Preface xv
Acknowledgments xvii
Contributors xix
Chapter 1. Nematicons 1
Gaetano Assanto, Alessandro Alberucci, and Armando Piccardi
1.1 Introduction 1
1.1.1 Nematic Liquid Crystals 1
1.1.2 Nonlinear Optics and Solitons 3
1.1.3 Initial Results on Light Self-Focusing in Liquid Crystals 3
1.2 Models 4
1.2.1 Scalar Perturbative Model 5
1.2.2 Anisotropic Perturbative Model 9
1.3 Numerical Simulations 13
1.3.1 Nematicon Profile 13
1.3.2 Gaussian Input 14
1.4 Experimental Observations 17
1.4.1 Nematicon-Nematicon Interactions 22
1.4.2 Modulational Instability 26
1.5 Conclusions 31
References 33
Chapter 2. Features of Strongly Nonlocal Spatial Solitons 37
Qi Guo, Wei Hu, Dongmei Deng, Daquan Lu, and Shigen Ouyang
2.1 Introduction 37
2.2 Phenomenological Theory of Strongly Nonlocal Spatial Solitons 38
2.2.1 The Nonlinearly Induced Refractive Index Change of Materials 38
2.2.2 From the Nonlocal Nonlinear Schr¨odinger Equation to the Snyder-Mitchell Model 39
2.2.3 An Accessible Soliton of the Snyder-Mitchell Model 42
2.2.4 Breather and Soliton Clusters of the Snyder-Mitchell Model 45
2.2.5 Complex-Variable-Function Gaussian Breathers and Solitons 46
2.2.6 Self-Induced Fractional Fourier Transform 47
2.3 Nonlocal Spatial Solitons in Nematic Liquid Crystals 49
2.3.1 Voltage-Controllable Characteristic Length of NLC 50
2.3.2 Nematicons as Strongly Nonlocal Spatial Solitons 52
2.3.3 Nematicon-Nematicon Interactions 54
2.4 Conclusion 61
Appendix 2.A: Proof of the Equivalence of the Snyder-Mitchell Model (Eq. 2.16) and the Strongly Nonlocal Model (Eq. 2.11) 61
Appendix 2.B: Perturbative Solution for a Single Soliton of the NNLSE (Eq. 2.4) in NLC 62
References 66
Chapter 3. Theoretical Approaches to Nonlinear Wave Evolution in Higher Dimensions 71
Antonmaria A. Minzoni and Noel F. Smyth
3.1 Simple Example of Multiple Scales Analysis 71
3.2 Survey of Perturbation Methods for Solitary Waves 77
3.3 Linearized Perturbation Theory for Nonlinear Schr¨odinger Equation 81
3.4 Modulation Theory: Nonlinear Schr¨odinger Equation 83
3.5 Radiation Loss 88
3.6 Solitary Waves in Nematic Liquid Crystals: Nematicons 91
3.7 Radiation Loss for The Nematicon Equations 96
3.8 Choice of Trial Function 101
3.9 Conclusions 105
Appendix 3.A: Integrals 106
Appendix 3.B: Shelf Radius 107
References 108
Chapter 4. Soliton Families in Strongly Nonlocal Media 111
Wei-Ping Zhong and Milivoj R. Beli¸c
4.1 Introduction 111
4.2 Mathematical Models 112
4.2.1 General 112
4.2.2 Nonlocality Through Response Function 113
4.3 Soliton Families in Strongly Nonlocal Nonlinear Media 115
4.3.1 One-Dimensional Hermite-Gaussian Spatial Solitons 115
4.3.2 Two-Dimensional Laguerre-Gaussian Soliton Families 116
4.3.3 Accessible Solitons in the General Model of Beam Propagation in NLC 118
4.3.4 Two-Dimensional Self-Similar Hermite-Gaussian Spatial Solitons 125
4.3.5 Two-Dimensional Whittaker Solitons 126
4.4 Conclusions 133
References 135
Chapter 5. External Control of Nematicon Paths 139
Armando Piccardi, Alessandro Alberucci, and Gaetano Assanto
5.1 Introduction 139
5.2 Basic Equations 140
5.3 Nematicon Control with External Light Beams 142
5.3.1 Interaction with Circular Spots 143
5.3.2 Dielectric Interfaces 145
5.3.3 Comments 146
5.4 Voltage Control of Nematicon Walk-Off 147
5.4.1 Out-of-Plane Steering of Nematicons 147
5.4.2 In-Plane Steering of Nematicon 149
5.5 Voltage-Defined Interfaces 152
5.6 Conclusions 156
References 156
Chapter 6. Dynamics of Optical Solitons in Bias-Free Nematic Liquid Crystals 159
Yana V. Izdebskaya, Anton S. Desyatnikov, and Yuri S. Kivshar
6.1 Summary 159
6.2 Introduction 159
6.3 From One to Two Nematicons 160
6.4 Counter-Propagating Nematicons 162
6.5 Interaction of Nematicons with Curved Surfaces 165
6.6 Multimode Nematicon-Induced Waveguides 167
6.7 Dipole Azimuthons and Charge-Flipping 170
6.8 Conclusions 172
References 173
Chapter 7. Interaction of Nematicons and Nematicon Clusters 177
Catherine Garc´ýa-Reimbert, Antonmaria A. Minzoni, and Noel F. Smyth
7.1 Introduction 177
7.2 Gravitation of Nematicons 179
7.3 In-Plane Interaction of Two-Color Nematicons 184
7.4 Multidimensional Clusters 190
7.5 Vortex Cluster Interactions 199
7.6 Conclusions 205
Appendix: Integrals 206
References 206
Chapter 8. Nematicons in Light Valves 209
Stefania Residori, Umberto Bortolozzo, Armando Piccardi, Alessandro Alberucci, and Gaetano Assanto
8.1 Introduction 209
8.2 Reorientational Kerr Effect and Soliton Formation in Nematic Liquid Crystals 210
8.2.1 Optically Induced Reorientational Nonlinearity 211
8.2.2 Spatial Solitons in Nematic Liquid Crystals 211
8.3 Liquid Crystal Light Valves 212
8.3.1 Cell Structure and Working Principle 213
8.3.2 Optical Addressing in Transverse Configurations 215
8.4 Spatial Solitons in Light Valves 216
8.4.1 Stable Nematicons: Self-Guided Propagation in the Longitudinal Direction 216
8.4.2 Tuning the Soliton Walk-Off 218
8.5 Soliton Propagation in 3D Anisotropic Media: Model and Experiment 220
8.5.1 Optical Control of Nematicon Trajectories 224
8.6 Soliton Gating and Switching by External Beams 224
8.7 Conclusions and Perspectives 227
References 229
Chapter 9. Propagation of Light Confined via Thermo-Optical Effect in Nematic Liquid Crystals 233
Marc Warenghem, Jean-Francois Blach, and Jean-Francois Henninot
9.1 Introduction 233
9.2 First Observation in NLC 235
9.3 Characterization and Nonlocality Measurement 240
9.4 Thermal Versus Orientational Self-Waveguides 246
9.5 Applications 248
9.5.1 Bent Waveguide 248
9.5.2 Fluorescence Recovery 249
9.6 Conclusions 250
References 252
Chapter 10. Discrete Light Propagation in Arrays of Liquid Crystalline Waveguides 255
Katarzyna A. Rutkowska, Gaetano Assanto, and Miroslaw A. Karpierz
10.1 Introduction 255
10.2 Discrete Systems 256
10.3 Waveguide Arrays in Nematic Liquid Crystals 258
10.4 Discrete Diffraction and Discrete Solitons 263
10.5 Optical Multiband Vector Breathers 265
10.6 Nonlinear Angular Steering 267
10.7 Landau-Zener Tunneling 268
10.8 Bloch Oscillations 270
10.9 Conclusions 272
References 273
Chapter 11. Power-Dependent Nematicon Self-Routing 279
Alessandro Alberucci, Armando Piccardi, and Gaetano Assanto
11.1 Introduction 279
11.2 Nematicons: Governing Equations 280
11.2.1 Perturbative Regime 282
11.2.2 Highly Nonlinear Regime 284
11.2.3 Simplified (1 + 1)D Model in a Planar Cell 285
11.3 Single-Hump Nematicon Profiles 287
11.3.1 (2 + 1)D Complete Model 288
11.3.2 (1 + 1)D Simplified Model 289
11.4 Actual Experiments: Role of Losses 290
11.4.1 BPM (1 + 1)D Simulations 291
11.4.2 Experiments 292
11.5 Nematicon Self-Steering in Dye-Doped NLC 293
11.6 Boundary Effects 298
11.7 Nematicon Self-Steering Through Interaction with Linear Inhomogeneities 302
11.7.1 Interfaces: Goos-H¨anchen Shift 303
11.7.2 Finite-Size Defects: Nematicon Self-Escape 304
11.8 Conclusions 305
References 306
Chapter 12. Twisted and Chiral Nematicons 309
Urszula A. Laudyn and Miroslaw A. Karpierz
12.1 Introduction 309
12.2 Chiral and Twisted Nematics 310
12.3 Theoretical Model 312
12.4 Experimental Results 314
12.4.1 Nematicons in a Single Layer 314
12.4.2 Asymmetric Configuration 315
12.4.3 Multilayer Propagation 317
12.4.4 Influence of an External Electric Field 317
12.4.5 Guiding Light by Light 319
12.4.6 Nematicon Interaction 319
12.5 Discrete Diffraction 321
12.6 Conclusions 323
References 323
Chapter 13. Time Dependence of Spatial Solitons in Nematic Liquid Crystals 327
Jeroen Beeckman and Kristiaan Neyts
13.1 Introduction 327
13.2 Temporal Behavior of Different Nonlinearities and Governing Equations 328
13.2.1 Reorientational Nonlinearity 328
13.2.2 Thermal Nonlinearity 331
13.2.3 Other Nonlinearities 333
13.3 Formation of Reorientational Solitons 333
13.3.1 Bias Voltage Switching Time 334
13.3.2 Soliton Formation Time 336
13.3.3 Experimental Observation of Soliton Formation 337
13.3.4 Influence of Flow Effects 341
13.4 Conclusions 344
References 344
Chapter 14. Spatiotemporal Dynamics and Light Bullets in Nematic Liquid Crystals 347
Marco Peccianti
14.1 Introduction 347
14.1.1 (2 + 1 + 1)D Nonlinear Wave Propagation in Kerr Media 348
14.2 Optical Propagation Under Multiple Nonlinear Contributions 349
14.2.1 Multiple Nonlinearities and Space-Time Decoupling of the Nonlinear Dynamics 349
14.2.2 Suitable Excitation Conditions 350
14.3 Accessible Light Bullets 351
14.3.1 From Nematicons to Spatiotemporal Solitons 351
14.3.2 Experimental Conditions for Accessible Bullets Observation 353
14.4 Temporal Modulation Instability in Nematicons 355
14.5 Soliton-Enhanced Frequency Conversion 355
14.6 Conclusions 357
References 358
Chapter 15. Vortices in Nematic Liquid Crystals 361
Antonmaria A. Minzoni, Luke W. Sciberras, Noel F. Smyth, and Annette L. Worthy
15.1 Introduction 361
15.2 Stabilization of Vortices in Nonlocal, Nonlinear Media 364
15.3 Vortex in a Bounded Cell 373
15.4 Stabilization of Vortices by Vortex-Beam Interaction 378
15.5 Azimuthally Dependent Vortices 382
15.6 Conclusions 387
References 389
Chapter 16. Dispersive Shock Waves in Reorientational and Other Optical Media 391
Tim R. Marchant
16.1 Introduction 391
16.2 Governing Equations and Modulational Instability 392
16.3 Existing Experimental and Numerical Results 394
16.4 Analytical Solutions for Defocusing Equations 396
16.5 Analytical Solutions for Focusing Equations 398
16.5.1 The 1 + 1 Dimensional Semianalytical Soliton 400
16.5.2 Uniform Soliton Theory 402
16.5.3 Comparisons with Numerical Solutions 403
16.6 Conclusions 406
References 407
Index 411
