Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals

A pioneer in the field presents the first book on nematicons,spatial optical solitons in nematic liquid crystals

Little more than a decade after they were first demonstrated,nematicons have become an important area of research worldwide.This wide–ranging introduction is the first comprehensivecollection of scientific work on optical spatial solitons innematic liquid crystals, where they have been observed at powers ofthe order of milliwatts or lower, and have shown remarkable andpeculiar features as compared to solitons in other media.

Nematicons: Spatial Optical Solitons in Nematic LiquidCrystals features information on the background physics andrelevant mathematics as well as numerical modeling, experimentalapproaches and results, and applications. It combines a review ofthe field with a considerable amount of original material as wellas background information on liquid crystals.

This one–of–a–kind resource:

Explores nematicons′ nonlocal features, including theirmodeling, and experimental results on their mutualinteractions Deals with models and numerics on light localization andspatial solitons, with emphasis on nematicons and their guidingproperties Discusses experimental as well as theoretical work onnematicons and their trajectories, which can be modified in variousways and in several configurations Illustrates the physics and properties of nonlinearreorientation and thermo–optically localized waves Underlines the role of electronic nonlinearities and theirsynergy with reorientation in nematic liquid crystals, pinpointingthe possibility of spatio–temporal solitons in these materials

Complete with a vast bibliography of all the existing work inspatial solitons in nematic liquid crystals as well as actualphotographs from experiments, Nematicons is a valuable toolfor scientists, students, and scholars working on lightlocalization and solitons, as well as on nonlinear optics in liquidcrystals.

Preface xv

Acknowledgments xvii

Contributors xix

Chapter 1. Nematicons 1
Gaetano Assanto, Alessandro Alberucci, and ArmandoPiccardi

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 Crystals3

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 Solitons37
Qi Guo, Wei Hu, Dongmei Deng, Daquan Lu, and ShigenOuyang

2.1 Introduction 37

2.2 Phenomenological Theory of Strongly Nonlocal SpatialSolitons 38

2.2.1 The Nonlinearly Induced Refractive Index Change ofMaterials 38

2.2.2 From the Nonlocal Nonlinear Schr¨odinger Equation tothe Snyder Mitchell Model 39

2.2.3 An Accessible Soliton of the Snyder Mitchell Model42

2.2.4 Breather and Soliton Clusters of the Snyder MitchellModel 45

2.2.5 Complex–Variable–Function Gaussian Breathers and Solitons46

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 theSnyder Mitchell Model (Eq. 2.16) and the Strongly NonlocalModel (Eq. 2.11) 61

Appendix 2.B: Perturbative Solution for a Single Soliton of theNNLSE (Eq. 2.4) in NLC 62

References 66

Chapter 3. Theoretical Approaches to Nonlinear Wave Evolutionin 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 NonlinearSchr¨odinger Equation 81

3.4 Modulation Theory: Nonlinear Schr¨odinger Equation83

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 Media111
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 Media115

4.3.1 One–Dimensional Hermite Gaussian Spatial Solitons115

4.3.2 Two–Dimensional Laguerre Gaussian Soliton Families116

4.3.3 Accessible Solitons in the General Model of BeamPropagation in NLC 118

4.3.4 Two–Dimensional Self–Similar Hermite GaussianSpatial 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 GaetanoAssanto

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 NematicLiquid 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 Clusters177
Catherine Garc´ýa–Reimbert, Antonmaria A. Minzoni, andNoel 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 NematicLiquid 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 theLongitudinal Direction 216

8.4.2 Tuning the Soliton Walk–Off 218

8.5 Soliton Propagation in 3D Anisotropic Media: Model andExperiment 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–OpticalEffect in Nematic Liquid Crystals 233
Marc Warenghem, Jean–Francois Blach, and Jean–FrancoisHenninot

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 LiquidCrystalline 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–Routing279
Alessandro Alberucci, Armando Piccardi, and GaetanoAssanto

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 LinearInhomogeneities 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 NematicLiquid Crystals 327
Jeroen Beeckman and Kristiaan Neyts

13.1 Introduction 327

13.2 Temporal Behavior of Different Nonlinearities and GoverningEquations 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 inNematic Liquid Crystals 347
Marco Peccianti

14.1 Introduction 347

14.1.1 (2 + 1 + 1)D Nonlinear Wave Propagation in Kerr Media348

14.2 Optical Propagation Under Multiple Nonlinear Contributions349

14.2.1 Multiple Nonlinearities and Space Time Decouplingof 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 BulletsObservation 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, andAnnette L. Worthy

15.1 Introduction 361

15.2 Stabilization of Vortices in Nonlocal, Nonlinear Media364

15.3 Vortex in a Bounded Cell 373

15.4 Stabilization of Vortices by Vortex Beam Interaction378

15.5 Azimuthally Dependent Vortices 382

15.6 Conclusions 387

References 389

Chapter 16. Dispersive Shock Waves in Reorientational andOther 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

GAETANO ASSANTO, PhD, is Professor of Optoelectronics at the University of Rome, where he heads the Nonlinear Optics and OptoElectronics Lab. He is Fellow of the Optical Society of America and a Senior Member of the IEEE Photonics Society.