Localized States in Physics: Solitons and Patterns (eBook)

Prof. Orazio Descalzi has edited in the last years the following books: Instabilities and Nonequilibrium Structures VII & VIII (Kluwer Academic Publishers, 2004)- Instabilities and Nonequilibrium Structures IX (Kluwer Academic Publishers, 2004) Nonequilibrium Statistical Mechanics and Nonlinear Physics (Elsevier, 2005)- Nonequilibrium Statistical Mechanics and Nonlinear Physics (American Institute of Physics, 2007

Localized States in Physics:Solitons and Patterns 3
Preface 5
Acknowledgements 7
Contents 9
List of Contributors 15
Part I Solitons, self-confined light and optical turbulence 19
Chapter 1 Light Self-trapping in Nematic Liquid Crystals 20
1.1 Introduction 20
1.2 Reorientational Self-focusing in Nematic Liquid Crystals 21
1.3 Spatial Optical Solitons in Purely Nematic Liquid Crystals 24
1.4 Spatial Optical Solitons in Chiral Nematic Liquid Crystals 28
1.5 Conclusions 31
References 32
Chapter 2 Photonic Plasma Instabilities and Soliton Turbulence in Spatially Incoherent Light 34
2.1 Introduction 35
2.2 Basic Theory and Formalism 36
2.2.1 Wigner Formalism 36
2.2.2 Initial Stages of Instability. Linear Perturbation Theory 38
2.2.3 Growth Rate and Conditions for Weak/Strong Turbulence 39
2.2.4 Debye Scaling 41
2.3 Quasi-Linear Approximation 42
2.3.1 General Derivation 43
2.3.2 Bump-on-Tail Dynamics 44
2.4 Numerical Analysis 45
2.4.1 Numerical Results for BOT Instability 45
2.4.2 Numerical Results for Multiple BOT Instability 46
2.5 Experimental Observation 47
2.5.1 Experimental Setup 47
2.5.2 Single Bump-on-Tail Instability 48
2.5.3 Holographic Readout of Dynamics 51
2.5.4 Multiple Bump-on-Tail Instability and Long-RangeTurbulence Spectra 52
2.6 Discussion and Conclusions 54
References 54
Chapter 3 Gap-Acoustic Solitons: Slowing and Stopping of Light 57
3.1 Introduction 58
3.2 Derivation of the Equations 61
3.2.1 Electromagnetic Field Equations with Phonon Perturbations 61
3.2.2 Acoustic Wave Equations with Electrostrictive Perturbations 64
3.2.2.1 Slowly-Varying Phonon Field 66
3.2.2.2 Brillouin Scattering—Phonon Fields at k ~ 2k0 66
3.2.3 The Bragg-Brillouin-Kerr System 67
3.3 Lagrangian, Hamiltonian, and Conserved Quantities 67
3.3.1 Dimensionless Variables 69
3.4 Gap-Acoustic Solitons 70
3.5 Soliton Stability and Instability 73
3.6 Summary and Conclusions 80
References 82
Chapter 4 Optical Wave Turbulence and Wave Condensation in a Nonlinear Optical Experiment 83
4.1 Introduction 84
4.2 Experimental setup 85
4.3 Theoretical Background 86
4.3.1 Evolution Equation 86
4.3.2 Long-Wave Model 87
4.3.3 The Fjørtoft Argument 88
4.3.4 Hamiltonian Formulation 90
4.3.5 Canonical Transformation 91
4.3.6 The Kinetic Wave Equation 92
4.3.7 Modulational Instability and the Creation of Solitons 95
4.4 Numerical Method 96
4.5 Experimental and Numerical Results 96
4.5.1 Direct cascade of energy 100
4.6 Conclusions 101
4.7 Acknowledgements 102
References 102
Part II Localized structures in pattern forming systems 104
Chapter 5 Localized Structures in the Liquid Crystal Light Valve Experiment 105
5.1 Introduction 106
5.2 The Liquid Crystal Light Valve Experiment 107
5.2.1 Description of the setup 107
5.2.2 The optical feedback: model equations 109
5.3 Experimental Observations of Optical Localized Structures 111
5.3.1 Round localized structures: interaction and dynamics 111
5.3.2 Triangular localized structures: bistability and phasesingularities 112
5.3.3 Bipatterns and localized peaks 114
5.3.4 1D spatially forced model 115
5.4 Control of Optical Localized Structures 116
5.4.1 Pinning range and localized structures 116
5.4.2 Controlled storage of localized structures matrices 117
5.5 Propagation Properties of Optical Localized Structures 119
5.6 Conclusions 121
References 121
Chapter 6 Convectons 123
6.1 Introduction 123
6.2 Convectons with periodic boundary conditions 126
6.3 Convectons with ICCBC 130
6.4 Multiconvectons 132
6.5 Localized traveling waves 133
6.6 Interpretation 134
6.7 Summary 137
References 138
Chapter 7 Morphological Characterization of Localized Hexagonal Patterns 140
7.1 Introduction 140
7.2 Prototypical Model for Hexagon Formation 142
7.3 Localized Hexagonal States: Geometrical Considerations and Morphological Characterizations 143
7.4 Heuristic Description of the Localization Process 146
7.5 The Case of a Localized Line of Cells 148
7.6 Conclusions and Perspective 150
References 151
Part III Localized structures for optical applications 152
Chapter 8 Cavity Solitons in Vertical Cavity Surface Emitting Lasers and their Applications 153
8.1 Introduction 154
8.2 CS motion 155
8.2.1 Numerical Analysis of CS motion in a constant phase gradient 156
8.2.2 Experimental Evidence of CS motion in a constant phase gradient 158
8.3 Applications of CS movement 162
8.3.1 CS drift in a constant gradient 162
8.3.2 Experimental realization of reconfigurable CS arrays 163
8.4 CS motion and device defects 167
8.4.1 CS force microscope 168
8.4.2 Modeling of an inhomogeneous device 169
8.4.3 Interaction between phase gradient and defects: the CS tap 170
8.4.3.1 Experiment 171
8.4.3.2 Numerical Simulations 173
8.5 Conclusions 177
References 178
Chapter 9 Cavity Soliton Laser based on coupledmicro-resonators 180
9.1 Introduction 181
9.2 Experimental Setup 182
9.3 Bistability regime 184
9.3.1 Multistable Regime 186
9.3.2 Local bifurcation diagram 187
9.3.3 Towards the whole bifurcation diagram 189
9.4 Coherence properties of laser solitons 192
9.4.1 Modal properties 193
9.5 Conclusions and Perspectives 195
References 196
Chapter 10 Cavity soliton laser based on a VCSEL with saturable absorber 198
10.1 Introduction 199
10.2 The model 201
10.2.1 Bistability 202
10.2.2 Plane wave instabilities 204
10.2.3 Pattern forming instabilities 204
10.3 CS switching techniques 205
10.3.1 Switching dynamics 207
10.3.1.1 Incoherent injection 208
10.3.1.2 Injection at the cavity frequency 210
10.3.1.3 Injection at the CS frequency 211
10.3.2 Switching energy 212
10.3.2.1 Injection at the cavity frequency 212
10.3.2.2 Injection at the CS frequency 214
10.4 Stability of the CS 214
10.5 Motion of the CS in a finite device 217
10.5.1 Circular pump 217
10.6 Conclusions 219
References 221
Chapter 11 Dynamic Control of Localized Structures in a Nonlinear Feedback Experiment 223
11.1 Introduction 224
11.2 Self-organized localized structures in feedback systems 225
11.3 Localized structures in a single-feedback system using a liquid crystal light valve as a nonlinearity 229
11.3.1 Formation of localized structures 231
11.4 Boundary-induced localized structures in LCLV 233
11.5 Dynamic and static position control of feedback localized states 237
11.6 Gradient induced motion control of feedback localized structures 241
11.7 Summary 246
References 246
Part IV Excitability and localized states 249
Chapter 12 Interaction of oscillatory and excitable localized states in a nonlinear optical cavity 250
12.1 Introduction 250
12.2 Model 251
12.3 Overview of the behavior of localized states 252
12.3.1 Hopf bifurcation 253
12.3.2 Saddle-loop bifurcation 253
12.3.3 Excitability 255
12.4 Interaction of two oscillating localized states 255
12.4.1 Full system 256
12.4.2 Simple model: two coupled Landau-Stuart oscillators 260
12.4.2.1 Estimation of parameters I 263
12.4.2.2 Estimation of parameters II: quenching experiments 263
12.4.2.3 Estimation of d 264
12.4.2.4 Results and dynamical regimes of the simple model 265
12.5 Interaction of excitable localized states: logical gates 268
12.6 Summary 272
References 272
Chapter 13 Lurching waves in thalamic neuronal networks 274
13.1 Introduction 274
13.2 The model 276
13.2.1 Smooth and Lurching waves 279
13.3 Exploration of parameter space and continuation 281
13.3.1 Direct time integration 281
13.3.2 Continuation using Newton method 283
13.3.3 Pseudo-arclength continuation 287
13.4 Discussion 288
References 289
Index 291