at 1550 nm. In two 2016 experiments [150,151], much longer pulses (close to 1 ns) from a Nd:YAG laser operating at 1064 nm were launched into GRIN fibers (core diameter 50 µm) of lengths up to 30 m. Fig. 14.23 shows the observed supercontinua for 12 and 30 m long fibers [151]. The supercontinuum extends from 500 to 2000 nm in the shorter fiber, with multiple peaks in the visible reason. It extends beyond 2500 nm in the longer fiber and becomes smoother as the peaks near 600 nm disappear. The peak power of Q-switched pulses exceeded 30 kW at the power levels used in this experiment. The important question is how the pulse spectrum evolves inside the fiber. This evolution can be easily simulated numerically but is difficult to observe experimentally.

Multimode fibers

Figure 14.23 Experimental spectra observed at the end of 12 m (top) and 30 m (bottom) long GRIN fibers using 0.9-ps input pulses at 1064 nm. Symbols G and G’ mark the locations of modulationinstability sidebands. (After Ref. [151] ©2016 OSA.)

One way is to employ multiple fibers of different lengths but it is difficult to maintain the same experimental conditions, including coupling to the fiber. In a better approach, a single fiber is used but its length is cut back in discrete steps, while recording output spectra each time. The results show that the spectrum broadens initially through SPM and SRS around the pump wavelength, but soon several widely separated sidebands appear on both sides of the pump in the visible and infrared regions [150,151]. Their origin lies in modulation instability discussed in Section 14.3.1 that sets in because of the relatively wide pump pulses used in the experiments. The frequencies of these sidebands agree with the predictions of Eq. (14.3.7). The sidebands nearest to the pump are shifted by about 125 THz from the pump’s central frequency (close to 285 THz) and appear at wavelengths near 730 nm and 1900 nm (marked as G and G’ in Fig. 14.23). These bands interact with each other and with the pump through XPM and FWM, and are also intense enough to create their own Stokes peaks through SRS. The net results is that more and more frequencies are generated nonlinearly and a supercontinuum begins to form if the GRIN fiber is long enough. As seen in Fig. 14.23, a broader and smoother supercontinuum is observed when the GRIN fiber is 30 m long. It should be clear from the preceding discussion that the physical mechanism behind the supercontinuum generation in GRIN fibers is quite different from those discussed in Sections 13.4 and 13.4, where anomalous GVD was required at the pump wavelengths for soliton effects to play a dominant role. The spectra in Fig. 14.23 were obtained using relatively long pulses (close to 1 ns) at a wavelength of 1064 nm, where GVD is normal for the low-order modes. In this situation, modulation instability occurring in the normal-GVD regime (because of periodic self-imaging of the beam) plays the dominant role. To quantify the spectro-temporal dynamics of the underlying physical process, the spectrogram shown in Fig. 14.24 was constructed experimentally at the end

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Figure 14.24 Experimental spectrum (top) and its corresponding spectrogram at the end of a 6-mlong GRIN fiber. Symbols mark the location of various spectral peaks; P corresponds to the input pump pulse. (After Ref. [151] ©2016 OSA.)

of a 6-m-long fiber by dispersing its output with a grating. It shows the temporal extent of different spectral peaks marked by the letters on the top. One can understand the results in Fig. 14.24 as follows. Peaks A to G correspond to the modulation-instability sidebands in the visible region. They represent pulses whose temporal extent is in the range of 100 to 200 ps and whose positions are shifted from the center of input pulse (T = 0) because of the wavelength dependence of the group velocity. The peak P corresponds to the pump pulse itself. Clearly, pulse wings remain nearly intact because of their relatively low intensity, but energy from the central part has shifted to three Stokes bands (R, S, and L) at longer wavelengths. These peaks represent a cascaded SRS process. Energy from the central part of the pump pulse is initially transferred to the first Stokes band, which creates the second and third sidebands in a cascaded fashion. Notice also that the G peak has created its own Stokes band. One can conclude that SRS plays an important role in producing the supercontinua seen in Fig. 14.23. To identify the role played by Kerr or Raman beam cleanup, it is important to consider spatial shapes of different spectral components. This can be easily done experimentally by imaging the output, after passing it through narrow-band spectral filters centered at different wavelengths [150,151]. The results show that the mode profiles at all wavelengths are relatively smooth (bell shaped), with a central bright spot and without any speckle-like structure. The Kerr beam cleanup occurring at the pump wavelength appears to apply across the entire bandwidth of the supercontinuum. These results suggest that the GRIN fibers can be used to make broadband sources that are spatially coherent. Because of their relatively large core area, such fibers can be pumped with high pump powers, resulting in a much brighter broadband source.

Multimode fibers

Figure 14.25 Top: simulated spectra out of a 20-cm-long GRIN fiber under the on-axis (gray) and offaxis (black) conditions shown in the two insets. Bottom: experimental spectra under the same two launch conditions out of a 1-m-long GRIN fiber in the infrared (left) and visible (right) regions. Spatial profiles are shown as insets with a 20-μm scale bar. (After Ref. [152] ©2017 OSA.)

In a 2017 study, 0.5-ps-wide pump pulses were launched into a GRIN fiber at a wavelength of 1550 nm at which the fiber’s GVD was anomalous [152]. The launch conditions were varied by varying the center of the input Gaussian beam with respect to the core center. Only the radially symmetric (LP0m ) modes are exited when these two are perfectly aligned. Fig. 14.25 compares the numerically simulated spectra at the end of a 20-cm-long fiber under the on-axis and off-axis launch conditions. Both spectra extend over 500 THz, covering a 0.5–2.5 µm wavelength range, but they differ considerably in the visible region. In contrast with the spectra in Fig. 14.23, the soliton effects play an important role in shaping these spectra. In particular, they are far from being spectrally uniform and exhibit a dominant broad peak centered at 1600 nm. This peak results from soliton fission that creates an ultrashort Raman soliton (width close to 10 fs), whose spectrum shifts toward the red side through intrapulse Raman scattering. The initial alignment of the input beam also affects the output spectrum. The broad peak of the Raman soliton shifts more toward the red side in the on-axis case. Also, the modulation-instability sidebands in the visible region are well separated in the case off-axis excitation but they merge together in the on-axis case. The lower part in Fig. 14.25 shows the experimental spectra in the infrared (left) and visible (right) regions at the end of a 1-m-long GRIN fiber for both the on-axis and off-axis launch conditions. The supercontinuum is wider and more uniform in the on-axis case, features that are in qualitative agreement with the numerical results. The

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spatial profiles of the beam (obtained without any filter) show that most of the pulse energy resides in the fundamental mode in the case of on-axis excitation. In the off-axis case, pulse energy remains distributed over multiple modes, resulting in a much wider and non-uniform spatial pattern.

14.6. Multicore fibers Multicore fibers whose cores support a single spatial mode can be thought of as an array of single-mode fibers. Nonlinear effects in such fiber arrays have been analyzed since the early 1990s [155–163]. With the advent of multicore fibers and their potential applications in space-division multiplexing, considerable work has been done since 2010 to further explore the nonlinear phenomena in such fibers [164–175]. We can use Eq. (14.2.20) for multicore fibers with some modifications. First, the Jones vector Am and the dispersion parameters β1m and β2m corresponds to the mth core. Second, the coupling coefficient κmn represents linear coupling between the modes in the cores identified by the integers m and n, a non-random quantity whose magnitude depends on the spacing between these two cores. Third, the overlap factors fmnpq are still defined by Eq. (14.2.11), but the modal distributions appearing there are centered at the location of individual cores. As a result, all overlap factors are relatively small except of the type fpppp corresponding to the SPM effect. If we assume that all cores are identical, only the x polarized modes are excited, the fiber has no birefringence, and each core is coupled only to its two nearest neighbors, Eq. (14.2.20) is reduced to   ∂ Am iβ2 ∂ 2 Am + = i κ A + A + i γ | Am |2 A m , m + 1 m − 1 ∂z 2 ∂T 2

(14.6.1)

where we introduced the reduced time as T = t − β1 z, used fmmmm = 1, and dropped the subscript m from β0m , β1m , and β2m as all cores are assumed to be identical. Eq. (14.6.1) is appropriate when all cores of a multicore fiber lie along a circle with equal spacing, as shown in Fig. 14.26(A). The parameter κ represents linear coupling between any two neighboring cores. If the cores are so far apart that κ ≈ 0, Eq. (14.6.1) reduces to the standard NLS equation and shows that each core of the fiber acts as a single-mode waveguide that supports soliton formation if a pulse experiences anomalous GVD inside it. The important question is how the linear coupling affects such solitons. To answer it, we introduce the soliton units and write Eq. (14.6.1) in the form i √

  ∂ um 1 ∂ 2 um 2 + | u | u + K u + u + m m m+1 m−1 = 0, ∂ξ 2 ∂τ 2

(14.6.2)

where um = γ LD Am and K = κ LD represents the linear coupling. The specific case of three-core fibers is interesting because the three coupled NLS equations permit analyt-

Multimode fibers

Figure 14.26 (A) A circular fiber array. (B) Discrete spatial solitons forming for three values of k when a CW beam excites five of the cores initially. (From Ref. [158]; ©1994 OSA.)

ical solution in both the CW and pulsed cases [155–157]. In particular, one can find soliton triplets that propagate without changing their shapes. Let us consider the CW case first and neglect the time-derivative term in Eq. (14.6.2). If we launch CW light into one core of a multicore fiber, some power will be transferred to the neighboring cores through linear coupling. The question we ask is how the power is distributed at the fiber output. Fig. 14.26(B) shows the power transfer along a 101-core array by solving Eq. (14.6.2) numerically for a CW beam launched initially with the amplitude [158] √

um = Ka sech[a/ 2(m − mc )] exp[−ik(m − mc )],

(14.6.3)

where mc = 51, a2 = 1.1, and five cores are exited initially at ξ = 0. The parameter k determines the initial phase difference among the excited cores. The beam remains confined to the same five cores when k = 0. However, when k = 0, the power is transferred to the neighboring cores as the CW beam propagates down the array. The remarkable feature is that the initial shape of the spatial envelope is maintained during this process. For this reason, such a structure is referred to as a discrete soliton [161]. It should be stressed that this is a kind of spatial soliton since the temporal term in Eq. (14.6.2) is not present for CW propagation. Such discrete solitons were studied extensively during the 1990s; see Chapter 11 of Ref. [58]. We have seen in Section 14.3 that modulation instability destabilizes the CW solution in a multimode fiber, and the same thing can occur in a multicore fiber. The case of two fibers was analyzed as early as 1989 in the context of a directional coupler [176]. The case of three-core fibers can also be treated analytically by using Eq. (14.6.1) with m = 1 to 3 after setting A0 = A3 and A4 = A1 . The results show that such a fiber supports three CW states with different features of modulation instability in each case [172]. The

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symmetric CW state has equal powers in each core with the amplitudes 

A1 = A2 = A3 = P /3 exp[i(γ P /3 + 2κ)z],

(14.6.4)

where P is the total power in all three cores. It becomes unstable only when the GVD is anomalous, and the gain spectrum of modulation instability exhibits features qualitatively similar to those of a single-mode fiber (see Section 5.1). The second CW solution of a three-core fiber is A1 = 0,



A2 = −A3 = P /2 exp[i(γ P /2 − κ)z].

(14.6.5)

It becomes unstable even in the case of normal GVD, and its gain spectrum exhibits two peaks on each side of the pump frequency. The second pair of sidebands has a relatively low gain but its shift from the pump frequency can exceed 50 THz at high pump powers. The third CW solution of a three-core fiber has amplitudes 

A1 = P1 e i ψ ,



A2 = A3 = P 2 e i ψ ,

(14.6.6)

with the total power P = P1 + 2P2 and a complicated expression for ψ [172]. It also exhibits modulation instability in normal-GVD region, and its gain spectrum has three peaks on each side of the pump frequency. These results suggest that gain spectrum of modulation instability is likely to exhibit multiple peaks in fibers with more than three cores, and the number of sideband pairs can be as large as the number of cores within the fiber. The situation becomes even more complex when the fiber has a central core in addition to the multiple cores arranged in a circle [165,166]. Consider next what happens when short optical pulses are launched into different cores of the fiber simultaneously (at the same or different wavelengths). Linearly arranged fiber arrays were studied as early as 1994 and were found to support soliton-like pulses [159]. In essence, energy is transferred among the cores such that several neighboring cores eventually carry pulses of different energies in a self-sustaining fashion. One way to understand the formation of such multicore solitons is to assume that the number of cores is so large that we can take the continuum limit. It is useful to write Eq. (14.6.2) in the form i

∂ um 1 ∂ 2 um + K (um+1 − 2um + um−1 ) + 2Kum + |um |2 um = 0. + ∂ξ 2 ∂τ 2

(14.6.7)

The linear term 2Kum can be removed through the transformation um = um e2iK ξ . In the continuum limit, we obtain a (2+1)D NLS equation [159] i

2 ∂ U 1 ∂ 2U 2∂ U + Kd + |U |2 U = 0, + ∂ξ 2 ∂τ 2 ∂ x2

(14.6.8)

Multimode fibers

Figure 14.27 Nonlinear energy transfer inside a fiber whose seven cores are arranged in a circle with equal spacing. The input amplitudes are modulated (b = 0.3) such that pulse energy is slightly enhanced in the seventh core. (From Ref. [173]; ©2016 APS.)

where x = md represents the position of the mth core along the linear array, d being the spacing between any two adjacent cores, and we replaced um (t) with U (x, t). In the CW case, this equation reduces to the standard NLS equation whose spatial solitons correspond to the discrete solitons discussed earlier. Eq. (14.6.8) does not apply to fibers whose cores are arranged in a circular fashion as seen in Fig. 14.26(A). The reason is that x is replaced with the angle φ that is defined only modulo 2π . In that case, one must solve Eq. (14.6.2) numerically with the periodic boundary condition uM +1 = u1 for a fiber with M core. In one study, Gaussian pulses were launched with their amplitudes modulated such that [173] um (t) = e−τ

2 /2

[1 + b cos(2π m/M )],

(14.6.9)

where b is the modulation depth. The results for two fibers with 7 and 19 cores show that energy is transferred from all cores such that a single core for which the amplitude is the largest initially carries most of the launched energy at the fiber output. An example is shown in Fig. 14.27 where 83.5% of total energy appears in the seventh core. From a practical perspective, the Kerr nonlinearity in multicore fibers can be used to create a single, energetic, compressed pulse by combining energies of several pulses. In a 2019 experiment, 370-fs pulses could be compressed down to 53 fs using a seven-core fiber [175]. Multicore fibers are useful for enhancing the capacity of optical communication systems [177]. Another application of a multicore fiber makes use of the property that when light is launched into a single core, it spreads to all cores if the input power is relatively low but the spreading is limited to a few nearest neighbors at power levels high enough that a discrete soliton is formed. As a result, the transmission loss is high at low powers but becomes low as the input power increases. Moreover, the transmitted pulse is narrower than the input pulse because only energy in the pulse wings (less intense

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part of the pulse) is transferred to other cores. In other words, a multicore fiber acts as a saturable absorber. This effect was quantified in a 2012 experiment by measuring the transmission loss and pulse widths as a function of input peak power when 60-fs pulses were launched into one core of a seven-core fiber [164]. It should be stressed that the study of nonlinear phenomena in multicore fibers is far from being complete. For example, one may dope some of the cores of such a fiber and pump them to provide amplification, while the remaining cores remain lossy. A multicore fiber may even be twisted along its length so that its cores follow a helical path. The parity-time (PT) symmetry in such fibers has attracted some attention [178]. One should expect further advances in the coming years.

Problems 14.1 Derive the eigenvalue equation (14.1.3) for the modes of a step-index fiber in the weakly guiding approximation. 14.2 Use the eigenvalue equation (14.1.3) to reproduce Figure (14.2) showing the effective mode index for the few low-order modes of a step-index fiber. 14.3 Derive the eigenvalue equation (14.1.13) for the modes of a graded-index fiber whose refractive index varies as indicated in Eq. (14.1.7) for all values of ρ . 14.4 Use the eigenvalue equation (14.1.13) and the concepts of mode groups to show that the effective mode index of the gth mode group of a graded-index fiber is given by Eq. (14.1.14). Identify all degenerate modes for g = 3 and 4. 14.5 Prove that only the radially symmetric LP0m modes of a multimode fiber are excited by a Gaussian beam launched such that its intensity peaks at the core center. 14.6 Derive the time-domain equation (14.2.6) starting from the frequency-domain equation (14.2.5). Hint: you need to expand βm (ω) around ω0 as a Taylor series. 14.7 Use the time-domain equation (14.2.6) to derive Eq. (14.2.8) assuming that the nonlinear response is much faster than the duration of optical pulses. 14.8 Starting from Eq. (14.2.8), derive Eqs. (14.2.13) and (14.2.14) indicating nonlinear coupling between the two specific modes of a fiber. 14.9 Use the variational technique to solve Eq. (14.2.23) and prove that a Gaussian beam evolves inside a GRIN fiber as B(r) = A0 F (r) with F (r) and w (z) given in Eqs. (14.2.24) and (14.2.25). You are allowed to consult Ref. [42]. 14.10 Solve Eq. (14.3.4) and prove that the frequencies of the modulation instability sidebands for a GRIN fiber are given by Eq. (14.3.7). Estimate the frequencies of the first two sidebands when β2 = −20 ps2 /km, zp = 5 mm, c0 = 2, and LNL = 1 cm. 14.11 Average Eq. (14.3.12) over the self-imaging period of a GRIN fiber and derive Eq. (14.3.14). Show that N is related to N as indicated in Eq. (14.3.15).

Multimode fibers

14.12 Consider FWM inside a GRIN fiber and show that the phase-matching condition is satisfied when the frequency shift s of the idler from the pump satisfies Eq. (14.4.3). You may consult Ref. [89]. 14.13 Plot the group index ng over the wavelength range 1 to 1.6 µm for the first six LP0m modes (m = 1–6) for a step-index fiber with a 50-µm core diameter and  = 5 × 10−3 . Can intermodal SRS be observed in this fiber? Explain your reasoning well. 14.14 Explain the concept of beam cleanup by a multimode fiber. What is the difference between the Kerr-induced and Raman-induced beam cleanup? How can the phenomenon of SBS be used for the beam cleanup? 14.15 Reproduce Fig. 14.27 by solving Eq. (14.6.2) numerically for a fiber with seven cores arranged in a circular fashion. Assume Gaussian pulses are launched into all cores with the initial amplitudes indicated in Eq. (14.6.9).

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683

APPENDIX A

System of units The international system of units (known as the SI, short for Système International) is used in this book. The three fundamental units in this system are meter (m), second (s), and kilogram (kg). A prefix can be added to each of them to change its magnitude by a multiple of 10. Mass units are rarely required in this book. On the other hand, measures of distance required in this text range from nanometers (10−9 m) to kilometers (103 m), depending on whether one is dealing with planar waveguides or optical fibers. Similarly, time measures range from femtoseconds (10−15 s) to a few seconds. Other common units used in this book are Watt (W) for optical power and W/m2 for optical intensity. They can be related to the fundamental units through energy because optical power represents the rate of energy flow (1 W = 1 J/s). The energy can be expressed in several other ways using E = hν = kB T = mc 2 , where h is the Planck constant, kB is the Boltzmann constant, and c is the speed of light. The frequency ν is expressed in hertz (1 Hz = 1 s−1 ). Of course, because of the large frequencies associated with optical waves, most frequencies in this book are expressed in GHz or THz units. Table A.1 lists the values of a few physical constants that may be needed for solving the problems listed at the end of each chapter. In both linear and nonlinear fiber optics it is common to make use of decibel units, abbreviated as dB and used by engineers in many different fields. Any ratio R can be converted into decibels using the general definition R (in dB) = 10 log10 R.

(A.1)

The logarithmic nature of the decibel scale allows a large ratio to be expressed as a much smaller number. For example, 10 9 and 10−9 correspond to 90 dB and −90 dB, Table A.1 Values of relevant physical constants. Physical constant Symbol Value

Vacuum permittivity Vacuum permeability Speed of light in vacuum Elementary charge Electron rest mass Planck constant Boltzmann constant

0 μ0

c e me h kB

8.85 × 10−12 F/m 4π × 10−7 H/m 2.998 × 108 m/s 1.602 × 10−19 C 9.109 × 10−31 Kg 6.626 × 10−34 J s 1.381 × 10−23 J/K

685

686

System of units

respectively. As R = 1 corresponds to 0 dB, ratios smaller than 1 are negative on the decibel scale. Furthermore, negative ratios cannot be expressed using decibel units. The most common use of the decibel scale occurs for power ratios. For instance, the fiber-loss parameter α appearing in Eq. (1.2.3) can be expressed in decibel units by noting that fiber losses decrease the optical power launched into an optical fiber from its value at the input end, and thus can be written as a power ratio. Eq. (1.2.4) shows how fiber losses can be expressed in units of dB/km. If a 1-mW signal reduces to 1 µW after transmission over 100 km of fiber, the power reduction by a factor of 1000 translates into a 30-dB loss from Eq. (A.1). Spreading this loss over the 100-km fiber length produces a loss of 0.3 dB/km. The same technique can be used to define the insertion loss of any component. For instance, a 1-dB loss of a fiber connector implies that the optical power is reduced by 1 dB (≈20%) when the signal passes through the connector. Examples of other quantities that are often quoted using the decibel scale include the signal-to-noise ratio and the amplification factor of an optical amplifier. If optical losses of all components in a fiber-optic communication system are expressed in decibel units, it is useful to express the transmitted and received powers also by using a decibel scale. This is achieved by using a derived unit, denoted as dBm and defined as power (in dBm) = 10 log10

 power 

1 mW

,

(A.2)

where the reference level of 1 mW is chosen for convenience; the letter m in dBm is a reminder of the 1-mW reference level. In this decibel scale for the absolute power, 1 mW corresponds to 0 dBm, whereas powers D -> 1/2N; first half step nonlinear temp = uu.*exp(abs(uu).^2.*hhz/2); for n=1:step_num

% note hhz/2

f_temp = ifft(temp).*dispersion; uu = fft(f_temp); temp = uu.*exp(abs(uu).^2.*hhz); end uu = temp.*exp(-abs(uu).^2.*hhz/2); % Final field %***************[ End of MAIN Loop ]************** %----Plot output pulse shape and spectrum temp = fftshift(ifft(uu));

% Fourier transform

spect = abs(temp).^2;

% output spectrum

spect = spect./max(spect);

% normalize

subplot(2,1,1) plot(tau, abs(uu).^2, ’-b’); hold off; subplot(2,1,2) plot(freq, spect, ’-b’); hold off;

APPENDIX E

List of acronyms Each scientific field has its own jargon, and the field of nonlinear fiber optics is not an exception. Although an attempt was made to avoid extensive use of acronyms, many still appear throughout the book. Each acronym is defined the first time it appears in a chapter so that the reader does not have to search the entire text to find its meaning. As a further help, all acronyms are listed here in alphabetical order. AM ASE CVD CW DCF DDF DFB DGD DSF EDFA FDTD FFT FM FOPA FROG FWHM FWM GVD GRIN HNLF LEAF LP MCVD MZI NLS NRZ NSR OAM PBG

amplitude modulation amplified spontaneous emission chemical vapor deposition continuous wave dispersion-compensating fiber dispersion-decreasing fiber distributed feedback differential group delay dispersion-shifted fiber erbium-doped fiber amplifier finite-difference time domain fast Fourier transform frequency modulation fiber-optic parametric amplifier frequency-resolved optical gating full width at half maximum four-wave mixing group-velocity dispersion graded-index highly nonlinear fiber large-effective-area fiber linearly polarized modified chemical vapor deposition Mach–Zehnder interferometer nonlinear Schrödinger nonreturn to zero nonsolitonic radiation orbital angular momentum photonic bandgap

695

696

List of acronyms

PCF PMD RIFS RIN RMS SBS SHG SNR SOP SPM SSFM SRS TDM THG TOD WDM XPM YAG ZDWL

photonic crystal fiber polarization-mode dispersion Raman-induced frequency shift relative intensity noise root mean square stimulated Brillouin scattering second-harmonic generation signal-to-noise ratio state of polarization self-phase modulation split-step Fourier method stimulated Raman scattering time-division multiplexing third-harmonic generation third-order dispersion wavelength-division multiplexing cross-phase modulation yttrium aluminum garnet zero-dispersion wavelength

Index

A Absorption coefficient, 29 Acoustic response function, 473 Acoustic velocity, 356, 363 Acoustic wave, 355, 375, 472 damping time of, 357 guided, 356, 454 Adiabatic perturbation theory, 160 Airy function, 69, 531 Akhmediev breather, 133, 611 All-optical sampling, 453 Amplifier Brillouin, 368–372 erbium-doped fiber, 163, 316, 426 multimode, 656 parabolic pulses in, 105 parametric, 413, 419–446, 656 phase-sensitive, 456 Raman, 163, 312–317, 332, 347, 422 semiconductor optical, 68, 314 SPM effects in, 105 Yb-doped fiber, 107, 332, 364, 375 Amplifier spacing, 162, 164 Angular momentum conservation, 434, 437, 443 Anisotropic stress, 189 Annihilation operator, 453 Anti-Stokes band, 306, 367, 389, 403, 413, 418, 561, 653 Attenuation constant, 5 Autocorrelation trace, 74, 131, 146, 336, 467 Avalanche photodiode, 574

B Babinet–Soleil compensator, 199 Backward-pumping configuration, 315 Baker–Hausdorff formula, 47 Bandwidth amplifier, 316, 422, 434 Brillouin-gain, 310 parametric amplifier, 421–432 pulse, 73 Raman-gain, 299, 312, 318, 408, 426 source, 73 spontaneous noise, 315

Stokes, 301 supercontinuum, 560 Beam cleanup Kerr, 666, 672 Raman, 657, 659, 672 SBS-induced, 664 spatial, 666 Beam collapse, 639 Beat length, 12, 190, 203, 208, 221 Bessel function, 30, 622 Biomedical imaging, 585 Birefringence, 416, 521, 549, 588 circular, 194, 210 elliptical, 193 fluctuating, 284, 362, 443 linear, 189, 192, 197, 202, 207, 283, 420 modal, 12, 189, 202, 416 modulated, 210 nonlinear, 189–194, 202, 207, 272, 282 pump-induced, 196 random, 236, 656 residual, 273, 285, 443 self-induced, 202 stress-induced, 419 temperature-induced, 419 XPM-induced, 275 Bloch equation, 285 Boltzmann constant, 685 Boundary condition, 387, 391, 406 Bragg condition, 137 Bragg diffraction, 356 Bragg fiber, 486 Bragg mirror, 486 Brillouin amplifier, 368–372 Brillouin gain, 17, 356–360, 663 Brillouin laser, 387–394 continuous-wave, 387 Fabry–Perot, 388 multiwavelength, 389 pulsed, 391 ring, 387, 391 self-seeded, 390 threshold of, 387 tunable, 389

697

698

Index

Brillouin scattering, 16, 472 guided-acoustic-wave, 356 spontaneous, 356, 361, 362, 386, 454 stimulated, 355–394, 426, 454 Brillouin shift, 356, 357, 363, 370, 385, 390, 662 Brillouin threshold, 360–365, 367, 384, 385, 388 polarization effects on, 361

C Cavity fiber, 533 high-Q, 533 ring, 533 Cavity soliton, see solitons Chalcogenide glass, see glass Chaos, 385, 536 feedback-induced, 386 polarization, 210 quasi-periodic route to, 386 SBS-induced, 386 Charge-delocalization model, 540 Charge-transport model, 540 Chemical vapor deposition, 4 Cherenkov radiation, 507, 572 Chirp definition of, 62 dispersion-induced, 62, 64, 110, 113 linear, 62, 106, 107 SPM-induced, 87, 92, 110, 113, 148 XPM-induced, 253, 260, 265, 268, 275, 286, 322, 324, 328, 341 Chirp parameter, 92, 149 Circulator, 389, 392 Coherence degradation of, 94, 593, 599 degree of, 93, 593 partial, 584 pump, 593 supercontinuum, 593–601 techniques for improving, 599 temporal, 93, 593, 595 Coherence function, 94 Coherence length, 359, 407, 411, 545 Coherent anti-Stokes Raman scattering, 306 Collision length, 167 Color center, 539, 541 Continuum radiation, see radiation Conversion efficiency, 366, 401, 425, 539, 540

Core–cladding index difference, 3, 39, 476, 546 Correlation length, 13, 237 Coupled-mode equations, 192, 631 Coupled-mode theory, 628 Coupler 3-dB, 370 directional, 387, 392, 675 fiber, 467, 476, 533, 628, 629, 637 Coupling coefficient, 628, 636 Coupling efficiency, 630 Crank–Nicholson scheme, 50 Cross-correlation, 75, 121, 329, 516, 574 Cross-phase modulation, 16, 189, 245–291, 418, 440, 517, 523 birefringence induced by, 190 coupling due to, 245 for measuring γ , 468 modulation instability by, 210, 248 nondispersive, 194 nonreciprocal nature of, 288 optical switching by, 270 phase shift induced by, 194 polarization effects on, 272 pulse compression due to, 267 SBS suppression with, 364 solitons formed by, 226, 252, 578 spectral changes due to, 258 supercontinuum and, 572 temporal changes due to, 263 wavelength conversion by, 271 wavelength shift by, 262 Crosstalk Brillouin-induced, 371 FWM-induced, 411 Raman-induced, 317 Cut-off wavelength, 31, 32, 623

D Darboux transformation, 614 Demultiplexing, 452 Depolarization, intrapulse, 285 Dielectric constant, 29, 472 nonlinear, 36 Differential group delay, 258, 625 Diffusion length, 286, 443 Dispersion anomalous, 279, 478 comb-like, 133, 153

Index

even-order, 511 flattening of, 559 fluctuations in, 431 fourth-order, 79, 121, 172, 448, 509, 510, 512, 527, 559 group-velocity, 10, 40, 95, 408, 412 higher-order, 42, 424, 503, 507, 559, 572 material, 9, 33, 410, 489 micro-management of, 599 odd-order, 511 origin of, 6–11 polarization-mode, 11, 236, 443, 523 third-order, 8, 44, 69–76, 78, 79, 104, 134, 171, 228, 478, 504, 527, 564, 580 waveguide, 9, 33, 410, 416, 489 Dispersion compensation, 76, 78, 422, 425 Dispersion length, 57, 161, 164, 194, 256, 279, 320, 334, 564, 567, 571 Dispersion management, 76–81, 155, 421 Dispersion parameter, 8, 9, 303, 415, 483, 489 Dispersion relation, 128, 211, 212, 214, 249, 287, 356, 580 Dispersion slope, 10, 79, 425, 523 Dispersion-compensating fiber, see fiber Dispersion-decreasing fiber, see fiber Dispersive waves, 149, 162, 240, 507–518, 568–588 Distributed amplification, 162, 315, 316 Distributed feedback, 68, 133, 341, 367, 420 Distributed fiber sensors, 371 Doppler shift, 356 Dual-pump configuration, 427–432, 435

E Effective mode area, 39, 198, 248, 361, 365, 405, 473, 477, 489, 492, 494, 567 spatially varying, 640 Eigenvalue equation, 30, 31, 477, 569, 621, 623, 651 Electrostriction, 355, 469, 472, 473 Electrostrictive constant, 357, 373, 473 Elliptic function, 203, 208, 255, 432, 434, 545 Ellipticity, 203, 591 Erbium-doped fiber amplifiers, 316, 389, 426 Error function, 259 Euler–Lagrange equation, 110, 159 Extinction ratio, 199 Extrusion technique, 481, 488, 489

F Fabry–Perot cavity, 309, 388, 389, 391, 449, 450, 465 Faraday mirror, 468 Fast axis, 12, 190, 197, 206, 207, 221, 416, 418, 521, 550, 589 Fast Fourier transform, 46 Fast light, 380 FDTD method, 52 Feedback, 386 distributed, 199 external, 367, 383, 386 fiber-end, 386 optical, 383, 386 reflection, 384 FFT algorithm, 47, 570 Fiber bimodal, 220 birefringent, 193, 282, 408, 416, 420, 517, 550, 588 bismuth-oxide, 492 Bragg, 486, 490 chalcogenide, 19, 199, 491, 567 characteristics of, 3–14 D-shaped, 541 dispersion-compensating, 10, 78, 358, 464, 467, 471, 474 dispersion-decreasing, 10, 80, 133, 153, 161, 519 dispersion-flattened, 10, 130, 172, 251, 279, 414, 474, 559 dispersion-oscillating, 137 dispersion-shifted, 10, 153, 270, 271, 341, 421, 464, 466, 471, 474, 560, 565, 574 dispersion-varying, 80 double-clad, 311 endlessly single-mode, 483 few-mode, 633 fluoride, 566 gas-filled, 604 graded-index, 3, 625, 638 high-birefringence, 190, 194, 213, 221, 287, 523 highly nonlinear, 40, 421, 426, 429, 474–497, 503–532 holey, 340, 599 hollow-core, 485, 603 hotonic bandgap, 526 isotropic, 215, 219, 588, 590, 592 large-core, 365

699

700

Index

large-mode-area, 599 lead-silicate, 19, 488, 566 low-birefringence, 190, 193, 201, 211, 220, 283 microstructured, 130, 340, 414, 480–493, 512, 518–532, 546, 560, 563, 581, 588, 602 modes of, 29–34, 621–631 multicore, 627, 674–678 multimode, 3, 543, 621–678 nano-, 602 narrow-core, 493 phosphosilicate, 311 photonic bandgap, 485 photonic crystal, 415, 480–493, 518–532, 546, 588 photosensitivity in, 539 polarization effects in, 189–240 polarization-maintaining, 13, 103, 189, 271, 315, 338, 342, 361, 364, 389, 416, 521, 565 preform for, 4, 193 random birefringence in, 236 step-index, 621 suspended-core, 603 tapered, 130, 414, 476–479, 515, 563, 586, 599, 610 tellurite, 567 twisted, 193, 210 two-core, 628 weakly guiding approximation, 621 ytterbium-doped, 108, 364 Fiber grating, see grating Fiber-grating compressor, 330, 332 Fiber-loop mirror, 153, 270 Finite-difference method, 46, 50 Finite-element method, 482, 489 Fixed points, 204, 206, 208, 210 Flip-flop circuit, 153 Forward-pumping configuration, 315 Four-wave mixing, 129, 215, 246, 251, 305, 367, 388, 401–419 applications of, 446–457 applied for measuring γ , 469 degenerate, 192, 403, 435 in GRIN fiber, 654 intermodal, 651–657 nearly phase-matched, 411 nondegenerate, 404, 427–432, 435 origin of, 401 PMD effect on, 443

polarization dependence of, 434–446 spontaneous, 455 SRS effects on, 408 supercontinuum and, 580 theory of, 403–409 ultrafast, 408 vector theory of, 435–446 Freak waves, see rogue waves Frequency chirp, 86, 145, 148, 260, 321, 325, 341, 564 Frequency comb, 532–537, 595, 604 Frequency shift cross-, 267 Raman-induced, 119, 176, 266, 335, 340, 512–532, 568–579 soliton self-, 175–182, 512 Frequency-resolved optical gating, 75 FROG technique, 75, 93, 333, 516, 574

G Gain bandwidth, 421, 424, 429, 443 Gain saturation, 106, 313, 368 Gain switching, 68 Gaussian Pulse, see pulse Geometric parametric instability, 644 Glass bismuth-oxide, 492 chalcogenide, 154, 199 lead-silicate, 488 SF57, 488 silica, 464 tellurite, 493 Gordon–Haus effect, 165 Grating array-waveguide, 558 Bragg, 311, 364 Brillouin, 372 dynamic, 372 fiber, 153, 311, 316, 364 index, 356, 669 Kerr-induced, 137, 668, 669 nonlinear index, 137, 668 stop band of, 364 Group delay, 627 Group index, 8, 379, 380 Group velocity, 8, 40, 220, 379, 408, 517, 571, 627 intensity-dependent, 115 matching of, 408, 660 Group-velocity dispersion, see GVD

Index

Group-velocity mismatch, 198, 214, 221, 231, 248, 250, 252, 257–267, 270, 305, 319, 329, 341, 383, 434, 449, 547, 575, 577 GVD anomalous, 11, 62, 96, 106, 129, 140, 210, 248, 250, 252, 254, 268, 332, 334, 414, 424, 489, 508, 525, 559, 563, 595 normal, 10, 62, 96, 106, 107, 113, 119, 129, 211, 248, 250–252, 254, 268, 281, 332, 409, 424, 479, 508, 525, 559, 575, 595 GVD parameter, 10, 40, 77, 128, 247, 559 Gyroscope fiber, 388 laser, 387, 388

H Harmonic generation, 538–551 Helmholtz equation, 29, 36 Heterodyne detection, 357, 454 Hirota method, 255 Homodyne detection, 371, 454

I Idler wave, 403–457 Inelastic scattering, 16 Inhomogeneous broadening, 393 Intensity discriminator, 200 Intrapulse Raman scattering, see Raman scattering Inverse scattering method, 46, 139–141, 150, 167, 172, 226, 240, 503, 647 Inversion symmetry, 540 Isolator, 367, 392

J Jones matrix, 236, 238, 272, 283, 344 Jones vector, 238, 273, 283, 436, 437, 590, 636

K Kagome lattice, 485 Kerr coefficient, 37, 39, 191 Kerr effect, 196, 200 Kerr frequency comb, 533 Kerr modulator, 196 Kerr nonlinearity, 154, 197 Kerr shutter, 196–200 Kramers–Kronig relation, 42, 344, 379 Kronecker delta function, 190 Kuznetsov–Ma soliton, 613

L Lagrangian, 110 Laguerre polynomial, 625 Langevin noise, 386 Lasers argon-ion, 357, 387, 388, 411, 464 Brillouin, see Brillouin laser CO2 , 476 color-center, 315, 338, 371, 449 DFB, 68, 133, 367, 370, 381, 420, 466, 470 distributed feedback, 315 double-clad, 311 dye, 329, 336, 365 erbium-doped fiber, 565, 595 external-cavity, 357, 370 fiber, 466 gain-switched, 199, 341 He–Ne, 387 mode-locked, 331, 338, 391, 448, 565, 593 modulation-instability, 448 Nd-fiber, 316 Nd:YAG, 199, 314, 327, 331, 337, 339, 340, 366, 367, 372, 376, 392, 413, 448, 581 Q-switched, 315, 375, 581 Raman, 309–312, 331 Raman soliton, 338 semiconductor, 199, 314, 315, 341, 357, 366, 370, 371, 389, 420, 466, 595 soliton, 338 synchronously pumped, 340 Ti:sapphire, 478, 563, 565, 588 xenon, 365 Yb-doped fiber, 107, 311, 364, 562, 564, 585 Yb-fiber, 587 Linear stability analysis, 127, 248, 383, 392 Lithographic technique, 541 Local oscillator, 371, 454 Logic gates, 225 Longitudinal modes, 370, 388, 391 Lorentzian spectrum, 357, 524 Loss Brillouin, 372 cavity, 387 fiber, 5, 160, 300, 409 microbending, 32 polarization-dependent, 237 round-trip, 387 Lugiato–Lefever equation, 534 Lumped-amplification scheme, 162, 163

701

702

Index

M Mach–Zehnder interferometer, 153, 198, 262, 271, 468 Mach–Zehnder modulator, 153 Magnetic dipole, 401, 539 Manakov equation, 445, 638 Markovian process, 284 Maxwell’s equations, 27, 482 Michelson interferometer, 269, 593 Mid-infrared region, 492 Mode acoustic, 373 degeneracy of, 624, 626 excitation of, 630 fundamental, 32, 546 guided, 29 higher-order, 546 hybrid, 31 leaky, 487, 547 linearly polarized, 32 radiation, 29 Mode coupling linear, 632, 634, 656, 674 nonlinear, 633 random linear, 635, 662 strength of, 637 Mode-locking active, 391, 451 passive, 451 self-induced, 391 Modulation instability, 127–138, 210–220, 338, 414, 419, 470, 534, 562, 582, 608, 611 critical power for, 214 effects of SRS, 328 experiments on, 131, 217, 250 gain spectrum of, 129, 212, 214, 216, 250, 643 graded-index fiber, 642 higher-order, 614 induced, 130, 131, 133, 583, 609 multicore fiber, 675 multimode fiber, 641 SBS-induced, 385 SHG-induced, 543 sidebands of, 220, 470, 582 SPM-induced, 132 spontaneous, 130, 583, 609, 613 supercontinuum and, 582 two-mode, 641 vector, 210

XPM-induced, 248–251, 329 Modulator amplitude, 391 electro-optic, 364, 381 LiNbO3 , 153 liquid-crystal, 79 Mach–Zehnder, 153 phase, 153, 364 spatial light, 631 Moment method, 108–113, 169 Momentum conservation, 402, 524 Multimode interference, 657 Multimode propagation equations, 631 Multiphoton ionization, 540 Multiplexing mode-division, 631, 638 space-division, 621, 674 wavelength-division, 10, 317 Multipole method, 482

N Neumann function, 30 NLS equation, 40, 45, 128, 466 algebraic solutions of, 612 asymptotic solution of, 106 coupled, 247, 256, 283, 408, 635 coupled vector, 283 cubic, 45 for GRIN fibers, 638 generalized, 45, 515, 524, 527, 569, 576, 584, 607 inhomogeneous, 156 inverse scattering method for, 139 moment method for, 108 multimode, 646, 668 numerical methods for, 46 periodic solutions of, 254, 611 quintic, 45 split-step method for, 46–50 variational method for, 110 vector form of, 237, 273 Noise intensity, 595, 599 quantum, 453, 594 Raman, 595 spontaneous-emission, 470 Noise figure, 316, 426 Nonlinear birefringence, 189, 190

Index

Nonlinear length, 57, 86, 129, 201, 221, 279, 320, 514, 571, 582, 594 Nonlinear parameter, 39, 45, 110, 285, 303, 405, 443, 445, 463–497 spatially varying, 640 Nonlinear phase shift, see phase shift Nonlinear polarization rotation, 191, 200, 206, 275, 345, 440, 588, 592 Nonlinear refraction, 15 Nonlinear response, 35, 41, 426, 570, 687 Nonlinear Schrödinger equation, see NLS equation Nonlinear-index coefficient, 15, 18, 404 measurements of, 463–474 Nonlinearity Kerr, 37, 303, 687 Raman, 303, 687 NRZ format, 153

O Optical bistability, 288, 534 Optical buffer, 537 Optical shock, 115 effect of dispersion on, 117 Optical spectral analyzer, 593 Optical switching, 199, 270, 453, 518 Optical wave breaking, see wave breaking Orbital angular momentum, 631 Outside vapor deposition, 4 Overlap integral, 247, 404, 405, 544

P Packet switching, 453 Parabolic pulse, see pulse Parametric amplification, 419–446 dual-pump, 427–432 multimode, 656 phase-sensitive, 456 polarization effects on, 434–446 single-pump, 423–427 vector theory of, 435–446 Parametric amplifier, see amplifier Parametric gain, 406, 407, 409, 421, 427, 434, 439 bandwidth of, 421 polarization dependence of, 437 Parametric oscillator tunable, 448, 449 Partially coherent light, 93–95 Pauli matrices, 238, 274, 283, 590 Pauli spin vector, 274, 284, 344, 440

Peregrine soliton, 613 Period-doubling route, 385 Phase conjugation, 406, 422, 446, 664 Phase matching, 406, 419, 428 birefringence-induced, 416, 562 Cherenkov-type, 547 condition for, 403, 410, 423 for THG, 546 in GRIN fiber, 654 multimode, 651–657 near zero-dispersion wavelength, 412 quasi, 541 requirement for, 403 SPM-induced, 413 techniques for, 409–419 Phase mismatch, 402, 422 effective, 406 FWM, 633 linear, 424, 427 nonlinear, 424, 428, 548 Phase modulation, 375 Phase shift nonlinear, 86, 87, 194, 214, 258, 276, 440, 468, 507 SPM-induced, 467, 488 time-dependent, 258 XPM-induced, 258, 270, 271, 276, 286, 364, 468, 578 Phase space, 204 Phase-matching condition, 215, 251, 305, 403, 409–432, 449, 470, 507, 508, 544, 546, 550, 559, 572, 580 Phase-retrieval algorithm, 574 Phonon lifetime, 357, 359, 372, 375 Photon-pair source, 454 Photonic bandgap, 485, 610 Photonic lantern, 631 Photosensitivity, 539 Photovoltaic effect, 540 Pitchfork bifurcation, 204 Planar lightwave circuit, 79 Planck constant, 685 PMD, 11, 236, 284 effect on four-wave mixing, 443 first-order, 236 pulse broadening induced by, 237 Raman amplification with, 346 second-order, 237 solitons and, 239, 240

703

704

Index

PMD parameter, 13, 237, 284, 443 Poincaré sphere, 204–208, 232, 239, 274, 275, 285, 344, 362, 440, 443, 592 Polarization degree of, 364 evolution of, 202–210, 273 induced, 34 intrapulse, 275 nonlinear, 34, 266, 401, 435, 687 static, 539 third-order, 272 Polarization attraction, 290 Polarization chaos, 210 Polarization domain wall, 290 Polarization ellipse, 191, 203 nonlinear rotation of, see nonlinear polarization rotation rotation of, 204, 206 Polarization instability, 207–210, 220 effect on solitons, 221 observation of, 208 origin of, 207 Polarization-dependent gain, 237, 346 Polarization-diversity loop, 430, 434 Polarization-division multiplexing, 240 Polarization-mode dispersion, see PMD Preform, 4, 193 rocking of, 210 spinning of, 215 Principal axis, 190, 588 Principal states of polarization, 236 Propagation constant, 7, 30 Pseudorandom bit pattern, 375, 426 Pseudospectral method, 46 Pulse chirped, 90, 109 chirped Gaussian, 62 Gaussian, 60, 63, 90, 109, 115, 148, 264 hyperbolic secant, 64, 143, 148 linearly chirped, 565 parabolic, 106, 107, 162, 332, 564 Q-switched, 364, 375, 455, 557 super-Gaussian, 65, 87, 91, 148 Pulse broadening dispersion-induced, 59–81, 120 PMD-induced, 236, 237 Pulse compression, 599 amplifier-induced, 106 soliton effect, 341

XPM-induced, 267, 341 Pulse-to-pulse fluctuations, 593 Pump depletion, 300, 302, 313, 324, 366, 368, 406, 432, 545 Pump-phase modulation, 426, 430 Pump-probe configuration, 260, 262, 264, 273, 278, 284, 468

Q Quadrupole moment, 401, 539 Quantum interference, 540 Quantum limit, 426 Quarter-wave plate, 197 Quasi-CW regime, 307, 404, 408, 473, 546 Quasi-monochromatic approximation, 34 Quasi-periodic route, 386 Quasi-phase matching, 541

R Radiation Cherenkov, 507 continuum, 148, 149 nonsolitonic, 507–518, 568–579 Raman amplification, 163, 312–317 distributed, 662 PMD effects on, 346 short-pulse, 332 vector theory of, 342 Raman amplifier, see amplifier Raman effect, 266, 297, 530, 604 Raman gain, 17, 42, 298, 407, 418, 431, 491, 560, 653 polarization dependence of, 342 spectrum of, 298, 570 Raman laser, 309–312, 338 synchronously pumped, 331 Raman response, 41, 44, 175, 229, 303, 304, 321, 326, 343, 472, 524, 570, 687 Raman scattering, 16, 472 coherent anti-Stokes, 561 intermodal, 659 interpulse, 267 intrapulse, 40, 42–44, 119–121, 133, 170, 175–182, 228, 266, 335, 505, 512–532, 564, 568, 569, 571, 585, 659 spontaneous, 297–300, 426, 455, 595 stimulated, 95, 162, 297–309, 486 Raman soliton, see solitons Raman threshold, 300, 302, 413, 486, 558

Index

Raman-induced index changes, 326 Raman-induced power transfer, 429, 431 Rayleigh scattering, 6, 368 Refractive index, 29, 464 effective, 391, 478 intensity-dependent, 258 nonlinear, 245, 463 Raman-induced, 304 SBS-induced, 378 SRS-induced, 326 Relaxation oscillations, 365, 383, 391 Ring cavity, 387, 449 Rise time, 66 Rogue waves L-shaped statistics of, 607 optical, 607–614

S Saddle point, 204 Sagnac interferometer, 270, 271, 389, 449, 467, 469 Sampling oscilloscope, 330 Saturable-absorber mirror, 451 SBS, 355–394 cascaded, 388, 389, 391 dynamics of, 372–387 experiments on, 365 gain spectrum of, 356–360 intermodal, 662–665 quasi-CW, 360–368 sensors based on, 371 suppression of, 363, 426, 431, 470 threshold of, 360–365, 367, 375, 426, 662 transient regime of, 374 SBS-induced index changes, 378 Scanning electron microscope, 481, 489 Schrödinger equation nonlinear, see NLS equation Second-harmonic generation, 74, 401, 538–546 Selection rule, 356, 437 Self-focusing, 638, 657, 665, 668 Self-frequency shift, 267 Self-imaging, 626, 639, 643, 646, 648 Self-phase modulation, 16, 40, 85–121, 171, 413, 418, 440, 465, 567 Self-pulsing, 392 Self-similarity, 107, 332 Self-steepening, 42, 114–118, 173–175

Sellmeier equation, 7, 477 Seperatrix, 204 Signal-to-noise ratio, 165, 372, 427 Similaritons, 156 Single-pump configuration, 423–427, 437 Slow axis, 12, 190, 197, 206, 207, 221, 416, 418, 521, 550, 589 Slow light, 327, 380 Slowly varying envelope approximation, 35, 51, 373 Solitary waves, see solitons Soliton dragging, 225 Soliton period, 144, 336 Soliton spectral tunneling, 529 Soliton trapping, 223, 225, 254, 517, 522, 575, 577 Solitons amplification of, 162 birefringence effects on, 223 bistable, 154, 155 black, 151, 152 bright, 150, 409 Brillouin, 378, 392 cavity, 532–537 chirp effects on, 148 collision of, 166, 167, 584, 585 dark, 150–153, 252 discrete, 675 dispersion-managed, 155 dissipative, 378, 409, 533 domain-wall, 290 experimental observation of, 146 fission of, 172, 175, 178, 503, 515, 525, 564, 568, 571, 592, 594 four-wave mixing, 408 fundamental, 141–143, 147, 341, 564, 571 gray, 151, 152 GRIN, 647 guiding-center, 164 higher-order, 143–147, 161, 172, 175, 178, 221, 269, 334, 341, 503–532, 564, 571, 592 history of, 138 impact of fiber losses, 160 interaction of, 165–169, 233 kink, 290 loss-managed, 647 multicomponent, 256 multicore, 676 multimode, 644–651 multipeak Raman, 530

705

706

Index

parametric, 408 peak power for, 143 period of, 144 perturbation methods for, 159 PMD effects on, 239, 240 polarization effects on, 221 Raman, 334, 336, 338, 512, 530, 547, 550, 565, 566, 573, 576, 578 second-order, 144, 335 self-frequency shift of, 175–182, 340, 512, 568 spatial, 675 spatio-temporal, 646 spectral incoherent, 596 spectral tunneling of, 529 stability of, 147 symbiotic, 252, 409 third-order, 144, 175 vector, 226, 230, 240, 256, 592 XPM-coupled, 252, 408, 577 Spatial light modulator, 631, 634 Spatio-temporal instability, 644 Spectral asymmetry, 260, 585 Spectral broadening, 85–93 asymmetric, 116, 260, 265 polarization-dependent, 276 SPM-induced, 116, 145, 171, 465, 491, 526, 545, 558, 561, 573, 581 XPM-induced, 259, 262, 577 Spectral filtering, 153, 331 Spectral fringes, 593 Spectral hole burning, 393 Spectral narrowing, 92, 112 Spectral recoil, 523, 524, 526 Spin decoherence, 285 Split-step Fourier method, 46–50, 99, 104, 108, 279, 323, 569, 570 symmetrized, 47 Spontaneous emission, 73, 164, 426, 454, 470 amplified, 389, 595 Spread-spectrum technique, 364 Squeezing, 453 SRS, 297–349 cascaded, 302, 307, 316, 561, 657 equations for, 302 fluctuations in, 309, 322 four-wave mixing effects on, 305 intermodal, 657–662 intrapulse, see Raman scattering quasi-CW, 307

single-pass, 307 soliton effects on, 334 threshold of, 299, 328, 657 ultrafast, 319–331 Steepest descent method, 301 Stimulated Brillouin scattering, see SBS Stimulated Raman scattering, see SRS Stokes band, 297, 302, 356, 367, 403, 408, 413, 418, 558, 561 Stokes parameters, 205, 238 Stokes vector, 205, 206, 239, 274, 285, 344, 440 Stop band, 610 Streak camera, 74, 251 Stress-induced anisotropy, 11 Sum-frequency generation, 516 Super-Gaussian pulse, see pulse Supercontinuum, 42, 181 coherence properties of, 593–601 ctave spanning, 565 CW pumping of, 581–588 femtosecond pumping of, 563–581 flat, 565 FWM effects on, 580 incoherent regime of, 596 mid-infrared, 604, 669 multimode, 668 noisy nature of, 593 numerical modeling of, 569 picosecond pumping of, 557–563 polarization effects on, 588 rogue waves in, 607–614 ultraviolet, 602 visible, 586 with GRIN fibers, 669 XPM effects on, 572 Supermode, 628 Susceptibility linear, 15 nonlinear, 401 Raman, 472, 660 second-order, 15, 401, 538 third-order, 15, 190, 246, 298, 401, 435, 463, 492, 546 Synchronous pumping, 310, 331, 391

T Taylor series, 423, 508, 548, 570 Terahertz wave, 543 Thermal poling, 541, 542

Index

Third-harmonic generation, 75, 401, 403, 546–551 Third-order dispersion, see dispersion Third-order susceptibility, see susceptibility Three-wave mixing, 403 Time-dispersion technique, 310, 331, 338 Time-division multiplexing, 452, 558 Time-resolved optical gating, 75 Timing jitter, 165, 340 Topological charge, 631 Total internal reflection, 1, 480 Tunable optical delay, 381 Two-photon absorption, 36, 75, 574

U Ultrafast signal processing, 451 Undepleted-pump approximation, 405

W Walk-off length, 11, 195, 248, 260, 267, 274, 279, 305, 320, 322, 323 Wave breaking, 100–103, 107 suppression of, 107 XPM-induced, 265, 281 Wave equation, 28, 30 Wave-vector mismatch, 407, 411, 416, 423, 539, 544 Wavelength conversion, 422, 426, 430, 434, 446, 493 Wavelength multicasting, 453 Wavelength-selective feedback, 309 WDM, see multiplexing WDM systems, 316, 317, 411, 558, 559 Weibull distribution, 608 Wiener–Khintchine theorem, 94

X V V parameter, 3, 31, 474, 623 Vacuum fluctuations, 130 Vacuum permeability, 28 Vacuum permittivity, 28, 464 Vapor-phase axial deposition, 4 Variational method, 110–113, 159, 638, 645

X-FROG technique, 516

Z Zero-dispersion wavelength, 8, 10, 11, 69, 171, 198, 254, 270, 336, 338, 409, 410, 412, 416, 419, 423–425, 428, 431, 478, 518, 523, 525, 563, 576, 586

707