Dedication ...................................................... v
Author Biography .............................................. vii
Preface ....................................................... xix
CHAPTER 1 Introduction ......................................... 1
1.1 Historical Perspective ..................................... 1
1.2 Fiber Characteristics ...................................... 3
1.2.1 Material and Fabrication ............................ 4
1.2.2 Fiber Losses ........................................ 5
1.2.3 Chromatic Dispersion ................................ 6
1.2.4 Polarization-Mode Dispersion ....................... 11
1.3 Fiber Nonlinearities ...................................... 15
1.3.1 Nonlinear Refraction ............................... 15
1.3.2 Stimulated Inelastic Scattering .................... 16
1.3.3 Importance of Nonlinear Effects .................... 18
1.4 Overview .................................................. 19
Problems .................................................. 21
References ................................................ 22
CHAPTER 2 Pulse Propagation in Fibers ......................... 21
2.1 Maxwell's Equations ....................................... 27
2.2 Fiber Modes ............................................... 30
2.2.1 Eigenvalue Equation ................................ 30
2.2.2 Single-Mode Condition .............................. 31
2.2.3 Characteristics of the Fundamental Mode ............ 32
2.3 Pulse-Propagation Equation ................................ 34
2.3.1 Nonlinear Pulse Propagation ........................ 34
2.3.2 Higher-Order Nonlinear Effects ..................... 39
2.3.3 Raman Response Function and its Impact ............. 41
2.3.4 Extension to Multimode Fibers ...................... 45
2.4 Numerical Methods ......................................... 47
2.4.1 Split-Step Fourier Method .......................... 47
2.4.2 Finite-Difference Methods .......................... 51
Problems .................................................. 52
References ................................................ 53
CHAPTER 3 Group-Velocity Dispersion ........................... 57
3.1 Different Propagation Regimes ............................. 57
3.2 Dispersion-Induced Pulse Broadening ....................... 59
3.2.1 Gaussian Pulses .................................... 60
3.2.2 Chirped Gaussian Pulses ............................ 62
3.2.3 Hyperbolic-Secant Pulses ........................... 64
3.2.4 Super-Gaussian Pulses .............................. 65
3.2.5 Experimental Results ............................... 67
3.3 Third-Order Dispersion .................................... 68
3.3.1 Evolution of Chirped Gaussian Pulses ............... 69
3.3.2 Broadening Factor .................................. 71
3.3.3 Arbitrary-Shape Pulses ............................. 74
3.3.4 Ultrashort-Pulse Measurements ...................... 76
3.4 Dispersion Management ..................................... 78
3.4.1 GVD-Induced Limitations ............................ 78
3.4.2 Dispersion Compensation ............................ 80
3.4.3 Compensation of Third-Order Dispersion ............. 81
Problems .................................................. 83
References ................................................ 84
CHAPTER 4 Self-Phase Modulation ............................... 87
4.1 SPM-Induced Spectral Changes .............................. 87
4.1.1 Nonlinear Phase Shift .............................. 88
4.1.2 Changes in Pulse Spectra ........................... 90
4.1.3 Effect of Pulse Shape and Initial Chirp ............ 93
4.1.4 Effect of Partial Coherence ........................ 96
4.2 Effect of Group-Velocity Dispersion ....................... 98
4.2.1 Pulse Evolution .................................... 98
4.2.2 Broadening Factor ................................. 100
4.2.3 Optical Wave Breaking ............................. 102
4.2.4 Experimental Results .............................. 105
4.2.5 Effect of Third-Order Dispersion .................. 106
4.2.6 SPM Effects in Fiber Amplifiers ................... 108
4.3 Semianalytic Techniques .................................. 111
4.3.1 Moment Method ..................................... 111
4.3.2 Variational Method ................................ 112
4.3.3 Specific Analytic Solutions ....................... 114
4.4 Higher-Order Nonlinear Effects ........................... 115
4.4.1 Self-Steepening ................................... 116
4.4.2 Effect of GVD on Optical Shocks ................... 119
4.4.3 Intrapulse Raman Scattering ....................... 121
Problems ................................................. 124
References ............................................... 125
CHAPTER 5 Optical Solitons ................................... 129
5.1 Modulation Instability ................................... 129
5.1.1 Linear Stability Analysis ......................... 130
5.1.2 Gain Spectrum ..................................... 131
5.1.3 Experimental Results .............................. 133
5.1.4 Ultrashort Pulse Generation ....................... 135
5.1.5 Impact on Lightwave Systems ....................... 137
5.2 Fiber Solitons ........................................... 139
5.2.1 Inverse Scattering Method ......................... 140
5.2.2 Fundamental Soliton ............................... 142
5.2.3 Second and Higher-Order Solitons .................. 144
5.2.4 Experimental Confirmation ......................... 147
5.2.5 Soliton Stability ................................. 148
5.3 Other Types of Solitons .................................. 151
5.3.1 Dark Solitons ..................................... 151
5.3.2 Bistable Solitons ................................. 154
5.3.3 Dispersion-Managed Solitons ....................... 156
5.3.4 Optical Similaritons .............................. 156
5.4 Perturbation of Solitons ................................. 159
5.4.1 Perturbation Methods .............................. 159
5.4.2 Fiber Losses ...................................... 161
5.4.3 Soliton Amplification ............................. 163
5.4.4 Soliton Interaction ............................... 166
5.5 Higher-Order Effects ..................................... 170
5.5.1 Moment Equations for Pulse Parameters ............. 170
5.5.2 Third-Order Dispersion ............................ 172
5.5.3 Self-Steepening ................................... 174
5.5.4 Intrapulse Raman Scattering ....................... 176
5.5.5 Propagation of Femtosecond Pulses ................. 181
Problems ................................................. 183
References ............................................... 184
CHAPTER 6 Polarization Effects ............................... 193
6.1 Nonlinear Birefringence ................................... 193
6.1.1 Origin of Nonlinear Birefringence ................. 194
6.1.2 Coupled-Mode Equations ............................ 196
6.1.3 Elliptically Birefringent Fibers .................. 197
6.2 Nonlinear Phase Shift .................................... 199
6.2.1 Nondispersive XPM ................................. 199
6.2.2 Optical Kerr Effect ............................... 200
6.2.3 Pulse Shaping ..................................... 204
6.3 Evolution of Polarization State .......................... 206
6.3.1 Analytic Solution ................................. 207
6.3.2 Poincare-Sphere Representation .................... 209
6.3.3 Polarization Instability .......................... 212
6.3.4 Polarization Chaos ................................ 214
6.4 Vector Modulation Instability ............................ 215
6.4.1 Low-Birefringence Fibers .......................... 215
6.4.2 High-Birefringence Fibers ......................... 218
6.4.3 Isotropic Fibers .................................. 220
6.4.4 Experimental Results .............................. 221
6.5 Birefringence and Solitons ............................... 224
6.5.1 Low-Birefringence Fibers .......................... 225
6.5.2 High-Birefringence Fibers ......................... 226
6.5.3 Soliton-Dragging Logic Gates ...................... 229
6.5.4 Vector Solitons ................................... 230
6.6 Random Birefringence ..................................... 233
6.6.1 Polarization-Mode Dispersion ...................... 233
6.6.2 Vector Form of the NLS Equation ................... 234
6.6.3 Effects of PMD on Solitons ........................ 236
Problems ................................................. 239
References ............................................... 240
CHAPTER 7 Cross-Phase Modulation ............................. 245
7.1 XPM-Induced Nonlinear Coupling ........................... 246
7.1.1 Nonlinear Refractive Index ........................ 246
7.1.2 Coupled NLS Equations ............................. 247
7.2 XPM-Induced Modulation Instability ....................... 248
7.2.1 Linear Stability Analysis ......................... 249
7.2.2 Experimental Results .............................. 251
7.3 XPM-Paired Solitons ...................................... 252
7.3.1 Bright-Dark Soliton Pair .......................... 252
7.3.2 Bright-Gray Soliton Pair .......................... 254
7.3.3 Periodic Solutions ................................ 255
7.3.4 Multiple Coupled NLS Equations .................... 256
7.4 Spectral and Temporal Effects ............................ 258
7.4.1 Asymmetric Spectral Broadening .................... 259
7.4.2 Asymmetric Temporal Changes ....................... 264
7.4.3 Higher-Order Nonlinear Effects ..................... 267
7.5 Applications of XPM ...................................... 268
7.5.1 XPM-Induced Pulse Compression ..................... 268
7.5.2 XPM-Induced Optical Switching ..................... 270
7.5.3 XPM-Induced Nonreciprocity ........................ 272
7.6 Polarization Effects ..................................... 274
7.6.1 Vector Theory of XPM .............................. 274
7.6.2 Polarization Evolution ............................ 275
7.6.3 Polarization-Dependent Spectral Broadening ........ 278
7.6.4 Pulse Trapping and Compression .................... 280
7.6.5 XPM-Induced Wave Breaking ......................... 282
7.7 XPM Effects in Birefringent Fibers ....................... 284
7.7.1 Fibers with Low Birefringence ..................... 284
7.7.2 Fibers with High Birefringence .................... 287
Problems ................................................. 289
References ............................................... 290
CHAPTER 8 Stimulated Raman Scattering ........................ 295
8.1 Basic Concepts ........................................... 295
8.1.1 Raman-Gain Spectrum ............................... 296
8.1.2 Raman Threshold ................................... 297
8.1.3 Coupled Amplitude Equations ....................... 300
8.1.4 Effect of Four-Wave Mixing ........................ 303
8.2 Quasi-Continuous SRS ..................................... 305
8.2.1 Single-Pass Raman Generation ...................... 305
8.2.2 Raman Fiber Lasers ................................ 307
8.2.3 Raman Fiber Amplifiers ............................ 310
8.2.4 Raman-Induced Crosstalk ........................... 315
8.3 SRS with Short Pump Pulses ............................... 316
8.3.1 Pulse-Propagation Equations ....................... 317
8.3.2 Nondispersive Case ................................ 318
8.3.3 Effects of GVD .................................... 320
8.3.4 Raman-Induced Index Changes ....................... 323
8.3.5 Experimental Results .............................. 325
8.3.6 Synchronously Pumped Raman Lasers ................. 328
8.3.7 Short-Pulse Raman Amplification ................... 330
8.4 Soliton Effects .......................................... 331
8.4.1 Raman Solitons .................................... 331
8.4.2 Raman Soliton Lasers .............................. 335
8.4.3 Soliton-Effect Pulse Compression .................. 338
8.5 Polarization Effects ..................................... 339
8.5.1 Vector Theory of Raman Amplification .............. 339
8.5.2 PMD Effects on Raman Amplification ................ 343
Problems ................................................. 346
References ............................................... 347
CHAPTER 9 Stimulated Brillouin Scattering .................... 353
9.1 Basic Concepts ........................................... 353
9.1.1 Physical Process .................................. 354
9.1.2 Brillouin-Gain Spectrum ........................... 354
9.2 Quasi-CWSBS .............................................. 358
9.2.1 Brillouin Threshold ............................... 358
9.2.2 Polarization Effects .............................. 359
9.2.3 Techniques for Controlling the SBS Threshold ...... 360
9.2.4 Experimental Results .............................. 363
9.3 Brillouin-Fiber Amplifiers ............................... 366
9.3.1 Gain Saturation ................................... 366
9.3.2 Amplifier Design and Applications ................. 367
9.4 SBS Dynamics ............................................. 370
9.4.1 Coupled Amplitude Equations ....................... 370
9.4.2 SBS with Q-Switched Pulses ........................ 372
9.4.3 SBS-Induced Index Changes ......................... 376
9.4.4 Relaxation Oscillations ........................... 380
9.4.5 Modulation Instability and Chaos .................. 382
9.5 Brillouin-Fiber Lasers ................................... 384
9.5.1 CW Operation ...................................... 384
9.5.2 Pulsed Operation .................................. 388
Problems ................................................. 391
References ............................................... 392
CHAPTER 10 Four-Wave Mixing ................................... 397
10.1 Origin of Four-Wave Mixing ............................... 397
10.2 Theory of Four-Wave Mixing ............................... 399
10.2.1 Coupled Amplitude Equations ....................... 400
10.2.2 Approximate Solution .............................. 401
10.2.3 Effect of Phase Matching .......................... 402
10.2.4 Ultrafast Four-Wave Mixing ........................ 404
10.3 Phase-Matching Techniques ................................ 405
10.3.1 Physical Mechanisms ............................... 405
10.3.2 Phase Matching in Multimode Fibers ................ 406
10.3.3 Phase Matching in Single-Mode Fibers .............. 409
10.3.4 Phase Matching in Birefringent Fibers ............. 414
10.4 Parametric Amplification ................................. 417
10.4.1 Review of Early Work .............................. 417
10.4.2 Gain Spectrum and Its Bandwidth ................... 418
10.4.3 Single-Pump Configuration ......................... 421
10.4.4 Dual-Pump Configuration ........................... 425
10.4.5 Effects of Pump Depletion ......................... 430
10.5 Polarization Effects ..................................... 431
10.5.1 Vector Theory of Four-Wave Mixing ................. 432
10.5.2 Polarization Dependence of Parametric Gain ........ 434
10.5.3 Linearly and Circularly Polarized Pumps ........... 437
10.5.4 Effect of Residual Fiber Birefringence ............ 439
10.6 Applications of Four-Wave Mixing ......................... 443
10.6.1 Parametric Oscillators ............................ 443
10.6.2 Ultrafast Signal Processing ....................... 445
10.6.3 Quantum Correlation and Noise Squeezing ........... 447
10.6.4 Phase-Sensitive Amplification ..................... 449
Problems ................................................. 451
References ............................................... 452
CHAPTER 11 Highly Nonlinear Fibers ............................ 457
11.1 Nonlinear Parameter ...................................... 457
11.1.1 Units and Values of n2 ............................ 458
11.1.2 SPM-Based Techniques .............................. 459
11.1.3 XPM-Based Technique ............................... 462
11.1.4 FWM-Based Technique ............................... 463
11.1.5 Variations in m Values ............................ 464
11.2 Fibers with Silica Cladding .............................. 467
11.3 Tapered Fibers with Air Cladding ......................... 469
11.4 Microstructured Fibers ................................... 474
11.4.1 Design and Fabrication ............................ 474
11.4.2 Modal and Dispersive Properties ................... 476
11.4.3 Hollow-Core Photonic Crystal Fibers ............... 478
11.4.4 Bragg Fibers ...................................... 480
11.5 Non-Silica Fibers ........................................ 481
11.5.1 Lead-Silicate Fibers .............................. 482
11.5.2 Chalcogenide Fibers ............................... 485
11.5.3 Bismuth-Oxide Fibers .............................. 486
11.6 Pulse Propagation in Narrow-Core Fibers .................. 487
11.6.1 Vectorial Theory .................................. 487
11.6.2 Frequency-Dependent Mode Profiles ................. 489
Problems ................................................. 491
References ............................................... 492
CHAPTER 12 Novel Nonlinear Phenomena .......................... 497
12.1 Soliton Fission and Dispersive Waves ..................... 497
12.1.1 Fission of Second- and Higher-Order Solitons ...... 498
12.1.2 Generation of Dispersive Waves .................... 501
12.2 Intrapulse Raman Scattering .............................. 506
12.2.1 Enhanced RTFS Through Soliton Fission ............. 506
12.2.2 Cross-correlation Technique ....................... 510
12.2.3 Wavelength Tuning through RTFS .................... 512
12.2.4 Effects of Birefringence .......................... 514
12.2.5 Suppression of Raman-Induced Frequency Shifts ..... 516
12.2.6 Soliton Dynamics Near a Zero-Dispersion
Wavelength ........................................ 520
12.2.7 Multipeak Raman Solitons .......................... 523
12.3 Four-Wave Mixing ......................................... 526
12.3.1 Role of Fourth-Order Dispersion ................... 526
12.3.2 Role of Fiber Birefringence ....................... 527
12.3.3 Parametric Amplifiers and Wavelength Converters ... 531
12.3.4 Tunable Fiber-Optic Parametric Oscillators ........ 532
12.4 Second-Harmonic Generation ............................... 534
12.4.1 Physical Mechanisms ............................... 534
12.4.2 Thermal Poling and Quasi-Phase Matching ........... 536
12.4.3 SHG Theory ........................................ 539
12.5 Third-Harmonic Generation ................................ 541
12.5.1 THG in Highly Nonlinear Fibers .................... 541
12.5.2 Effects of Group-Velocity Mismatch ................ 543
12.5.3 Effects of Fiber Birefringence .................... 545
Problems ................................................. 546
References ............................................... 547
CHAPTER 13 Supercontinuum Generation .......................... 553
13.1 Pumping with Picosecond Pulses ........................... 553
13.1.1 Nonlinear Mechanisms .............................. 554
13.1.2 Experimental Progress After 2000 .................. 556
13.2 Pumping with Femtosecond Pulses .......................... 559
13.2.1 Microstructured Silica Fibers ..................... 559
13.2.2 Microstructured Nonsilica Fibers .................. 563
13.3 Temporal and Spectral Evolutions ......................... 566
13.3.1 Numerical Modeling of Supercontinuum .............. 566
13.3.2 Role of Cross-Phase Modulation .................... 570
13.3.3 XPM-Induced Trapping .............................. 573
13.3.4 Role of Four-Wave Mixing .......................... 577
13.4 CW or Quasi-CW Pumping ................................... 579
13.4.1 Nonlinear Mechanisms .............................. 579
13.4.2 Experimental Progress ............................. 582
13.5 Polarization Effects ..................................... 585
13.5.1 Birefringent Microstructured Fibers ............... 586
13.5.2 Nearly Isotropic Fibers ........................... 587
13.5.3 Nonlinear Polarization Rotation in Isotropic
Fibers ............................................ 589
13.6 Coherence Properties ..................................... 590
13.6.1 Spectral-Domain Degree of Coherence ............... 591
13.6.2 Techniques for Improving Coherence ................ 594
13.6.3 Spectral Incoherent Solitons ...................... 596
13.7 Optical Rogue Waves ...................................... 598
13.7.1 L-Shaped Statistics of Pulse-to-Pulse
Fluctuations ...................................... 599
13.7.2 Techniques for Controlling Rogue-Wave Statistics .. 600
13.7.3 Modulation Instability Revisited .................. 602
Problems ................................................. 606
References ............................................... 607
Appendix A System of Units .................................... 613
Appendix В Numerical Code for the NLS Equation ................ 615
Appendix С List of Acronyms ................................... 619
Index ......................................................... 621
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