Preface xiii
List of Abbreviations xv
1 Introduction 1
References 5
2 Waves Called Solitons 9
2.1 Linear and Nonlinear Effects of a Wave 9
2.2 Solitary Waves and Solitons 11
2.3 Solitons in Optical Fibers 13
2.4 Dissipative Optical Solitons 15
References 16
3 Fiber Dispersion and Nonlinearity 19
3.1 Fiber Chromatic Dispersion 19
3.1.1 Gaussian Input Pulses 21
3.2 Fiber Nonlinearity 25
3.2.1 The Nonlinear Refractive Index 25
3.2.2 Relevance of Nonlinear Effects in Fibers 26
3.3 The Pulse Propagation Equation 28
3.3.1 The Normalized NLSE 29
3.3.2 Propagation in the Absence of Dispersion and Nonlinearity 30
3.3.3 Effect of Dispersion Only 30
3.3.4 Effect of Nonlinearity Only 32
References 33
4 Nonlinear Effects in Optical Fibers 35
4.1 Self-Phase Modulation 35
4.1.1 Modulation Instability 39
4.2 Cross-Phase Modulation 40
4.3 Four-Wave Mixing 42
4.4 Stimulated Raman Scattering 45
4.5 Stimulated Brillouin Scattering 49
References 52
5 Optical Amplification 57
5.1 General Concepts on Optical Amplifiers 57
5.2 Erbium-Doped Fiber Amplifiers 59
5.2.1 Two-Level Model 60
5.3 Fiber Raman Amplifiers 63
5.4 Fiber Parametric Amplifiers 68
5.5 Lumped versus Distributed Amplification 72
5.6 Parabolic Pulses 74
References 76
6 Solitons in Optical Fibers 81
6.1 The Fundamental Soliton Solution 81
6.2 Higher-Order Solitons 83
6.3 Soliton Units 86
6.4 Dark Solitons 87
6.5 Bistable Solitons 88
6.6 XPM-Paired Solitons 89
6.7 Optical Similaritons 90
6.8 Numerical Solution of the NLSE 92
6.9 The Variational Approach 94
6.10 The Method of Moments 97
References 98
7 Soliton Transmission Systems 101
7.1 Soliton Perturbation Theory 101
7.2 Effect of Fiber Losses 102
7.3 Soliton Amplification 103
7.3.1 Lumped Amplification 104
7.3.2 Distributed Amplification 105
7.4 Soliton Interaction 107
7.5 Timing Jitter 110
7.5.1 Gordon-Haus Jitter 110
7.5.2 Polarization-Mode Dispersion Jitter 113
7.5.3 Acoustic Jitter 113
7.5.4 Soliton Interaction Jitter 114
7.6 WDM Soliton Systems 114
7.6.1 Lossless Soliton Collisions 114
7.6.2 Soliton Collisions in Perturbed Fiber Spans 116
7.6.3 Timing Jitter 117
References 117
8 Soliton Transmission Control 121
8.1 Fixed-Frequency Filters 121
8.1.1 Control of Timing Jitter 122
8.1.2 Control of Soliton Interaction 123
8.1.3 Background Instability 125
8.2 Sliding-Frequency Filters 125
8.2.1 Evolution of Soliton Parameters 126
8.2.2 Control of Timing Jitter 129
8.2.3 Control of Soliton Interaction 131
8.3 Synchronous Modulators 132
8.4 Amplifiers with Nonlinear Gain 133
8.4.1 Stationary Solutions 134
8.4.2 Control of Soliton Interaction 137
References 139
9 Propagation of Ultrashort Solitons 141
9.1 Generalized NLSE 141
9.1.1 Third-Order Dispersion 142
9.1.2 Self-Steepening 143
9.1.3 Intrapulse Raman Scattering 144
9.2 Timing Jitter of Ultrashort Solitons 145
9.3 Bandwidth-Limited Amplification of Ultrashort Solitons 147
9.4 Transmission Control Using Nonlinear Gain 151
9.4.1 Stationary Solutions 151
9.4.2 Linear Stability Analysis 153
References 157
10 Dispersion-Managed Solitons 161
10.1 Dispersion Management 161
10.2 Characteristics of the Dispersion-Managed Soliton 163
10.3 The Variational Approach to DM Solitons 167
10.3.1 Generic Ansatz 167
10.3.2 Gaussian Pulses 168
10.3.3 Stationary Solutions 169
10.4 Interaction Between DM Solitons 170
10.5 The Gordon–Haus Effect for DM Solitons 172
10.6 Effects of a Spectral Filter 173
10.6.1 Timing Jitter Control 174
10.7 Effects of an Amplitude Modulator 175
10.8 WDM with DM Solitons 177
References 179
11 Polarization Effects 183
11.1 Fiber Birefringence and Polarization Mode Dispersion 183
11.1.1 PMD in Long Fiber Spans 185
11.1.2 PMD-Induced Pulse Broadening in Linear Systems 187
11.1.3 PMD Compensation 188
11.2 Coupled Nonlinear Schrodinger Equations 190
11.3 Solitons in Fibers with Constant Birefringence 191
11.4 Vector Solitons 195
11.5 Solitons in Fibers with Randomly Varying Birefringence 196
11.6 PMD-Induced Soliton Pulse Broadening 197
11.7 Dispersion-Managed Solitons and PMD 200
References 202
12 Stationary Dissipative Solitons 207
12.1 Balance Equations for the CGL Equation 207
12.2 Exact Analytical Solutions 210
12.2.1 Solutions of the Cubic CGLE 210
12.2.2 Solutions of the Quintic CGLE 212
12.3 Numerical Stationary Soliton Solutions 213
12.4 High-Energy Dissipative Solitons 216
12.5 Soliton Bound States 221
12.6 Impact of Higher-Order Effects 225
References 229
13 Pulsating Dissipative Solitons 233
13.1 Dynamic Models for CGLE Solitons 233
13.1.1 The Variational Approach 234
13.1.1.1 Sech Ansatz 235
13.1.1.2 Gaussian Ansatz 235
13.1.2 The Method of Moments 236
13.2 Plain Pulsating Solitons 238
13.2.1 Impact of Higher-Order Effects 239
13.3 Creeping Solitons 241
13.3.1 Impact of Higher-Order Effects 242
13.4 Chaotic Solitons 244
13.5 Erupting Solitons 247
13.5.1 Impact of Higher-Order Effects 251
13.5.2 Experimental Observation of Soliton Explosions 253
References 256
14 Soliton Fiber Lasers 259
14.1 The First Soliton Laser 259
14.2 Fundamentals of Fiber Soliton Lasers 260
14.3 Mode-Locking Techniques 262
14.3.1 Active Mode-Locking 262
14.3.2 Passive Mode-Locking 262
14.3.3 Nonlinear Optical Loop Mirrors 263
14.3.4 Figure-Eight Laser 264
14.3.5 Nonlinear Polarization Rotation 265
14.3.6 Hybrid Mode-Locking 265
14.4 High-Energy Soliton Fiber Lasers 266
14.5 Modeling of Soliton Fiber Lasers 268
14.6 Polarization Effects 272
14.7 Dissipative Soliton Molecules 273
14.8 Experimental Observation of Pulsating Solitons 274
References 279
15 Other Applications of Optical Solitons 285
15.1 All-Optical Switching 285
15.1.1 The Fiber Coupler 285
15.1.2 The Sagnac Interferometer 286
15.2 2R Optical Regeneration 288
15.3 Pulse Compression 290
15.3.1 Grating-Fiber Compression 290
15.3.2 Higher-Order Soliton-Effect Compression 291
15.3.3 Compression of Fundamental Solitons 293
15.3.4 Dissipative Soliton Compression 295
15.4 Solitons in Fiber Gratings 298
15.4.1 Pulse Compression Using Fiber Gratings 300
15.4.2 Fiber Bragg Solitons 302
References 305
16 Highly Nonlinear Optical Fibers 309
16.1 Highly Nonlinear Silica Fibers 309
16.1.1 Tapered Fibers 310
16.2 Microstructured Optical Fibers 311
16.3 Non-Silica Fibers 318
16.4 Soliton Fission and Dispersive Waves 320
16.5 Four-Wave Mixing 324
16.6 Hollow-Core Microstructured Fibers 325
References 332
17 Supercontinuum Generation 337
17.1 Pumping with Femtosecond Pulses 337
17.2 Modeling the Supercontinuum 341
17.3 Pumping with Picosecond Pulses 344
17.4 Continuous Wave Supercontinuum Generation 347
17.5 Mid-IR Supercontinuum Generation 350
17.6 Supercontinuum Coherence 352
17.6.1 Spectral Incoherent Solitons 354
17.7 Supercontinuum Generation in Hollow-Core Kagome Fibers 356
References 365
Index 369
