Title Details

List of Contributors XIII

Foreword XXIII

1 Introduction1
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