Preface ................................................. XVII
List of Contributors ................................... XXIII
1 Non-Diffracting Waves: An Introduction ..................... 1
Erasmo Recami, Michel Zamboni-Rached, Hugo E. Hernández-
Figueroa, and Leonardo A. Ambrosio
1.1 A General Introduction ..................................... 1
1.1.1 A Prologue .......................................... 1
1.1.2 Preliminary, and Historical, Remarks ................ 3
1.1.3 Definition of Non-Diffracting Wave (NDW) ............ 6
1.1.4 First Examples ...................................... 8
1.1.5 Further Examples: The Non-Diffracting Solutions ..... 9
1.2 Eliminating Any Backward Components: Totally Forward NDW
Pulses .................................................... 13
1.2.1 Totally Forward Ideal Superluminal NDW Pulses ...... 14
1.3 Totally Forward, Finite-Energy NDW Pulses ................. 17
1.3.1 A General Functional Expression for Whatever
Totally-Forward NDW Pulses ......................... 20
1.4 Method for the Analytic Description of Truncated Beams .... 21
1.4.1 The Method ......................................... 21
1.4.2 Application of the Method to а ТВ Beam ............. 24
1.5 Subluminal NDWs (or Bullets) .............................. 25
1.5.1 A First Method for Constructing Physically
Acceptable, Subluminal Non-Diffracting Pulses ...... 26
1.5.2 Examples ........................................... 29
1.5.3 A Second Method for Constructing Subluminal Non-
Diffracting Pulses ................................. 32
1.6 "Stationary" Solutions with Zero-Speed Envelopes: Frozen
Waves ..................................................... 33
1.6.1 A New Approach to the Frozen Waves ................. 35
1.6.2 Frozen Waves in Absorbing Media .................... 38
1.6.3 Experimental Production of the Frozen Waves ........ 38
1.7 On the Role of Special Relativity and of Lorentz
Transformations ........................................... 38
1.8 Non-Axially Symmetrie Solutions: The Case of Higher-
Order Bessel Beams ........................................ 42
1.9 An Application to Biomedical Optics: NDWs and the GLMT
(Generalized Lorenz-Mie Theory) ........................... 44
1.10 Soliton-Like Solutions to the Ordinary Schroedinger
Equation within Standard Quantum Mechanics (QM) ........... 50
1.10.1 Bessel Beams as Non-Diffracting Solutions (NDS)
to the Schroedinger Equation ....................... 52
1.10.2 Exact Non-Diffracting Solutions to the
Schroedinger Equation .............................. 54
1.10.3 A General Exact Localized Solution ................. 58
1.11 A Brief Mention of Further Topics ......................... 59
1.11.1 Airy and Airy-Type Waves ........................... 59
1.11.2 "Soliton-Like" Solutions to the Einstein
Equations of General Relativity and Gravitational
Waves .............................................. 60
1.11.3 Super-Resolution ................................... 60
Acknowledgments ........................................... 60
References ................................................ 60
2 Localized Waves: Historical and Personal Perspectives ..... 69
Richard W. Ziolkowski
2.1 The Beginnings: Focused Wave Modes ........................ 69
2.2 The Initial Surge and Nomenclature ........................ 71
2.3 Strategic Defense Initiative (SDI) Interest ............... 71
2.4 Reflective Moments ........................................ 72
2.5 Controversy and Scrutiny .................................. 73
2.6 Experiments ............................................... 75
2.7 What's in a Name: Localized Waves ......................... 76
2.8 Arizona Era ............................................... 76
2.9 Retrospective ............................................. 78
Acknowledgments ........................................... 78
References ................................................ 78
3 Applications of Propagation Invariant Light Fields ........ 83
Michael Mazilu and Kishan Dholakia
3.1 Introduction .............................................. 83
3.2 What Is a "Non-Diffracting" Light Mode? ................... 83
3.2.1 Linearly Propagating "Non-Diffracting" Beams ....... 84
3.2.2 Accelerating "Non-Diffracting" Beams ............... 87
3.2.3 Self-Healing Properties and Infinite Energy ........ 88
3.2.4 Vectorial "Non-Diffracting" Beams .................. 88
3.3 Generating "Non-Diffracting" Light Fields ................. 91
3.3.1 Bessel and Mathieu Beam Generation ................. 91
3.3.2 Airy Beam Generation ............................... 93
3.4 Experimental Applications of Propagation Invariant Light
Modes ..................................................... 93
3.4.1 Microscopy, Coherence, and Imaging ................. 94
3.4.2 Optical Micromanipulation with Propagation
Invariant Fields ................................... 97
3.4.3 Propagation Invariant Beams for Cell
Nanosurgery ....................................... 102
3.5 Conclusion ............................................... 104
Acknowledgment ........................................... 104
References ............................................... 104
4 X-Type Waves in Ultrafast Optics ......................... 109
Peeter Saari
4.1 Introduction ............................................. 109
4.2 About Physics of Superluminal and Subluminal,
Accelerating and Decelerating Pulses ..................... 110
4.2.1 Remarks on Some Persistent Issues ................. 110
4.2.1.1 Group Velocity: Plane Waves versus
Three-Dimensional Waves .................. 110
4.2.1.2 Group Velocity: Superluminal versus
Subluminal Cylindrically Symmetric
Wavepackets .............................. 111
4.2.1.3 Group Velocity versus Energy Transport
Velocity ................................. 116
4.2.1.4 Group Velocity versus Signal Velocity .... 117
4.2.1.5 Cherenkov Radiation versus Superluminal
X-Type Waves and Causality versus
Acausality ............................... 118
4.2.2 Accelerating and Decelerating Quasi-Bessel-X
Pulses ............................................ 120
4.2.3 "Technology Transfer" to Quantum Optics ........... 121
4.3 Overview of Spatiotemporal Measurements of Localized
Waves by SEA TADPOLE Technique ........................... 122
4.3.1 Spatiotemporal Measurement of Light Fields ........ 122
4.3.2 New Results on Bessel-X Pulse ..................... 123
4.3.3 Grating-Generated Bessel Pulses ................... 124
4.3.4 Lens-Generated Accelerating and Decelerating
Quasi-Bessel-X Pulses ............................. 125
4.3.5 Boundary Diffraction Wave as a Decelerating
Quasi-Bessel-X Pulse .............................. 127
4.4 Conclusion ............................................... 129
Acknowledgments .......................................... 130
References ............................................... 131
5 Limited-Diffraction Beams for High-Frame-Rate Imaging .... 135
Jian-yu Lu
5.1 Introduction ............................................. 135
5.2 Theory of Limited-Diffraction Beams ...................... 138
5.2.1 Generalized Solutions to Wave Equation ............ 138
5.2.2 Bessel Beams and X Waves .......................... 140
5.2.2.1 Bessel Beams ............................. 140
5.2.2.2 X Waves .................................. 140
5.2.3 Limited-Diffraction Array Beams ................... 141
5.3 Received Signals ......................................... 142
5.3.1 Pulse-Echo Signals and Relationship with
Imaging ........................................... 142
5.3.2 Limited-Diffraction Array Beam Aperture
Weighting and Spatial Fourier Transform of Echo
Signals ........................................... 143
5.3.3 Special Case for 2D Imaging ....................... 144
5.4 Imaging with Limited-Diffraction Beams ................... 144
5.4.1 High-Frame-Rate Imaging Methods ................... 145
5.4.1.1 Plane-Wave HFR Imaging without
Steering ................................. 145
5.4.1.2 Steered Plane-Wave Imaging ............... 145
5.4.1.3 Limited-Diffraction Array Beam Imaging ... 146
5.4.2 Other Imaging Methods ............................. 147
5.4.2.1 Two-Way Dynamic Focusing ................. 147
5.4.2.2 Multiple Steered Plane Wave Imaging ...... 148
5.5 Mapping between Fourier Domains .......................... 148
5.5.1 Mapping for Steer Plane Wave Imaging .............. 149
5.5.2 Mapping for Limited-Diffraction-Beam Imaging ...... 150
5.5.2.1 General Case ............................. 150
5.5.2.2 Special Case ............................. 151
5.6 High-Frame-Rate Imaging Techniques-Their Improvements
and Applications ......................................... 151
5.6.1 Aperture Weighting with Square Functions to
Simplify Imaging System ........................... 151
5.6.1.1 Applied to Transmission .................. 151
5.6.1.2 Applied to Reception ..................... 152
5.6.2 Diverging Beams with a Planar Array Transducer
to Increase Image Frame Rate ...................... 153
5.6.3 Diverging Beams with a Curved Array Transducer
to Increase Image Field of View ................... 153
5.6.4 Other Studies on Increasing Image Field of View ... 153
5.6.5 Coherent and Incoherent Superposition to Enhance
Images and Increase Image Field of View ........... 153
5.6.6 Nonlinear Image Processing for Speckle
Reduction ......................................... 154
5.6.7 Coordinate Rotation for Reduction of
Computation ....................................... 154
5.6.8 Reducing Number of Elements of Array Transducer ... 154
5.6.9 A Study of Trade-Off between Image Quality and
Data Densification ................................ 154
5.6.10 Masking Method for Improving Image Quality ........ 155
5.6.11 Reducing Clutter Noise by High-Pass Filtering ..... 155
5.6.12 Obtaining Flow or Tissue Velocity Vectors for
Functional Imaging ................................ 155
5.6.13 Strain and Strain Rate Imaging to Obtain Tissue
Parameters or Organ Functions ..................... 156
5.6.14 High-Frame-Rate Imaging Systems ................... 156
5.7 Conclusion ............................................... 156
References ............................................... 156
6 Spatiotemporally Localized Null Electromagnetic Waves .... 163
Ioannis M. Besieris and Amr M. Shaarawi
6.1 Introduction ............................................. 161
6.2 Three Classes of Progressive Solutions to the 3D Scalar
Wave Equation ............................................ 162
6.2.1 Luminal Localized Waves ........................... 163
6.2.1.1 Luminal .................................. 163
6.2.1.2 Modified Luminal ......................... 165
6.2.2 Superluminal Localized Waves ...................... 165
6.2.2.1 Superluminal ............................. 165
6.2.2.2 Hybrid Superluminal ...................... 166
6.2.2.3 Modified Hybrid Superluminal ............. 167
6.2.3 Subluminal Localized Waves ........................ 168
6.3 Construction of Null Electromagnetic Localized Waves ..... 169
6.3.1 Riemann-Silberstein Vector ........................ 169
6.3.2 Null Riemann-Silberstein Vector ................... 170
6.3.3 The Whittaker-Bateman Method ...................... 173
6.4 Illustrative Examples of Spatiotemporally Localized
Null Electromagnetic Waves ............................... 173
6.4.1 Luminal Null Electromagnetic Localized Waves ...... 173
6.4.2 Modified Luminal Null Electromagnetic Localized
Waves ............................................. 175
6.4.3 Superluminal Null Electromagnetic Localized
Waves ............................................. 176
6.4.4 Hybrid Superluminal Null Electromagnetic
Localized Waves ................................... 179
6.4.5 Modified Hybrid Superluminal Null
Electromagnetic Localized Waves ................... 181
6.4.6 A Note on Subluminal Null Electromagnetic
Localized Waves ................................... 182
6.5 Concluding Remarks ....................................... 183
References ............................................... 185
7 Linearly Traveling and Accelerating Localized Wave
Solutions to the Schrцdinger and Schrцdinger-Like
Equations ................................................ 189
Ioannis M. Besieris, Amr M. Shaarawi, and Richard
W. Ziolkowski
7.1 Introduction ............................................. 189
7.2 Linearly Traveling Localized Wave Solutions to the 3D
Schrцdinger Equation ..................................... 193
7.2.1 MacKinnon-Type, Infinite-Energy, Localized,
Traveling Wave Solutions .......................... 192
7.2.2 Extensions to MacKinnon-Type, Infinite-Energy,
Localized, Traveling Wave Solutions ............... 193
7.2.3 Finite-Energy, Localized, Traveling Wave
Solutions ......................................... 196
7.3 Accelerating Localized Wave Solutions to the 3D
Schrödinger Equation ..................................... 198
7.4 Linearly Traveling and Accelerating Localized Wave
Solutions to Schrцdinger-Like Equations .................. 199
7.4.1 Anomalous Dispersion .............................. 200
7.4.1.1 Linearly Traveling Localized Wave
Solutions ................................ 200
7.4.1.2 Accelerating Localized Wave Solutions .... 201
7.4.2 Normal Dispersion ................................. 202
7.4.2.1 Linearly Traveling X-Shaped Localized
Waves .................................... 202
7 A.2.2 Accelerating Localized Waves ............. 204
7.5 Concluding Remarks ....................................... 206
References ............................................... 206
8 Rogue X-Waves ............................................ 211
Audrius Dubietis, Daniele Faccio, and Gintaras Valiulis
8.1 Introduction ............................................. 211
8.2 Ultrashort Laser Pulse Filamentation ..................... 212
8.3 The X-Wave Model ......................................... 215
8.4 Rogue X-Waves ............................................ 219
8.5 Conclusions .............................................. 226
Acknowledgments .......................................... 227
References ............................................... 227
9 Quantum X-Waves and Applications in Nonlinear Optics ..... 231
Claudio Conti
9.1 Introduction ............................................. 231
9.2 Derivation of the Paraxial Equations ..................... 232
9.3 The X-Wave Transform and X-Wave Expansion ................ 234
9.4 Quantization ............................................. 235
9.5 Optical Parametric Amplification ......................... 237
9.6 Kerr Media ............................................... 239
9.7 Conclusions .............................................. 242
Acknowledgments .......................................... 243
References ............................................... 243
10 ТЕ and TM Optical Localized Beams ........................ 247
Pierre Hillion
10.1 Introduction ............................................. 247
10.2 ТЕ Optical Beams ......................................... 248
10.2.1 We First Suppose k,r ≤ 1 .......................... 248
10.2.2 We Now Suppose k,r > 1 ............................ 249
10.2.3 Approximations .................................... 250
10.3 Energetics of the ТЕ Optical Beam ........................ 251
10.4 Discussion ............................................... 253
10.5 Appendix ................................................. 254
References ............................................... 255
11 Spatiotemporal Localization of Ultrashort-Pulsed Bessel
Beams at Extremely Low Light Level ....................... 257
Martin Bock and Ruediger Grunwald
11.1 Introduction ............................................. 257
11.2 Non-Diffracting Young's Interferometers .................. 258
11.3 Non-Diffracting Beams at Low Light Level ................. 259
11.4 Experimental Techniques and Results ...................... 260
11.5 Retrieval of Temporal Information ........................ 263
11.6 Wave Function and Fringe Contrast ........................ 264
11.7 Conclusions .............................................. 267
Acknowledgments .......................................... 267
References ............................................... 267
12 Adaptive Shaping of Nondiffracting Wavepackets for
Applications in Ultrashort Pulse Diagnostics ............. 271
Martin Bock, Susanta Kumar Das, Carsten Fischer,
Michael Diehl, Peter Boemer, and Ruediger Grunwald
12.1 Introduction ............................................. 271
12.2 Space-Time Coupling and Spatially Resolved Pulse
Diagnostics .............................................. 272
12.3 Shack-Hartmann Sensors with Microaxicons ................. 273
12.4 Nonlinear Wavefront Autocorrelation ...................... 275
12.5 Spatially Resolved Spectral Phase ........................ 276
12.6 Adaptive Shack-Hartmann Sensors with Localized Waves ..... 277
12.7 Diagnostics of Ultrashort Wavepackets .................... 278
12.7.1 Time-Wavefront Sensing ............................ 278
12.7.2 Travel-Time Mapping ............................... 280
12.7.3 Optical Angular Momentum of Few-Cycle
Wavepackets ....................................... 281
12.8 Conclusions .............................................. 281
Acknowledgments .......................................... 282
References ............................................... 283
13 Localized Waves Emanated by Pulsed Sources: The
Riemann-Volterra Approach ................................ 287
Andrei B. Uikin
13.1 Introduction ............................................. 287
13.2 Basics of the Riemann-Volterra Approach .................. 289
13.2.1 Problem Posing .................................... 289
13.2.2 Riemann-Volterra Solution ......................... 290
13.3 Emanation from Wavefront-Speed Source Pulse of Gaussian
Transverse Variation: Causal Clipped Brittingham's
Focus Wave Mode .......................................... 291
13.4 Emanation from a Source Pulse Moving Faster than the
Wavefront: Droplet-Shaped Waves .......................... 297
13.4.1 General Solution for the Superluminal
(Supersonic) Motion ............................... 297
13.4.2 Droplet-Shaped Waves as Causal Counterparts of
the X-Shaped Waves ................................ 302
13.5 Conclusive Remarks ....................................... 302
References ............................................... 304
14 Propagation-Invariant Optical Beams and Pulses ........... 307
Kimmo Saastamoinen, Ari T. Friberg, and Jari Turunen
14.1 Introduction ............................................. 307
14.2 Theoretical Background ................................... 308
14.3 General Propagation-Invariant Solutions .................. 309
14.3.1 Conditions for Propagation Invariance ............. 310
14.3.2 Plane-Wave Representation of Nonstationary
Fields ............................................ 311
14.3.3 Solutions in the Space-Frequency Domain ........... 312
14.3.4 Solutions in the Space-Time Domain ................ 313
14.4 Classification in Terms of Spectral and Angular
Coherence ................................................ 314
14.5 Stationary Propagation-Invariant Fields .................. 315
14.5.1 Coherent Fields ................................... 316
14.5.2 Partially Coherent Fields ......................... 318
14.6 Nonstationary Propagation-Invariant Fields ............... 319
14.6.1 Coherent Fields ................................... 320
14.6.2 Partially Coherent Fields ......................... 321
14.7 Conclusions .............................................. 324
References ............................................... 325
15 Diffractionless Nanobeams Produced by Multiple-
Waveguide Metallic Nanostructures ........................ 327
Matyas Mechler and Sergei V. Kukhlevsky
15.1 Introduction ............................................. 327
15.2 Concept of Diffractionless Subwavelength-В earn Optics
on Nanometer Scale ....................................... 328
15.3 Diffractionless Nanobeams Produced by Multiple-
Waveguide Metallic Nanostructures ........................ 331
15.4 Summary and Conclusions .................................. 335
Acknowledgments .......................................... 335
References ............................................... 336
16 Low-Cost 2D Collimation of Real-Time Pulsed Ultrasonic
Beams by X-Wave-Based High-Voltage Driving of Annular
Arrays ................................................... 339
Antonio Ramos, Luis Castellanos, and Héctor Calás
16.1 Introduction ............................................. 339
16.2 Classic Electronic Procedures to Improve Lateral
Resolutions in Emitted Beams for Ultrasonic Detection:
Main Limitations ......................................... 341
16.3 An X-Wave-Based Option for Beam Collimation with Bessel
Arrays ................................................... 343
16.3.1 Design of Bessel Arrays ........................... 344
16.3.1.1 Bases for Designing the Bessel
Transducers .............................. 344
16.3.1.2 A Design Example: Bessel Transducer
with 10 Annuli and 50 mm in Diameter ..... 345
16.3.2 Modeling and Characterization of the Bessel
Annular Arrays .................................... 345
16.3.2.1 Transducers' Complex Electric Impedance
around the Resonance Frequency ........... 346
16.3.2.2 Characterization of Emission Transfer
Functions and Impulsive Responses ........ 347
16.3.3 Some Characterization Results ..................... 348
16.3.4 Broadband X-Wave Pulses for Deriving the Bessel
Array Excitations ................................. 353
16.4 Low-Cost Circuits for Efficient Rectangular Driving of
Annular Piezoelectric Transducers ........................ 356
16.5 Comparative Excitation and Field Results Calculated for
X-Beams .................................................. 357
16.6 Conclusions .............................................. 360
Acknowledgments .......................................... 361
References ............................................... 361
17 Localized Beams and Localized Pulses: Generation Using
the Angular Spectrum ..................................... 363
Colin Sheppard
17.1 Bessel Beams ............................................. 363
17.2 The Bessel-Gauss Beam .................................... 365
17.3 Pulsed Bessel Beams ...................................... 367
17.4 Applications in Biomedical Imaging ....................... 375
References ............................................... 376
18 Lossy Light Bullets ...................................... 379
Miguel A. Porras
18.1 Introduction ............................................. 379
18.2 Lossy Light Bullets in Self-Focusing Media with
Nonlinear Losses ......................................... 380
18.3 The Structured Profile of Lossy Light Bullets and
their Energy Reservoir ................................... 381
18.3.1 The Most Lossy Light Bullet in a Nonlinear
Dissipative Medium ................................ 384
18.4 Propagation Properties of Physically Realizable Lossy
Light Bullets ............................................ 384
18.5 Self-Reconstruction Property ........................ 386
18.6 Stability Properties ................................ 387
18.6.1 The Most Lossy Light Bullet as an Attractor
of the Self-Focusing Dynamics with
Nonlinear Losses ............................. 388
18.6.2 Stability Under Small Perturbations .......... 392
18.7 Conclusions ......................................... 395
Acknowledgments .......................................... 396
References ............................................... 396
19 Beyond the Diffraction Limit: Composed Pupils ............ 399
Anedio Ranfagni and Daniela Mugnai
19.1 Introduction ............................................. 399
19.2 Theoretical Description .................................. 401
19.2.1 Analytical Details ................................ 402
19.3 Super Resolving Pupils ................................... 405
19.3.1 Amplitude Measurements: Transversal Dependence .... 405
19.3.2 Amplitude Measurements: Axial Dependence .......... 409
19.3.2.1 The Shadow's Theorem ..................... 411
19.4 Conclusions .............................................. 413
Acknowledgments .......................................... 415
References ............................................... 415
20 Experimental Generation of Frozen Waves in Optics:
Control of Longitudinal and Transverse Shape of Optical
Non-diffracting Waves .................................... 417
Tárcio A. Vieira, Marcos R.R. Gesualdi, and Michel
Zamboni-Rached
20.1 Introduction ............................................. 417
20.2 Frozen Waves: Theoretical Description .................... 417
20.3 Frozen Waves: Experimental Generation .................... 418
20.3.1 Holographic Experimental Setup .................... 420
20.3.2 Results ........................................... 421
20.3.2.1 Example One .............................. 422
20.3.2.2 Example Two .............................. 424
20.3.2.3 Examples Three and Four .................. 425
20.3.2.4 Example Five ............................. 426
20.3.2.5 Example Six .............................. 426
20.3.2.6 Example Seven ............................ 427
20.4 Conclusions .............................................. 430
Acknowledgments .......................................... 430
References ............................................... 430
21 Airy Shaped Waves ........................................ 433
Kleber Zuza Nóbrega, Cesar Augusto Dartora, and Michel
Zamboni-Rached
21.1 Introduction ............................................. 433
21.2 Airy Beams ............................................... 435
21.2.1 Ideal Airy Beam ................................... 436
21.3 Maximum Invariance Depth, Zma% ........................... 438
21.4 Analytical Description of Truncated Airy-Type Beams ...... 441
21.4.1 Theoretical Framework ............................. 442
21.4.2 Examples .......................................... 444
21.5 Airy Pulses Considerations ............................... 447
21.6 Conclusions .............................................. 448
Acknowledgments .......................................... 448
References ............................................... 448
22 Solitons and Ultra-Short Optical Waves: The Short-Pulse
Equation Versus the Nonlinear Schrцdinger Equation ....... 451
Jose Nathan Kutz and Edward Farnum
22.1 Introduction ............................................. 451
22.2 Maxwell's Equations ...................................... 453
22.3 Linear Propagation ....................................... 454
22.3.1 Center-Frequency Asymptotics ...................... 455
22.3.2 Short-Pulse Asymptotics ........................... 457
22.4 Nonlinear Propagation: Instantaneous Nonlinear Response .. 458
22.4.1 Center-Frequency Asymptotics ...................... 459
22.4.2 Short-Pulse Asymptotics ........................... 459
22.4.3 Soliton Solutions ................................. 460
22.5 Nonlinear Propagation: Time-dependent Nonlinear
Response ................................................. 461
22.5.1 Center-Frequency Asymptotics ...................... 462
22.5.2 Short-Pulse Asymptotics ........................... 462
22.6 Application: Mode-Locked Lasers .......................... 463
22.6.1 Haus Master Mode-locking Equation ................. 463
22.6.2 SPE Master Equation ............................... 465
22.7 Conclusions .............................................. 468
References ............................................... 469
Index .................................................... 473
|
