Spatial Solitons

By: Stefano Trillo (Editor), William Torruellas (Editor)

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Hardcover | 1 January 2001

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482 Pages

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Soliton-based concepts open the road for newly designed laser sources, new frequency converters and high-intensity laser-material interactions. Optical solitons as stable spatial patterns of complex nonlinear systems allow for the control of the diffraction of optical beams. Solitons also prevent unwanted chaotic behavior. Thus, solitary wave physics plays a significant role from modern optical physics to optical communication, optical switching, and optical storage. The book gives an updated overview of optical solitons and can serve as a reference and guide for advanced students and scientists working in the field and related areas of science where solitons are observed.

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"This book is the volume 82 in Springer Series 'Optical Sciences', which includes invited contributions from the majority of world leading experts and pioneers in the field. ... The content of the book covers a wide spectrum of issues concerning the propagation of spatial solitons, ranging from the most challenging theoretical problems to the most exciting experimental achievements to date." (Andrei Mourachkine, Physicalia, Vol. 38 (4), 2002)

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Non-Fiction Science Physics Classical Mathematics Wave Mechanics including Vibration & Acoustics Optical Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Engineering & Technology Other Technologies & Applied Sciences Applied Optics Laser Technology & Holography

List of Contributors	p. XVII
Introduction to Spatial Solitons	p. 1
Introduction	p. 1
Solitons in one Transverse Dimension	p. 2
Experiments	p. 7
New Directions	p. 8
Acknowledgments	p. 16
References	p. 16
Materials for Spatial Solitons	p. 19
Introduction	p. 19
Third-Order Kerr and Kerr-Like Saturating Nonlinearities	p. 20
Higher-Order Kerr-Like Nonlinearities	p. 25
Photorefractive Materials	p. 27
Second-Order Nonlinearities for Quadratic Solitons	p. 29
Conclusions	p. 32
	p. 33
One-Dimensional Spatial Solitons in Kerr Media	p. 37
Introduction	p. 37
Early and Miscellaneous Experiments	p. 38
Spatial Solitons in Homogeneous Semiconductor Waveguides	p. 40
Homogeneous AlGa As Passive Waveguides	p. 41
Weak-Beam-Spatial-Soliton Interactions	p. 43
Dragging and Trapping of Orthogonally Polarized Spatial Solitons	p. 44
Generation of Manakov Solitons	p. 47
Stability of Orthogonally Polarized Solitons	p. 48
Soliton Reconfigurable Interconnect	p. 48
Spatial Solitons in Semiconductor Optical Amplifiers	p. 50
Discrete Solitons	p. 52
Summary	p. 57
References	p. 58
Photorefractive Spatial Solitons	p. 61
Introduction	p. 61
Photorefractive Beam Nonlinearity	p. 61
Space-Charge Field	p. 62
Beam Propagation	p. 63
Self-Trapping Mechanisms	p. 64
Quasi-Steady-State Solitons	p. 64
Screening Solitons	p. 67
Screening Slab-Solitons	p. 71
Screening Needle Solitons	p. 73
Self-bending	p. 75
Photovoltaic Solitons	p. 75
Self-Trapping in Semiconductors	p. 76
Self-Trapping in Paraelectrics	p. 76
Material Nonlinearities and Solitons	p. 78
Nonlinear Beam Interactions	p. 78
Applications	p. 79
Acknowledgments	p. 80
References	p. 80
Incoherent Solitons: Self-Trapping of Weakly Correlated Wavepackets	p. 87
Introduction, First Experiments, and Intuition	p. 87
The Modal Theory of Incoherent Solitons	p. 92
The Coherent-Density Theory and the Propagation of Mutual Coherence	p. 97
Dark Incoherent Solitons	p. 101
Coherence Properties of Incoherent Solitons	p. 107
Modulational Instability and Pattern Formation with Incoherent Wavepackets	p. 112
Effects of Anisotropic Coherence and Propagation Dynamics	p. 120
Prospects	p. 122
Dedication and Acknowledgments	p. 123
References	p. 123
Quadratic Solitons	p. 127
Introduction and Historical Remarks	p. 127
Summary of Theoretical Results	p. 128
Basic Models for Quadratic Solitons	p. 128
(1 + 1)-Dimensional Quadratic Solitons	p. 131
Solitons in the Cascading Limit	p. 131
Two-Wave Bright Solitons	p. 132
Two-Wave Dark Solitons	p. 134
(2 + 1)-Dimensional Quadratic Solitons	p. 136
General Remarks	p. 136
Fundamental Cylindrical Solitons	p. 137
Soliton Scattering and Fusion	p. 139
Important Generalizations and Related Problems	p. 141
Walking Solitons	p. 141
Three-Wave Quadratic Solitons	p. 142
Competing Nonlinearities	p. 143
Doughnut Modes and Vortex Solitons	p. 144
Quadratic Light Bullets	p. 145
Discrete x$((2)$) Systems and Nonlinear Photonic Modes	p. 146
Summary of Experimental Results	p. 148
Freely Propagating Spatial Solitons	p. 149
(1 + 1)-Dimensional Spatial Solitons	p. 149
(2 + 1)-Dimensional Spatial Solitons	p. 150
Temporal Solitons and Light Bullets	p. 158
Quadratic Solitons in Cavities	p. 159
Soliton Interaction	p. 160
Transverse Effects	p. 160
Concluding Remarks	p. 162
Acknowledgments	p. 164
References	p. 164
Bragg Solitons: Theory and Experiments	p. 169
Introduction	p. 169
Theory of Periodic Structures	p. 175
Coupled-Mode Theory	p. 175
Dispersion Relation and Eigenfunctions	p. 178
CW Solutions to the Coupled-Mode Equations	p. 180
Nonlinear-Schrödinger-Equation Limit	p. 181
Bragg Grating Solitons and Gap Solitons	p. 182
Experimental Considerations: Grating Fabrication	p. 183
Optical-Fiber Bragg Gratings	p. 183
Semiconductors	p. 186
Chalcogenide Glasses	p. 187
Experimental Apparatus	p. 187
Summary of Experimental Results	p. 189
Continuous-Wave Experiments	p. 189
Pulse Experiments	p. 191
Further Developments	p. 195
Optical Push Broom	p. 195
Frequency Conversion and Generation	p. 197
Nonuniform Gratings	p. 198
Novel Nonlinear Grating-Based Compressors	p. 198
Polarization Effects	p. 199
Optical AND Gate	p. 199
Polarization Rotation and Polarization Instability	p. 200
Copropagating Geometries	p. 202
Gratings with Quadratic Nonlinearities	p. 202
Outlook	p. 203
Acknowledgments	p. 204
References	p. 205
Stability of Spatial Optical Solitons	p. 211
Introduction	p. 211
Linear Eigenvalue Problem	p. 213
Soliton Internal Modes and Stability	p. 214
Stability Criterion for Fundamental Solitons	p. 218
Marginal Stability Point: Asymptotic Analysis	p. 222
Nonlinear Guided Waves and Impurity Modes	p. 223
Model and General Remarks	p. 223
Stability of Symmetric and Asymmetric Modes	p. 225
Symmetric Modes	p. 225
Asymmetric Modes	p. 226
Example: Competing Power-Law Nonlinearities	p. 226
Multicomponent Solitary Waves	p. 228
General Theory: N Coupled NLS Equations	p. 228
Example: Two Coupled NLS Equations	p. 231
Multihump Solitary Waves	p. 234
Oscillatory Instabilities	p. 237
Symmetry-Breaking Instabilities	p. 239
Concluding Remarks	p. 242
Acknowledgments	p. 243
References	p. 243
Nonlinear Wave Collapse	p. 247
Introduction	p. 247
General Results on the Collapse of a Single Component	p. 250
Interaction Regimes of Several Collapsing Components	p. 253
Collapse with Four-Wave Mixing and Walk-Off	p. 258
Interplay between Quadratic and Cubic Nonlinearities	p. 261
Conclusion	p. 265
Acknowledgments	p. 265
References	p. 265
Discrete Solitons	p. 269
Introduction	p. 269
Basic Mathematical Models	p. 271
Discrete Diffraction and Modulational Instability	p. 273
Linear Plane Waves and Discrete Diffraction	p. 273
Nonlinear Plane Waves and Their Modulational Instability	p. 275
Cubic Nonlinearity	p. 275
Quadratic Nonlinearity	p. 276
Solitons in Cubic Nonlinear Media	p. 278
Types of Soliton Solutions	p. 278
Even (Intersite) Bright Solitons	p. 278
Odd (On-Site) Bright Solitons	p. 279
Topological and Flat-Top Solitons	p. 280
Stability Analysis	p. 281
Bright Solitons	p. 282
Topological and Flat-Top Solitons	p. 283
Solitons in Quadratic Nonlinear Media	p. 285
Types of Soliton Solutions	p. 285
Even Bright Solitons	p. 285
Odd Bright Solitons	p. 286
Topological and Flat-Top Solitons	p. 286
Stability Analysis	p. 287
Bright Solitons	p. 287
Topological and Flat-Top Solitons	p. 288
Conclusions	p. 290
Acknowledgments	p. 290
References	p. 290
Optical Vortex Solitons	p. 293
Introduction	p. 293
Vortex Wavefunctions	p. 294
Vortex Wavevectors	p. 295
r-Vortex Solutions	p. 297
Hankel Transform into Vortex Modes	p. 298
Point Vortex	p. 299
Multiple Vortices	p. 300
Nonlinear Refraction	p. 303
Optical Vortex Solitons	p. 304
Pinched Optical Vortex	p. 306
Conservation of Momentum	p. 307
Closing Thoughts	p. 308
References	p. 309
Solitons of the Complex Ginzburg-Landau Equation	p. 311
Introduction	p. 311
Balance Equations and Perturbation Theory	p. 315
Exact Solutions	p. 317
Solutions of the Cubic CGLE	p. 318
The Solution with Arbitrary Amplitude	p. 320
Chirp-Free Soliton	p. 320
Dark-Soliton Solution of the Cubic Equation	p. 321
Solitons of the Quintic CGLE	p. 322
Relation Between Coefficients	p. 322
Solutions with Fixed Amplitude	p. 322
The Solution with Arbitrary Amplitude	p. 323
Flat-Top Solitons	p. 323
Rational Solution	p. 324
Chirp-Free Soliton	p. 324
Reduction to a Set of Ordinary Differential Equations	p. 324
Stability Issues	p. 326
Stable Solitons	p. 327
Multiplicity of Solutions	p. 330
Composite Solitons	p. 330
Moving Solitons	p. 331
Coexistence of Solitons	p. 332
Collisions	p. 333
Soliton Bound States	p. 334
Conclusions	p. 337
References	p. 337
Existence, Stability, and Properties of Cavity Solitons	p. 343
Introduction	p. 343
History	p. 344
Mean-Field Models and Cavity Solitons	p. 346
Cavity Solitons in Semiconductor Microresonators	p. 351
Stationary Solutions and Stability	p. 352
Applications	p. 355
Conclusion	p. 355
Acknowledgments	p. 356
References	p. 356
Parametric Solitons in Passive Structures with Feedback	p. 359
Introduction	p. 359
General Models for Quadratic Passive Cavities	p. 360
Mean-Field Models	p. 361
Order-Parameter Equations	p. 363
Competing Instabilities in Intracavity Parametric Mixing	p. 366
Optical Parametric Oscillators	p. 370
Localized Phase Defects or Domain Walls in DOPOs	p. 370
Below-Threshold Bright Signal Solitons in DOPOs	p. 374
Solitons in NDOPOs	p. 376
Role of Walk-Off	p. 376
Intracavity Second-Harmonic Generation	p. 378
Pattern Formation and Localized Structures	p. 378
Vectorial (Type II) Effects	p. 381
Parametric Gap Solitons in Distributed-Feedback Structures	p. 383
Summary and Further Developments	p. 387
Acknowledgments	p. 389
References	p. 389
Spatial Solitons in Resonators	p. 395
Introduction	p. 395
Solitons in a Laser with a Nonlinear Absorber	p. 397
Parametric-Mixing Solitons	p. 404
Nonlinear Semiconductor Resonators	p. 408
Acknowledgments	p. 415
References	p. 415
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Spatial Solitons

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