Nematicons Spatial Optical Solitons in Nematic Liquid Crystals Wiley Series in Pure and Applied Optics Series

The first book of its kind to introduce the fundamentals, basic features and models, potential applications and novel phenomena and its important applications in liquid crystal technology.

Recognized leader in the field Gaetano Assanto outlines the peculiar characteristics of nematicons and the promise they have for the future growth of this captivating new field.

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

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.