Self-focusing: Past and Present (eBook)

Robert W. Boyd Prof. Boyd was born in Buffalo, NY. He received the B.S. degree in physics from the Massachusetts Institute of Technology and the Ph.D. degree in physics in 1977 from the University of California at Berkeley. His Ph.D. thesis was supervised by Professor Charles H. Townes and involves the use of nonlinear optical techniques in infrared detection for astronomy. Professor Boyd joined the faculty of the Institute of Optics of the University of Rochester in 1977 and since 1987 has held the position of Professor of Optics. Since July 2001 he has also held the position of the M. Parker Givens Professor of Optics, and since July 2002 has also held the position of Professor of Physics. His research interests include studies of "slow" and "fast" light propagation, quantum imaging techniques, nonlinear optical interactions, studies of the nonlinear optical properties of materials, the development of photonic devices including photonic biosensors, and studies of the quantum statistical properties of nonlinear optical interactions. Professor Boyd has written two books, including widely used text "Nonlinear optics", co-edited two anthologies, published over 230 research papers, and been awarded five patents. He is a fellow of the American Physical Society and the Optical Society of America and is a past chair of the Division of Laser Science of the American Physical Society. Svetlana G. Lukishova Dr. Lukishova was born in Moscow, Russia. She received her M.S. degree in Physics (with high honors) and Ph.D. degree (1977) from the Moscow Institute of Physics and Technology (FizTech). Her M.S. and Ph.D. research was performed at the P.N. Lebedev Physical Institute of the USSR Academy of Sciences. Her Ph.D. thesis was supervised by P.P. Pashinin and Nobel Prize winner A.M. Prokhorov and involved spatial beam-profile and temporal pulse-shape control in laser-fusion systems. After holding research positions at the I.V. Kurchatov Nuclear Power Institute, Troitsk branch TRINITI (Moscow Region), the Institute of Radioengineering and Electronics of the Russian Academy of Sciences (Moscow), and the Liquid Crystal Institute (Kent, Ohio), she joined the Institute of Optics, University of Rochester in 1999 where she holds the position of Senior Scientist. She has received a Long-Term Grant from the International Science (G. Soros) Foundation and Grants from the Russian Government and the Russian Foundation for Basic Research for her work on nonlinear optics. Dr. Lukishova’s research interests include both optical material and optical radiation properties. She has more than 30 years experience with the development of high-power laser systems and the interaction of laser radiation with matter. Currently her main research areas are nonlinear optics and photonic quantum information systems. She has near 170 scientific publications and awarded one US patent and 3 USSR Inventor Certificates. Yuen-Ron Shen Y. R. Shen received his BS degree from the National Taiwan University in 1956 and his Ph.D. from Harvard University in 1963 under the supervision of Nicolaas Bloembergen. After a year of postdoctoral work at Harvard, he was appointed to the Physics faculty of the University of California at Berkeley where he has been ever since. He has also been associated with the Lawrence Berkeley National Laboratory since 1966. Shen’s research interest is in the broad area of interaction of light with matter. He was involved in the early development of nonlinear optics, searching for basic understanding of various nonlinear optical phenomena. He is the author of the widely used text "The Principles of Nonlinear Optics". He contributed to the early accurate determination of band structures of semiconductors by developing a high-resolution wavelength-modulation spectroscopic technique. He initiated the field of nonlinear optics in liquid crystals and applications of nonlinear optics to characterization of liquid crystals. He pioneered the development of optical second harmonic generation and sum-frequency generation as powerful spectroscopic tools for surface and interface studies and their applications to many neglected, but important, areas of surface science. More recently, he has devoted himself to the development of sum-frequency generation as a novel sensitive probe for molecular chirality. Shen has received numerous prestigious awards including the 1986 Charles H. Townes Award of the OSA, the 1992 Arthur L. Schawlow Prize and the 1998 Frank Isakson Prize of the APS, and the 1996 Max-Planck Research Award. He is a member of the American Academy of Arts and Sciences, the National Academy of Sciences, and the Academia Sinica. He is also a foreign member of the Chinese Academy of Sciences.

Preface 6
Terminology in the Classical Case 7
Importance of Self-Focusing 8
Intent and Outline of This Book 9
Part I: Self-Focusing in the Past 10
Part II: Self-Focusing in the Present 12
Conclusion 15
References 16
Contents 20
Contributors 23
Part I: Self-focusing in the Past 28
Self-Focusing and Filaments of Light: Past and Present 29
1.1 Introduction 29
1.2 Early History of Self-Focusing and Filaments of Light 30
1.3 Quasi-Steady-State Self-Focusing and Moving Focus 32
1.4 Effects of Transient Response and Dynamic Self-Focusing 35
1.5 Self-Focusing of Femtosecond Laser Pulses 40
1.6 Conclusion 42
References 43
Notes on Early Self-Focusing Papers 46
References 49
Self-focusing: Theory 50
Optical Self-Focusing: Stationary Beams and Femtosecond Pulses 127
3.1 Introduction 127
3.2 Nonlinear Parabolic Equation 128
3.3 Homogeneous Wave Beams 131
3.4 Filamentation and Modulation Instabilities of Supercritical Beams 132
3.5 Self-Similar Solutions of the Self-Focusing Equation: Lens Transform 135
3.6 Averaged Description of Self-Focusing Beams (Method of Moments). Sufficient Condition of Self-Focusing. Self-Focusing of Supercritical Beams 137
3.7 Self-Focusing of Wave Beams in Periodic Systems 141
3.8 Field Structure in the Vicinity of the Nonlinear Focus Self-Similar Collapse
3.9 Nonstationary Self-Focusing Distributed Collapse
3.10 Spectral Broadening 145
3.11 Self-Action of Femtosecond Pulses 146
3.12 Conclusions 148
References 148
Self-Focusing of Optical Beams 152
4.1 Introduction 152
4.2 Early History 153
4.3 Nonlinear Polarization and the Nonlinear Refractive Index 153
4.4 The Nonlinear Schrödinger Equation 155
4.5 Four-Wave Mixing, Weak-Wave Retardation, Instability 155
4.6 Spatial Self-Phase Modulation and Estimating the Beam Self-Focusing Distance 157
4.7 Self-Focusing Intensity Singularity and Beam Collapse 159
4.8 Limitations on Blow-Up and Collapse 161
4.9 Beam Breakup and Multiple Filament Formation 162
4.10 Self-Focusing of Pulses: Light Bullets 163
4.11 Self-Trapping, Spatial Solitons 164
References 164
Multi-Focus Structure and Moving Nonlinear Foci: Adequate Models of Self-Focusing of Laser Beams in Nonlinear Media 167
5.1 Introduction 167
5.2 Review of the Theory of MFS-MNLF Models 168
5.3 Self-Focusing of Super-Gaussian Laser Beams 172
5.4 Experimental Verification of MFS-MNLF Models 172
5.5 Comments on Self-focusing of Femtosecond Laser Pulses in Air and Possible Future Directions of This Field 174
References 176
Small-Scale Self-focusing 178
6.1 Introduction 178
6.2 Modulational Instability 179
6.2.1 Linearized Theory 181
6.2.2 Experimental Confirmation of the Modulation Instability 183
6.3 Beam Shape, Polarization, and Pulse Duration Effects 187
6.4 Filamentation in Lasers 189
6.5 Applications/Impact 190
References 191
Wave Collapse in Nonlinear Optics 195
7.1 Introduction 195
7.2 Solitons Versus Collapses 197
7.3 Collapse 201
7.3.1 Virial Theorem 201
7.3.2 Strong Collapse 202
7.3.3 Weak Collapse 202
7.3.4 Black Hole Regime 205
7.4 Role of Dispersion in Collapse 205
7.5 Conclusion 208
References 208
Beam Shaping and Suppression of Self-focusing in High-Peak-Power Nd:Glass Laser Systems 211
8.1 Introduction 211
8.2 Fresnel Diffraction on Apertures in Linear Media 214
8.2.1 Fresnel Diffraction on Hard-Edge Apertures 214
8.2.2 Fresnel Diffraction by Soft Apertures and Propagation of Super-Gaussian Beams 217
8.2.3 Preparation of Super-Gaussian Beams 219
8.3 Whole-Beam Self-focusing 220
8.4 Small-Scale Self-focusing Effects in High-Peak-Power Nd:Glass Laser Systems 225
8.5 Methods of Suppression of Self-focusing 229
8.5.1 Application of Spatial Filters and Relay Imaging Optics 230
8.5.2 Elimination of Fresnel Diffraction Effects and Small-Scale Self-focusing Using Apodizing Devices 232
8.5.3 Application of Divergent Beams 235
8.5.4 Partitioning of the Active Medium 236
8.5.5 Using Circular Polarization 236
8.5.6 Coherence Limiting of Laser Radiation 237
8.6 Summary 239
References 240
Self-focusing, Conical Emission, and Other Self-action Effects in Atomic Vapors 250
9.1 Introduction 250
9.2 Self-focusing and Self-trapping in Atomic Vapors 252
9.3 Conical Emission 253
9.3.1 Conical Emission with a Single Pump Beam and One-Photon Resonant Transition 254
9.3.1.1 Multiple Filamentation in Conical Emission 261
9.3.2 Conical Emission with a Single Pump Beam and Multi-Photon Resonant Transitions 263
9.3.3 Conical Emission Generation Using Two (or More) Pump Beams 264
9.4 Self-focusing and Pattern Formation 264
9.5 Concluding Remarks 267
References 267
Periodic Filamentation and Supercontinuum Interference 271
10.1 Introduction 271
10.2 Self-phase Modulation and Conical Emission 272
10.3 Periodic Filamentation and Supercontinuum Generation from Diffraction 275
10.4 Conclusion 279
References 280
Reprints of Papers from the Past 282
Effects of the Gradient of a StrongElectromagnetic Beam on Electrons and Atoms 283
On Self-focusing of Electromagnetic Wavesin Nonlinear Media 287
References 291
Laser-induced Damage in Transparent Media 292
Part II: Self-focusing in the Present 307
Self-focusing and Filamentation of Femtosecond Pulses in Air and Condensed Matter: Simulations and Experiments 308
12.1 Introduction 308
12.2 Models 311
12.2.1 Self-trapping 311
12.2.2 Moving Focus 312
12.2.3 Saturation of Self-focusing, Self-channeling 313
12.2.4 X-Waves 314
12.2.5 Numerical Simulations 315
12.2.6 Typical Simulation of Filamentation 317
12.3 Self-phase Modulation and Pulse Mode Cleaning 319
12.4 Single Cycle Pulse Generation by Filamentation 320
12.4.1 Single-Cycle Pulse Generation in Low Pressure Gas Cells 321
12.4.2 Single Cycle Pulse Generation in a Pressure Gradient 321
12.5 Amplification of Filaments 325
12.6 Organization of Multiple Filamentation 326
12.7 Conclusion 330
References 330
Self-organized Propagation of Femtosecond Laser Filamentation in Air 334
13.1 Introduction 334
13.2 Mechanism of Filamentation 335
13.3 Diagnostics of Filamentation 336
13.3.1 Imaging of Beam Cross-Section 337
13.3.2 Resistivity Measurement 337
13.3.3 Acoustic Diagnostics 338
13.3.4 Fluorescence Detection 339
13.3.5 Interferometry 340
13.4 Nonlinear Interactions in the Filamentation 340
13.4.1 Spatial Evolution of Filamentation 340
13.4.2 Third-Harmonic Generation (THG) 341
13.5 Experimental Studies for Potential Applications 342
13.5.1 Lifetime Prolongation of the Filaments 342
13.5.2 Optimization of Multiple Filaments (MF) 344
13.5.3 Laser-Guided Discharge 345
13.5.4 Laser Propulsion 348
13.6 Long-Distance Filamentation 349
13.6.1 Chirp-Dependent Propagation of Filamentation 349
13.6.2 Divergence Angle-Dependent Propagation of Filamentation 351
13.6.3 Energy Reservoir 352
13.7 Comparison of Filamentation in Focused and Unfocused Laser Beams 354
13.8 Conclusions 355
References 356
The Physics of Intense Femtosecond Laser Filamentation 359
14.1 Introduction 359
14.2 Slice-by-Slice Self-focusing of Laser Pulse 360
14.2.1 Basic Idea 360
14.2.2 Experimental Proofs 361
14.2.3 Pulse Duration Dependence of the Critical Power 362
14.3 Intensity Clamping 363
14.3.1 The Physics of Plasma Generation 363
14.3.2 Plasma Defocusing Effect Cancels Self-Focusing 364
14.4 White Light Laser (or Supercontinuum Generation) and Conical Emission 366
14.4.1 White Light Laser 366
14.4.2 Conical Emission 368
14.4.3 Band Gap Dependence of Supercontinuum in Condensed Matters 369
14.5 Background Energy Reservoir 370
14.5.1 Proof of the Existence of Energy Reservoir 370
14.5.2 Multiple Refocusing 371
14.6 Multiple Filamentation Competition 373
14.7 Applications 376
14.8 Summary 376
References 377
Self-focusing and Filamentation of Powerful Femtosecond Laser Pulses 381
15.1 Introduction 381
15.2 Femtosecond Filamentation of a Laser Pulse and the Moving-Focus Model 383
15.2.1 The Dynamic Moving-Focus Model 383
15.2.2 Refocusing and the Multiple Foci Model 386
15.2.3 Filamentation and Transverse Energy Flows 387
15.3 Quasi-Stationary Model of Filament Origination 388
15.3.1 Initial Stage of Filamentation 388
15.3.2 Quasi-Stationary Estimation of Critical Power 389
15.3.3 Generalized Marburger Formula 390
15.4 Modulational Instability and Multifilamentation 391
15.4.1 Origin of Filaments from Initial Intensity Perturbations 391
15.4.2 ‘‘Energy’’ Competition between Initial Perturbations 393
15.4.3 Spatial Regularization of Filaments 394
15.5 Femtosecond Pulse Filamentation in the Atmosphere 396
15.5.1 Filament Wandering 396
15.5.2 Bunch of Chaotic Filaments in the Turbulent Atmosphere 397
15.5.3 Scattering in Aerosols and Formation of Filaments 397
15.5.4 Regularization of Filaments in the Atmosphere by a Lens Array 399
15.6 Spatio-Temporal Picture of Femtosecond Filamentation 400
15.6.1 Dynamic Model 400
15.6.2 Chirped Pulse 401
15.6.3 Dynamic Multiple Filament Competition 402
15.7 Conclusions 403
References 404
Spatial and Temporal Dynamics of Collapsing Ultrashort Laser Pulses 409
16.1 Introduction 409
16.2 Spatial Collapse Dynamics 410
16.2.1 Self-similar Collapse 410
16.2.2 Modulational Instability versus Townes Collapse 412
16.2.3 Self-focusing with Non-Gaussian Beams 412
16.3 Spatio-Temporal Collapse Dynamics 413
16.3.1 Self-Focusing in the Normal-GVD Regime 414
16.3.2 Optical ‘‘Shock’’ Formation and Supercontinuum Generation 415
16.3.3 Filamentation and Light Strings 416
16.3.4 Self-focusing in the Anomalous-Dispersion Regime 417
16.4 Conclusions 418
References 418
Some Modern Aspects of Self-focusing Theory 422
17.1 Introduction 422
17.2 Some Pre-1975 Results 423
17.2.1 Effect of a Lens 424
17.2.2 The R (Townes) Profile 425
17.3 Critical Power 425
17.3.1 Hollow Waveguides (Bounded Domains) 426
17.4 The Universal Blowup Profile yR 427
17.4.1 New Blowup Profiles 427
17.5 Super-Gaussian Input Beams 428
17.6 Partial Beam Collapse 430
17.6.1 Common Misinterpretations of the Variance Identity 431
17.7 Blowup Rate 431
17.7.1 The Loglog Law 431
17.7.2 A Square-Root Law 433
17.8 Self-focusing Distance 433
17.8.1 Effect of a Lens 434
17.9 Multiple Filamentation 434
17.9.1 Noise-Induced Multiple Filamentation 434
17.9.2 Deterministic Multiple Filamentation 437
17.10 Perturbation Theory: Effect of Small Mechanisms Neglected in the NLS Model 438
17.10.1 Unreliability of Aberrationless Approximation and Variational Methods 438
17.10.2 Modulation Theory 440
17.11 Effect of Normal Group Velocity Dispersion 441
17.12 Nonparaxiality and Backscattering 442
17.13 Final Remarks 444
References 445
X-Waves in Self-Focusing of Ultra-Short Pulses 448
18.1 Introduction 448
18.2 Experimental Results 450
18.2.1 Historical Preamble: The Key Role of Angular Dispersion 450
18.2.2 Spontaneous X Waves in Second-Harmonic Generation 451
18.2.3 X Waves in Kerr-Like Media 454
18.3 Theory 456
18.3.1 Linear X Waves 456
18.3.2 Nonlinear Models 458
18.3.3 Instability and Generation 460
18.4 Perspectives and Other Systems 461
18.5 Conclusions 462
References 462
On the Role of Conical Waves in Self-focusing and Filamentation of Femtosecond Pulses with Nonlinear Losses 466
19.1 Introduction 466
19.2 Light Filaments Supported by a Conical Wave 469
19.2.1 Model Equations 469
19.2.2 Single-Filament Formation 471
19.2.3 Filament Reconstruction 473
19.2.4 X-Waves Generated from Femtosecond Filaments 477
19.3 Multiple Filaments Supported by a Conical Wave 479
19.3.1 Multiple Filaments from Femtosecond Pulses in Water 480
19.3.2 Multiple Filaments from Femtosecond Pulses in Fused Silica 482
19.4 Conclusions 485
References 486
Self-focusing and Self-defocusing of Femtosecond Pulses with Cascaded Quadratic Nonlinearities 489
20.1 Nonlinear Phase Shifts in Quadratic Media 489
20.1.1 Self-focusing Versus Self-defocusing 491
20.1.2 Analytical Framework 491
20.2 Ultrashort Pulse Shaping 493
20.3 Extension of the Cascaded Quadratic Nonlinearity to Femtosecond Pulses 494
20.4 Saturable Self-focusing: Space-Time Solitons 497
20.5 Self-defocusing Nonlinearities: Applications to Ultrashort Pulse Generation 501
20.5.1 Compensation for Self-focusing 502
20.5.2 Pulse Compression with Self-defocusing Nonlinear Phase Shifts 504
20.6 Few-Cycle Pulses and the Non-stationary Regime of the Cascaded Quadratic Nonlinearity 506
20.6.1 Controllable Raman-Like Nonlinearity 506
20.6.2 Beyond the Slowly Varying Envelope Approximation 510
20.7 Conclusion 512
References 512
Effective Parameters of High-Power Laser Femtosecond Radiation at Self-focusing in Gas and Aerosol Media 515
21.1 The Evolution of Effective Parameters of High-Power Femtosecond Laser Radiation in the Air 515
21.1.1 Integral Characteristics of a Light Pulse 516
21.1.2 Beam Effective Parameters Evolution 517
21.2 Filamentation of Ultrashort Laser Pulse in the Presence of an Aerosol Layer 519
21.2.1 Experiment 520
21.2.2 Theoretical Simulations 521
21.3 Conclusions 523
References 524
Diffraction-Induced High-Order Modes of the (2 + 1)D Nonparaxial Nonlinear Schrödinger Equation 525
22.1 Introduction 525
22.2 Spatial Modulational Instability in Nonlinear Media: Analytical Approach 526
22.3 Nonparaxiality and Filamentation of Intense Beams 530
22.4 Higher-Order Modes of the Nonparaxial Nonlinear Schrödinger Equation 533
22.5 Ring and Filament Formation of Beams in Self-Focusing Media: Numerical Study 535
22.5.1 Modulational Instability of Optical Beams 535
22.5.2 Modes of the Nonparaxial NLSE 542
22.5.3 Small-Scale Self-Focusing 545
22.6 Conclusions 551
References 552
Self-Focusing and Solitons in Photorefractive Media 554
23.1 Introduction 554
23.2 Self-Trapping in Photorefractives 555
23.3 Nonlinear Mechanism 557
23.3.1 Photorefraction 557
23.3.2 Light-Induced Space-Charge Field 558
23.3.3 Nonlinear Index Change 559
23.3.4 The Soliton-Supporting Nonlinear Equation 560
23.3.5 Soliton Waveforms and Existence Curve 561
23.3.6 Experiments and Theory 562
23.4 Two-Dimensional Solitons 563
23.5 Temporal Effects and Quasi-Steady-State Dynamics 566
23.5.1 The Transition from a Diffracting Wave to a Soliton 567
23.5.2 External Modulation of Soliton Parameters 567
23.5.3 Quasi-Steady-State Solitons 568
23.5.4 Response Change in Beams That Approximately Do Not Evolve in Time 568
23.6 Non-screening Self-Trapping Mechanisms 569
23.7 Materials 569
23.8 Soliton Interaction-Collisions 569
23.9 Vector and Composite Solitons 570
23.10 Incoherent (Random-Phase) Solitons 571
23.11 Applications 572
23.12 Concluding Remarks 572
References 573
Measuring Nonlinear Refraction and Its Dispersion 580
24.1 Introduction 580
24.2 Beam Propagation 581
24.3 Z-Scan 584
24.4 Measuring Nonlinear Dispersion 588
24.5 Physical Mechanisms Leading to Nonlinear Refraction 590
24.6 Conclusion 595
References 596
Index 599