Localized Waves

Contributors xiii

Preface xv

1 Localized Waves: A Historical and Scientific Introduction 1
Erasmo Recami, Michel Zamboni-Rached, and Hugo E. Hernández-Figueroa

1.1 General Introduction 2

1.2 More Detailed Information 6

1.2.1 Localized Solutions 9

Appendix: Theoretical and Experimental History 17

Historical Recollections: Theory 17

X-Shaped Field Associated with a Superluminal Charge 20

A Glance at the Experimental State of the Art 23

References 34

2 Structure of Nondiffracting Waves and Some Interesting Applications 43
Michel Zamboni-Rached, Erasmo Recami, and Hugo E. Hernández-Figueroa

2.1 Introduction 43

2.2 Spectral Structure of Localized Waves 44

2.2.1 Generalized Bidirectional Decomposition 46

2.3 Space–Time Focusing of X-Shaped Pulses 54

2.3.1 Focusing Effects Using Ordinary X-Waves 55

2.4 Chirped Optical X-Type Pulses in Material Media 57

2.4.1 Example: Chirped Optical X-Type Pulse in Bulk Fused Silica 62

2.5 Modeling the Shape of Stationary Wave Fields: Frozen Waves 63

2.5.1 Stationary Wave Fields with Arbitrary Longitudinal Shape in Lossless Media Obtained by Superposing Equal-Frequency Bessel Beams 63

2.5.2 Stationary Wave Fields with Arbitrary Longitudinal Shape in Absorbing Media: Extending the Method 70

References 76

3 Two Hybrid Spectral Representations and Their Applications to the Derivations of Finite-Energy Localized Waves and Pulsed Beams 79
Ioannis M. Besieris and Amr M. Shaarawi

3.1 Introduction 79

3.2 Overview of Bidirectional and Superluminal Spectral Representations 80

3.2.1 Bidirectional Spectral Representation 81

3.2.2 Superluminal Spectral Representation 83

3.3 Hybrid Spectral Representation and Its Application to the Derivation of Finite-Energy X-Shaped Localized Waves 84

3.3.1 Hybrid Spectral Representation 84

3.3.2 (3 + 1)-Dimensional Focus X-Wave 85

3.3.3 (3 + 1)-Dimensional Finite-Energy X-Shaped Localized Waves 86

3.4 Modified Hybrid Spectral Representation and Its Application to the Derivation of Finite-Energy Pulsed Beams 89

3.4.1 Modified Hybrid Spectral Representation 89

3.4.2 (3 + 1)-Dimensional Splash Modes and Focused Pulsed Beams 89

3.5 Conclusions 93

References 93

4 Ultrasonic Imaging with Limited-Diffraction Beams 97
Jian-yu Lu

4.1 Introduction 97

4.2 Fundamentals of Limited-Diffraction Beams 99

4.2.1 Bessel Beams 99

4.2.2 Nonlinear Bessel Beams 101

4.2.3 Frozen Waves 101

4.2.4 X-Waves 101

4.2.5 Obtaining Limited-Diffraction Beams with Variable Transformation 102

4.2.6 Limited-Diffraction Solutions to the Klein–Gordon Equation 103

4.2.7 Limited-Diffraction Solutions to the Schrödinger Equation 106

4.2.8 Electromagnetic X-Waves 108

4.2.9 Limited-Diffraction Beams in Confined Spaces 109

4.2.10 X-Wave Transformation 114

4.2.11 Bowtie Limited-Diffraction Beams 115

4.2.12 Limited-Diffraction Array Beams 115

4.2.13 Computation with Limited-Diffraction Beams 115

4.3 Applications of Limited-Diffraction Beams 116

4.3.1 Medical Ultrasound Imaging 116

4.3.2 Tissue Characterization (Identification) 116

4.3.3 High-Frame-Rate Imaging 116

4.3.4 Two-Way Dynamic Focusing 116

4.3.5 Medical Blood-Flow Measurements 117

4.3.6 Nondestructive Evaluation of Materials 117

4.3.7 Optical Coherent Tomography 117

4.3.8 Optical Communications 117

4.3.9 Reduction of Sidelobes in Medical Imaging 117

4.4 Conclusions 117

References 118

5 Propagation-Invariant Fields: Rotationally Periodic and Anisotropic Nondiffracting Waves 129
Janne Salo and Ari T. Friberg

5.1 Introduction 129

5.1.1 Brief Overview of Propagation-Invariant Fields 130

5.1.2 Scope of This Chapter 133

5.2 Rotationally Periodic Waves 134

5.2.1 Fourier Representation of General RPWs 135

5.2.2 Special Propagation Symmetries 135

5.2.3 Monochromatic Waves 136

5.2.4 Pulsed Single-Mode Waves 138

5.2.5 Discussion 142

5.3 Nondiffracting Waves in Anisotropic Crystals 142

5.3.1 Representation of Anisotropic Nondiffracting Waves 143

5.3.2 Effects Due to Anisotropy 146

5.3.3 Acoustic Generation of NDWs 148

5.3.4 Discussion 149

5.4 Conclusions 150

References 151

6 Bessel X-Wave Propagation 159
Daniela Mugnai and Iacopo Mochi

6.1 Introduction 159

6.2 Optical Tunneling: Frustrated Total Reflection 160

6.2.1 Bessel Beam Propagation into a Layer: Normal Incidence 160

6.2.2 Oblique Incidence 164

6.3 Free Propagation 169

6.3.1 Phase, Group, and Signal Velocity: Scalar Approximation 169

6.3.2 Energy Localization and Energy Velocity: A Vectorial Treatment 172

6.4 Space–Time and Superluminal Propagation 180

References 181

7 Linear-Optical Generation of Localized Waves 185
Kaido Reivelt and Peeter Saari

7.1 Introduction 185

7.2 Definition of Localized Waves 186

7.3 The Principle of Optical Generation of LWs 191

7.4 Finite-Energy Approximations of LWs 193

7.5 Physical Nature of Propagation Invariance of Pulsed Wave Fields 195

7.6 Experiments 198

7.6.1 LWs in Interferometric Experiments 198

7.6.2 Experiment on Optical Bessel X-Pulses 200

7.6.3 Experiment on Optical LWs 203

7.7 Conclusions 211

References 213

8 Optical Wave Modes: Localized and Propagation-Invariant Wave Packets in Optically Transparent Dispersive Media 217
Miguel A. Porras, Paolo Di Trapani, and Wei Hu

8.1 Introduction 217

8.2 Localized and Stationarity Wave Modes Within the SVEA 219

8.2.1 Dispersion Curves Within the SVEA 221

8.2.2 Impulse-Response Wave Modes 222

8.3 Classification of Wave Modes of Finite Bandwidth 224

8.3.1 Phase-Mismatch-Dominated Case: Pulsed Bessel Beam Modes 226

8.3.2 Group-Velocity-Mismatch-Dominated Case: Envelope Focus Wave Modes 227

8.3.3 Group-Velocity-Dispersion-Dominated Case: Envelope X- and Envelope O-Modes 229

8.4 Wave Modes with Ultrabroad Bandwidth 231

8.4.1 Classification of SEWA Dispersion Curves 233

8.5 About the Effective Frequency, Wave Number, and Phase Velocity of Wave Modes 236

8.6 Comparison Between Exact, SEWA, and SVEA Wave Modes 238

8.7 Conclusions 240

References 240

9 Nonlinear X-Waves 243
Claudio Conti and Stefano Trillo

9.1 Introduction 243

9.2 NLX Model 245

9.3 Envelope Linear X-Waves 247

9.3.1 X-Wave Expansion and Finite-Energy Solutions 250

9.4 Conical Emission and X-Wave Instability 252

9.5 Nonlinear X-Wave Expansion 255

9.5.1 Some Examples 255

9.5.2 Proof 256

9.5.3 Evidence 257

9.6 Numerical Solutions for Nonlinear X-Waves 257

9.6.1 Bestiary of Solutions 259

9.7 Coupled X-Wave Theory 262

9.7.1 Fundamental X-Wave and Fundamental Soliton 264

9.7.2 Splitting and Replenishment in Kerr Media as a Higher-Order Soliton 264

9.8 Brief Review of Experiments 265

9.8.1 Angular Dispersion 265

9.8.2 Nonlinear X-Waves in Quadratic Media 265

9.8.3 X-Waves in Self-Focusing of Ultrashort Pulses in Kerr Media 266

9.9 Conclusions 266

References 267

10 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 273
Sergei V. Kukhlevsky

10.1 Introduction 273

10.2 Natural Spatial and Temporal Broadening of Light Waves 275

10.3 Diffraction-Free Optics in the Overwavelength Domain 281

10.4 Diffraction-Free Subwavelength-Beam Optics on a Nanometer Scale 286

10.5 Conclusions 292

Appendix 292

References 293

11 Self-Reconstruction of Pulsed Optical X-Waves 299
Ruediger Grunwald, Uwe Neumann, Uwe Griebner, Günter Steinmeyer, Gero Stibenz, Martin Bock, and Volker Kebbel

11.1 Introduction 299

11.2 Small-Angle Bessel-Like Waves and X-Pulses 300

11.3 Self-Reconstruction of Pulsed Bessel-Like X-Waves 303

11.4 Nondiffracting Images 306

11.5 Self-Reconstruction of Truncated Ultrabroadband Bessel–Gauss Beams 307

11.6 Conclusions 310

References 311

12 Localization and Wannier Wave Packets in Photonic Crystals Without Defects 315
Stefano Longhi and Davide Janner

12.1 Introduction 315

12.2 Diffraction and Localization of Monochromatic Waves in Photonic Crystals 317

12.2.1 Basic Equations 317

12.2.2 Localized Waves 319

12.3 Spatiotemporal Wave Localization in Photonic Crystals 324

12.3.1 Wannier Function Technique 325

12.3.2 Undistorted Propagating Waves in Two- and Three-Dimensional Photonic Crystals 329

12.4 Conclusions 334

References 335

13 Spatially Localized Vortex Structures 339
Zdeněk Bouchal, Radek Č elechovsk, and Grover A. Swartzlander, Jr.

13.1 Introduction 339

13.2 Single and Composite Optical Vortices 342

13.3 Basic Concept of Nondiffracting Beams 346

13.4 Energetics of Nondiffracting Vortex Beams 350

13.5 Vortex Arrays and Mixed Vortex Fields 352

13.6 Pseudo-nondiffracting Vortex Fields 354

13.7 Experiments 357

13.7.1 Fourier Methods 357

13.7.2 Spatial Light Modulation 358

13.8 Applications and Perspectives 361

References 363

Index 367