Solitons in Optical Fiber Systems

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 xiiiList of Abbreviations xv1 Introduction 1References 52 Waves Called Solitons 92.1 Linear and Nonlinear Effects of a Wave 92.2 Solitary Waves and Solitons 112.3 Solitons in Optical Fibers 132.4 Dissipative Optical Solitons 15References 163 Fiber Dispersion and Nonlinearity 193.1 Fiber Chromatic Dispersion 193.1.1 Gaussian Input Pulses 213.2 Fiber Nonlinearity 253.2.1 The Nonlinear Refractive Index 253.2.2 Relevance of Nonlinear Effects in Fibers 263.3 The Pulse Propagation Equation 283.3.1 The Normalized NLSE 293.3.2 Propagation in the Absence of Dispersion and Nonlinearity 303.3.3 Effect of Dispersion Only 303.3.4 Effect of Nonlinearity Only 32References 334 Nonlinear Effects in Optical Fibers 354.1 Self-Phase Modulation 354.1.1 Modulation Instability 394.2 Cross-Phase Modulation 404.3 Four-Wave Mixing 424.4 Stimulated Raman Scattering 454.5 Stimulated Brillouin Scattering 49References 525 Optical Amplification 575.1 General Concepts on Optical Amplifiers 575.2 Erbium-Doped Fiber Amplifiers 595.2.1 Two-Level Model 605.3 Fiber Raman Amplifiers 635.4 Fiber Parametric Amplifiers 685.5 Lumped versus Distributed Amplification 725.6 Parabolic Pulses 74References 766 Solitons in Optical Fibers 816.1 The Fundamental Soliton Solution 816.2 Higher-Order Solitons 836.3 Soliton Units 866.4 Dark Solitons 876.5 Bistable Solitons 886.6 XPM-Paired Solitons 896.7 Optical Similaritons 906.8 Numerical Solution of the NLSE 926.9 The Variational Approach 946.10 The Method of Moments 97References 987 Soliton Transmission Systems 1017.1 Soliton Perturbation Theory 1017.2 Effect of Fiber Losses 1027.3 Soliton Amplification 1037.3.1 Lumped Amplification 1047.3.2 Distributed Amplification 1057.4 Soliton Interaction 1077.5 Timing Jitter 1107.5.1 Gordon-Haus Jitter 1107.5.2 Polarization-Mode Dispersion Jitter 1137.5.3 Acoustic Jitter 1137.5.4 Soliton Interaction Jitter 1147.6 WDM Soliton Systems 1147.6.1 Lossless Soliton Collisions 1147.6.2 Soliton Collisions in Perturbed Fiber Spans 1167.6.3 Timing Jitter 117References 1178 Soliton Transmission Control 1218.1 Fixed-Frequency Filters 1218.1.1 Control of Timing Jitter 1228.1.2 Control of Soliton Interaction 1238.1.3 Background Instability 1258.2 Sliding-Frequency Filters 1258.2.1 Evolution of Soliton Parameters 1268.2.2 Control of Timing Jitter 1298.2.3 Control of Soliton Interaction 1318.3 Synchronous Modulators 1328.4 Amplifiers with Nonlinear Gain 1338.4.1 Stationary Solutions 1348.4.2 Control of Soliton Interaction 137References 1399 Propagation of Ultrashort Solitons 1419.1 Generalized NLSE 1419.1.1 Third-Order Dispersion 1429.1.2 Self-Steepening 1439.1.3 Intrapulse Raman Scattering 1449.2 Timing Jitter of Ultrashort Solitons 1459.3 Bandwidth-Limited Amplification of Ultrashort Solitons 1479.4 Transmission Control Using Nonlinear Gain 1519.4.1 Stationary Solutions 1519.4.2 Linear Stability Analysis 153References 15710 Dispersion-Managed Solitons 16110.1 Dispersion Management 16110.2 Characteristics of the Dispersion-Managed Soliton 16310.3 The Variational Approach to DM Solitons 16710.3.1 Generic Ansatz 16710.3.2 Gaussian Pulses 16810.3.3 Stationary Solutions 16910.4 Interaction Between DM Solitons 17010.5 The Gordon–Haus Effect for DM Solitons 17210.6 Effects of a Spectral Filter 17310.6.1 Timing Jitter Control 17410.7 Effects of an Amplitude Modulator 17510.8 WDM with DM Solitons 177References 17911 Polarization Effects 18311.1 Fiber Birefringence and Polarization Mode Dispersion 18311.1.1 PMD in Long Fiber Spans 18511.1.2 PMD-Induced Pulse Broadening in Linear Systems 18711.1.3 PMD Compensation 18811.2 Coupled Nonlinear Schrödinger Equations 19011.3 Solitons in Fibers with Constant Birefringence 19111.4 Vector Solitons 19511.5 Solitons in Fibers with Randomly Varying Birefringence 19611.6 PMD-Induced Soliton Pulse Broadening 19711.7 Dispersion-Managed Solitons and PMD 200References 20212 Stationary Dissipative Solitons 20712.1 Balance Equations for the CGL Equation 20712.2 Exact Analytical Solutions 21012.2.1 Solutions of the Cubic CGLE 21012.2.2 Solutions of the Quintic CGLE 21212.3 Numerical Stationary Soliton Solutions 21312.4 High-Energy Dissipative Solitons 21612.5 Soliton Bound States 22112.6 Impact of Higher-Order Effects 225References 22913 Pulsating Dissipative Solitons 23313.1 Dynamic Models for CGLE Solitons 23313.1.1 The Variational Approach 23413.1.1.1 Sech Ansatz 23513.1.1.2 Gaussian Ansatz 23513.1.2 The Method of Moments 23613.2 Plain Pulsating Solitons 23813.2.1 Impact of Higher-Order Effects 23913.3 Creeping Solitons 24113.3.1 Impact of Higher-Order Effects 24213.4 Chaotic Solitons 24413.5 Erupting Solitons 24713.5.1 Impact of Higher-Order Effects 25113.5.2 Experimental Observation of Soliton Explosions 253References 25614 Soliton Fiber Lasers 25914.1 The First Soliton Laser 25914.2 Fundamentals of Fiber Soliton Lasers 26014.3 Mode-Locking Techniques 26214.3.1 Active Mode-Locking 26214.3.2 Passive Mode-Locking 26214.3.3 Nonlinear Optical Loop Mirrors 26314.3.4 Figure-Eight Laser 26414.3.5 Nonlinear Polarization Rotation 26514.3.6 Hybrid Mode-Locking 26514.4 High-Energy Soliton Fiber Lasers 26614.5 Modeling of Soliton Fiber Lasers 26814.6 Polarization Effects 27214.7 Dissipative Soliton Molecules 27314.8 Experimental Observation of Pulsating Solitons 274References 27915 Other Applications of Optical Solitons 28515.1 All-Optical Switching 28515.1.1 The Fiber Coupler 28515.1.2 The Sagnac Interferometer 28615.2 2R Optical Regeneration 28815.3 Pulse Compression 29015.3.1 Grating-Fiber Compression 29015.3.2 Higher-Order Soliton-Effect Compression 29115.3.3 Compression of Fundamental Solitons 29315.3.4 Dissipative Soliton Compression 29515.4 Solitons in Fiber Gratings 29815.4.1 Pulse Compression Using Fiber Gratings 30015.4.2 Fiber Bragg Solitons 302References 30516 Highly Nonlinear Optical Fibers 30916.1 Highly Nonlinear Silica Fibers 30916.1.1 Tapered Fibers 31016.2 Microstructured Optical Fibers 31116.3 Non-Silica Fibers 31816.4 Soliton Fission and Dispersive Waves 32016.5 Four-Wave Mixing 32416.6 Hollow-Core Microstructured Fibers 325References 33217 Supercontinuum Generation 33717.1 Pumping with Femtosecond Pulses 33717.2 Modeling the Supercontinuum 34117.3 Pumping with Picosecond Pulses 34417.4 Continuous Wave Supercontinuum Generation 34717.5 Mid-IR Supercontinuum Generation 35017.6 Supercontinuum Coherence 35217.6.1 Spectral Incoherent Solitons 35417.7 Supercontinuum Generation in Hollow-Core Kagomé Fibers 356References 365Index 369