Boyd / Lukishova / Shen Self-focusing: Past and Present

Fundamentals and Prospects
1. Auflage 2008
ISBN: 978-0-387-34727-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 114, 605 Seiten

Reihe: Topics in Applied Physics

Self-focusing has been an area of active scientific investigation for nearly 50 years. This book presents a comprehensive treatment of this topic and reviews both theoretical and experimental investigations of self-focusing. This book should be of interest to scientists and engineers working with lasers and their applications. From a practical point of view, self-focusing effects impose a limit on the power that can be transmitted through a material medium. Self-focusing also can reduce the threshold for the occurrence of other nonlinear optical processes. Self-focusing often leads to damage in optical materials and is a limiting factor in the design of high-power laser systems. But it can be harnessed for the design of useful devices such as optical power limiters and switches. At a formal level, the equations for self-focusing are equivalent to those describing Bose-Einstein condensates and certain aspects of plasma physics and hydrodynamics. There is thus a unifying theme between nonlinear optics and these other disciplines. One of the goals of this book is to connect the extensive early literature on self-focusing, filament-ation, self-trapping, and collapse with more recent studies aimed at issues such as self-focusing of fs pulses, white light generation, and the generation of filaments in air with lengths of more than 10 km. It also describes some modern advances in self-focusing theory including the influence of beam nonparaxiality on self-focusing collapse. This book consists of 24 chapters. Among them are three reprinted key landmark articles published earlier. It also contains the first publication of the 1964 paper that describes the first laboratory observation of self-focusing phenomena with photographic evidence.

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.

Autoren/Hrsg.

Boyd, Robert W.

Lukishova, Svetlana G.

Shen, Y. R.

Inhaltsverzeichnis

1;Preface;6
1.1;Terminology in the Classical Case;7
1.2;Importance of Self-Focusing;8
1.3;Intent and Outline of This Book;9
1.4;Part I: Self-Focusing in the Past;10
1.5;Part II: Self-Focusing in the Present;12
1.6;Conclusion;15
1.7;References;16
2;Contents;20
3;Contributors;23
4;Part I: Self-focusing in the Past;28
4.1;Self-Focusing and Filaments of Light: Past and Present;29
4.1.1;1.1 Introduction;29
4.1.2;1.2 Early History of Self-Focusing and Filaments of Light;30
4.1.3;1.3 Quasi-Steady-State Self-Focusing and Moving Focus;32
4.1.4;1.4 Effects of Transient Response and Dynamic Self-Focusing;35
4.1.5;1.5 Self-Focusing of Femtosecond Laser Pulses;40
4.1.6;1.6 Conclusion;42
4.1.7;References;43
4.2;Notes on Early Self-Focusing Papers;46
4.2.1;References;49
4.2.2;Self-focusing: Theory;50
4.3;Optical Self-Focusing: Stationary Beams and Femtosecond Pulses;127
4.3.1;3.1 Introduction;127
4.3.2;3.2 Nonlinear Parabolic Equation;128
4.3.3;3.3 Homogeneous Wave Beams;131
4.3.4;3.4 Filamentation and Modulation Instabilities of Supercritical Beams;132
4.3.5;3.5 Self-Similar Solutions of the Self-Focusing Equation: Lens Transform;135
4.3.6;3.6 Averaged Description of Self-Focusing Beams (Method of Moments). Sufficient Condition of Self-Focusing. Self-Focusing of Supercritical Beams;137
4.3.7;3.7 Self-Focusing of Wave Beams in Periodic Systems;141
4.3.8;3.8 Field Structure in the Vicinity of the Nonlinear Focus; Self-Similar Collapse;141
4.3.9;3.9 Nonstationary Self-Focusing; Distributed Collapse;143
4.3.10;3.10 Spectral Broadening;145
4.3.11;3.11 Self-Action of Femtosecond Pulses;146
4.3.12;3.12 Conclusions;148
4.3.13;References;148
4.4;Self-Focusing of Optical Beams;152
4.4.1;4.1 Introduction;152
4.4.2;4.2 Early History;153
4.4.3;4.3 Nonlinear Polarization and the Nonlinear Refractive Index;153
4.4.4;4.4 The Nonlinear Schrödinger Equation;155
4.4.5;4.5 Four-Wave Mixing, Weak-Wave Retardation, Instability;155
4.4.6;4.6 Spatial Self-Phase Modulation and Estimating the Beam Self-Focusing Distance;157
4.4.7;4.7 Self-Focusing Intensity Singularity and Beam Collapse;159
4.4.8;4.8 Limitations on Blow-Up and Collapse;161
4.4.9;4.9 Beam Breakup and Multiple Filament Formation;162
4.4.10;4.10 Self-Focusing of Pulses: Light Bullets;163
4.4.11;4.11 Self-Trapping, Spatial Solitons;164
4.4.12;References;164
4.5;Multi-Focus Structure and Moving Nonlinear Foci: Adequate Models of Self-Focusing of Laser Beams in Nonlinear Media;167
4.5.1;5.1 Introduction;167
4.5.2;5.2 Review of the Theory of MFS-MNLF Models;168
4.5.3;5.3 Self-Focusing of Super-Gaussian Laser Beams;172
4.5.4;5.4 Experimental Verification of MFS-MNLF Models;172
4.5.5;5.5 Comments on Self-focusing of Femtosecond Laser Pulses in Air and Possible Future Directions of This Field;174
4.5.6;References;176
4.6;Small-Scale Self-focusing;178
4.6.1;6.1 Introduction;178
4.6.2;6.2 Modulational Instability;179
4.6.2.1;6.2.1 Linearized Theory;181
4.6.2.2;6.2.2 Experimental Confirmation of the Modulation Instability;183
4.6.3;6.3 Beam Shape, Polarization, and Pulse Duration Effects;187
4.6.4;6.4 Filamentation in Lasers;189
4.6.5;6.5 Applications/Impact;190
4.6.6;References;191
4.7;Wave Collapse in Nonlinear Optics;195
4.7.1;7.1 Introduction;195
4.7.2;7.2 Solitons Versus Collapses;197
4.7.3;7.3 Collapse;201
4.7.3.1;7.3.1 Virial Theorem;201
4.7.3.2;7.3.2 Strong Collapse;202
4.7.3.3;7.3.3 Weak Collapse;202
4.7.3.4;7.3.4 Black Hole Regime;205
4.7.4;7.4 Role of Dispersion in Collapse;205
4.7.5;7.5 Conclusion;208
4.7.6;References;208
4.8;Beam Shaping and Suppression of Self-focusing in High-Peak-Power Nd:Glass Laser Systems;211
4.8.1;8.1 Introduction;211
4.8.2;8.2 Fresnel Diffraction on Apertures in Linear Media;214
4.8.2.1;8.2.1 Fresnel Diffraction on Hard-Edge Apertures;214
4.8.2.2;8.2.2 Fresnel Diffraction by Soft Apertures and Propagation of Super-Gaussian Beams;217
4.8.2.3;8.2.3 Preparation of Super-Gaussian Beams;219
4.8.3;8.3 Whole-Beam Self-focusing;220
4.8.4;8.4 Small-Scale Self-focusing Effects in High-Peak-Power Nd:Glass Laser Systems;225
4.8.5;8.5 Methods of Suppression of Self-focusing;229
4.8.5.1;8.5.1 Application of Spatial Filters and Relay Imaging Optics;230
4.8.5.2;8.5.2 Elimination of Fresnel Diffraction Effects and Small-Scale Self-focusing Using Apodizing Devices;232
4.8.5.3;8.5.3 Application of Divergent Beams;235
4.8.5.4;8.5.4 Partitioning of the Active Medium;236
4.8.5.5;8.5.5 Using Circular Polarization;236
4.8.5.6;8.5.6 Coherence Limiting of Laser Radiation;237
4.8.6;8.6 Summary;239
4.8.7;References;240
4.9;Self-focusing, Conical Emission, and Other Self-action Effects in Atomic Vapors;250
4.9.1;9.1 Introduction;250
4.9.2;9.2 Self-focusing and Self-trapping in Atomic Vapors;252
4.9.3;9.3 Conical Emission;253
4.9.3.1;9.3.1 Conical Emission with a Single Pump Beam and One-Photon Resonant Transition;254
4.9.3.1.1;9.3.1.1 Multiple Filamentation in Conical Emission;261
4.9.3.2;9.3.2 Conical Emission with a Single Pump Beam and Multi-Photon Resonant Transitions;263
4.9.3.3;9.3.3 Conical Emission Generation Using Two (or More) Pump Beams;264
4.9.4;9.4 Self-focusing and Pattern Formation;264
4.9.5;9.5 Concluding Remarks;267
4.9.6;References;267
4.10;Periodic Filamentation and Supercontinuum Interference;271
4.10.1;10.1 Introduction;271
4.10.2;10.2 Self-phase Modulation and Conical Emission;272
4.10.3;10.3 Periodic Filamentation and Supercontinuum Generation from Diffraction;275
4.10.4;10.4 Conclusion;279
4.10.5;References;280
4.11;Reprints of Papers from the Past;282
4.11.1;Effects of the Gradient of a StrongElectromagnetic Beam on Electrons and Atoms;283
4.11.2;On Self-focusing of Electromagnetic Wavesin Nonlinear Media;287
4.11.2.1;References;291
4.11.3;Laser-induced Damage in Transparent Media;292
5;Part II: Self-focusing in the Present;307
5.1;Self-focusing and Filamentation of Femtosecond Pulses in Air and Condensed Matter: Simulations and Experiments;308
5.1.1;12.1 Introduction;308
5.1.2;12.2 Models;311
5.1.2.1;12.2.1 Self-trapping;311
5.1.2.2;12.2.2 Moving Focus;312
5.1.2.3;12.2.3 Saturation of Self-focusing, Self-channeling;313
5.1.2.4;12.2.4 X-Waves;314
5.1.2.5;12.2.5 Numerical Simulations;315
5.1.2.6;12.2.6 Typical Simulation of Filamentation;317
5.1.3;12.3 Self-phase Modulation and Pulse Mode Cleaning;319
5.1.4;12.4 Single Cycle Pulse Generation by Filamentation;320
5.1.4.1;12.4.1 Single-Cycle Pulse Generation in Low Pressure Gas Cells;321
5.1.4.2;12.4.2 Single Cycle Pulse Generation in a Pressure Gradient;321
5.1.5;12.5 Amplification of Filaments;325
5.1.6;12.6 Organization of Multiple Filamentation;326
5.1.7;12.7 Conclusion;330
5.1.8;References;330
5.2;Self-organized Propagation of Femtosecond Laser Filamentation in Air;334
5.2.1;13.1 Introduction;334
5.2.2;13.2 Mechanism of Filamentation;335
5.2.3;13.3 Diagnostics of Filamentation;336
5.2.3.1;13.3.1 Imaging of Beam Cross-Section;337
5.2.3.2;13.3.2 Resistivity Measurement;337
5.2.3.3;13.3.3 Acoustic Diagnostics;338
5.2.3.4;13.3.4 Fluorescence Detection;339
5.2.3.5;13.3.5 Interferometry;340
5.2.4;13.4 Nonlinear Interactions in the Filamentation;340
5.2.4.1;13.4.1 Spatial Evolution of Filamentation;340
5.2.4.2;13.4.2 Third-Harmonic Generation (THG);341
5.2.5;13.5 Experimental Studies for Potential Applications;342
5.2.5.1;13.5.1 Lifetime Prolongation of the Filaments;342
5.2.5.2;13.5.2 Optimization of Multiple Filaments (MF);344
5.2.5.3;13.5.3 Laser-Guided Discharge;345
5.2.5.4;13.5.4 Laser Propulsion;348
5.2.6;13.6 Long-Distance Filamentation;349
5.2.6.1;13.6.1 Chirp-Dependent Propagation of Filamentation;349
5.2.6.2;13.6.2 Divergence Angle-Dependent Propagation of Filamentation;351
5.2.6.3;13.6.3 Energy Reservoir;352
5.2.7;13.7 Comparison of Filamentation in Focused and Unfocused Laser Beams;354
5.2.8;13.8 Conclusions;355
5.2.9;References;356
5.3;The Physics of Intense Femtosecond Laser Filamentation;359
5.3.1;14.1 Introduction;359
5.3.2;14.2 Slice-by-Slice Self-focusing of Laser Pulse;360
5.3.2.1;14.2.1 Basic Idea;360
5.3.2.2;14.2.2 Experimental Proofs;361
5.3.2.3;14.2.3 Pulse Duration Dependence of the Critical Power;362
5.3.3;14.3 Intensity Clamping;363
5.3.3.1;14.3.1 The Physics of Plasma Generation;363
5.3.3.2;14.3.2 Plasma Defocusing Effect Cancels Self-Focusing;364
5.3.4;14.4 White Light Laser (or Supercontinuum Generation) and Conical Emission;366
5.3.4.1;14.4.1 White Light Laser;366
5.3.4.2;14.4.2 Conical Emission;368
5.3.4.3;14.4.3 Band Gap Dependence of Supercontinuum in Condensed Matters;369
5.3.5;14.5 Background Energy Reservoir;370
5.3.5.1;14.5.1 Proof of the Existence of Energy Reservoir;370
5.3.5.2;14.5.2 Multiple Refocusing;371
5.3.6;14.6 Multiple Filamentation Competition;373
5.3.7;14.7 Applications;376
5.3.8;14.8 Summary;376
5.3.9;References;377
5.4;Self-focusing and Filamentation of Powerful Femtosecond Laser Pulses;381
5.4.1;15.1 Introduction;381
5.4.2;15.2 Femtosecond Filamentation of a Laser Pulse and the Moving-Focus Model;383
5.4.2.1;15.2.1 The Dynamic Moving-Focus Model;383
5.4.2.2;15.2.2 Refocusing and the Multiple Foci Model;386
5.4.2.3;15.2.3 Filamentation and Transverse Energy Flows;387
5.4.3;15.3 Quasi-Stationary Model of Filament Origination;388
5.4.3.1;15.3.1 Initial Stage of Filamentation;388
5.4.3.2;15.3.2 Quasi-Stationary Estimation of Critical Power;389
5.4.3.3;15.3.3 Generalized Marburger Formula;390
5.4.4;15.4 Modulational Instability and Multifilamentation;391
5.4.4.1;15.4.1 Origin of Filaments from Initial Intensity Perturbations;391
5.4.4.2;15.4.2 ‘‘Energy’’ Competition between Initial Perturbations;393
5.4.4.3;15.4.3 Spatial Regularization of Filaments;394
5.4.5;15.5 Femtosecond Pulse Filamentation in the Atmosphere;396
5.4.5.1;15.5.1 Filament Wandering;396
5.4.5.2;15.5.2 Bunch of Chaotic Filaments in the Turbulent Atmosphere;397
5.4.5.3;15.5.3 Scattering in Aerosols and Formation of Filaments;397
5.4.5.4;15.5.4 Regularization of Filaments in the Atmosphere by a Lens Array;399
5.4.6;15.6 Spatio-Temporal Picture of Femtosecond Filamentation;400
5.4.6.1;15.6.1 Dynamic Model;400
5.4.6.2;15.6.2 Chirped Pulse;401
5.4.6.3;15.6.3 Dynamic Multiple Filament Competition;402
5.4.7;15.7 Conclusions;403
5.4.8;References;404
5.5;Spatial and Temporal Dynamics of Collapsing Ultrashort Laser Pulses;409
5.5.1;16.1 Introduction;409
5.5.2;16.2 Spatial Collapse Dynamics;410
5.5.2.1;16.2.1 Self-similar Collapse;410
5.5.2.2;16.2.2 Modulational Instability versus Townes Collapse;412
5.5.2.3;16.2.3 Self-focusing with Non-Gaussian Beams;412
5.5.3;16.3 Spatio-Temporal Collapse Dynamics;413
5.5.3.1;16.3.1 Self-Focusing in the Normal-GVD Regime;414
5.5.3.2;16.3.2 Optical ‘‘Shock’’ Formation and Supercontinuum Generation;415
5.5.3.3;16.3.3 Filamentation and Light Strings;416
5.5.3.4;16.3.4 Self-focusing in the Anomalous-Dispersion Regime;417
5.5.4;16.4 Conclusions;418
5.5.5;References;418
5.6;Some Modern Aspects of Self-focusing Theory;422
5.6.1;17.1 Introduction;422
5.6.2;17.2 Some Pre-1975 Results;423
5.6.2.1;17.2.1 Effect of a Lens;424
5.6.2.2;17.2.2 The R (Townes) Profile;425
5.6.3;17.3 Critical Power;425
5.6.3.1;17.3.1 Hollow Waveguides (Bounded Domains);426
5.6.4;17.4 The Universal Blowup Profile yR;427
5.6.4.1;17.4.1 New Blowup Profiles;427
5.6.5;17.5 Super-Gaussian Input Beams;428
5.6.6;17.6 Partial Beam Collapse;430
5.6.6.1;17.6.1 Common Misinterpretations of the Variance Identity;431
5.6.7;17.7 Blowup Rate;431
5.6.7.1;17.7.1 The Loglog Law;431
5.6.7.2;17.7.2 A Square-Root Law;433
5.6.8;17.8 Self-focusing Distance;433
5.6.8.1;17.8.1 Effect of a Lens;434
5.6.9;17.9 Multiple Filamentation;434
5.6.9.1;17.9.1 Noise-Induced Multiple Filamentation;434
5.6.9.2;17.9.2 Deterministic Multiple Filamentation;437
5.6.10;17.10 Perturbation Theory: Effect of Small Mechanisms Neglected in the NLS Model;438
5.6.10.1;17.10.1 Unreliability of Aberrationless Approximation and Variational Methods;438
5.6.10.2;17.10.2 Modulation Theory;440
5.6.11;17.11 Effect of Normal Group Velocity Dispersion;441
5.6.12;17.12 Nonparaxiality and Backscattering;442
5.6.13;17.13 Final Remarks;444
5.6.14;References;445
5.7;X-Waves in Self-Focusing of Ultra-Short Pulses;448
5.7.1;18.1 Introduction;448
5.7.2;18.2 Experimental Results;450
5.7.2.1;18.2.1 Historical Preamble: The Key Role of Angular Dispersion;450
5.7.2.2;18.2.2 Spontaneous X Waves in Second-Harmonic Generation;451
5.7.2.3;18.2.3 X Waves in Kerr-Like Media;454
5.7.3;18.3 Theory;456
5.7.3.1;18.3.1 Linear X Waves;456
5.7.3.2;18.3.2 Nonlinear Models;458
5.7.3.3;18.3.3 Instability and Generation;460
5.7.4;18.4 Perspectives and Other Systems;461
5.7.5;18.5 Conclusions;462
5.7.6;References;462
5.8;On the Role of Conical Waves in Self-focusing and Filamentation of Femtosecond Pulses with Nonlinear Losses;466
5.8.1;19.1 Introduction;466
5.8.2;19.2 Light Filaments Supported by a Conical Wave;469
5.8.2.1;19.2.1 Model Equations;469
5.8.2.2;19.2.2 Single-Filament Formation;471
5.8.2.3;19.2.3 Filament Reconstruction;473
5.8.2.4;19.2.4 X-Waves Generated from Femtosecond Filaments;477
5.8.3;19.3 Multiple Filaments Supported by a Conical Wave;479
5.8.3.1;19.3.1 Multiple Filaments from Femtosecond Pulses in Water;480
5.8.3.2;19.3.2 Multiple Filaments from Femtosecond Pulses in Fused Silica;482
5.8.4;19.4 Conclusions;485
5.8.5;References;486
5.9;Self-focusing and Self-defocusing of Femtosecond Pulses with Cascaded Quadratic Nonlinearities;489
5.9.1;20.1 Nonlinear Phase Shifts in Quadratic Media;489
5.9.1.1;20.1.1 Self-focusing Versus Self-defocusing;491
5.9.1.2;20.1.2 Analytical Framework;491
5.9.2;20.2 Ultrashort Pulse Shaping;493
5.9.3;20.3 Extension of the Cascaded Quadratic Nonlinearity to Femtosecond Pulses;494
5.9.4;20.4 Saturable Self-focusing: Space-Time Solitons;497
5.9.5;20.5 Self-defocusing Nonlinearities: Applications to Ultrashort Pulse Generation;501
5.9.5.1;20.5.1 Compensation for Self-focusing;502
5.9.5.2;20.5.2 Pulse Compression with Self-defocusing Nonlinear Phase Shifts;504
5.9.6;20.6 Few-Cycle Pulses and the Non-stationary Regime of the Cascaded Quadratic Nonlinearity;506
5.9.6.1;20.6.1 Controllable Raman-Like Nonlinearity;506
5.9.6.2;20.6.2 Beyond the Slowly Varying Envelope Approximation;510
5.9.7;20.7 Conclusion;512
5.9.8;References;512
5.10;Effective Parameters of High-Power Laser Femtosecond Radiation at Self-focusing in Gas and Aerosol Media;515
5.10.1;21.1 The Evolution of Effective Parameters of High-Power Femtosecond Laser Radiation in the Air;515
5.10.1.1;21.1.1 Integral Characteristics of a Light Pulse;516
5.10.1.2;21.1.2 Beam Effective Parameters Evolution;517
5.10.2;21.2 Filamentation of Ultrashort Laser Pulse in the Presence of an Aerosol Layer;519
5.10.2.1;21.2.1 Experiment;520
5.10.2.2;21.2.2 Theoretical Simulations;521
5.10.3;21.3 Conclusions;523
5.10.4;References;524
5.11;Diffraction-Induced High-Order Modes of the (2 + 1)D Nonparaxial Nonlinear Schrödinger Equation;525
5.11.1;22.1 Introduction;525
5.11.2;22.2 Spatial Modulational Instability in Nonlinear Media: Analytical Approach;526
5.11.3;22.3 Nonparaxiality and Filamentation of Intense Beams;530
5.11.4;22.4 Higher-Order Modes of the Nonparaxial Nonlinear Schrödinger Equation;533
5.11.5;22.5 Ring and Filament Formation of Beams in Self-Focusing Media: Numerical Study;535
5.11.5.1;22.5.1 Modulational Instability of Optical Beams;535
5.11.5.2;22.5.2 Modes of the Nonparaxial NLSE;542
5.11.5.3;22.5.3 Small-Scale Self-Focusing;545
5.11.6;22.6 Conclusions;551
5.11.7;References;552
5.12;Self-Focusing and Solitons in Photorefractive Media;554
5.12.1;23.1 Introduction;554
5.12.2;23.2 Self-Trapping in Photorefractives;555
5.12.3;23.3 Nonlinear Mechanism;557
5.12.3.1;23.3.1 Photorefraction;557
5.12.3.2;23.3.2 Light-Induced Space-Charge Field;558
5.12.3.3;23.3.3 Nonlinear Index Change;559
5.12.3.4;23.3.4 The Soliton-Supporting Nonlinear Equation;560
5.12.3.5;23.3.5 Soliton Waveforms and Existence Curve;561
5.12.3.6;23.3.6 Experiments and Theory;562
5.12.4;23.4 Two-Dimensional Solitons;563
5.12.5;23.5 Temporal Effects and Quasi-Steady-State Dynamics;566
5.12.5.1;23.5.1 The Transition from a Diffracting Wave to a Soliton;567
5.12.5.2;23.5.2 External Modulation of Soliton Parameters;567
5.12.5.3;23.5.3 Quasi-Steady-State Solitons;568
5.12.5.4;23.5.4 Response Change in Beams That Approximately Do Not Evolve in Time;568
5.12.6;23.6 Non-screening Self-Trapping Mechanisms;569
5.12.7;23.7 Materials;569
5.12.8;23.8 Soliton Interaction-Collisions;569
5.12.9;23.9 Vector and Composite Solitons;570
5.12.10;23.10 Incoherent (Random-Phase) Solitons;571
5.12.11;23.11 Applications;572
5.12.12;23.12 Concluding Remarks;572
5.12.13;References;573
5.13;Measuring Nonlinear Refraction and Its Dispersion;580
5.13.1;24.1 Introduction;580
5.13.2;24.2 Beam Propagation;581
5.13.3;24.3 Z-Scan;584
5.13.4;24.4 Measuring Nonlinear Dispersion;588
5.13.5;24.5 Physical Mechanisms Leading to Nonlinear Refraction;590
5.13.6;24.6 Conclusion;595
5.13.7;References;596
6;Index;599