Solitons in Optical Fiber Systems

Discover a robust exploration of the main properties and behaviors of solitons in fiber systems

In Solitons in Optical Fiber Systems, distinguished researcher Dr. Mário F. S. Ferreira delivers a thorough treatment of the main characteristics of solitons in optical fiber communication systems and fiber devices, paying special attention to stationary and pulsating dissipative soliton pulses. The book discusses the technical aspects associated with the physical background and the theoretical description of soliton characteristics under different conditions.

The author employs numerical analyses and variational approaches to describe soliton evolution and describes the phenomenon of supercontinuum generation and various solitonic effects observed in highly nonlinear fibers, like photonic crystal fibers.

Readers will learn about different applications of fiber solitons in transmission systems, fiber lasers, couplers, and pulse compression schemes, as well as complex Ginzburg-Landau equations, which are used to model different types of dissipative systems.

The book also includes:

• A thorough introduction to solitons, including the linear and nonlinear effects of a wave, the discovery of solitary waves, and the discovery of solitons in optical fibers
• An exploration of fiber dispersion and nonlinearity, including optical fiber dispersion, the pulse propagation equation, and the impact of fiber dispersion
• Practical discussions of nonlinear effects in optical fibers, including self-phase modulation, cross-phase modulations, four-wave mixing, and stimulated raman scattering
• In-depth treatments of solitons in optical fibers, including modulation instability, dark solitons, bistable solitons, XPM-paired solitons, and the variational approach

Perfect for senior undergraduate and graduate students in courses dealing with fiber-optics technology, Solitons in Optical Fiber Systems is also an ideal resource for engineers and technicians in the fiber-optics industry and researchers of nonlinear fiber optics.

Mário F. S. Ferreira, PhD, is a Senior Member of the Optical Society of America (OSA) and SPIE - The International Society for Optical and Photonics. He is also a member of the IEEE Photonics Society and the Portuguese Physical Society. He is a Travelling Lecturer of the OSA and SPIE.

Preface xiii

List of Abbreviations xv

1 Introduction 1

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

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

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

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

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

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

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

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

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

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 Schrödinger 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

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

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

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

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

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

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 Kagomé Fibers 356

References 365

Index 369

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