By recirculating light in a nonlinear propagation medium, the nonlinear optical cavity allows for countless options of light transformation and manipulation. In passive media, optical bistability and frequency conversion are central figures. In active media, laser light can be generated with versatile underlying dynamics. Emphasizing on ultrafast dynamics, the vital arena for the information technology, the soliton is a common conceptual keyword, thriving into its modern developments with the closely related denominations of dissipative solitons and cavity solitons. Recent technological breakthroughs in optical cavities, from micro-resonators to ultra-long fiber cavities, have entitled the exploration of nonlinear optical dynamics over unprecedented spatial and temporal orders of magnitude. By gathering key contributions by renowned experts, this book aims at bridging the gap between recent research topics with a view to foster cross-fertilization between research areas and stimulating creative optical engineering design.
Table of Contents:
List of Contributors XIII
Foreword XXIII
1 Introduction 1
Philippe Grelu
References 8
2 Temporal Cavity Solitons in Kerr Media 11
Stéphane Coen andMiro Erkintalo
2.1 Introduction 11
2.2 Mean-Field Equation of Coherently Driven Passive Kerr Resonators 13
2.3 Steady-State Solutions of the Mean-Field Equation 15
2.4 Existence and Characteristics of One-Dimensional Kerr Cavity Solitons 18
2.5 Original Experimental Observation of Temporal Kerr Cavity Solitons 21
2.6 Interactions of Temporal CSs 25
2.7 Breathing Temporal CSs 29
2.8 Emission of DispersiveWaves by Temporal CSs 31
2.9 Conclusion 34
References 34
3 Dynamics and Interaction of Laser Cavity Solitons in Broad-Area Semiconductor Lasers 41
Thorsten Ackemann, Jesus Jimenez, Yoann Noblet, Neal Radwell, Guangyu Ren, Pavel V. Paulau, Craig McIntyre, Gian-Luca Oppo, Joshua P. Toomey, and Deborah M. Kane
3.1 Introduction 41
3.2 Devices and Setup 43
3.2.1 Devices 43
3.2.2 Experimental Setup 44
3.3 Basic Observations and Dispersive Optical Bistability 45
3.3.1 Basic Observation of Spatial Solitons 45
3.3.2 Interpretation as Dispersive Optical Bistability 47
3.3.3 Comparison to Absorptive Case 49
3.4 Modelling of LS and Theoretical Expectations in Homogenous System 50
3.4.1 Model Equations 50
3.4.2 Interaction of Laser Solitons in a Homogenous System 52
3.5 Phase and Frequency Locking of Trapped Laser Cavity Solitons 54
3.5.1 Basic Observation 54
3.5.2 Experiments on Locking Phase 55
3.5.3 Adler Locking: Theory 59
3.6 Dynamics of Single Solitons 60
3.6.1 Transient Dynamics 62
3.6.2 Outlook on Asymptotic Dynamics 65
3.7 Summary and Outlook 68
Acknowledgments 70
References 70
4 Localized States in SemiconductorMicrocavities, from Transverse to Longitudinal Structures and Delayed Systems 77
Stéphane Barland, Massimo Guidici, Julien Javaloyes, and Giovanna Tissoni
4.1 Introduction 77
4.2 Lasing Localized States 80
4.2.1 Transverse Localized States in Coupled Microcavities 80
4.2.2 Time-Localized Structures in Passive Mode-Locked Semiconductor Laser 82
4.3 Localized States in Nonlinear Element with Delayed Retroaction 87
4.3.1 Front Pinning in Bistable System with Delay 88
4.3.2 Topological Dissipative Solitons in Excitable System with Delay 92
4.4 Conclusion and Outlook 98
Acknowledgements 99
References 99
5 Dynamics of Dissipative Solitons in Presence of Inhomogeneities and Drift 107
Pedro Parra-Rivas, Damià Gomila, Lendert Gelens, Manuel A. Matías, and Pere Colet
5.1 Introduction 107
5.2 General Theory: Swift–Hohenberg Equation with Inhomogeneities and Drift 108
5.3 Excitability Regimes 113
5.4 Fiber Cavities and Microresonators:The Lugiato–Lefever model 116
5.5 Periodically Pumped Ring Cavities 119
5.6 Effects of Drift in a Periodically Pumped Ring Cavity 120
5.7 Summary 125
Acknowledgments 125
References 125
6 Dissipative Kerr Solitons in Optical Microresonators 129
Tobias Herr, Michael L. Gorodetsky, and Tobias J. Kippenberg
6.1 Introduction to Optical Microresonator Kerr-Frequency Combs 129
6.2 Resonator Platforms 131
6.2.1 Ultra High-Q (MgF2) Crystalline Microresonators 131
6.2.2 Integrated Photonic Chip Microring Resonators 132
6.3 Physics of the Kerr-comb Formation Process 132
6.3.1 Nonlinear Coupled Mode Equations 135
6.3.2 Degenerate Hyperparametric Oscillations 138
6.3.3 Primary Sidebands 140
6.4 Dissipative Kerr Solitons in Optical Microresonators 141
6.4.1 AnalyticalTheory of Dissipative Kerr Solitons 141
6.5 Signatures of Dissipative Kerr Soliton Formation in Crystalline Resonators 145
6.6 Laser Tuning into the Dissipative Kerr Soliton States 147
6.7 Simulating Soliton Formation in Microresonators 148
6.8 Characterization of Temporal Dissipative Solitons in Crystalline Microresonators 149
6.9 Resonator Mode Structure and Soliton Formation 151
6.10 Using Dissipative Kerr solitons to Count the Cycles of Light 152
6.11 Temporal Solitons and Soliton-Induced Cherenkov Radiation in an Si3N4 Photonic Chip 155
6.12 Summary 157
References 158
7 Dynamical Regimes in Kerr Optical Frequency Combs: Theory and Experiments 163
Aurélien Coillet, Nan Yu, Curtis R. Menyuk, and Yanne K. Chembo
7.1 Introduction 163
7.2 The System 164
7.3 The Models 166
7.3.1 Modal Expansion Model 166
7.3.2 Spatiotemporal Model 167
7.3.3 Stability Analysis 168
7.4 Dynamical States 171
7.4.1 Primary Combs 171
7.4.2 Solitons 176
7.4.3 Chaos 179
7.5 Conclusion 183
7.6 Acknowledgments 184
References 184
8 Nonlinear Effects in Microfibers and Microcoil Resonators 189
Muhammad I.M. Abdul Khudus, Rand Ismaeel, Gilberto Brambilla, Neil G. R. Broderick, and Timothy Lee
8.1 Introduction 189
8.2 Linear Optical Properties of Optical Microfibers 191
8.3 Linear Properties of Optical Microcoil Resonators 193
8.4 Bistability in Nonlinear Optical Microcoil Resonators 195
8.4.1 Broken Microcoil Resonators 197
8.4.2 Polarization Effects in Nonlinear Optical Microcoil Resonators 198
8.4.3 Possible Experimental Verification 199
8.5 Harmonic Generation in Optical Microfibers and Microloop Resonators 200
8.5.1 Mathematical Modeling and Efficiency ofThird Harmonic Generation 201
8.5.2 Third Harmonic Generation in Microloop Resonators 204
8.5.3 Second-Harmonic Generation 208
8.6 Conclusions and Outlook 209
References 209
9 Harmonic Laser Mode-Locking Based on Nonlinear Microresonators 213
Alessia Pasquazi, Marco Peccianti, David J. Moss, Sai Tac Chu, Brent E. Little, and Roberto Morandotti
9.1 Introduction 213
9.2 Modeling 215
9.3 Experiments 219
9.3.1 Short Cavity, Unstable Laser Oscillation 223
9.3.2 Short Cavity, Stable Laser Oscillation 224
9.3.3 Short Cavity, Dual-Line Laser Oscillation 226
9.4 Conclusions 228
References 229
10 Collective Dissipative Soliton Dynamics in Passively Mode-Locked Fiber Lasers 231
François Sanchez, Andrey Komarov, Philippe Grelu, Mohamed Salhi, Konstantin Komarov, and Hervé Leblond
10.1 Introduction 231
10.1.1 Dissipative Solitons and Mode-Locked Lasers 231
10.1.2 Multiple Pulses and Their Interactions 232
10.2 Multistability and Hysteresis Phenomena 234
10.2.1 Multiple Pulsing 234
10.2.2 Multistability Observations 235
10.2.3 Modeling Multiple Pulsing and Hysteresis 236
10.3 Soliton Crystals 238
10.3.1 From Soliton Molecules to Soliton Crystals 238
10.3.2 Soliton Crystal Experiments 239
10.3.3 Modeling Soliton Crystal Formations 240
10.3.4 Soliton Crystal Instability 243
10.4 Toward the Control of Harmonic Mode-Locking by Optical Injection 244
10.5 Complex Soliton Dynamics 247
10.5.1 Unfolding Complexity 247
10.5.2 Analogy Between Soliton Patterns and the States of Matter 247
10.5.3 Soliton Rain Dynamics 250
10.5.4 Chaotic Pulse Bunches 252
10.6 Summary 256
Acknowledgments 257
References 257
11 Exploding Solitons and RogueWaves in Optical Cavities 263
Wonkeun Chang and Nail Akhmediev
11.1 Introduction 263
11.2 Passively Mode-Locked Laser Model 266
11.3 The Results of Numerical Simulations 268
11.4 Probability Density Function 270
11.5 Conclusions 272
11.6 Acknowledgements 272
References 273
12 SRS-Driven Evolution of Dissipative Solitons in Fiber Lasers 277
Sergey A. Babin, Evgeniy V. Podivilov, Denis S. Kharenko, Anastasia E. Bednyakova, Mikhail P. Fedoruk, Olga V. Shtyrina, Vladimir L. Kalashnikov, and Alexander A. Apolonski
12.1 Introduction 277
12.2 Generation of Highly Chirped Dissipative Solitons in Fiber Laser Cavity 279
12.2.1 Modeling 279
12.2.1.1 Analytical Solution of CQGLE in the High Chirp Limit 281
12.2.1.2 Comparison of Analytics with Numerics 284
12.2.2 Experiment and its Comparison with Simulation 286
12.2.3 NPE Overdriving and its Influence on Dissipative Solitons 288
12.3 Scaling of Dissipative Solitons in All-Fiber Configuration 290
12.3.1 DifferentWays to Increase Pulse Energy, Limiting Factors 290
12.3.2 SRS Threshold for Dissipative Solitons at Cavity Lengthening 292
12.4 SRS-Driven Evolution of Dissipative Solitons in Fiber Laser Cavity 297
12.4.1 NSE-Based Model in Presence of SRS 297
12.4.1.1 Model Details 298
12.4.1.2 Simulation, Comparison with Experiment 299
12.4.2 Generation of Stokes-Shifted Raman Dissipative Solitons 302
12.4.2.1 Proof-of-Principle Experiment 304
12.4.3 Characteristics of Raman dissipative Solitons 306
12.4.3.1 Variation of the Soliton Spectra with Filter Parameters 306
12.4.3.2 Variation of the Soliton Spectra with the Raman Feedback Parameters 307
12.4.4 Generation of Multicolor Soliton Complexes and Their Characteristics 307
12.5 Conclusions and Future Developments 310
References 312
13 Synchronization in Vectorial Solid-State Lasers 317
Marc Brunel, Marco Romanelli, and Marc Vallet
13.1 Introduction 317
13.2 Self-Locking in Dual-Polarization Lasers 318
13.2.1 Vectorial Description of the Cavity 318
13.2.2 Self-Pulsing in Lasers with Crossed Loss and Phase Anisotropies 319
13.2.3 Polarization Self-Modulated Lasers 321
13.2.4 Mode-Locked Dual-Polarization Lasers 323
13.2.4.1 Phase Locking at c/4L 325
13.3 Dynamics of Solid-State Lasers Submitted to a Frequency-Shifted Feedback 327
13.3.1 Description of the System 327
13.3.1.1 Experimental Setup 328
13.3.2 Lang–Kobayashi Rate Equations 330
13.3.2.1 Phase Dynamics 331
13.3.2.2 Time-Scaled Rate Equations 331
13.3.3 Phase Locking 332
13.3.3.1 Continuous-Wave Case 332
13.3.3.2 Passive Q-Switching Case 333
13.3.4 Bounded Phase Dynamics 334
13.3.4.1 Intensity Bifurcation Diagram 334
13.3.4.2 Phase Bifurcation Diagram 336
13.3.4.3 Phasors 337
13.3.4.4 Role of the Coupling in the Active Medium 338
13.3.5 Measure of the Synchronization in the Bounded Phase Regime 339
13.4 Conclusion 341
Acknowledgments 341
References 341
14 Vector Patterns and Dynamics in Fiber Laser Cavities 347
StefanWabnitz, Caroline Lecaplain, and Philippe Grelu
14.1 Introduction 347
14.1.1 Pulsed Vector Dynamics with a Saturable Absorber 347
14.1.2 Vector DynamicsWithout a Saturable Absorber 348
14.2 Fiber Laser Models 349
14.2.1 The Scalar Cubic Ginzburg–Landau Equation 350
14.2.2 Vector Ginzburg–Landau Equations 352
14.2.3 Vector Nonlinear Schrödinger Equation 355
14.2.4 Numerical Simulations 357
14.3 Experiments of Vector Dynamics 357
14.3.1 The Anomalous GVD: From Chaos to Antiphase Dissipative Dynamics 359
14.3.2 The Normal GVD: Polarization-DomainWalls 362
14.4 Summary 364
Acknowledgments 364
References 364
15 Cavity Polariton Solitons 369
Oleg A. Egorov and Falk Lederer
15.1 Introduction 369
15.2 Mathematical Model 371
15.3 One-Dimensional Bright Cavity Polariton Solitons 373
15.3.1 Amplitude Equation in the Polaritonic Basis 374
15.3.2 CPSs Beyond the “Magic Angle” and Their Stability 376
15.3.3 Multi-Hump Cavity Polariton Solitons 378
15.4 Two-Dimensional Parametric Polariton Solitons 380
15.4.1 Amplitude Equations for the ParticipatingWaves 380
15.4.2 Families of Parametric Polariton Solitons 382
15.4.3 Excitation and Dynamics of PPSs 385
15.5 Two-Dimensional Moving Bright CPSs 387
15.6 Summary 389
Acknowledgments 389
References 390
16 Data Methods and Computational Tools for Characterizing Complex Cavity Dynamics 395
J. Nathan Kutz, Steven L. Brunton, and Xing Fu
16.1 Introduction 395
16.2 Data Methods 396
16.2.1 Dimensionality-Reduction: Principal Components Analysis 397
16.2.2 Search Algorithms and Library Building 398
16.2.3 Sparse Measurements and Compressive Sensing 400
16.2.4 Sparse Representation and Classification 401
16.3 Adaptive, Equation-Free Control Architecture 402
16.4 Prototypical Example: Self-Tuning Mode-Locked Fiber Lasers 403
16.4.1 Governing Equations 404
16.4.2 Jones Matrices forWaveplates and Polarizers 404
16.4.3 Performance Monitoring and Objective Function 405
16.4.4 Sparse Representation for Birefringence Classification 405
16.4.5 Self-Tuning Laser 406
16.5 Broader Applications of Self-Tuning Complex Systems 409
16.5.1 Phased Array Antennas 409
16.5.2 Coherent Laser Beam Combining 411
16.5.3 Neuronal Stimulation 412
16.6 Conclusions and Technological Outlook 413
Acknowledgments 415
References 415
17 Conclusion and Outlook 419
Philippe Grelu
References 421
Index 423
