Transverse Patterns in Nonlinear Optical Resonators

Kestutis Staliunas, V.J. Sánchez-Morcillo9783540004349, 3540004343

This book is devoted to the formation and dynamics of localized structures (vortices, solitons) and of extended patterns (stripes, hexagons, tilted waves) in nonlinear optical resonators such as lasers, optical parametric oscillators, and the like. Theoretical analysis is performed by deriving order parameter equations, and also through numerical integration of microscopic models of the systems underinvestigation. Experimental observations, and possible technological implementations of transverse optical patterns are also discussed. A comparison with patterns found in other nonlinear systems, both optical systems and extended systems in general, is given.

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Table of contents :
1 Introduction……Page 5
1.1 Historical Survey……Page 6
1.2 Patterns in Nonlinear Optical Resonators……Page 8
1.2.1 Localized Structures: Vortices and Solitons……Page 10
1.2.2 Extended Patterns……Page 12
1.3.1 Mirrorless Configuration……Page 15
1.3.3 Optical Feedback Loops……Page 16
1.4 The Contents of this Book……Page 19
References……Page 23
2 Order Parameter Equations for Lasers……Page 36
2.1 Model of a Laser……Page 37
2.2 Linear Stability Analysis……Page 39
2.3.1 Adiabatic Elimination……Page 44
2.3.2 Multiple-Scale Expansion……Page 49
References……Page 51
3.1 Optical Parametric Oscillators……Page 53
3.2.1 Linear Stability Analysis……Page 54
3.2.2 Scales……Page 55
3.2.3 Derivation of the OPE……Page 56
3.3 The Complex Swift-Hohenberg Equation for OPOs……Page 57
3.3.1 Linear Stability Analysis……Page 58
3.3.3 Derivation of the OPE……Page 59
3.4.1 Description and Model……Page 61
3.4.2 Adiabatic Elimination and Operator Inversion……Page 62
3.5 Phenomenological Derivation of Order Parameter Equations……Page 63
References……Page 65
4.1 Hydrodynamic Form……Page 67
4.2 Optical Vortices……Page 69
4.2.1 Strong Diffraction……Page 70
4.2.2 Strong Diffusion……Page 73
4.2.3 Intermediate Cases……Page 74
4.3 Vortex Interactions……Page 76
References……Page 81
5 Finite Detuning: Vortex Sheets and Vortex Lattices……Page 82
5.1 Vortices “Riding” on Tilted Waves……Page 83
5.2 Domains of Tilted Waves……Page 85
5.3 Square Vortex Lattices……Page 88
References……Page 91
6 Resonators with Curved Mirrors……Page 92
6.1 Weakly Curved Mirrors……Page 93
6.2 Mode Expansion……Page 94
6.2.1 Circling Vortices……Page 95
6.2.2 Locking of Transverse Modes……Page 96
6.3 Degenerate Resonators……Page 98
References……Page 103
7.1 The Model……Page 104
7.2 Single Vortex……Page 106
7.3 Vortex Lattices……Page 109
7.3.1 “Optical” Oscillation Mode……Page 110
7.3.2 Parallel translation of a vortex lattice……Page 111
7.4.1 Mode Expansion……Page 112
7.4.2 Phase-Insensitive Modes……Page 114
7.4.3 Phase-Sensitive Modes……Page 115
References……Page 116
8.1 Subcritical Versus Supercritical Systems……Page 117
8.2 Mechanisms Allowing Soliton Formation……Page 118
8.2.1 Supercritical Hopf Bifurcation……Page 119
8.2.2 Subcritical Hopf Bifurcation……Page 120
8.3 Amplitude and Phase Domains……Page 122
8.4 Amplitude and Phase Spatial Solitons……Page 123
References……Page 124
9.1 Model and Order Parameter Equation……Page 125
9.2 Amplitude Domains and Spatial Solitons……Page 127
9.3.1 Soliton Formation……Page 129
9.3.2 Soliton Manipulation: Positioning, Propagation, Trapping and Switching……Page 132
9.4 Experiments……Page 133
References……Page 138
10.1 Analysis of the Homogeneous State. Nonlinear Resonance……Page 139
10.2.1 One-Dimensional Case……Page 141
10.2.2 Two-Dimensional Case……Page 144
References……Page 146
11.1 Patterns in Systems with a Real-Valued Order Parameter……Page 147
11.2 Phase Domains……Page 148
11.3.1 Variational Approach……Page 150
11.3.2 Two-Dimensional Domains……Page 152
11.4 Phase Solitons……Page 155
11.5 Nonmonotonically Decaying Fronts……Page 157
11.6 Experimental Realization of Phase Domains and Solitons……Page 160
11.7 Domain Boundaries and Image Processing……Page 163
References……Page 166
12.1 The Turing Mechanism in Nonlinear Optics……Page 168
12.2 Laser with Diffusing Gain……Page 170
12.2.1 General Case……Page 171
12.2.2 Laser with Saturable Absorber……Page 173
12.2.3 Stabilization of Spatial Solitons by Gain Diffusion……Page 175
12.3 Optical Parametric Oscillator with Diffracting Pump……Page 179
12.3.1 Turing Instability in a DOPO……Page 180
12.3.2 Stochastic Patterns……Page 183
12.3.3 Spatial Solitons Influenced by Pump Diffraction……Page 186
References……Page 190
13.1 The Synchronously Pumped DOPO……Page 192
13.1.1 Order Parameter Equation……Page 193
13.2 Patterns Obtained from the 3D Swift-Hohenberg Equation……Page 195
13.3 The Nondegenerate OPO……Page 199
13.4.1 Tunability of a System with a Broad Gain Band……Page 200
References……Page 201
14 Patterns and Noise……Page 203
14.1 Noise in Condensates……Page 204
14.1.1 Spatio-Temporal Noise Spectra……Page 205
14.1.2 Numerical Results……Page 208
14.1.3 Consequences……Page 212
14.2 Noisy Stripes……Page 214
14.2.1 Spatio-Temporal Noise Spectra……Page 215
14.2.2 Stochastic Drifts……Page 219
14.2.3 Consequences……Page 221
References……Page 222
Index……Page 0