Nonlinear Fiber Optics

Nonlinear Fiber Optics

Third Edition

OPTICS AND PHOTONICS

(formerly Quantum Electronics)

Series Editors

PAUL L. KELLEY

Tufts University

Medford, Massachusetts

IVAN P. KAMINOW

Lucent Technologies

Holmdel, New Jersey

GOVIND P. AGRAWAL

University of Rochester

Rochester, New York

Recently Published Books in the Series:

Jean-Claude Diels and Wolfgang Rudolph, Ultrashort Laser Pulse Phenomena:

Fundamentals, Techniques, and Applications on a Femtosecond Time Scale

Eli Kapon, editor, Semiconductor Lasers I: Fundamentals

Eli Kapon, editor, Semiconductor Lasers II: Materials and Structures

P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers:

Fundamentals and Technology

Raman Kashyap, Fiber Bragg Gratings

Katsunari Okamoto, Fundamentals of Optical Waveguides

Govind P. Agrawal, Applications of Nonlinear Fiber Optics

A complete list of titles in this series appears at the end of this volume.

Nonlinear Fiber Optics

Third Edition

GOVIND P. AGRAWAL

The Institute of Optics

University of Rochester

OPTICS AND PHOTONICS

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Contents

Preface xv

1 Introduction 1

1.1 Historical Perspective ...................... 1

1.2 Fiber Characteristics ...................... 3

1.2.1 Material and Fabrication ................ 4

1.2.2 Fiber Losses ...................... 5

1.2.3 Chromatic Dispersion ................. 7

1.2.4 Polarization-Mode Dispersion . . . . . . . . . . . . . 13

1.3 Fiber Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.1 Nonlinear Refraction . . . . . . . . . . . . . . . . . . 17

1.3.2 Stimulated Inelastic Scattering . . . . . . . . . . . . . 19

1.3.3 Importance of Nonlinear Effects . . . . . . . . . . . . 20

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Pulse Propagation in Fibers 31

2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Fiber Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Eigenvalue Equation . . . . . . . . . . . . . . . . . . 34

2.2.2 Single-Mode Condition . . . . . . . . . . . . . . . . . 36

2.2.3 Characteristics of the Fundamental Mode . . . . . . . 37

2.3 Pulse-Propagation Equation . . . . . . . . . . . . . . . . . . . 39

2.3.1 Nonlinear Pulse Propagation . . . . . . . . . . . . . . 39

2.3.2 Higher-Order Nonlinear Effects . . . . . . . . . . . . 45

2.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 51

vii

viii Contents

2.4.1 Split-Step Fourier Method . . . . . . . . . . . . . . . 51

2.4.2 Finite-Difference Methods . . . . . . . . . . . . . . . 55

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Group-Velocity Dispersion 63

3.1 Different Propagation Regimes . . . . . . . . . . . . . . . . . 63

3.2 Dispersion-Induced Pulse Broadening . . . . . . . . . . . . . 66

3.2.1 Gaussian Pulses . . . . . . . . . . . . . . . . . . . . . 67

3.2.2 Chirped Gaussian Pulses . . . . . . . . . . . . . . . . 69

3.2.3 Hyperbolic-Secant Pulses . . . . . . . . . . . . . . . 71

3.2.4 Super-Gaussian Pulses . . . . . . . . . . . . . . . . . 72

3.2.5 Experimental Results . . . . . . . . . . . . . . . . . . 75

3.3 Third-Order Dispersion . . . . . . . . . . . . . . . . . . . . . 76

3.3.1 Changes in Pulse Shape . . . . . . . . . . . . . . . . 77

3.3.2 Broadening Factor . . . . . . . . . . . . . . . . . . . 79

3.3.3 Arbitrary-Shape Pulses . . . . . . . . . . . . . . . . . 82

3.3.4 Ultrashort-Pulse Measurements . . . . . . . . . . . . 85

3.4 Dispersion Management . . . . . . . . . . . . . . . . . . . . 86

3.4.1 GVD-Induced Limitations . . . . . . . . . . . . . . . 86

3.4.2 Dispersion Compensation . . . . . . . . . . . . . . . 88

3.4.3 Compensation of Third-Order Dispersion . . . . . . . 90

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4 Self-Phase Modulation 97

4.1 SPM-Induced Spectral Broadening . . . . . . . . . . . . . . . 97

4.1.1 Nonlinear Phase Shift . . . . . . . . . . . . . . . . . 98

4.1.2 Changes in Pulse Spectra . . . . . . . . . . . . . . . . 100

4.1.3 Effect of Pulse Shape and Initial Chirp . . . . . . . . . 104

4.1.4 Effect of Partial Coherence . . . . . . . . . . . . . . . 106

4.2 Effect of Group-Velocity Dispersion . . . . . . . . . . . . . . 109

4.2.1 Pulse Evolution . . . . . . . . . . . . . . . . . . . . . 109

4.2.2 Broadening Factor . . . . . . . . . . . . . . . . . . . 113

4.2.3 Optical Wave Breaking . . . . . . . . . . . . . . . . . 115

4.2.4 Experimental Results . . . . . . . . . . . . . . . . . . 118

4.2.5 Effect of Third-Order Dispersion . . . . . . . . . . . . 120

4.3 Higher-Order Nonlinear Effects . . . . . . . . . . . . . . . . 122

Contents ix

4.3.1 Self-Steepening . . . . . . . . . . . . . . . . . . . . . 123

4.3.2 Effect of GVD on Optical Shocks . . . . . . . . . . . 126

4.3.3 Intrapulse Raman Scattering . . . . . . . . . . . . . . 128

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5 Optical Solitons 135

5.1 Modulation Instability . . . . . . . . . . . . . . . . . . . . . 136

5.1.1 Linear Stability Analysis . . . . . . . . . . . . . . . . 136

5.1.2 Gain Spectrum . . . . . . . . . . . . . . . . . . . . . 138

5.1.3 Experimental Observation . . . . . . . . . . . . . . . 140

5.1.4 Ultrashort Pulse Generation . . . . . . . . . . . . . . 142

5.1.5 Impact on Lightwave Systems . . . . . . . . . . . . . 144

5.2 Fiber Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.2.1 Inverse Scattering Method . . . . . . . . . . . . . . . 147

5.2.2 Fundamental Soliton . . . . . . . . . . . . . . . . . . 149

5.2.3 Higher-Order Solitons . . . . . . . . . . . . . . . . . 152

5.2.4 Experimental Confirmation . . . . . . . . . . . . . . . 154

5.2.5 Soliton Stability . . . . . . . . . . . . . . . . . . . . 156

5.3 Other Types of Solitons . . . . . . . . . . . . . . . . . . . . . 159

5.3.1 Dark Solitons . . . . . . . . . . . . . . . . . . . . . . 159

5.3.2 Dispersion-Managed Solitons . . . . . . . . . . . . . 164

5.3.3 Bistable Solitons . . . . . . . . . . . . . . . . . . . . 165

5.4 Perturbation of Solitons . . . . . . . . . . . . . . . . . . . . . 166

5.4.1 Perturbation Methods . . . . . . . . . . . . . . . . . . 167

5.4.2 Fiber Losses . . . . . . . . . . . . . . . . . . . . . . 169

5.4.3 Soliton Amplification . . . . . . . . . . . . . . . . . . 171

5.4.4 Soliton Interaction . . . . . . . . . . . . . . . . . . . 176

5.5 Higher-Order Effects . . . . . . . . . . . . . . . . . . . . . . 180

5.5.1 Third-Order Dispersion . . . . . . . . . . . . . . . . . 181

5.5.2 Self-Steepening . . . . . . . . . . . . . . . . . . . . . 183

5.5.3 Intrapulse Raman Scattering . . . . . . . . . . . . . . 186

5.5.4 Propagation of Femtosecond Pulses . . . . . . . . . . 190

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

x Contents

6 Polarization Effects 203

6.1 Nonlinear Birefringence . . . . . . . . . . . . . . . . . . . . 204

6.1.1 Origin of Nonlinear Birefringence . . . . . . . . . . . 204

6.1.2 Coupled-Mode Equations . . . . . . . . . . . . . . . 206

6.1.3 Elliptically Birefringent Fibers . . . . . . . . . . . . . 208

6.2 Nonlinear Phase Shift . . . . . . . . . . . . . . . . . . . . . . 210

6.2.1 Nondispersive XPM . . . . . . . . . . . . . . . . . . 210

6.2.2 Optical Kerr Effect . . . . . . . . . . . . . . . . . . . 211

6.2.3 Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . 216

6.3 Evolution of Polarization State . . . . . . . . . . . . . . . . . 218

6.3.1 Analytic Solution . . . . . . . . . . . . . . . . . . . . 219

6.3.2 Poincar´e-Sphere Representation . . . . . . . . . . . . 221

6.3.3 Polarization Instability . . . . . . . . . . . . . . . . . 224

6.3.4 Polarization Chaos . . . . . . . . . . . . . . . . . . . 227

6.4 Vector Modulation Instability . . . . . . . . . . . . . . . . . . 228

6.4.1 Low-Birefringence Fibers . . . . . . . . . . . . . . . 229

6.4.2 High-Birefringence Fibers . . . . . . . . . . . . . . . 231

6.4.3 Isotropic Fibers . . . . . . . . . . . . . . . . . . . . . 234

6.4.4 Experimental Results . . . . . . . . . . . . . . . . . . 235

6.5 Birefringence and Solitons . . . . . . . . . . . . . . . . . . . 238

6.5.1 Low-Birefringence Fibers . . . . . . . . . . . . . . . 239

6.5.2 High-Birefringence Fibers . . . . . . . . . . . . . . . 240

6.5.3 Soliton-Dragging Logic Gates . . . . . . . . . . . . . 243

6.5.4 Vector Solitons . . . . . . . . . . . . . . . . . . . . . 244

6.6 Random Birefringence . . . . . . . . . . . . . . . . . . . . . 246

6.6.1 Polarization-Mode Dispersion . . . . . . . . . . . . . 246

6.6.2 Polarization State of Solitons . . . . . . . . . . . . . . 248

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7 Cross-Phase Modulation 260

7.1 XPM-Induced Nonlinear Coupling . . . . . . . . . . . . . . . 261

7.1.1 Nonlinear Refractive Index . . . . . . . . . . . . . . . 261

7.1.2 Coupled NLS Equations . . . . . . . . . . . . . . . . 263

7.1.3 Propagation in Birefringent Fibers . . . . . . . . . . . 264

7.2 XPM-Induced Modulation Instability . . . . . . . . . . . . . . 265

7.2.1 Linear Stability Analysis . . . . . . . . . . . . . . . . 265

7.2.2 Experimental Results . . . . . . . . . . . . . . . . . . 268

Contents xi

7.3 XPM-Paired Solitons . . . . . . . . . . . . . . . . . . . . . . 270

7.3.1 Bright–Dark Soliton Pair . . . . . . . . . . . . . . . . 270

7.3.2 Bright–Gray Soliton Pair . . . . . . . . . . . . . . . . 272

7.3.3 Other Soliton Pairs . . . . . . . . . . . . . . . . . . . 272

7.4 Spectral and Temporal Effects . . . . . . . . . . . . . . . . . 274

7.4.1 Asymmetric Spectral Broadening . . . . . . . . . . . 275

7.4.2 Asymmetric Temporal Changes . . . . . . . . . . . . 281

7.4.3 Higher-Order Nonlinear Effects . . . . . . . . . . . . 284

7.5 Applications of XPM . . . . . . . . . . . . . . . . . . . . . . 286

7.5.1 XPM-Induced Pulse Compression . . . . . . . . . . . 286

7.5.2 XPM-Induced Optical Switching . . . . . . . . . . . . 289

7.5.3 XPM-Induced Nonreciprocity . . . . . . . . . . . . . 290

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8 Stimulated Raman Scattering 298

8.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 298

8.1.1 Raman-Gain Spectrum . . . . . . . . . . . . . . . . . 299

8.1.2 Raman Threshold . . . . . . . . . . . . . . . . . . . . 300

8.1.3 Coupled Amplitude Equations . . . . . . . . . . . . . 304

8.2 Quasi-Continuous SRS . . . . . . . . . . . . . . . . . . . . . 306

8.2.1 Single-Pass Raman Generation . . . . . . . . . . . . . 306

8.2.2 Raman Fiber Lasers . . . . . . . . . . . . . . . . . . 309

8.2.3 Raman Fiber Amplifiers . . . . . . . . . . . . . . . . 312

8.2.4 Raman-Induced Crosstalk . . . . . . . . . . . . . . . 318

8.3 SRS with Short Pump Pulses . . . . . . . . . . . . . . . . . . 320

8.3.1 Pulse-Propagation Equations . . . . . . . . . . . . . . 320

8.3.2 Nondispersive Case . . . . . . . . . . . . . . . . . . . 321

8.3.3 Effects of GVD . . . . . . . . . . . . . . . . . . . . . 324

8.3.4 Experimental Results . . . . . . . . . . . . . . . . . . 327

8.3.5 Synchronously Pumped Raman Lasers . . . . . . . . . 332

8.4 Soliton Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 333

8.4.1 Raman Solitons . . . . . . . . . . . . . . . . . . . . . 334

8.4.2 Raman Soliton Lasers . . . . . . . . . . . . . . . . . 339

8.4.3 Soliton-Effect Pulse Compression . . . . . . . . . . . 341

8.5 Effect of Four-Wave Mixing . . . . . . . . . . . . . . . . . . 343

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

xii Contents

9 Stimulated Brillouin Scattering 355

9.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 355

9.1.1 Physical Process . . . . . . . . . . . . . . . . . . . . 356

9.1.2 Brillouin-Gain Spectrum . . . . . . . . . . . . . . . . 357

9.2 Quasi-CW SBS . . . . . . . . . . . . . . . . . . . . . . . . . 359

9.2.1 Coupled Intensity Equations . . . . . . . . . . . . . . 360

9.2.2 Brillouin Threshold . . . . . . . . . . . . . . . . . . . 360

9.2.3 Gain Saturation . . . . . . . . . . . . . . . . . . . . . 362

9.2.4 Experimental Results . . . . . . . . . . . . . . . . . . 364

9.3 Dynamic Aspects . . . . . . . . . . . . . . . . . . . . . . . . 367

9.3.1 Coupled Amplitude Equations . . . . . . . . . . . . . 367

9.3.2 Relaxation Oscillations . . . . . . . . . . . . . . . . . 368

9.3.3 Modulation Instability and Chaos . . . . . . . . . . . 371

9.3.4 Transient Regime . . . . . . . . . . . . . . . . . . . . 373

9.4 Brillouin Fiber Lasers . . . . . . . . . . . . . . . . . . . . . . 375

9.4.1 CW Operation . . . . . . . . . . . . . . . . . . . . . 375

9.4.2 Pulsed Operation . . . . . . . . . . . . . . . . . . . . 377

9.5 SBS Applications . . . . . . . . . . . . . . . . . . . . . . . . 380

9.5.1 Brillouin Fiber Amplifiers . . . . . . . . . . . . . . . 380

9.5.2 Fiber Sensors . . . . . . . . . . . . . . . . . . . . . . 383

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

10 Parametric Processes 389

10.1 Origin of Four-Wave Mixing . . . . . . . . . . . . . . . . . . 389

10.2 Theory of Four-Wave Mixing . . . . . . . . . . . . . . . . . . 392

10.2.1 Coupled Amplitude Equations . . . . . . . . . . . . . 392

10.2.2 Approximate Solution . . . . . . . . . . . . . . . . . 394

10.2.3 Effect of Phase Matching . . . . . . . . . . . . . . . . 396

10.2.4 Ultrafast FWM . . . . . . . . . . . . . . . . . . . . . 397

10.3 Phase-Matching Techniques . . . . . . . . . . . . . . . . . . 399

10.3.1 Physical Mechanisms . . . . . . . . . . . . . . . . . . 399

10.3.2 Phase Matching in Multimode Fibers . . . . . . . . . 400

10.3.3 Phase Matching in Single-Mode Fibers . . . . . . . . 404

10.3.4 Phase Matching in Birefringent Fibers . . . . . . . . . 408

10.4 Parametric Amplification . . . . . . . . . . . . . . . . . . . . 412

10.4.1 Gain and Bandwidth . . . . . . . . . . . . . . . . . . 412

10.4.2 Pump Depletion . . . . . . . . . . . . . . . . . . . . 414

Contents xiii

10.4.3 Parametric Amplifiers . . . . . . . . . . . . . . . . . 416

10.4.4 Parametric Oscillators . . . . . . . . . . . . . . . . . 417

10.5 FWM Applications . . . . . . . . . . . . . . . . . . . . . . . 418

10.5.1 Wavelength Conversion . . . . . . . . . . . . . . . . 419

10.5.2 Phase Conjugation . . . . . . . . . . . . . . . . . . . 420

10.5.3 Squeezing . . . . . . . . . . . . . . . . . . . . . . . . 422

10.5.4 Supercontinuum Generation . . . . . . . . . . . . . . 424

10.6 Second-Harmonic Generation . . . . . . . . . . . . . . . . . 427

10.6.1 Experimental Results . . . . . . . . . . . . . . . . . . 427

10.6.2 Physical Mechanism . . . . . . . . . . . . . . . . . . 429

10.6.3 Simple Theory . . . . . . . . . . . . . . . . . . . . . 431

10.6.4 Quasi-Phase-Matching Technique . . . . . . . . . . . 434

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Appendix A Decibel Units 445

Appendix B Nonlinear Refractive Index 447

Appendix C Acronyms 454

Index 457

Preface

Since the publication of the first edition of this book in 1989, the field of

nonlinear fiber optics has virtually exploded. A major factor behind such a

tremendous growth was the advent of fiber amplifiers, made by doping silica

or fluoride fibers with rare-earth ions such as erbium and neodymium. Such

amplifiers revolutionized the design of fiber-optic communication systems, in￾cluding those making use of optical solitons whose very existence stems from

the presence of nonlinear effects in optical fibers. Optical amplifiers permit

propagation of lightwave signals over thousands of kilometers as they can com￾pensate for all losses encountered by the signal in the optical domain. At the

same time, fiber amplifiers enable the use of massive wavelength-division mul￾tiplexing (WDM) and have led to the development of lightwave systems with

capacities exceeding 1 Tb/s. Nonlinear fiber optics plays an increasingly im￾portant role in the design of such high-capacity lightwave systems. In fact,

an understanding of various nonlinear effects occurring inside optical fibers is

almost a prerequisite for a lightwave-system designer.

The third 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. An attempt was made to include recent research results on all topics

relevant to the field of nonlinear fiber optics. Such an ambitious objective

increased the size of the book to the extent that it was necessary to split it

into two separate books. This book will continue to deal with the fundamental

aspects of nonlinear fiber optics. A second book Applications of Nonlinear

Fiber Optics is devoted to its applications; it is referred to as Part B in this text.

Nonlinear Fiber Optics, 3rd edition, retains most of the material that ap￾peared in the first edition, with the exception of Chapter 6, which is now de￾voted to the polarization effects relevant for light propagation in optical fibers.

Polarization issues have become increasingly more important, especially for

high-speed lightwave systems for which the phenomenon of polarization-mode

xv

xvi Preface

dispersion (PMD) has become a limiting factor. It is thus necessary that stu￾dents learn about PMD and other polarization effects in a course devoted to

nonlinear fiber optics.

The potential readership is likely to consist of senior undergraduate stu￾dents, graduate students enrolled in the M. S. and Ph. D. degree programs, en￾gineers and technicians involved with the telecommunication industry, and sci￾entists 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

that are designed to provide mastery of the fundamental aspects. Some uni￾versities 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 com￾pletion of the third edition. I am thankful to all of them, especially to my stu￾dents whose curiosity led to several improvements. Several of my colleagues

have helped me in preparing the third edition. I thank them for reading drafts

and making helpful suggestions. I am grateful to many readers for their occa￾sional feedback. Last, but not least, I thank my wife, Anne, and my daughters,

Sipra, Caroline, and Claire, for understanding why I needed to spend many

weekends on the book instead of spending time with them.

Govind P. Agrawal

Rochester, NY

Chapter 1

Introduction

This introductory chapter is intended to provide an overview of the fiber char￾acteristics 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 nonlin￾ear effects probed by using ultrashort optical pulses. Section 1.3 introduces

various nonlinear effects resulting from the intensity dependence of the refrac￾tive 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

the text 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 discovery of

this phenomenon [1]. Although uncladded glass fibers were fabricated in the

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 charac￾1