Non-diffracting waves (Weinheim, 2014). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаNon-diffracting waves / ed. by H.E.Hernéndez-Figueroa, E.Recami, M.Zamboni-Rached. - Weinheim: Wiley-VCH, 2014. - xxvi, 481 p.: ill. - Bibliogr. at the end of the chapters. - Ind.: p.473-481. - ISBN 978-3-527-41195-5
 

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Оглавление / Contents
 
     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


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