PROGRESS IN OPTICS VOLUME XXXI
EDITORIAL ADVISORY BOARD G. S. AGARWAL,
Hyderabad, India
C. COHEN-TANNOUDJI, Paris. France V. L. GINZBURG,
Moscow, Russia
F. GORI,
Rome, Italy
A. KUJAWSKI,
Warsaw. Poland
J. PE~INA,
Olomouc, Czech Republic
R. M. SILLITTO,
Edinburgh, Scotland
J . TSUJIUCHI,
Chiba. Japan
H . WALTHER,
Garching, Germany
B. ZEL’DOVICH,
Chelyabinsk , Russia
PROGRESS IN OPTICS VOLUME XXXI
EDITED BY
E. WOLF University oJ Rochesier, N . Y . , U.S . A .
Contributors G . DATTOLI, N . K. DUTTA, L. GIANNESSI, P. W. MILONNI, E. POPOV, D. PSALTIS, Y. QIAO, A. RENIERI, J. R. SIMPSON, R. J. C. SPREEUW, B. SUNDARAM, A. TORRE, J. P. WOERDMAN
1993
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P R I N T E D ON ACID-FREE PAPER
P R I N I t I) I N rtTi N E T H L R L A N D S
CONTENTS OF PREVIOUS VOLUMES
VOLUME I(1961) The Modern Development of Hamiltonian Optics. R. J . PEGIS. . . . . . Wave Optics and Geometrical Optics in Optical Design. K . MIYAMOTO . The Intensity Distribution and Total Illumination of Aberration-Free Diffraction Images. R . BARAKAT . . . . . . . . . . . . . . . . . . . . . IV . Light and Information. D . GABOR. . . . . . . . . . . . . . . . . . V . On Basic Analogies and Principal Differences between Optical and Electronic Information. H . WOLTER . . . . . . . . . . . . . . . . . . . . . . v1. Interference Color. H . KUBOTA . . . . . . . . . . . . . . . . . . . VII . Dynamic Characteristics of Visual Processes. A . FIORENTINI . . . . . . VIII . Modern Alignment Devices. A . C . S . V A N HEEL . . . . . . . . . . . . I. 11. I11 .
I - 29 31- 66
67- 108 109-153 155-210 211-251 253-288 289-329
VOLUME I 1 (1963)
I. I1. III . IV .
V. VI .
Ruling. Testing and Use ofoptical Gratings for High-resolution Spectroscopy. G. W . STROKE . . . . . . . . . . . . . . . . . . . . . . . . . . The Metrological Applications of Diffraction Gratings. J . M . BURCH . . . Diffusion Through Non-Uniform Media. R . G . GIOVANELLI . . . . . . . Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering. J . TSUJIUCHI . . . . . . . . . . . . . . . . . . Fluctuations of Light Beams. L. MANDEL. . . . . . . . . . . . . . . Methods for Determining Optical Parameters of Thin Films. F . A B E L ~ S.
1-72 73-108 109-129 131-180 181-248 249-288
V O L U M E 111 ( 1 9 6 4 ) 1. 11. 111.
The Elements of Radiative Transfer. F . KOTTLER . . . . . . . . . . . Apodisation. P . JACQUINOT. B. ROIZEN-DOSSIER . . . . . . . . . . . Matrix Treatment of Partial Coherence. H . GAMO . . . . . . . . . . .
1-28 29-186 187-332
V O L U M E IV ( 1 9 6 5 ) 1.
I1. 111. IV . V. VI . VII .
1-36 Higher Order Aberration Theory. J . FOCKE. . . . . . . . . . . . . . . . . . . . . . 37- 83 Applications of Shearing Interferometry. 0. BRYNGDAHL 85-143 Surface Deterioration of Optical Glasses. K . KlNOSlTA . . . . . . . . . Optical Constants of Thin Films. P . ROUARD.P . BOUSQUET . . . . . . 145-197 The Miyamoto-Wolf Diffraction Wave. A . RUBINOWICZ. . . . . . . . 199-240 Aberration Theory of Gratings and Grating Mountings. W . T. WELFORD . 241-280 Diffraction at a Black Screen. Part I: Kirchhoffs Theory. F. KOTTLER . . 281-314 V
VI
CON I1.N I S 0 1 I’RI.VIOIIS VOI.IIMI:S
VOLUME V (1966) Optical Pumping. C. COHEN.TANNOLII)JI. A . KASI,I.EK. . . . . . . . . Non-Linear Optics, P . S . PPRSHAN . . . . . . . . . . . . . . . . . Two-Beam Interferometry, W . H . STEEL . . . . . . . . . . . . . . . Instruments for the Measuring of Optical Transfer Functions, K . MURATA. Light Reflection from Films of Continuously Varying Refractive Index, R . JACOBSSON . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H . LIPSON, c. A . TAYLOR. . . . . . . . . . . . . . . . . . . . . . . VII . The Wave of a Moving Classical Electron, J . PICHT . . . . . . . . . . 1.
I1. 111. IV . V.
1-81 83-144 145-197 199-245 247-286 287-350 351-370
V O L U M E V I (1967) Recent Advances in Holography. E. N . LEITH.J . UPATNIEKS . . . . . . Scattering of Light by Rough Surfaces. P . BECKMANN. . . . . . . . . Measurement of the Second Order Degree of Coherence. M . FRANCON.S. MALLlCK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1v. Design of Zoom Lenses. K . YAMAJI . . . . . . . . . . . . . . . . . V. Some Applications of Lasers to Interferometry. D . R. HERRIOTT. . . . . VI . Experimental Studies of Intensity Fluctuations in Lasers. J . A . ARMSTRONG. A . W. SMITH . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Fourier Spectroscopy. G . A . VANASSE. H . SAKAI . . . . . . . . . . . VIII . Diffraction at a Black Screen. Part 11: Electromagnetic Theory. F. KOTTLER 1.
I1. 111.
1-52 53- 69 71-104 105-170 171-209 21 1-257 259-330 331-377
V O L U M E VI1 ( 1 9 6 9 ) Multiple-Beam Interference and Natural Modes in Open Resonators. G . KOPPELMAN. . . . . . . . . . . . . . . . . . . . . . . . . . . I1 . Methods of Synthesis for Dielectric Multilayer Filters. E. DELANO.R . J . PEGIS I11. Echoes and Optical Frequencies. I . D . ABELLA . . . . . . . . . . . . IV . Image Formation with Partially Coherent Light. B. J . THOMPSON . . . . V. Quasi-Classical Theory of Laser Radiation. A. L. MIKAELIAN. M . L. TERMlKAELlAN . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . The Photographic Image. S. OOUE. . . . . . . . . . . . . . . . . . VII . Interaction of Very Intense Light with Free Electrons. J . H . EBERLY . . . I.
1-66 67-137 139-168 169-230 231-297 299-358 359-415
VOLUME VIII (1970) Synthetic-Aperture Optics. J . W . GOODMAN. . . . . . . . . . . . . I1. The Optical Performance of the Human Eye. G . A . FRY . . . . . . . . I11. Light Beating Spectroscopy. H . Z . CUMMINS. H . L. SWINNEY. . . . . . IV . Multilayer Antireflection Coatings. A . MUSSET.A . THELEN. . . . . . . V . Statistical Properties of Laser Light. H . RISKEN . . . . . . . . . . . . VI . Coherence Theory of Source-Size Compensation in Interference Microscopy. T . YAMAMOTO . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Vision in Communication. H . LEVI. . . . . . . . . . . . . . . . . . VIII . Theory of Photoelectron Counting. C . L. MEHTA. . . . . . . . . . . . 1.
1-50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
CONTENTS OF PREVIOUS VOLUMES
VII
VOLUME IX (1971) Gas Lasers and their Application to Precise Length Measurements. A. L. I- 30 BLOOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31- 71 I1 . Picosecond Laser Pulses. A . J . DEMARIA. . . . . . . . . . . . . . . I11. Optical Propagation Through the Turbulent Atmosphere. J . W. STROHBEHN 73-122 . . . . . . . 123-177 IV . Synthesis of Optical Birefringent Networks. E. 0. AMMANN V . Mode Locking in Gas Lasers. L. ALLEN.D . G . C. JONES . . . . . . . . 179-234 V . L. GINZBURG 235-280 VI . Crystal Optics with Spatial Dispersion. V . M . AGRANOVICH. VII . Applications of Optical Methods in the Diffraction Theory of Elastic Waves. K . GNIADEK. J . PETYKIEWICZ. . . . . . . . . . . . . . . . . . . 281-310 VIII . Evaluation. Design and Extrapolation Methods for Optical Signals. Based on Use of the Prolate Functions. B . R. FRIEDEN. . . . . . . . . . . . . 31 1-407 I.
VOLUME X (1972) Bandwidth Compression of Optical Images. T . S . HUANG. . . . . . . . I1. The Use of Image Tubes as Shutters. R. W. SMITH . . . . . . . . . . I11. Tools of Theoretical Quantum Optics. M. 0. SCULLY. K . G . WHITNEY. . IV . Field Correctors for Astronomical Telescopes. C. G . WYNNE . . . . . . Optical Absorption Strength of Defects in Insulators. D . Y . SMITH.D. L. V. DEXTER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Elastooptic Light Modulation and Deflection. E. K . S I ~ I G. . . . . . . VII . Quantum Detection Theory. C. W. HELSTROM . . . . . . . . . . . . 1.
1-44 45- 87 89-135 137-164 165-228 229-288 289-369
VOLUME XI (1973) Master Equation Methods in Quantum Optics. G. S . AGARWAL. . . . . Recent Developments in Far Infrared Spectroscopic Techniques. H . YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Interaction of Light and Acoustic Surface Waves. E . G . LEAN . . . . . . IV . Evanescent Waves in Optical Imaging. 0. BRYNGDAHL . . . . . . . . . V . Production of Electron Probes Using a Field Emission Source. A. V . CREWE VI . Hamiltonian Theory of Beam Mode Propagation. J . A. ARNAUD. . . . . VII . Gradient Index Lenses. E. W. MARCHAND. . . . . . . . . . . . . .
I. I1.
1-76 77-122 123-166 167-221 223-246 247-304 305-337
VOLUME XI1 (1974) I.
I1. I11. IV . V.
VI .
Se1f.Focusing. Self.Trapping. and Self-phase Modulation of Laser Beams. 0. SVELTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Induced Transparency. R . E. SLUSHER. . . . . . . . . . . . . . Modulation Techniques in Spectrometry. M . HARWIT. J . A. DECKERJR. . Interaction of Light with Monomolecular Dye Layers. K . H . DREXHAGE. The Phase Transition Concept and Coherence in Atomic Emission. R. GRAHAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam-Foil Spectroscopy. S . BASHKIN . . . . . . . . . . . . . . . .
1-51 53-100 101-162 163-232 233-286 287-344
Vlll
CONTENTS OF PREVIOUS VOLUMES
V O L U M E XI11 ( 1 9 7 6 ) 1. 11. 111.
IV. V. VI.
On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment, H. P. BALTES . . . . . . . . . . . . . . . 1- 25 The Case For and Against Semiclassical Radiation Theory, L. MANDEL . 27- 68 Objective and Subjective Spherical Aberration Measurements of the Human J. L. CHRISTENSEN, . . . . . . . . . . . . 69- 91 Eye, W. M. ROSENBLUM, lnterferometric Testing of Smooth Surfaces, G. SCHULZ,J. SCHWIDER. . 93-167 Self Focusing of Laser Beams in Plasmas and Semiconductors, M. S. SODHA, . . . . . , . , . . . . . . . . . . . 169-265 A. K. GHATAK,V. K. TRIPATHI Aplanatism and Isoplanatism, W. T. WELFORD , . . . . . . . . . . . 267-292
V O L U M E XIV (1977) The Statistics of Speckle Patterns, J. C. DAINTY. . . . . . . . . . . . High-Resolution Techniques in Optical Astronomy, A. LABEYRIE . . . . Relaxation Phenomena in Rare-Earth Luminescence, L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. The Ultrafast Optical Kerr Shutter, M. A. DUGUAY . . . . . . . . . . Holographic Diffraction Gratings, G . SCHMAHL, D. RUDOLPH. . . , . . V. VI. Photoemission, P. J. VERNIER. . . . . . . . . . . . . . . . . . . . . . . . . . VII. Optical Fibre Waveguides - A Review, P. J. B. CLARRICOATS 1. 11. 111.
1- 46 47- 87
89-159 161-193 195-244 245-325 327-402
V O L U M E XV ( 1 9 7 7 )
I. 11. 111.
IV. V.
Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical Properties of Thin Metal Films, P. ROUARD,A. MEESSEN . . . . Projection-Type Holography, T. OKOSHI . . . . . . . . . . . . . . . Quasi-Optical Techniques of Radio Astronomy, T. W. COLE . . . . . . . Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, .I.VAN KRANENDONK, J. E. SIPE . , . . . . . . . . . . . . . . . .
1-75 77-137 139-185 187-244 245-350
VOLUME XVI (1978) Laser Selective Photophysics and Photochemistry, V. S. LETOKHOV . . . Recent Advances in Phase Profiles Generation, J. J. CLAIR,C. I. ABITBOL. 111. Computer-Generated Holograms: Techniques and Applications, W.-H. LEE IV. Speckle Interferometry, A. E. ENNOS. , . . . . . . . , . . . . . . . V. Deformation Invariant, Space-Variant Optical Recognition, D. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLY Ill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . . . . . . . 1.
11.
.
1- 69 71-1 17 119-232 233-288
289-356 357-411 413-448
CONI'ENTS OF PREVIOUS VOLUMES
IX
VOLUME XVII (1980) 1. 11.
111. IV. V.
Heterodyne Holographic Interferometry, R. DANDLIKER. . . . . . . . Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, B. CAGNAC . . The Mutual Dependence Between Coherence Properties of Light and NonB. WlLHELMl . . . . . . . . . linear Optical Processes, M. SCHUBERT, Michelson Stellar Interferometry, W. J. TANGO, R. Q . Twlss . . , , . . Self-Focusing Media with Variable Index of Refraction, A. L. MIKAELIAN .
1- 84 85- I62 163-238 239-278 279-345
V O L U M E X V I I I (1980)
I. 11.
111.
IV.
Graded Index Optical Waveguides: A Review, A. GHATAK,K. THYAGARAJAN 1- 126 Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media, J. P E R l N A . . . . . . . . , . . . . . . . . . . . . . 127-203 Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, v . I. TATARSKII, v. u . ZAVOROTNYI . . . . . . . . . . . . . 204-256 Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, M. V. BERRY,C. UPSTILL . , . . . . . . . . . . . . , . , . . , . 257-346 V O L U M E X I X (1981)
1.
111. IV. V.
Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence, B. R. MOLLOW . . . . . . . . . . , . . . . . . . . . Surface and Size Effects on the Light Scattering Spectra of Solids, D. L. MILLS,K. R. SUBBASWAMY. . . . . . . . . . . . . . . . . . . . Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, S. USHIODA , . . . . . . . . . . . . . , . . . . . . . . . . . . . Principles of Optical Data-Processing, H. J. BUTTERWECK . . . . . . . The Effects of Atmospheric Turbulence in Optical Astronomy, F. RoDDlER
1- 43
45-137 139-210 21 1-280 281-376
V O L U M E X X (1983)
I.
11. 111. IV. V.
Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects, G . COURTl?3, P. CRUVELLIER, M. DETAILLE, M.
SAYSSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-62 Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . , . . . . . . . . . . . . . 63-154 Multi-Photon Scattering Molecular Spectroscopy, S. KlELlCH . . , . . . 155-262 Colour Holography, P. HARIHARAN . . . , . . . . . . . . , . . , . 263-324 Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, B. P. STOICHEFF . . . . . . . . . . . . . . . . . . . . . . . . . 325-380
V O L U M E X X I (1984)
I. 11. 111. IV. V
Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE. . . . . Theory of Optical Bistability, L. A. LUGIATO , . . . . . . , . . . . . The Radon Transform and its Applications, H. H. BARRETT. . . . . . . Zone Plate Coded Imaging: Theory and Applications, N. M. CEGLIO, D. W. SWEENEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J. C. ENGLUND, R. R. SNAPP,W. C. SCHIEVE. . . . . . . . . .
1- 68 69-21.6 217-286
287-354 355-428
X
CONTENTS OF PREVIOUS VOLUMES
V O L U M E X X I I (1985)
I. 11.
111.
IV. V. VI.
Optical and Electronic Processing of Medical Images, D. MALACARA . . . Quantum Fluctuations in Vision, M. A. BOUMAN,w. A. VAN DE GRIND,P. ZUIDEMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A. V. MASALOV . . . . . . . . . . . . . . . . . . . . . . . . . . , . Holographic Methods of Plasma Diagnostics, G. V. OSTROVSKAYA, Yu. I. OSTROVSKY. . . . . . . . . . . . . . . . . . . . . . . . . . . Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAMACJUCHI. . . . . . . . . . . . . . . . . . . . . . . Wave Propagation in Random Media: A Systems Approach, R. L. FANTE.
1- 76
77-144 145-196 197-270 271-340 341-398
V O L U M E X X I I I (1986) 1. 11. 111.
1v. V.
Analytical Techniques for Multiple Scattering from Rough Surfaces, J. A. DESANTO,G. S. BROWN . . . . . . . . . . . . . . . . . . . . . . Paraxial Theory in Optical Design in Terms ofGaussian Brackets, K. TANAKA Optical Films Produced by Ion-Based Techniques, P. J. MARTIN,R. P. NETTERFIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron Holography, A. TONOMURA. . . . . . . . . . . . . . , . Principles of Optical Processing with Partially Coherent Light, F. T. S. Yu .
1- 62 63-1 12
113-182 183-220 221-276
V O L U M E XXIV (1987)
I. 11. 111.
1V. V.
Micro Fresnel Lenses, H. NISHIHARA, T. SUHARA. . . . . . . . . . . Dephasing-Induced Coherent Phenomena, L. ROTHBERG . . . . . . . . Interferometry with Lasers, P. HARIHARAN . . . . . . . . . . . . , . Unstable Resonator Modes, K. E. OUGHSTUN. . . . . . . . . . . . . Information Processing with Spatially Incoherent Light, I. GLASER. . . .
1- 38 39-102 103-164 165-388 389-510
V O L U M E XXV (1988) I. 11. 111.
IV.
Dynamical Instabilities and Pulsations in Lasers, N. B. ABRAHAM,P. MANDEL,L. M. NARDUCCI. . . . . . . . . . . . . . . . . . . . . Coherence in Semiconductor Lasers, M. OHTSU,T. TAKO . . . . . . . Principles and Design of Optical Arrays, WANGSHAOMIN,L. RONCHI . . Aspheric Surfaces, G. SCHULZ . . . . . . . . . . . . . . . . . . .
1-190 191-278 279-348 349-416
V O L U M E XXVI (1988)
I. 11. 111.
IV. V.
Photon Bunching and Antibunching, M. C. TEICH, B. E. A. SALEH . . . . Nonlinear Optics of Liquid Crystals, I. C. KHOO . . . . . . . . . . . . Single-Longitudinal-Mode Semiconductor Lasers, G. P. AGRAWAL. . . . Rays and Caustics as Physical Objects, Yu. A. KRAVTSOV . . . . . . . Phase-Measurement Interferometry Techniques, K. CREATH. . . . . . .
1-104 105-161 163-225 227-348 349-393
CONTENTS OF PREVIOUS VOLUMES
XI
VOLUME XXVII (1989) 1. 11. 111.
IV V
The Self-Imaging Phenomenon and Its Applications, K. PA.roRSKl Axicons and Meso-Optical Imaging Devices, L. M. SOROKO. . . . . . . Noniniaging Optics for Flux Concentration. 1. M. BASSETT,W. T. WI:I.FOKD, R. WINSTON. , . . . . . . . . . . . . . , . . . . . . . . . . , Nonlinear Wave Propagation in Planar Structures, D. Mihalache, M. BEKTOLOT.II, C. S I B I I . I A . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R. P. POKTIK . . . . . . . . . . . . . . . . . . .
1-108 109-160 161-226 227-313 315-397
VOLUME XXVIII (1990)
I. I1
111.
IV. V.
F. Digital Holography - ComputerGenerated Holograms, 0. BKYNGDAIII., WYROWSKI Quantum Mechanical Limit in Optical Precision Measurement and CommuniU ,M , 4 C l l l n ~S. , S.4ITO.N. IMOTO,T.YANAGAWA, M. cation, Y. Y A M A M O I S. KITAGAWA, G. BJORK . . . . . . . . . . . . . . . . . . . , . . . The Quantum Coherence Properties of Stimulated Raman Scattering, M. G. RAYMER, L A . WALMSI.IY. . . . . . . . . . . . . . . . . , . . . Advanced Evaluation Techniques in Interferometry, J. SCtiWlDEK . . . . Quantum Jumps, R. J. COOK . . . . . . . . . . . . . . . . . . . .
I- 86
87-179 181-270 271-359 361-416
VOLUME XXIX (1991)
I. 11. 111. IV. V.
Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, D.G.HALL . . . . . . . . . . . . . . . . . . . . . . . . . . . V , A. Enhanced Backscattering in Optics, Yu. N. B A K A B A N E N K OYO. KRAVTSOV, V. D. O Z R I NA. , I. SAICHEV . . . . . . . . . . . . . . . Generation and Propagation of Ultrashort Optical Pulses, 1. P. C H K I S I ~ O .V Triple-Correlation Imaging in Optical Astronomy, G. WE1GEi.r . . . . . Nonlinear Optics in Composite Materials. I . Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS, F. HActiE, M. C. K L E I ND. , RI. . . . . . . . . . . . . . . . . . . . . . CARD,P I I . ROLISSIGNOI.
1-63 65-197 199-291 293-319
321-41 I
VOLUME XXX (1992)
I. 11. 111.
IV.
V.
Quantum Fluctuations in Optical Systems, S. REYNALID, A. H E I D M A NE. N, G I A C O B I NC. O , F A B R E. . . . . . . . . . . . . . . . . . . . . . . Correlation Holographic and Speckle Interferometry, Yu. 1. OSIH O V S K Y , V.P. SIICHEPINOV . . . . . . . . . . . . . . . . . . . . . . . . . Localization ofWaves i n Media with One-Dimensional Disorder, V. D. FKt.1I . I K ~ I ES. R ,A. G R E D L S K ~. J I. . . . . . . . . . . . . . . . . . . . Theoretical Foundation of Optical-Soliton concept in Fibers, Y. KODAMA, A. HASLGAWA . . . . , , . . , , . , , , . . . . . . . . . . . . . Cavity Quantum Optics and the Quantum Measurement Process. P. MrYsrRE . , . . . . . . . . . , , . . . . . . . . . . . , . . .
1- 85
87-135 137-203 205-259 261-355
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PREFACE This volume contains six review articles covering a broad range of topics. The first article, by P. W. Milonni and B. Sundaram, presents a thorough review of recent investigations concerning multiphoton ionization of atoms in intense radiation fields. Among the main topics discussed are “above threshold ionization”, generation of higher-order harmonics of an intense field interacting with a gaseous medium and the role of chaotic dynamics in the inleraction of atoms with monochromatic radiation. A tutorial section on chaotic behavior is also included. The second article, by E. Popov, presents a review of modern developments regarding properties of light diffracted by gratings. Both a phenomenological treatment and a macroscopic analysis are presented. In the following article, N. K. Dutta and J . R. Simpson review developments relating to optical amplifiers, especially those which use semiconductors and optical fibers. The article covers the operating principles, fabrication and pcrforniance characteristics. The next article, by D. Psaltis and Y . Qiao, reviews recent research on a rather promising new class of neural networks, the so-called adaptive multilayer optical networks. Although still in the early stages of development, these devices offer the possibility of implementing optical interconnections in three dimensions and they can be functionally equivalent to several thousand chips. The fifth article, by R. J. C. Spreeuw and J . P. Woerdman, deals with idealized but rather useful models of some atomic systems, namely two-levcl and four-level atoms. The analogy between a quantum two-level atom and a classical model consisting of two coupled optical modes is discussed. Extension of these considerations to optical band structure and to four-level systems is also treated. The concluding articlc, by G. Dattoli, L. Giannessi, A. Renieri and A. Torre, is concerned with a relatively new source of coherent radiation, the so-called Compton free electron laser. The basic elements of such a laser are an accelerator which provides a high-energy electron beam and a suitable periodic arrangement of magnets (an undulator). Systems of this kind are becoming practical sources of coherent radiation. One of their important features is tunability.
XIV
PREFACE
The variety of topics discussed in this volume demonstrates once again that optics continues to be a very active field of endeavor, which contributes in a substantial way to modern science and technology. EMILWOLF Department of Physics and Astronomy Uiiiversity of Rochester Rochester. New York 14627, USA
CONTENTS I . ATOMS IN STRONG FIELDS: PHOTOIONIZATION A N D CHAOS by P . W . M I L O N Nand I B. SUNDARAM
$ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 $ 2 . CLASSICAL PHENOMENOLOGICAL THEORY OF GASBREAKDOWN BY A LASER . 4 $ 3 . PERTURBATION THEORYO F MULTIPHOTON IONIZATION . . . . . . . . . . . 7 3 . I . One-photon ionization (the photoelectric effect) . . . . . . . . . . . . . 9 3.2 Multiphoton ionization . . . . . . . . . . . . . . . . . . . . . . . 11 3.3. Computation of multiphoton ionization rates . . . . . . . . . . . . . . 14 3.4. Field statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 $ 4 . BtYOND LOWEST-ORDER PERTURBATION THEORY:INTERMEDIATE RESONANCES 18 $ 5. VOLKOV STATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 $ 6. THEKELDYSHAPPROXIMATION . . . . . . . . . . . . . . . . . . . . . 25 6.1. Digression on the form of the interaction Hamiltonian . . . . . . . . . . 26 6.2. Strong-field perturbation theory . . . . . . . . . . . . . . . . . . . 30 6.3. Limitations of the Keldysh theory . . . . . . . . . . . . . . . . . . . 36 7 . ABOVE-THRESHOLD IONIZATION: EXPERIMENTS . . . . . . . . . . . . . . 39 $ 8. ABOVE-THRESHOLD IONIZATION: THEORY . . . . . . . . . . . . . . . . . 45 8.1. Predictions of Keldysh-Reiss theory: AT1 peaks and polarization effects . . 46 8.2. The ponderomotive potential . . . . . . . . . . . . . . . . . . . . . 51 8.3. Numerical experiments on simplified models . . . . . . . . . . . . . . 56 $ 9 . HIGH-ORDER H A R M O N IGENERATION C . . . . . . . . . . . . . . . . . . 61 $ 1 0. DISCUSSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 $ I 1 . WHATI S CHAOS? . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 1 1.1. Preliminary notions . . . . . . . . . . . . . . . . . . . . . . . . 75 11.2. Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . 78 1 1.3. Integrability, tori and quasiperiodicity . . . . . . . . . . . . . . . . 80 11.4. The KAM theorem . . . . . . . . . . . . . . . . . . . . . . . . 84 11.5. Resonance overlap . . . . . . . . . . . . . . . . . . . . . . . . 85 11.6. Resonance overlap in driven systems . . . . . . . . . . . . . . . . 91 f 12. QUESTIONS OF CHAOSI N ATOMICPHYSICS. . . . . . . . . . . . . . . . 96 12.1. Is there any quantum chaos? . . . . . . . . . . . . . . . . . . . . 96 12.2. Regular and irregular spectra . . . . . . . . . . . . . . . . . . . . 102 12.3. Quantum systems can mimic classical chaos . . . . . . . . . . . . . 103 $ 13. MICROWAVE IONIZATION OF HYDROGEN: EXPERIMENTS A N D CLASSICAL THEORY 109 13.1. Ionization experiments . . . . . . . . . . . . . . . . . . . . . . 109 13.2. Resonance overlap for the classical, one-dimensional hydrogen atom . . . I I4 13.3. Comparison of classical theory with ionization experiments . . . . . . . 118 13.4. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 $ 14. MICROWAVE IONIZATIONOF HYDROGEN: QUANTUM THEORY . . . . . . . . 124 $ 15. S U M M A RAYN D OPENQUESTIONS . . . . . . . . . . . . . . . . . . . . 132 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xv
XVI
CONTENTS
I1. LIGHT DIFFRACTION BY RELIEF GRATINGS: A MACROSCOPIC AND MICROSCOPIC VIEW by E . POPOV(SOFIA.BULGARIA) $ 1. 1NTRODUCTlON
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Grating anomalies . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Grating properties and physical intuition . . . . . . . . . . . . . . . . I .3. Theoretical approaches to grating properties . . . . . . . . . . . . . . PROPERTY OF GRATINGS . . . . . . . $ 2. QUASIPERIODICITY: A FUNDAMENTAL 2.1. Statement of the problem . . . . . . . . . . . . . . . . . . . . . . 2.2. Reflection grating supporting two diffraction orders . . . . . . . . . . . 2.2.1. Littrow mount . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Non-Littrow mount . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Surface waves on corrugated metallic surfaces . . . . . . . . . . . 2.3. Grating supporting a single diffraction order . . . . . . . . . . . . . . 2.3. I . Perfectly conducting grating . . . . . . . . . . . . . . . . . . 2.3.2. Total absorption of light by metallic gratings . . . . . . . . . . . 2.4. Dielectric gratings . . . . . . . . . . . . . . . . . . . . . . . . . $ 3. PHENOMENOLOGICAL APPROACH: A STEP TOWARD THE PHYSICAL INTERPRETATION OF GRATING PROPERTIES . . . . . . . . . . . . . . . . . . . . . . 3.1. Resonance anomalies . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Nonresonance anomalies . . . . . . . . . . . . . . . . . . . . . . $ 4. MICROSCOPIC PROPERTIES OF LIGHT DIFFRACTED BY RELIEF GRATINGS . . . 4.1. Perfectly conducting grating in Littrow mount . . . . . . . . . . . . . 4.1.1. Flat surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Shallow gratings . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Perfect blazing in Littrow mount . . . . . . . . . . . . . . . . 4.1.4. Antiblazing of gratings . . . . . . . . . . . . . . . . . . . . . 4.1.5. Very deep gratings . . . . . . . . . . . . . . . . . . . . . . 4.2. Perfectly conducting grating supporting a single diffraction order . . . . . 4.3. Plasmon surface wave along a metallic grating . . . . . . . . . . . . . 4.4. Resonant total absorption of light by metallic gratings . . . . . . . . . . 4.5. Nonresonant total absorption of light . . . . . . . . . . . . . . . . . 4.6. Total internal reflection by dielectric gratings . . . . . . . . . . . . . . 4.7. Light refraction by deep transmission gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141 141 142 145 147 147 149 149 151 152 153 153 154 156 158 159 164 168 169 169 170 172 173 173 174 174 176 181 182 184 185 185
Ill . OPTICAL AMPLIFIERS by N . K . D U ~ and A J . R . SIMPSON (MURRAYHILL,USA) $ I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. SEMICONDUCTOR OPTICALAMPLIFIERS . . . . . . . . . . . . . . . . . 2.1. Impact of facet reflectivity . . . . . . . . . . . . . . . . . . . . . . 2.2. Amplifier designs . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. I . Low-reflectivity coatings . . . . . . . . . . . . . . . . . . . . 2.2.2. Buried-facet amplifiers . . . . . . . . . . . . . . . . . . . . . 2.2.3. Tilted-facet amplifiers . . . . . . . . . . . . . . . . . . . . .
.
191 191 194 196 196 197 202
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XVll
2.3. Multiquantum well amplifiers . . . . . . . . . . . . . . . . . . . . . 2.4. Integrated laser amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 3 . F I B E RAMPLIFIERS 3.1. Energy levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Fiber design and fabrication . . . . . . . . . . . . . . . . . . . . . 3.2.1. Fiber fabrication . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Amplifier design . . . . . . . . . . . . . . . . . . . . . . . 3.3. Fiber amplifier performance . . . . . . . . . . . . . . . . . . . . . 3.3.1. Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Commercial erbium fiber amplifiers . . . . . . . . . . . . . . . § 4 . LIGHTWAVE T R A N S M I S S I SYSTEM ON SILJDIES. . . . . . . . . . . . . . . 4.1. Direct-detection transmission . . . . . . . . . . . . . . . . . . . . 4.2. Coherent transmission . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Soliton transmission . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Video transmission . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204 205 207 207 210 210 211 212 212 215 216 216 216 219 222 222
IV . ADAPTIVE MULTILAYER OPTICAL NETWORKS by D . PSALTISand Y. QIAO (PASADENA. CA. USA)
$ 1. IKTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 2. OPTICAL MUI.'TIL.AYER NETWORK. . . . . . . . . . . . . . . . . . . . . 2.1. System architecture . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Character recognition application . . . . . . . . . . . . . . . . . . . 2.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . $ 3. IhlPLEMENTATlON OF FULLYADAPTIVE L E A R N I NALGORITIIMS G . . . . . . . 3.1. Anti-Hebbian local learning algorithm . . . . . . . . . . . . . . . . . 3.2. Weight decay and hologram copying . . . . . . . . . . . . . . . . . 3.3. Phase coherence of the holographic gratings . . . . . . . . . . . . . . 3.3. I . Temporal response derivation . . . . . . . . . . . . . . . . . . 3.3.2. Experimental demonstration . . . . . . . . . . . . . . . . . . 3.3.3. Multiple reference beams . . . . . . . . . . . . . . . . . . . . $ 4. DISCUSSION A N D CONCL.USIONS . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
229 231 232 236 240 243 245 248 250 251 255 258 259 260 260
V . OPTICAL ATOMS by R. J . C . SPREEUW and J . P . WOERDMAN (LEIDEN.T H E NETHERLANDS)
$ 1. INTRODUCTION . . . . . . . . . . . . . . . . 5 2. TWO-LEVEL SYSTEMS WIT11 CONSTANT COUPLING . 2.1. Avoided optical crossings . . . . . . . . . . 2.2. Coupled modes and two-level systems . . . . . 2.3. Eigenstates . . . . . . . . . . . . . . . . 2.4. The pseudospin picture . . . . . . . . . . . 2.5. Conservative and dissipative coupling . . . . . $ 3. OPTICALBANDSTRCJCTIIRE . . . . . . . . . . .
. . . . . . . . . . . . 265 . . . . . . . . . . . . 266 . . . . . . . . . . . . 267 . . . . . . . . . . . . 270 . . . . . . . . . . . . 271 . . . . . . . . . . . . 214 . . . . . . . . . . . . 277 . . . . . . . . . . . . 219
CONTENTS
XVlll
$ 4. FOUR-LEVEL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . $ 5. DYNAMICAL BEHAVIOR OF THE OPTICAL ATOM . . . . . . . . . . . . . . 5. I . Rabi oscillation in the rotating-wave approximation . . . . . . . . . . . 5.1.1. Rabi experiments in the time domain . . . . . . . . . . . . . . 5.1.2. Rabi experiments in the frequency domain . . . . . . . . . . . . 5.2. Violation of the rotating-wave approximation . . . . . . . . . . . . . . 5.2.1. Distorted Rabi oscillation . . . . . . . . . . . . . . . . . . . 5.2.2. Bloch-Siegert shifts and multiphoton transitions . . . . . . . . . 5.2.3. The optical atom beyond the rotating-wave approximation . . . . . 5.3. Landau-Zener dynamics . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Adiabatic limit . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Multiphoton resonances . . . . . . . . . . . . . . . . . . . . 5.3.3. Diabatic limit . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Passive and active ring cavities . . . . . . . . . . . . . . . . . . . . 5.5. Two-level atoms and electric-dipole coupling . . . . . . . . . . . . . . $ 6. TliE D R I V E N OPTICAL RINGRESONATOR A S A MODELFOR MICROSCOPIC SYSTEMS 6.1. Can one simulate spontaneous decay of the optical atom? . . . . . . . . 6.2. Landau-Zener crossing problems . . . . . . . . . . . . . . . . . . . 6.3. Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . 6.4. Driven top . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Quantum limit of the driven top . . . . . . . . . . . . . . . . . . . 6.6. Hybrid nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . $ 7. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
280 283 284 287 289 290 290 292 294 297 297 300 301 302 303 307 307 309 310 313 314 316 317 317 318
VI . THEORY O F COMPTON FREE ELECTRON LASERS by G. DATTOLI.L . GIANNESSI. A . RENIERI and A . TORRE(ROME.ITALY) $ 1 . INTRODUCTION.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
$ 2. SPONTANEOUS EMISSION B Y RELATIVISTIC ELECTRONS MOVINGI N A N UNDULATOR MAGNET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Qualitative introduction . . . . . . . . . . . . . . . . . . . . . . . 2.2. Spectral brightness calculation of undulator magnet radiation . . . . . . . 2.3. Inhomogeneous broadening . . . . . . . . . . . . . . . . . . . . . $ 3. T H EFEL GAIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Low-gain regime . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. High-gain regime . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Very high gain regime . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Gain degradation induced by inhomogeneous broadening . . . . . . . . . $ 4. TRANSVERSE MODED Y N ~ M I C .S . . . . . . . . . . . . . . . . . . . . 4.1. Analytical approach . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Numerical results for a transversally uniform electron beam . . . . . . . $ 5. LONGITUDINAL D Y N A M I C S. . . . . . . . . . . . . . . . . . . . . . . $ 6. FEL OSCILLATOR REGIME A N D . T I I I. PWSE PROPAGATION PROBLEM. . . . . 6. I . Preliminary considerations . . . . . . . . . . . . . . . . . . . . . . 6.2. Quantitative analysis . . . . . . . . . . . . . . . . . . . . . . . . § 7. FEL SATURATION. . . . . . . . . . . . . . . . . . . . . . . . . . . $ 8. A S I M P L I F I EVDI E W OF FEL STORAGE R I N GD Y N A M I C S. . . . . . . . . . § 9. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
323 333 334 340 346 350 350 354 359 360 364 364 369 370 376 376 379 387 393 396
CONTENTS
XIX
APPENDIX A . OPTICALCAVITY FOR T H E FEL . . . A.1. Ray matrix and stability condition
. . . . . . . . . . . . . . . . . . . . . . . . . . . . A . 2 . Modes of a stable resonator free of diffraction losses . . . . . . A.3. Diffraction integral and ray matrix . . . . . . . . . . . . . . APPENDIX B . UNDULATOR MAGNETSFOR THE FEL . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
396 398 402 405 406 411
AUTHORINDEX . . . . . . . . . . . SUBJECTINDEX . . . . . . . . . . . CUMULATIVE INDEX. VOLUMESI-XXXI
413 423 427
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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E. WOLF, PROGRESS IN OPTICS XXXl 0 1993 ELSEVIER SCIENCE PUBLISHERS B.V.
I
ATOMS IN STRONG FIELDS: PHOTOIONIZATION AND CHAOS* BY
PETERW. MILONNI and BALA SUNDARAMt Theoretical Division Los Alamos National Laboratory Los Alamos. New Mexico 87545. USA
*
This article is dedicated to the memory of our friend and colleague Jay Richard Ackerhalt
(1947-1992).
Present address: Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA. 1
CONTENTS PAGE
$ 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . .
3
$ 2. CLASSICAL PHENOMENOLOGICAL THEORY O F GAS BREAKDOWN BY A LASER . . . . . . . . . . . . . .
4
$ 3. PERTURBATION THEORY OF MULTIPHOTON IONIZATION . . . . . . . . . . . . . . . . . . . . . . . .
7
Q 4 . BEYOND LOWEST-ORDER PERTURBATION THEORY: INTERMEDIATE RESONANCES . . . . . . . . . . . 18 $ 5 . VOLKOVSTATES . . . . . . . . . . . . . . . . . . . 22 $ 6. THE KELDYSH APPROXIMATION
. . . . . . . . . . 25
$ 7. ABOVE-THRESHOLD IONIZATION: EXPERIMENTS . .
39
$ 8. ABOVE-THRESHOLDIONIZATI0N:THEORY . . . . . 45 $ 9. HIGH-ORDER HARMONIC GENERATION
. . . . . . 61
$ 1 0. DISCUSSION . . . . . . . . . . . . . . . . . . . . .
69
Q 11. WHAT IS CHAOS? . . . . . . . . . . . . . . . . . . 74 $ 12. QUESTIONSOFCHAOS IN ATOMICPHYSICS
. . . . 96
$ 13. MICROWAVE IONIZATION OF HYDROGEN: EXPERIMENTS AND CLASSICAL THEORY . . . . . . . . . . 109 $ 14. MICROWAVE IONIZATION OF HYDROGEN: QUANTUM THEORY . . . . . . . . . . . . . . . . . . . .
124
$ 15. SUMMARY AND OPEN QUESTIONS . . . . . . . . . 132
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 133 REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . 2
133
#
1. Introduction
The possibility of multiphoton atomic transitions has been recognized since the early 1930s. However, the observation of stimulated multiphoton transitions generally requires high field intensities and experimental studies of multiphoton absorption were only realized with the advent of the laser some 30 years later. This chapter reviews recent developments in the understanding of the multiphoton ionization of atoms in intense radiation fields. Of particular interest will be the so-called above-threshold ionization, or ATI. We shall summarize some experimental results and introduce various concepts, such as the Keldysh approximation and Volkov states, which have been at the forefront of theoretical research into ATI. We also discuss the generation of high-order harmonics of an intense quasimonochromatic field incident upon a gaseous medium. This method of generating coherent vacuum ultraviolet and shorter-wavelength radiation is potentially very useful, and we shall summarize the current experimental situation and theoretical methodologies that have been employed thus far to account for the experimental observations. A second major theme of this review concerns the role of chaotic dynamics in the interaction of atoms with monochromatic radiation, in which the principal paradigm is the microwave ionization of hydrogen prepared in highly excited (Rydberg) states. We shall preface our discussion of experiment and theory with a brief tutorial on chaos and “quantum chaos”. Section 2 briefly reviews some phenomenological results for laser-induced breakdown to remind the reader that gas breakdown is a complicated problem to which our basic study in this chapter is not directly applicable in most instances. In $ 3 we bring more coals to Newcastle by reviewing the perturbation theory of multiphoton ionization, and $ 4 examines extensions of perturbation theory to account for intermediate resonances, transition linewidths, etc. These two sections deal with broad and well-established areas of research, and are intended to provide either a gentle theoretical introduction or a cursory review. Only a few remarks are made about the comparison of theory and experiment. Volkov states and the Keldysh approximation have loomed large in attempts 3
4
ATOMS IN STRONG FIELDS
[I, § 2
to construct a nonperturbative theory of multiphoton ionization, and they are discussed in 0 5 and 0 6. respectively. Although the (lowest-order) Keldysh theory remains a valuable benchmark, we shall see that its “nonperturbative” character is illusory. Sections 7 and 8 are devoted to above-threshold ionization and especially to the comparison of theory and experiment for the photoelectron energy distributions. Section 9 summarizes the experimental observations and some of the theoretical work in high-order harmonic generation. Section 10 elaborates on the earlier discussions and introduces the concept of stabilization of atoms that is predicted to occur at sufficiently large intensities. Sections 11 and 12 give a brief overview of some key concepts in nonlinear and chaotic dynamics that have been found useful in the theory of the microwave ionization of hydrogen atoms. The classical and quantum theories of this effect, and an overview of the experimental results, are examined in 5 13 and $ 14. Section 15 summarizes the review. We wish to emphasize that our main objective is to present the basic concepts and physical ideas in the field, and not to proffer a detailed technical analysis of various viewpoints, results, or possible applications. In particular, we hope this will be a useful introductory survey for readers who have not been actively involved in research on strong-field interactions. It is useful to indicate, at the outset, that throughout the review the terms “strong” or “intense” field interactions refer to any situation where a fieldinduced transition frequency, such as the Rabi frequency, is of the order of, or greater than, an internal transition frequency, such as an energy level spacing. This definition clearly demonstrates the relative nature of the concept. When dealing with atoms in the ground state, for which the internal frequencies are large, the strong-field regime may involve field strengths of the order of or greater than lo6 V cm When treating Rydberg atoms, the onset of the strong-field regime, and the effects discussed in the latter part of this review, occurs at field strengths of a few volts per centimeter. ~
8 2.
’.
Classical Phenomenological Theory of Gas Breakdown by a Laser
We shall be concerned principally with a single atom in the field of an intense laser, first considering briefly the classical phenomenological theory of the laser-induced breakdown of a gas. The ionization of a gas in the path of an intense laser beam has been studied since the early 1960s, when it was discovered that there is a threshold intensity below which breakdown does not occur. The problem continues to be of
1 9 8 21
5
G A S RKI!AKDOWN BY 4 LASER
interest for a variety of reasons. For instance, in the propagation of intense laser beams in the atmosphere the plasma formed at breakdown can grow rapidly and strongly attenuate the laser beam. According to our present understanding, gas breakdown by a laser can occur in at least two ways. For low pressures and short (subnanosecond) pulses, multiphoton ionization appears to be the dominant breakdown mechanism. Although the cross section is very low, the process has a strong intensity dependence according to perturbation theory, namely I” for an n-photon ionization if there are no intermediate resonances ( 0 4). For longer pulses and larger pressures, an avalanche (cascade) effect comes into play, lowering the intensity threshold for breakdown (Zel’dovich and Raizer [ 19651). The basic idea behind the avalanche theory is that a few “seed” electrons pick up energy from the laser beam due to inverse bremsstrahlung in the presence of heavy particles, which are necessary for energymomentum conservation. When their energy is sufficient for electron-impact ionization of the heavy particles, they give rise to more electrons, which in turn can take energy from the laser and lead to further impact ionization. The initial seed electrons may be produced by multiphoton ionization. In some situations, however, many seed electrons may be around to trigger the avalanche process. In air, for instance, there are something like lo3-lo4 electrons/cm3. A simple classical description of the avalanche ionization process is based on the electron equation of motion,
e d2x/dt2 + v, d x l d t = - E e m
(2.1)
io‘,
where v, is the electron-atom collision rate and represents a dephasing of the (average) electron motion, which otherwise is perfectly in phase with the oscillating field. The rate of energy gain by the electron is therefore ~w e - Fv =
eEi
=
e 2 E 2 v , / [ 2 m ( 0 2+ v:)]
=
4xe21vc/mc(02+ v:)] ,
dt
(2.2) where I is the field intensity. Now the electron number density n follows the continuity equation
n
=
vin - van - vRn2 - ( D / A 2 ) n,
(2.3)
where vi, v,, and vR are, respectively, the ionization, attachment and recombination rates, D is the diffusion coefficient, and A is a characteristic diffusion
6
[I. § 2
ATOMS IN STRONG FIELDS
length. Ignoring recombination, which is only significant at electron densities larger than the nominal value for breakdown given below, we have
n(z,)
=
noexp[(vi - v, - D/A’)z,]
(2.4)
for the electron density at the end of a square laser pulse of duration zp; no is the initial electron number density. Using (2.4) we can write the ionization rate as vi
= T;
log[n(z,)/n,]
+ V, + D / A 2 .
(2.5)
We define breakdown (BD) as the point at which the rate of gain of electron energy is equal to the ionization rate times the ionization potential I,, of the gas: d WJdt = viIo. This leads us to define the breakdown intensity as I,,
=
[rnc(w2+ v,Z)I,/4ne2v,] {z;
z (nrnc310/e2vc12){z;
I
log[n(z,)/n,]
log[n(z,)/n,]
+ v, + D / n 2 )
+ v, + D / A 2 }
(2.6)
for w = 2 n c / l B v,. If we assume a final electron density n ( z , ) x 2.7 x loi9 c m - 3 (equal to the atomic density at standard temperature and pressure), and no x lo4 ~ m - then ~ , log[n(z,)/n,] x 36. For short pulses, such that the first term in brackets in (2.6) is dominant over the other two, the breakdown condition is fluence-dependent, i.e. (fluence),,
=
I,,Z,
x (nrnc310/e2v,12)log[n(z,)/n,]
.
(2.7)
For I = 10.6 pm, I , = 15 eV, and v, = 4 x 10l2 s - I , we estimate a breakdown fluence of roughly 7 J c m - 2 ; this breakdown fluence scales with wavelength as I - ’ . This short-pulse regime is applicable for pulses shorter than about 100 ns at atmospheric pressures. For pulses longer than about 1 ps the breakdown intensity for clean air is around 3 x 10i’I-’ W cm-’, where 1 is the wavelength in microns. The breakdown intensity is independent of zp in this regime. For “dirty” air, breakdown occurs at much lower intensities due to vaporization and ionization of aerosols. This is obviously a complicated subject, especially when one tries to interpret the results of experiments. (See, for instance, Smith 119701 and references therein.) Our interest is in the more basic problem of photoionization of a single atom, but it must be understood that various complications (like the presence of aerosols) often thwart the application of our theory to “real-world’’ problems like gas breakdown.
1,
s 31
PERTURBATION THEORY OF MULTIPHOTON IONIZATION
I
$ 3. Perturbation Theory of Multiphoton Ionization The electric field E = e / u i acting on an electron in the first Bohr orbit of hydrogen is about 5 x 10’ V cm - I , which is equivalent to an intensity of about 3 x 10l6 W c m - 2 for a monochromatic field of the same field strength. This represents a photon flux of about cm - s - for , I= 5000 A. This provides a rough estimate of the field intensity below which a bound electron is “weakly” perturbed by the field. Obviously, however, this intensity decreases if the electron is in a more weakly bound, excited state; it also depends on the particular type of atom being considered. If the applied field strength is small compared with the binding field, we can expect perturbation theory to provide a reasonably accurate description of multiphoton ionization. However, ‘)perturbation theory” here must be understood to include higher orders thun the lowest order necessury to “see” the process in the theory. Let us write the condition E < ejui in the form ea,E < e2/uo. This inequality is roughly equivalent to
where SZ is the optical “Rabi frequency” and wo is of the order of a transition frequency from the ground state, with associated transition matrix element p ( zeao). The condition that the Rabi frequency is small compared with the transition frequency of a “two-level atom” is an almost universal assumption in the physics of resonant atom-field interactions, and is the basis of the rotating-wave approximation (RWA). Such resonant interactions generally require us to go beyond first-order perturbation theory. We shall return later to the question of higher-order corrections in multiphoton ionization, but we begin with a review of the lowest-order theory and some calculational problems it raises . For strong fields we can treat the applied field classically and write the Hamiltonian in the form H = HA + H,(t), where H A is the Hamiltonian for the unperturbed atom and H , is the atom-field interaction term. (Some authors appear to be more comfortable with a quantized-field approach, but as a practical matter precious little is to be gained here by field quantization.) Let us write the time evolution operator in the form U ( t ) = Uo(t)u(t), where Uo(t)= exp( - iH,t/h) and ihli(t)
=
UJ(t)H,(t)U,(t)u(t)= h,(t)u(t) ,
(3.2)
h,(t) and u(t) are the interaction Hamiltonian and time evolution operators,
8
[I, § 3
ATOMS IN STRONG FIELDS
respectively, in the interaction picture. The operator u ( t ) satisfies the equation u(t) = 1
+
(-k)
JOrdt‘h , ( t ‘ ) u ( t ’ )
(3.3)
where in the second line we have written out the first few terms of the Dyson series for u(t). The probability amplitude A,(t) for the atom to be in state I f ) at time t, having started out at t = 0 in state l i ) , is
=
exp(-iE,t/h)
[
(-i> + (-ir
(flil)
+
j O r d t f(f IhI(t’)l i >
j O r d t Jj O r ’ d t ”( f l h , ( t ’ ) h , ( t ” ) l i )
+
*
1
(3.4)
The transition probability Pfi(t) = IA,(t)I2 is thus
+
(IT jOf’ jOtdt’
d t “ exp[i(E,t’ - Eit”)/h]
where ha, = E, - E,, and we assume (fI i } = 0. The lowest-order contribution to the transition probability is dt’ exp(iw,t’)
(f IHI(t’)l i )
I
2
.
(3.6)
For an atom in a linearly polarized monochromatic field we take the interaction Hamiltonian to be
.
9
PERTURBATION THEORY OF MULTIPHOTON IONIZATION
where p is the transition dipole operator in the direction of the applied electric field. Thus, exp(iwfit) (fIH,(t)/ i >
= +
-(pfiEo/2){exp[i(ofi + o)tI i- exP[i(Ufi - (p,iEo/2) ~ X [i(mfi P -
w)tI
-
w>tI)
(3.8)
in the approximation of ignoring rapidly oscillating terms. (We assume that ofi > 0, i.e. Ef > Ei.) Then
This leads in the usual way to the Fermi golden rule for the transition rate when we argue that there is a continuum of possible final states: make the replacement
(3.10) where we have used the delta-function property of the function sin2xt/x2 for large t. Thus the transition rate according to the golden rule is
(3.11) where I is the (cycle-averaged) field intensity.
3.1. ONE-PHOTON IONIZATION (THE PHOTOELECTRIC EFFECT)
Let us now apply the golden rule to one-photon ionization. In this case the final electron state belongs to the continuous spectrum. We shall assume that the final electron state is just a free plane wave, box-normalized in a volume V : II/,(r)= V -
exp(iq * r ) = V - ‘I2 exp(ip * r / h ) .
(3.12)
10
ATOMS IN STRONG FIELDS
[I. § 3
This assumption ignores the atom-electron interaction in the final state, and can be expected to be valid if the photoelectron energy is large compared with the ionization potential. This also means that the photon energy is large compared with the energy necessary to ionize the atom (but still small compared with mc’, in order to justify a nonrelativistic approach). We assume for simplicity that the initial (bound) electron state is the 1 s state of hydrogen: tji(r)
= (KU,’)
-
exp ( - r/u,) .
‘1’
Then pfi = e(7c Vu:)-
I/’
s
(3.13)
d3rexp( - iq. r ) x exp( - r/uo)
(3.14)
if the field is taken to be linearly polarized in the x-direction. A straightforward calculation yields
pfi 2 -32ie(7c/l/a,5)1~2q-5~.2,
(3.15)
where the caret ( * ) denotes a unit vector. This approximation is valid for quo p 1, which is just the condition mentioned earlier that the photoelectron
energy is large compared with the ionization potential. Now for plane-wave states of the form (3.12) we have
[ w 4 3 1 q 2dqdQ2,
+@ &d
(3.16)
where d Q is a differential element of solid angle. From (3.1 l), therefore,
R‘”
=
I 29e2 Acu,S
s
dQ(4.2)’
s
dqq’q-”
6(h2q2/2m - Ei- ho)
s
2 ( 2 m / h ) 1 ~ 2 ( 1 6 e 2 h 2 / c m 4 u ~ ) I wd62(4*2)’, -9~2
(3.17)
where we have used the approximation h2q2/2m 2 ho,implicit in the planewave assumption (3.12) for the final electron state. Now the cross section for the process may be defined as o = h o R ( ’ ) / I ,which, on using (3.17), leads to the differential cross section being given by da/dQ = ( 2 m / h ) 1 / 2 ( 1 6 e 2 A 3 / c m 4 u ~ ) w (4.2)’. -7~2
(3.18)
The total cross section is obtained by integration over solid angle, which amounts to replacing (8.2)’ by 4n/3 in (3.18): CT =
=
( 2 m / h ) ’ / 2 ( 6 4 7 c e 2 h 3 / 3 ~ m 4 ~= ~ )(2567c/3) ~ - 7 i 2 (au,’) ( I , / ~ w ) ~ / ~ 5.5 x 10-
l7
(10/ho)7/2cm2,
(3.19)
11
PERTLIRDATION THEORY OF MULTIPHOTON IONIZATION
where a is the fine structure constant and I , = e 2 / 2 a , = 13.6 eV is the ionization potential for the ground state. Equation (3.19) is a standard result (Bethe and Salpeter [ 1973]), but we have obtained it in a somewhat nonstandard way, using the - p E form of the interaction Hamiltonian instead of the A * p form. (See the discussion in Q 6.) Before going on to the case of multiphoton ionization, let us recall some salient features and consequences of the calculation leading to eqs. (3.18) and (3.19). (1) The calculation assumes that the electron energy h2q2/2m and the photon energy hw are both large compared with I,,. Near threshold ( h w E lo) the energy of the freed electron is not large compared with the Coulomb field, and we must use a distorted wave for the final state. ( 2 ) The above-threshold cross section varies with photon frequency as - 712.
(3) The angular distribution of the photoelectron is given by (Q f)2.It peaks in the directions parallel to the field polarization, and vanishes in the directions parallel to the field propagation direction. This result, which is true as long as hw 4 mc2, is independent of the plane-wave approximation made for & ( r ) .
3.2. MULTIPHOTON IONIZATION
Consider first the case of two-photon ionization. In this case the transition probability of interest is given by the second term in (3.5): P$’(t)
=
i(irjof lor’ dt’
d t ” exp[i(E’t’
x
- E,t”)/h]
I
2
(f IH,(t’)U,(t’)UJ(t‘’)H,(t”)l i) . (3.20)
To evaluate the matrix element in this expression we insert between the operators Uo(t’) and U J ( t ” )a complete set of states (C, l a ) (a1 = 1)
and proceed as above to evaluate the matrix elements of the interaction
12
ATOMS IN S I R O N G F l l I D S
(2) 1
[I,
2
=
pfupui exp[i(E, - h w ) t ” / h ]exp[ -i(Eu
+ ho)t’/h]
83
(3.23)
0
and
P$)(r)
=
(zy; 1
pfupcliS ‘ d f exp[i(wfa t w ) t ‘ ]
l2
dr” exp[i(wOi- o)t“]
0
(3.24) Once again we argue that there is a continuum of possible final states and use the delta-function property of sin2xt/x2for large t to obtain a golden-rule rate:
(3.26) The generalization to an n-photon transition rate is straightforward:
1, I 31
PERTURBATION THEORY OF MULTIPHOTON IONIZATION
13
These expressions are also applicable to multiphoton absorption, i.e. to multiphoton transitions between bound, discrete states. In this case the density of final states is determined by the field spectral distribution in the case of broad-band light, or by the atomic lineshape in the case of quasimonochromatic radiation. In the latter case
(3.29) if the n-photon transition i+ f is assumed to be homogeneously broadened with a Lorentzian lineshape of width p. Then
R(")+ (2/p) (EO/2h)'" lp$) I
(3.30)
is the n-photon absorption rate given by lowest-order perturbation theory when the resonance condition n w = ufi is satisfied exactly. For multiphoton ionization it is somewhat conventional to write (3.27) in the form derived by Bebb and Gold [ 19661:
(3.31) Here 0: is the fine structure constant, F = I / h w is the photon flux, q is the magnitude of the photoelectron wave vector, and 9%) is as defined in (3.28) but with the electron coordinate r . f replacing the dipole moment p = e r . 2 . The form (3.31) is obtained from (3.27) by replacing p(E') by (l/87c3)q2dQ(dq/dE,) = ( m q / 8 x 3 h 2 d) o , which follows from E, = h2q2/2m. The most obvious prediction of eq. (3.3 1) is that the n-photon ionization rate should vary as the nth power of the intensity. This prediction has been accurately confirmed experimentally for the noble gases by Lompre, Mainfray, Manus, Repoux and Thebault [ 19761, who used 30 ps pulses of intensity I % 1015 W c m - 2 from a mode-locked Nd: YAG laser. They reported experimental values ofn(exp) = a log Ni/a IogI, with Nithe ion yield, shown in table 1 . Although the agreement between theory and experiment found in these experiments is obviously quite good, the I" dependence predicted by lowestorder perturbation theory is often violated.
14
ATOMS IN STRONG FIELDS
[I, § 3
TABLE1 Values of n(exp) = a IogNJa logl, where N, is the ion yield, measured by LomprC, Mainfray, Manus, Repoux and Thebault [1976]; n(theory) is the prediction obtained by dividing the ionization potential by the photon energy. Atom
n(exP)
n(theory)
Xe Kr Ar Ne He
1 1 f 0.5 13 f 0.5 14 & 0.5 20 f 2 23 & 2
11 13 14 19 22
3.3. COMPUTATION OF MULTIPHOTON IONIZATION RATES
The difficulty in calculating multiphoton absorption and ionization rates based on (3.31) is twofold. First, of course, is the fact that the matrix elements p,,, required in (3.28) are known exactly only for hydrogen. Note that a knowledge of the oscillator strengths (proportional to I pabl’) for all transitions is not sufficient: we also require the phases of the pub. Various approximations have been used to estimate these matrix elements. One such approximation, for instance, is the quantum defect method (Zon, Manakov and Rapoport [ 19701) based on the assumption that the dominant contribution to pubis from regions sufficiently far from the nucleus that an electron sees an effectively hydrogenic Coulomb field due to the charge of the residual ion. The problem of obtaining approximate wave functions is, of course, an old one in atomic theory, and we shall not discuss it further here. The second problem lies in performing the summation over the infinite number of states required in (3.28). Even for hydrogen this is not easy. We shall now survey briefly a few of the methods used to evaluate b$) in the case of photoionization, where the final state f belongs to the continuous spectrum. The first calculations of multiphoton ionization rates were presented by Bebb and Gold in the mid-1960s (Bebb and Gold [1966], Bebb [1966, 19671). To outline their approximate method of evaluating fi$), consider first the two-photon case: (3.32)
1.
s 31
PERTURBATION ?HEORY OF MULTIPHOTON IONIZATION
15
Define an average frequency 55 such that
F$'
=
(0o)-' C ( f i x 1 a ) ( a 1x1 i ) a
=
(0o)-'( f I x ' 1 i) ,
(3.33)
where the second equality follows from the completeness relation l a ) (a1 = 1. This appears to reduce the problem to the calculation of a single matrix element (f I x2 I i ) , but W is unknown except for the definition (3.33). For b$), similarly, an average frequency is defined by writing
ca
p y = (,I: n (W-vo)
)-I
x
c c *.. c an
a2
a,,-
(SIX1
an-,
) * * *
(a2
1x1 Q , > ( a l 1x1 i >
I
(3.34) Bebb and Gold argued that the frequency corresponding to the first excited state is a reasonable approximation to the average frequency 0 (assuming i is the ground state), and used this value in their numerical calculations for hydrogen. They also reported calculations for the noble gases, using hydrogenic wave functions in which the Bohr radii were scaled by means of an effective charge parameter 2. The average frequency approximation was later applied to the two- and three-photon ionization of the alkali atoms using quantum-defect wave functions. In the light of the more accurate methods that have been used since the work of Bebb and Gold, it seems fair to say that their method provides reasonable order-of-magnitude estimates of multiphoton ionization rates in many cases. A comparison with a more accurate calculation is shown in fig. 1. The same seems true of the approximation introduced soon afterwards by Morton [1967], in which the average frequency of Bebb and Gold is replaced by an "average" energy level; each intermediate level uj in 6%)is replaced by a single effective level. Based on the data presented by Morton [ 19671, this method appears to be comparable in accuracy with the method of Bebb and Gold. It is noteworthy that Morton found the r * E form of the interaction (Power and Zienau [ 19591, Milonni [ 19761, Ackerhalt and Milonni [ 19841) to be more preferable to the A * p form for calculational purposes, although of course the two forms are related by a unitary transformation. Another way of performing a summation like (3.32) for hydrogen is based on the introduction of a certain function from which fi$) follows after an
16
[I. § 3
ATOMS IN STRONG FIELDS
t
t 430)
68b8
Fig. 1. Solid curves represent six- and eight-photon ionization rates for hydrogen computed by Gontier and Trahin [ 19711; dashed curves represent results of their previous computations. Dot-dashes curves show the results of the Bebb-Gold average-frequency approach. (From Gontier and Trahin [1971].)
integration involving this function. The Laplace transform of this function satisfies an ordinary differential equation that must be solved numerically; however, the sum over states is then “contained” exactly in this solution. This essentially exact method was applied by Zernik [ 19641 to the two-photon ionization of hydrogen initially in the 2s state. For n > 2 this approach leads to a set of coupled differential equations that must be solved numerically. Figure 1 shows the results computed in this way by Gontier and Trahin [ 19711. Also shown for comparison are results of Bebb and Gold. (Note how wide the widths of the maxima are compared with the ordinary one-photon resonance curves.) Analytical expressions involving hypergeometric functions have been obtained for the multiphoton transition amplitudes for hydrogen (Karule [ 197 1 I), and comparison with the calculations of Gontier and Trahin show very good agreement. It thus appears that the theoretical situation for hydrogen is
1.
I 31
PERTURBATION 'THEORY OF MULTIPHOTON IONIZATION
17
satisfactory, although experimental difficulties have precluded any detailed corroboration of the theory. For multielectron atoms the most frequently used approximations are based on quantum-defect theory in combination with Green function methods for performing the summations over states (Zon, Manakov and Rapoport [ 1970, 1971, 19721). It should also be mentioned that the approximation of including only a relatively small number of low-lying states in the summation over intermediate states is fairly accurate (Bebb and Gold [ 19661, Lambropoulos [ 19761, also see Lambropoulos and Teague [ 1976a,b]).
3.4. FIELD STATISTICS
The comparison between theory and experiment for multiphoton ionization rates can now be considered encouraging compared with the situation 20 years ago, although in most instances detailed quantitative comparisons are difficult to make. In essence the standard type of experiment involves the detection of ions following the irradiation of a gas or atomic beam by a laser pulse. Ions rather than electrons are detected to discriminate against the ionization of residual or impurity molecules (Chin [ 19701). Discrimination against the residual ions is accomplished by time-of-flight detection, so that signals due to ions of different mass can be measured separately. According to the earlier discussion on perturbation theory, the signal associated with n-photon ionization should be proportional to I". In an experiment, however, the intensity I is not constant but varies with both time and position in the interaction region, i.e. I = I(r, t). Thus the measured n-photon ionization rate will be proportional to
(I")
=
joTdt
1
d3r I"(r, t ) .
(3.35)
V
For n = 1 this is just the average intensity. For n > 1, however, ( I " ) # ( I ) " in general, and the details of the spatial and temporal variations of the intensity must be known to make an accurate comparison of theory and experiment. Suppose, for simplicity, that the intensity is spatially uniform. Even in this case ( I , , ) = ( I ) " only in the case of a fully coherent field. For incoherent radiation with fluctuations characteristic of thermal radiation, ( I " ) = n! ( I ) " (Loudon [ 19731). Since the multimode radiation from a free-running high-power laser can usually be assumed to approximate such incoherent radiation in its temporal fluctuations, we can expect n-photon ionization rates
18
ATOMS IN STRONG FIELDS
[I. § 4
to be a factor !z n! larger than those calculated according to perturbation theory under the assumption of a fully coherent field. Indeed, experimentally determined n-photon ionization cross sections are often much larger, by an order of magnitude or more, than those predicted by perturbation theory under the assumption of fully coherent radiation, and furthermore the discrepancy increases with increasing n. The field statistics effect may be blamed for at least part of this discrepancy. It is possible experimentally to realize the coherent and incoherent limits approximately by using single-mode and (nonmode-locked) multimode laser output, respectively. For instance, Lecompte, Mainfray, Manus and Sanchez [ 1974, 19751 found an enhancement of the 11-photon ionization of Xe in the multimode case by a factor of about lo7 over the single-mode case. Note that 11! z 107.6. Considerable effort has been devoted over the past decade or so to different models of laser bandwidth and phase fluctuations, and their effect on multiphoton processes. For instance, an observed asymmetry in the curves of the three-photon ionization versus frequency was interpreted by Georges and Lambropoulos [ 1978, 19791as the result of a non-Lorentzian laser bandshape. Of course, the detailed temporal shape of the laser pulse also has an effect on the experimental observations, whereas the aforementioned perturbation theory assumed a perfectly monochromatic wave. Another important consideration in the comparison of theory and experiment concerns effects beyond the domain of the simple, lowest-order perturbation theory outlined in this section. We now turn our attention to some of these.
8 4. Beyond Lowest-Order Perturbation Theory: Intermediate Resonances Substantial deviations from simple perturbation theory arise when there are intermediate resonances in addition to the n-photon, bound-free resonance that occurs whenever n h o exceeds the ionization potential. Such an intermediate resonance occurs when one of the frequency denominators oUli- m u in (3.28) vanishes, corresponding to an allowed m-photon transition between states i and aj. Obviously the lowest-order perturbation theory of the preceding section is inapplicable when such a resonance occurs. Of course this “blow-up” at a resonance is not peculiar to multiphoton processes, but is a general shortcoming of any low-order perturbation theory. The remedy is well known: we should include the linewidths of the resonances, which typically means we replace waIi - m o by - m u - iyuli in (3.28). The width yu,, might arise from the fact that one of the levels of the resonant
BEYOND LOWEST-ORDER PERTURBATION THEORY
1, I 41
19
transition has a finite lifetime. Then, when exp [ - i(wa,I - mw)t] occurs in perturbation theory, we should replace it by exp [ - i(wa,I - m o - i~,,~)l],or in effect
oat,- m u + oat,- m u - iy,,, in (3.28). This leads to a Lorentzian form lo,,, - m ~ - i ~ a , i I - [(ma,, ~ = -mmI2+
YZ~I-'
for the curve of ionization rate versus frequency near the intermediate resonance. The width of the intermediate resonance is, in general, due not only to spontaneous emission or inelastic, state-changing collisions but also to elastic collisions that "dephase" the oscillations of the off-diagonal density matrix elements associated with the resonance. In general, we should also account for a shift in the transition frequency of an intermediate resonance, in addition to a broadening. Both the shift and width of the resonance have intensity-dependent parts, which means that the proportionality of the n-photon ionization rate R(")to I" will be modified. In fact this deviation from the I" prediction is a frequent experimental observation. We can obtain an expression for an intensity-dependent level shift by beginning with the model of a two-level atom describing a one-photon resonance. For a two-level atom detuned by A = w2' - o from the field frequency o we have the RWA optical Bloch equations (Allen and Eberly [ 19751) -Av,
(4.1a)
b = AU + Q w ,
(4.1b)
w=
(4. lc)
M =
-fill,
where 61 = p , 2 E , / h is the Rabi frequency (3.1). In writing (4.1) we are ignoring the damping terms that contribute to the transition linewidth. For 61 approximately constant it follows that u + (A2 + R2)o = 0, implying a field-dependent detuning A' = (A2 + For the exact resonance case A = 0, we have A' = 2 61. In other words, a splitting of the resonance frequency occurs which is proportional to the Rabi frequency (electric field). In resonance fluorescence this Rabi splitting manifests itself in the form of (amplitude-modulation) sidebands at w +_ 61 in the resonantly scattered light (see Knight and Milonni [ 19801). Far from resonance, on the other hand,
A'
=
(A2 + 612)"2
=
d(l
+ 612/A2)1/2 z A + Q2/2A.
(4.2)
For a two-level atom the upper and lower levels are shifted by equal amounts
20
[I, § 4
ATOMS IN STRONG FIELDS
but in opposite directions; (4.2) results from upper- and lower-level shifts of hQ2/4A and - hQ2/4A, respectively. Thus the level shift of the lower level is AEl
=
-
AQ2/4A = - (p?2E;/4h)
(02,
-
w)-’ ,
(4.3)
and the generalization to the level shift of state i of a multilevel system is AEi= - -E,’ 4h
C 1pLijI2[(wji- w ) - ’ + (wji + w)-’]
(4.4a)
.i
(4.4b) In (4.4a) we have added a term (wji + o ) - I for the multilevel system. If i is the ground state, so that oji> 0, this term is a nonresonant (non-RWA) contribution to the level shift. Equation (4.4), of course, is applicable if there are no one-photon resonances (oji z o)at the field frequency o. AEi is the quadratic Stark shift of level i due to the applied field E, cos ot (Bonch-Bruevich and Khodovoi [ 19681). It is sometimes also called the AC Stark shift, or just the “light shift”. We can write AEi in the form - icti(w)E,Z, where oli(w) is the polarizability of an atom in state i, or equivalently
AE; =
-
’
0.063[ni(o) - 1]/(MW/cm2) cm- ,
(4.5)
where n,(w) is the refractive index at STP associated with the atoms in state i. Obviously the shift will be small under most circumstances, but it is not negligible at high intensities, and it has been observed (directly or indirectly) in many experiments. Note that AEiitselfcan be resonantly enhanced if o z oji for some intermediate state j . A clear observation of an AC Stark shift in a two-photon absorption process was reported by Liao and Bjorkholm [1975]. In their experiment two counter-propagating beams from two cw dye lasers were used for the Dopplerfree, two-photon pumping of the 4D level of sodium from the 3 s ground level. The two-photon transition was monitored by measuring the fluorescence at the 4P + 3 s transition, which results from the 4D 44P radiative decay, as one of the laser frequencies was varied. The lasers were tuned so that either the 3P,,, or 3P,,, intermediate state was nearly resonant and made the dominant contribution to the two-photon matrix element pg). The frequency denominators in (4.4) were such that the 3 s state was Stark-shifted primarily by one laser,
21
BEYOND LOWEST-ORDER PERTURBATION THEORY
whereas the 4D state was shifted mainly by the second laser which had a different wavelength. Figure 2 shows the experimental results for the level shift of the 3 S ( F = 2) level as a function of (a) the laser intensity and (b) the frequency detuning from the 3S(F = 2) + 3P,,, intermediate resonance. The dispersion shape shown in fig. 2b is a consequence of the frequency dependence in (4.4). In fact, the solid lines in fig. 2 are theoretical fits (using one intermediate 3P state) with no adjustable parameters. In the context of perturbation theory the most obvious effect of fielddependent level shifts and widths is to introduce field-dependent terms in the denominators appearing in p$). This means that the n-photon ionization rate R(") will no longer be simply proportional to I", as noted earlier. The intermediate resonances then also complicate the field statistics effect, because we must average a more complicated function of intensity than I".
IN T ENS1T Y ( kW/cmz 1
(b)
+ I000
Gv(MHz1
+500
-15
-10
-5
I:
L m +5
+10
+15
_,
+20
; +25
Fig. 2. (a) Level shift of the 3 S ( F = 2) state of sodium versus laser intensity, and (b) level shift versus detuning. Solid curves are theoretical results and dots are experimental points. (From Liao and Bjorkholm [1975].)
22
ATOMS IN STRONG FIELDS
[I, I 5
Various formalisms have been developed to handle the intermediate resonances systematically (see Lambropoulos [ 19761, and to explain the frequent experimental observation that R(”)is not always proportional to ( I “ ) . For various aspects of atomic multiphoton processes we refer the reader to the review by Eberly and Krasinski [ 19841 and the literature cited therein.
6 5. Volkov States We have already seen the effects of approximating the final (continuum) electron state by a plane wave corresponding to a free electron. A logical extension of this leads to another approximation where the final state is assumed to be that of an electron in the applied field, with the effect of the residual ion being neglected. This approximation is made in the Keldysh theory described in the following section. First we consider the wave function for the electron in the applied field, which may be obtained exactly in both relativistic and nonrelativistic theory. In either case the states are called Volkov states. Volkov [ 19351 solved the Dirac equation for an electron in a plane-wave monochromatic or polychromatic field, but we shall restrict ourselves to the nonrelativistic theory for an electron in a plane monochromatic wave. We shall also ignore electron spin, which is a reasonable thing to do for interactions involving optical fields. Consider an electron in a plane monochromatic field described by the vector potential A = A, cos at (with corresponding electric field E = - (l/c) a A / a t = ( o A , / c ) sin ot = E, sin at). We can use the standard “ A * p” Hamiltonian H
=
(l/2m) ( p - eA/c)*
=
(1/2rn) [ p - (eA,/c) cos o t I 2
=
p2/2m - (e/mc)A, p cos a t
-
+ (e2Ai/2mc2)cos2 a t .
(5.1)
Up to now we have used instead the “ r e E” form (3.7) for the interaction of the electron with the applied field. These two forms of the Hamiltonian are, in fact, unitarily equivalent, as discussed in the next section, and we use (5.1) now mainly because it seems to be the most commonly used form in the present context. The Schrodinger equation ih at,b/at = H J / , with H given by (5.1), has a solution
-
$(r, t ) = exp { - i[p2t/2m - p r
+ $(t)]/h} ,
(5.2)
1, I 51
23
VOLKOV STATES
where $(t) =
jOr
d t ’ [ - ( e / m c ) A ( t ’ )p.
+ ( e 2 / 2 m c 2 ) A ( t ’ ) 2. ]
(5.3)
The wave function (5.2) reduces to a free plane wave when A + 0 . For A = A , coscot we have the Volkov state
$(r, t ) = exp [r
- i{(p2/2m
+ e2A:/4mc2)t - [r
-
rc(t))* p
+ (e2Ag/8mwc2)sin2ot}/h] ,
(5.4)
where r J t ) = - eA, (sin cot)/moc. The term e2Ag/4mc2= e 2 E i / 4 m w 2is associated with the “mass shift” in the interaction of unbound electrons with intense radiation (Eberly [ 19691). In the context of ionization it is associated with the ponderomotive potential, which has been the subject of much discussion in the theory of above-threshold ionization. To interpret this term, let us consider the Stark shift (4.4) in the free-electron limit wli 4 w :
where the second step follows from the dipole sum rule. Thus we can regard the ponderomotive potential as simply the free-electron limit of a quadratic Stark shift. This is discussed in more detail in the following section. The displacement r,(t), which may be written as - (eE,/mco2)cos cot, will be recognized as the classical displacement for an electron in the field E, cos cot, i.e. a solution of the equation d2r/dt2 = (eE,/m) cos cot. Using the identity eix sin 8
-
00
Jn(x)ein”,
(5.6)
-aJ
where the Jnare ordinary Bessel functions, we may write
c 00
exp [ - irc(t) * p / h ] = n=
Jn(eA, * p/rnchw) einor.
(5.7)
-a
We shall see later that the use of Volkov states in the theory of above-threshold ionization leads in similar fashion to sums over Bessel functions. In fact, the multiphoton absorption and emission associated with these Bessel functions also occur in free-free transitions, a brief discussion of which follows. In the scattering of an electron by an atom or molecule, the field from the
24
ATOMS IN STRONG FIELDS
[I, I 5
heavy particle imparts an acceleration to the electron and so causes it to radiate (bremsstrahlung). The continuous part of the X-ray spectrum from an X-ray tube, for instance, is due to this bremsstrahlung by high-energy electrons. For low-energy electrons the rate at which this “spontaneous” radiation process produces detectable photons is very small. However, the process can be stimulated by high-intensity radiation and of course the possibility of stimulated absorption of radiation also exists as the electron undergoes a free-free transition between two continuum states; multiphoton free-free transitions are also possible. Such multiphoton processes were discussed by Kroll and Watson [ 19731, and the first experimental observations were made in 1977 (Weingartshofer, Holmes, Caudle, Clarke and Kruger [ 19771, Weingartshofer, Clarke, Holmes and Jung [ 19791). Let do,/dQ be the differential cross section for the scattering of an electron from a state of linear momentum p o to a final momentum p ( o ) with the emission ( n > 0) or absorption (n < 0) of n photons of frequency o,so that the initial and final electron energies are related by p ( ~ ) ~ / 2=r pn 3 2 m - n h o .
(5.8)
Kroll and Watson showed that for photon energies much smaller than the electron energy or for weak electron-target interactions,
where da,,/dQ is an electron-atom cross section for elastic scattering in the absence of the field, and the dimensionless parameter x
=
-
eAo [ p ( w ) - po]/mchw
(5.10)
is a measure of the change in the electron-field interaction energy compared with the photon energy. The appearance of this parameter could have been anticipated from (5.7). Equation (5.9) predicts that the energy spectrum of the scattered electrons should have peaks about the initial energy at integral multiples of the photon energy h cc), corresponding to multiphoton absorptive or emissive free-free transitions. Such energy spectra were first observed by Weingartshofer, Holmes, Caudle, Clarke and Kruger [ 19771 using a 10.6 pm CO, (multimode) laser with pulses of energy of about 15 J and a duration of about 5 ps. Figure 3 shows the basic geometry of their experiment. The target particles in the gas beam were Ar or H,, and the electron energies were varied from around 4 to 80 eV, with widths of about 60-150 rneV from the electron gun. The
1, § 61
25
THE KELDYSH APPROXIMATION
I
ELECTRON DETECTOR
POLARIZATION VECTOR
GAS BEAM
I SCATTERMG
PLANE
ELECTRON GUN
Fig. 3. Basic scattering geometry in the experiment of Weingartshofer et al. on free-free multiphoton transitions. (From Weingartshofer, Holmes, Caudle, Clarke and Kruger [ 19771.) 250
100 In
: 50
200
5
a m Y m. c 0
g
150
0 100
100
50
50
b
L
m + C
2
0
0 - 5 4 3 2 1 01 2 3 4 5 - t
0 -54321012 345-t-21012t
Energy in units of l a s e r photons
Fig. 4. Typical data of Weingartshofer et al. showing multiphoton peaks in the electron energy spectra. (From Weingartshofer, Clarke, Holmes and Jung [ 19791.)
scattered electrons were detected as functions of the incident electron energy (Ei),the electron scattering angle (e), and the angle ($) between the laser beam and p o , as indicated in fig, 3. The observations were in good qualitative accord with the Kroll-Watson theory. Typical data displaying the predicted peaks in the electron energy spectra are shown in fig. 4.
8 6. The Keldysh Approximation The perturbation theory of multiphoton ionization discussed earlier involves an expansion in powers of the field, and as such can be intractable or at best
26
ATOMS IN STRONG FIELDS
[I, § 6
very cumbersome when one is concerned with really intense fields. The Keldysh approximation (Keldysh [ 19651) is an alternative approach to intense-field ionization in which the binding potential rather than the field is regarded as a perturbation. It is characterized by the treatment of the photoelectron as an otherwise free electron in the applied field; the detached electron is thus described by a Volkov state. In this section the Keldysh approximation will be introduced via a perturbation expansion in the binding potential.
6.1. DIGRESSION ON THE FORM OF THE INTERACTION HAMILTONIAN
Keldysh used the r * E form of the electron-field interaction, i.e. he worked with the Hamiltonian
H'
=
p2/2m + V ( r ) - era E(t)
(6.1)
instead of the A * p form
H
=
-
p2/2m + V ( r ) - (e/mc)A(t) p
+ (e2/2mc2)A2(t).
(6.2)
Since the question of the interaction Hamiltonian has been a source of considerable confusion in this and other contexts, we will briefly review the connection between the forms ( 6 . 1 ) and (6.2). For simplicity we shall assume that the applied field is a prescribed, classical field. The fully quantum-electrodynamical theory relating ( 6 . 1 ) and ( 6 . 2 ) is not difficult (Power and Zienau [ 19591, Milonni [ 1976]), but it is unnecessary for our purposes. The Hamiltonian
H
=
( 1 / 2 m ) [ p - (e/c)AI2 + V ( r ) ,
(6.3)
with V * A = 0 (the Coulomb gauge), is introduced in classical electrodynamics because it generates the correct equation of motion for a charged particle acted on by the Lorentz force F = e E + (ev/c) x B. This Hamiltonian follows from the Lagrangian e
L
=
+mi.2+ - A i. - V ( r ) C
(6.4)
when we recall the general formula H = p s i . - L , with p = aLjai = mi. + ( e / c ) Athe canonical momentum for a charged particle in a field. We can always add a total time derivative to the Lagrangian to generate a canonical transformation. Consider, for instance, the Lagrangian
THE KELDYSH APPROXIMATION
1. § 61
L’
=
;mi2
+e
-
C
=
-
e
-
21
e d A - i . - V(r) - - - ( A * r ) c dt A . r - V(r)
=
fmL2
C
+ ei.. E - V(r).
(6.5)
In this case the transformed Hamiltonian is H’
=
p
a
i. - L’
=
p2/2m + V(r) - era E ,
(6.6)
where now p = aL’/ai. = mi.. Thus in classical electrodynamics the forms (6.1) and (6.2) of the Hamiltonian are simply related by a canonical transformation, and of course they provide equivalent descriptions of the electron’s dynamics. The quantum analogue of such a canonical transformation is a unitary transformation in which the function whose total time derivative is added to the classical Lagrangian is exponentiated. In the example just considered the unitary transformation of the state vector is I$‘)
=
exp(-ier.A/hc)
= St I $ ) ,
I$)
(6.7)
where ih
a -
at
I$)
(6.8)
=HI$)
is the SchrOdinger equation with the Hamiltonian (6.2). The equation for I $’ ) is
a
ih - ( S I $’ )) at
=
HS 1 $’ )
,
(6.9)
or
a I
(6.10)
er-E
(6.11)
=er.ESI$’) +ihS
-
at
and therefore ih
~
a I$’)
at
=
(StHS) I$’)
-
28
[I, $ 6
ATOMS IN SlRONG FIELDS
since r and S commute. Finally,
s+s+v(v)s=-
2m
=
2m l (
:>’
s ~ ~ sA - -+ V(r)
p2/2m + V(r) ,
(6.12)
where the last step follows from the general operator identity eABe - A = B
+ [ A , B ] + (1/2!) [ A , [ A , B ] ] +
.
(6.13)
Thus (6.11) becomes
a I$’)
ih
-
at
= ( p 2 / 2 m + V(r)-er.E)I$’)
=H‘l$‘),
(6.14)
which is the SchrBdinger equation when the Hamiltonian (6.1) is used. Therefore we can use either (6.1) or (6.2) as our fundamental Hamiltonian in the dipole approximation. The two forms of the Hamiltonian are equivalent, provided we relate the corresponding state vectors by the unitary transformation (6.7). Actually a different Hamiltonian with the same form as (6.2) can be derived by writing the original Hamiltonian (6.1) in terms of new canonical variables. In this case it is not necessary to transform the state vectors (see Ackerhalt and Milonni [ 19841). Consider, for instance, the Volkov state (5.2), which is an eigenstate of the Hamiltonian (6.2) with V = 0. According to our discussion, the Volkov state corresponding to the Hamiltonian (6.1) with V = 0 should be $’(r, t )
-
=
exp [ - ier A(t)/hc]$(r, t )
=
exp i ( p - f
[
A(t))
- r/h] exp [ - i jOrdt’ ( p -
A(t’)>’/2mh] (6.15)
Thus for E=
E , C O S ~ ~ ,
A
=
-1~ , s i n w t ,
-
E, sin w t ) r/h]
w
(6.16)
we have
[( +
$‘(r, t ) = exp i p
x exp[ - i I o r d l . ( p + -0 e E,sinwt‘ >’/2mh]
for the r * E Volkov state. This is the Volkov state used by Keldysh.
(6.17)
1, § 61
29
THE KELDYSH APPROXIMATION
As a practical matter, it is often possible to use effectively either Hamiltonian
without transforming the state vectors. Consider, for instance, the transition amplitude appearing in (3.6) for a bound-bound transition: :
rr
Afl(t) = - h
o
'J
=
d t ' exp(ioflt') (fl
(ie/h) (fl
r li)
for
H , ( t ' ) li)
d t ' exp(iwflt')E(t')
(6.18)
if we use the Hamiltonian (6.1). Using (6.2), on the other hand, we obtain the transition amplitude B,i(t) =
-ih j o ' d t ' exp(ico/,t') ie
-~
hmc
(fl
p li)
-
for
( f l - ( e / m c ) A ( t ' ) - p + (e2/2mc2)A2(t')li)
d t ' exp(iwflt')A(t')
(6.19) Now an integration by parts yields
S,:
dt' exp(iwfit')A(t')
d t ' exp(io+t')A(t') = -
-
for
d t ' exp(iwfit')E(t'),
(6.20)
'Ufi
provided A ( t ) = A ( 0 ) = 0, i.e. that the field is turned on and then off in some way between time 0 (or - 00) and time t (or + co). Thus, ie Bfl(t) = - ( f l
h
r li)
-
d t ' exp(iw&)E(t')
=
A,(t)
(6.21)
Under the assumption of a truly monochromatic field the situation is not necessarily so simple, because the field cannot be assumed to be turned on and off. Then we must be concerned with the transformation of the state vectors and sometimes with questions as to which set of eigenstates is more appropriate physically. For a monochromatic field exactly resonant with a transition, however, we can again effectively ignore the unitary transformation of state
30
ATOMS IN STRONG FIELDS
[I, 8 6
vectors (Power and Zienau [ 1959]), which is why we obtained the standard result (3.19) with either Hamiltonian and without transforming state vectors.
6.2. STRONG-FIELD PERTURBATION THEORY
For sufficiently strong fields it makes sense to treat the binding potential V ( v ) rather than the interaction with the applied field as a perturbation. Let us therefore write the Hamiltonian in the form
where H,(t)
=
p2/2rn + H,(t)
(6.23)
and H,(t) is either - er * E ( t ) or - (e/rnc)A(t)* p + (e2/2rnc2)A2(t).We now regard H,(t) as the “unperturbed” Hamiltonian and V as the perturbation. As in 0 3, we write the time evolution operator as U ( t ) = U,(t)u(t), where now (6.24)
ift irO(t)= H,(t) ~ , ( r ) ,
ihu(t) =
uJ(t)~ ~ , ( t ) u (.t )
(6.25)
Let l i ) be a bound state and If) a free-electron plane-wave state. The field is assumed to be adiabatically turned on and off at t = - co and t = co, respectively. The bound-free transition amplitude may be taken to be (6.26) Now 1 +i(t’)) = U ( t ’ ) l i ) is the state to which the initial bound state evolves at time t ’ . In the style of low-order perturbation theory we assume that the probability of the atom being removed from its initial state I i) is small, so that
I $ i ( t ’ ) ) z exp( -iEit’/h)
li)
=
exp(iZ,t’/h) li) ,
(6.27)
where 1, is the ionization potential for bound state li). This approximation also ignores any level shift of l i ) . Then :
rr (6.28)
THE KELDYSH APPROXIMATION
1, § 61
31
From (6.29) it follows that
or ,4,(t) z
j‘
I h
dt‘
(fI
i h aU,t/at’
+ U $ ( t ’ ) p 2 / 2 mI $ ; ( t ’ ) )
(6.31)
--w
after a partial integration and dropping a term that stays bounded for all t and therefore makes no contribution to a transition rate. Then it follows from (6.24) and (6.23) that (6.32) The state Uo(t)If) is the state to which the plane-wave state exp(ip-r/h) evolves at time t ‘ under the action of the unperturbed Hamiltonian (6.23). Therefore,
I $7 ( 0 ) = Uo(t) If>
(6.33)
is a Volkov state, which takes the form (6.17) when the r E Hamiltonian is employed, and we can write (6.32) as A,(t) E - h
j‘
dt‘
($7 0‘11 H & ’ )
I $;(t’)> .
(6.34)
--w
This is the Keldysh approximation. Note that the Keldysh approximation to the transition amplitude brings in the applied field twice, namely in the Volkov state I ( t ’ ) ) and in H,(t’). Note also that if I $;(t’)) is replaced by a free-electron plane-wave state, we recover the transition amplitude of conventional perturbation theory to first order in the applied field. However, in spite of such appearances the Keldysh approximation is not afirst-order expansion in the appliedjield. It is the first term in an expansion in the bindingpotential V, as we have just shown.
$7
32
[I. § 6
ATOMS IN STRONG FIELDS
To evaluate the transition amplitude (6.34), we shall use the form (6.30) and write the integrand in that expression as
($7 ( t ’ ) l -1, = -
=
-
S
- p2/2m I$i(o)>
ex~(il,r’/h)
d3p’ ($; ([’)I I , + p2/2tn I P’ ) ( P ’I
j
$j(O)>
exP(iI,t’/h)
($7
d3p‘ (1, + ~ ‘ ~ / 2 m ) ( f )I p ’ ) ( p ‘ I $j/i(0)>exp(iI,t’/h). (6.35)
Now for the r - E form of the Hamiltonian, the appropriate Volkov state is (6.17), and then the preceding matrix element is e
l x (p \
+e
-
E,, sin wt‘
Y] 1,“ exp (i
dt” ( p
+e
-
Eo sin wt”
W
E , sinwt“ 1 IG;.(O)
(6.36)
o
-
If we use the A p form of the Hamiltonian, on the other hand, the appropriate Volkov state is (5.2), and instead of (6.36) we derive the expression -(I,
+ p2/2m)( p i
IG;.(O)) exp(iI,t’/h) exp{i[p2t’/2m + $ ( f ‘ ) ] / h }(6.37)
where +(t) is defined by (5.3). Now ( p i &(O)) can, of course, be written as a Fourier transform of the bound-state wave function $j(r) = (rI $ i ( 0 ) ) :
( p i &(O))
=
(2nh)-”’S d3rexp(-ip.r/h)lG;.(r)= $ i ( p ) .
(6.38)
Thus we incur either cjj(p ) or & ( p + ( e / w ) E ,sinwt’), depending on whether we use A p ( + A’) or r *E, respectively. For this reason the ensuing algebra will be simpler if we use the A p form. In this case
x
jl
dt‘ exp {i
So‘’
dt”
+
(p -
A ( t ” ) y ] / h ] . (6.40)
1, § 61
33
THE KELDYSH APPROXIMATION
To summarize: we have arrived at the expression (6.40) for the transition amplitude by treating the binding potential as a perturbation to the dynamics of the electron in the strong field. This expression is essentially the Keldysh approximation, except that we have employed the A * p form of the interaction instead of the Y - E form used by Keldysh. Equation (6.40) gives the amplitude for the transition from the initial bound state I i ) to the continuum state of the electron momentum p . Note that the only information about the initial bound state needed in (6.40) is the Fourier transform &(p), which is the projection of the initial bound state onto the final continuum state I p ) . It should not be surprising, therefore, that some general conclusions based on (6.40) will be more or less insensitive to the details of the initial bound state. All of the “atomic physics” in the Keldysh approximation is contained in $+(p ) . Before discussing further the basis for the Keldysh approximation, we will carry through some of the algebra in (6.40) to see where the approximation leads. Define, for A(t) = A, C O S W ~ ,
=
exp [(i/h) (I,
+ p2/2m + e2Ai/4mc2)t]exp [
-
i(eA,-p/hwmc) sin wr]
x exp[i(e2Ai/8hwmc2)sin2wtI =
1 q
Jq(eA,- p/hwmc)Jr(e2Ai/8homc2) r
x exp{i[I, =
+ p2/2m + e2Ai/4mc2 - (q - 2r)hw]t/h)
1 1 JN+2n(eAO-p/homc)J,,(e2Ai/8hwmc2) n
N
x exp[i(I,
+ p2/2m + e2Ai/4mc2 - N h w ) t / h ] ,
(6.41)
where we have used the general identity (5.6). The transition rate may be written, for t + 00, as
J
-
x:
34
ATOMS IN STRONG FIELDS
x S(Io + p 2 / 2 m + e2Ai/4mc2- Nhw) IFN(p)I2b(I,
=
+ p 2 / 2 m + eZAi/4mc2- Nhw),
(6.42)
N
where F N ( p ) = (27t/h)’/2(Io + p2/2m)’$Ap) x
c JN+2n(eA0- p / h w m c ) J n ( e 2 A i / 8 h o m c 2 ) .
(6.43)
n
We can draw an important conclusion from (6.42):the energy distribution of the ejected electrons will have peaks at p2/2m = N h o - I , - e 2 A $ / 4 m c 2 .
(6.44)
Such peaks in the electron energy distribution, separated by the photon energy A w, are characteristic of strong-field above-threshold ionization, as discussed in the following sections. Equation (6.42) is the form derived by Reiss [ 19801 (see also Faisal [ 19731). Following along the lines of the original Keldysh paper, we use the integral representation n
dOexp[i(NO+ x sinO+ysin2O)] (6.45) n = -a,
and some simple changes of variables to cast F N ( p ) in (6.42) in the form
-
y-’JGp.Psint?+
(1/4y2)sin20]),
(6.46)
where y is the Keldysh adiabatic tunneling parameter:
and P is the direction of the electric field vector ( E = EoP sin 02, E, = wA,/c). The total ionization rate is obtained by summing over all possible momentum
1, § 61
THE KELDYSH APPROXIMATION
35
states of the ejected electron: (6.48 a) N
R,
=
s
d3p I F , ( P ) ~6(pz/2m ~
+ fa + eZAi/4mc2- NAw) .
(6.48b)
R , is the ionization rate associated with the N-photon ionization process in which the electron has the kinetic energy (6.44). We can understand the physical significance of the Keldysh parameter y, and see what it has to do with tunneling as follows. In classical terms eE, is the maximum amplitude of the force exerted on the electron by the field, which leads us to define I = Io/eEo as the width of an effective potential barrier to we ionization. Since in this classical picture the electron velocity u = ,/-, define a tunneling frequency w, = v/21= eEo/&
(6.49)
in terms of which y
=
w/w, .
(6.50)
Thus for y 4 1 the field frequency o is small compared with w,, and the electron has plenty of time to tunnel through the potential barrier during a cycle of the field. For y % 1, on the other hand, the electron does not have time to cross the barrier during a cycle of the field. The Keldysh parameter can be written in another useful form based on the Bohr model, namely, (6.5 1) where w,, is the orbital frequency of the electron, Fat is the force on the electron due to the nucleus, and F,,, = eEo. From this discussion we can refer to y 4 1 as the tunneling regime, whereas for yB 1 the ionization is dominated by multiphoton ionization. As noted by Keldysh, however, “the nature of these two effects is essentially the same”. Indeed, in the low-frequency limit (y 4 1) Keldysh obtained an ionization probability w - exp[ -(21,/Aw) (2y/3)] = exp[ -(4,/%1~/2/3eAE,)],
(6.52)
which is a standard result dating back to Oppenheimer (Oppenheimer
36
ATOMS I N STRONG FIELDS
[I, § 6
[ 19281, Landau and Lifshitz [ 19651). Keldysh [ 19651 derived the ionization probability as a function of y by applying the method of steepest descent to the evaluation of an integral like (6.46). For details and an elaboration of this approach the reader is referred to Perelomov, Popov and Terent’ev [ 19661 and Brandi, Davidovich and Zagury [ 19811. The preceding analysis can easily be extended to the case of a circularly polarized field, in which case ~ ( t=) ( ~ / J Z ) A , ( ~ cos wt 2 9 sin at)
(6.53)
and A(Q2 = $4;. (The factor of 1/2 is introduced so that A(t)2 here is equal to the cycle-averaged value of A(t)2 in the case of linear polarization.) Then, from
(5.3) $(t)
=
-
( e A , p , /Jzmwc> sin(ot T
c1)
+ (e2~;/4mc2)t,
(6.54a) (6.54b)
a = tan -
’ (P,/P,) ,
(6.54~)
and the appropriate Volkov state for a circularly polarized field follows from (5.2). The important difference from the case of linear polarization is that the term involving sin2wt in (5.4) is absent in the case of circular polarization. It is then easy to see that the Bessel function with argument e2A;/8homc2 in (6.43) does not appear in the transition amplitude for circular polarization, which makes the case of circular polarization considerably simpler in terms of the algebra. Reiss [1980] compared the two cases in both the low- and high-intensity limits. In addition to using the A * p rather than r - E form of the interaction Hamiltonian, Reiss does not proceed immediately to a high-multiphoton-order approximation, as does Keldysh. One result is that in Reiss’s approach the pondermotive potential appears explicitly in expressions like (6.43) and (6.44). In the Keldysh-Reiss approximation, in addition to the Keldysh parameter y, an intensity parameter appears that Reiss denotes by z (see 0 8). The Keldysh approximation should be understood to include Reiss’s modifications in the following discussion.
6.3. LIMITATIONS OF THE KELDYSH THEORY
A casual reading of some of the literature might give the impression that the Keldysh theory is “nonperturbative”. Our derivation of the approximation,
THE KELDYSH APPROXIMATION
1, § 61
31
however, shows that it is actually the first term in a perturbation expansion in the binding potential V. It could be argued that the Keldysh theory is “nonperturbative in the applied field”, but our derivation required the assumption that the probability of the electron being removed from its initial state is small. This raises the important question of how reliable the Keldysh theory is under conditions of strong ionization, when the probability of the electron being removed from its initial bound state is not negligible. Numerical experiments on simple models indicate that the Keldysh approximation may not be very accurate under conditions of strong ionization, where the ionization can occur after just a few periods of the applied field. These numerical studies are discussed in $ 8. It should also be remembered that the Volkov state used in the Keldysh theory ignores all the Coulomb interactions of the photoelectron. Keldysh [ 19651 pointed to this approximation in connection with a failure of his approach to produce the correct pre-exponential factor in the tunneling ionization probability (6.52). (See also Perelomov, Popov and Terent’ev [ 19661.) A more pernicious problem with the Keldysh approximation as modified by Reiss relates to gauge invariance. Let us recall briefly the basic gauge invariance of the nonrelativistic Schrddinger equation (6.55) at
where A and 9 are the vector and scalar potentials E = - (l/c) a A / a t - V$, B = V x A . The gauge transformation
such that
A+A’=A+Vx,
(6.56a)
$ --, 9’
(6.56b)
=
$ - ( i / c )w a t ,
leaves E and B unchanged and therefore should not affect physical predictions. In fact we can put the Schrddinger equation (6.55) into the same form after the transformation of the field as before:
ih
)’)11
-
a
at
=
-
[Z:m(
p - - A
+e$‘+ V
]I$’),
(6.57)
where
This form invariance of the Schrddinger equation means it is gauge invariant. (According to contemporary physics, gauge invariance, like Lorentz invariance,
38
[I, § 6
ATOMS IN STRONG FIELDS
is a fundamental requirement to be satisfied by any correct theory.) Notice that the change from the A * p Hamiltonian to the r * E form discussed earlier amounts to a gauge transformation with gauge function x = - r * A . (It is not, strictly speaking, a gauge transformation, since V x A = 0 in the dipole approximation employed.) Now consider the (exact) transition amplitude Afi(0 =
(fl W )l i >
=
(flICl(t)> =
(f(0)I
W)>.
(6.59)
After a gauge transformation we obtain Aji(t) = (S'(0)I $'(t)> = (f(0)I ~ X [P - i e ~ ( rO)/hI , ~ X [iex(r, P t)/hI I + ( t ) ) =
(f(0)I
W ) ) = Afi(4
(6.60)
I
if we assume, for simplicity, that x vanishes at the initial and final times. In this case we can effectively forget about having to transform the state vector. (Recall the discussion at the end of Q 6.1.) Thus the exact transition amplitude is gauge invariant, without the need to transform the state vector. This is not true of the approximate amplitude (6.34) (Antunes Net0 and Davidovich [ 19841). To see this as simply as possible, let us return to (6.28), from which we derived (6.34). It is easy to show that for the gauge transformations under consideration the evolution operator U,,(t) transforms to U;(t) = exp(ieX/h)U,(t), and so Afi(t)+ Aji(t) = -i
h
{
OU
dt'
(fl
UAt(t')V 1 t,bj(t'>)
-m
+ AfiW .
(6.61)
This lack of gauge invariance in this sense explains why different results are obtained when the Keldysh approximation is implemented with the A p and r * E forms of the interaction Hamiltonian (Antunes Net0 and Davidovich [ 19841, Milonni [ 19881, Milonni and Ackerhalt [ 19891). For weak fields the evolution operator U, may effectively be removed from (6.32), which is the same as replacing the Volkov state in (6.34) by the free-electron, plane-wave state I f ) . In this limit the Keldysh approximation reduces to conventional lowest-order perturbation theory in the interaction Hamiltonian If,, and no difficulty occurs with gauge invariance. An alternative to the lowest-order Keldysh approximation for atoms in strong fields is to iterate (6.25) consistently in the usual fashion of perturbation theory
1. I 71
ABOVE-THRESHOLD IONIZATION: EXPERIMENTS
39
in the interaction picture. The resulting “Dyson expansion” for the transition amplitude is
x (,f Uo(t) UJ(t’)V”(t’)UJ(t’’)VU,(t”) ti) +
. . (6.62)
This conventional form of perturbation theory is implicit in the Kroll-Watson theory for the scattering of electrons in the presence of strong fields (Kroll and Watson [ 19731). It is simple to check that the perturbation theory based on (6.62) is gauge invariant at every order in the Dyson expansion. In fact, by manipulations involving different exact forms of the time evolution operator, it is possible to show that the Keldysh amplitude (6.34) can actually be cancelled identically in a consistent perturbation theory leading to (6.62) (Antunes Net0 and Davidovich [ 19841, Milonni [ 19881, Milonni and Ackerhalt [ 19891). The Keldysh approximation thus appears to have some serious shortcomings. Nevertheless, it remains a valuable benchmark in the theory of abovethreshold ionization, to which we next turn our attention. We emphasize again that the Keldysh approximation assumes that the atom remains in the ground state with a high probability, that it ignores in effect all excited states, and that it treats the outgoing electron as a free particle, unperturbed by the residual ion.
8 7. Above-Threshold Ionization: Experiments The traditional sort of photoionization experiment has been based on the detection and counting of ions, as discussed in 5 3. During the past decade, however, experiments have been performed in which the photoelectrons produced by very intense fields have been analyzed. In particular, the energy spectra of the photoelectrons have been measured as a function of laser intensity. A prominent feature of the observations is that the electron energy spectra have peaks at integral multiples of the photon energy h a , as predicted by eq. (6.44). As can be inferred from that equation, each peak is associated with a different order of multiphoton absorption. Because the electron evidently absorbs more photons than it needs just for the ionization threshold, this peak
40
ATOMS IN STRONG FIELDS
[I. § 7
structure gives rise to the term above-threshold ionization (ATI); in a sense the electron continues to absorb photons after it has reached the continuum. This section will summarize some of the main experimental data on ATI. The first observation of what is now called AT1 was reported by Agostini, Fabre, Mainfray, Petite and Rahman [1979]. In these experiments the output from a Q-switched Nd:glass laser (12 ns, 5 mJ pulses) was amplified to a pulse energy of about 2 J and focused. The second harmonic of the 1.06 pm radiation (ho= 1.17 eV) was (sometimes) generated by passing the pulse through a KDP crystal. A magnetically shielded interaction chamber contained Xe at a pressure z 5 x 10 Torr, and the energy spectrum of the electrons produced by the intense pulse of radiation in the focal region was analyzed. The electron energy distribution was determined using a retarding-potential technique, the essence of which is as follows. The electrons produced at the focal region diffuse inside the interaction chamber, and some escape through a high-transparency mesh. The chamber was set to a (variable) voltage VR with respect to a second, grounded mesh, thus introducing a potential barrier VR for the electrons to cross before reaching a high-gain electron multiplier. The signal from the electron multiplier tube for a retarding potential VR was thus proportional to d E N(E), and the electron energy distribution N(E) was then determined by numerical differentiation of the data. Figure 5 shows electron energy spectra determined both with and without ~
Fig. 5. Photoelectron energy spectra measured by Agostini et al. for photon energies of 1.17 eV (triangles) and 2.34 eV (circles). The second peak in the 2.34 eV case is associated with abovethreshold ionization. (From Agostini, Fabre, Mainfray, Petite and Rahman [ 19791.)
1, I 71
ABOVE-THRESHOLD IONIZATION: EXPERIMENTS
41
frequency-doubling the 1.06 pm pulse. In the frequency-doubled case a peak occurs near 2 eV, corresponding to the energy of an electron produced by six-photon ionization of Xe: 6 x (2.34 eV) - (ionization potential 12.127 eV for Xe) = 1.91 eV. However, a peak also occurs at around 4.5 eV, close to the energy 7 x (2.34) - 12.127 = 4.25 eV and corresponding to photon absorption above the six-photon ionization threshold (“above-threshold ionization”). The data in fig. 5 without frequency doubling show a broad peak near 4 eV, but no additional peak as in the frequency-doubled case. The 4eV peak correlated well with the maximum energy expected to be acquired by a freed electron due to the ponderomotive force ( 0 5). In the second-harmonic case, by contrast, the intensity was considerably lower, and consequently the energy associated with the ponderomotive force was much smaller (estimated by Agostini, Fabre, Mainfray, Petite and Rahman [ 19791 to be only about 0.25 eV). This difference was invoked to explain why two peaks could be resolved in the frequency-doubled case. The relative heights of these peaks correlated well with the prediction of the Kroll-Watson theory of stimulated free-free transitions (cf. eq. (5.9)), thus supporting the idea that the abovethreshold photon absorption could be assigned to the free electrons in the presence of the residual ions. Many more AT1 peaks were reported by Kruit, Kimman, Muller and van der Wiel [ 19831. In their experiments the electron energies were measured by a time-of-flight technique using a confining magnetic field in a 50 cm tube. A Nd :YAG laser was used with second- and third-harmonic generation to produce photons with energies of 2.34 and 3.51 eV as well as 1.17 eV. Once again Xe, because of its large ionization potential, was used as the target atom. Figure 6 shows photoelectron energy spectra for I = 1.06 pm and a succession of different pulse energies. The peak assignments indicated at the top of the figure correspond to the ionization potentials for the P3,2 core (12.127 eV) and the P,,, core (13.44 eV) of Xe. Note that the peaks correspond to 11-photon to 19-photon absorption. Note also that with increasing pulse energy not only do more peaks occur, but the first few peaks become suppressed. This suppression was confirmed in the experiments of Lompre, L’Huillier, Mainfray and Manus [1985], who found that the first 30ATI peaks were suppressed when He was ionized with 1.06 pm radiation with an intensity of about l O I 5 W ern-,. The suppression is related to the ponderomotive potential, as discussed in the following section. Another effect could suppress the low-energy peaks. The high-energy photoelectrons quickly leave the focal region, leaving behind slower electrons that could be trapped or otherwise deviated in their motion by the positive ions
42
ATOMS IN STRONG FIELDS
PI12
I
I
1
lJ1
'41
1
1
15 p3/2
"=I2,
I
1
I1
I6 1
1
I 18
I
I
19
1
I
20 1
I
3
-
v)
I-2 z
? I m a
< -0 A
<
33 c
v)
2 1
0
0 ELECTRON ENERGY (eV 1 Fig. 6. AT1 spectra measured by Kruit et at. (see text). Note the suppression of the low-energy peaks with increasing intensity. (From Kruit, Kimman, Muller and van der Wiel [ 19831.)
1. J 71
ABOVE-THRESHOLD IONIZATION. EXPERIMENTS
43
(Crance [ 19861). This “space charge” effect, however, was effectively eliminated in experiments by Yergeau, Petite and Agostini [1986] in which the charge density in the focal region was very small. Even in the absence of significant space charge, suppression of low-order peaks was observed, as in the experiments of Kruit, Kimman, Muller and van der Wiel [ 19831. Suppression of the first peak was found at an intensity of about 5 x 10l2W cm-2, leading to the conclusion that the suppression could not be purely a space charge effect. (For circularly polarized radiation the suppression was observed at slightly lower values of the intensity than for linear polarization.) Since the intensity at which the suppression is observed is relatively small, it was also concluded that the suppression could not be related to a simple “depletion saturation” of the ionization signal as the ionization probability approaches unity. Experimental studies by McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871 and Lompre, Mainfray, Manus and Kupersztych [ 19871 clarified some aspects of ATI, especially in connection with the ponderomotive force and the effect of laser polarization. They studied the energy and angular distributions of the photoelectrons as a function of laser power and polarization under conditions where the spatial and temporal profiles of the pulses were well characterized. Figure 7 shows the energy spectra obtained for linearly and
-
(a)
N
FJ \
3
0
1.1
-1
1.9
-25
c .-
-100
-100 . ;
;4.8
<-
2‘
2.5
3.2
v ~
2
6.4
$150 0
2
4
6
8
1012
-150
electron energy (ev)
-1
-50 +loo +300
a.
i.
-440 10
15
-700
electron energy (ev)
Fig. 7. AT1 spectra measured by McIlrath et al. for (a) linear and (b) circular polarization. (From McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871.)
44
ATOMS IN STRONG FIELDS
[I, § 7
circularly polarized 1.06 pm radiation. Note the substantial suppression of the low-energy peaks in the case of circular polarization. It can also be concluded from figs. 6 and 7 that any intensity-dependent Stark shifts in the peaks are small on an eV energy scale (Kruit, Kimman, Muller and van der Wiel [ 19831, McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871). With increasing intensity the first peak broadens and is eventually swamped by a rising continuum background. Since even a pulse with high peak intensity has low-intensity spatial wings, the low-energy electrons are still present in the high-intensity case, but their presence is masked by the background, i.e., “the low-energy electrons do not disappear in any absolute sense” (McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871). The increasing prominence of the high-energy peaks as the intensity increases is also seen in fig. 8, in which the saturation behavior, with a saturation intensity of about 2.8 x l O I 3 W cm-*, is presumably due to ion depletion in the focal region (depletion saturation). Increasing the intensity tends to broaden all the peaks, especially the lower ones, and to increase the peaks’ relative sizes. However, the relative sizes of the peaks shown in fig. 7 seem to remain about the same. McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871 measured angular distributions of the photoelectrons as a function of laser polarization, using the detection geometry sketched in fig. 9 and rotating the polarization direction. At low intensities this distribution is strongly peaked in the direction
10‘
I
, , , , , , .T1
Fig. 8. Electron signal versus laser energy measured by McIlrath et al. with the radiation polarized linearly along the detector axis. (From McIlrath, Bucksbaum, Freeman and Bashkansky [1987].)
1, § 81
ABOVE-THRESHOLD IONIZATION: THEORY
45
r
Fig. 9. Detection geometry used by McIlrath et al. to measure the angular distribution of AT1 photoelectrons as a function of laser polarization. (From McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871.)
of the field polarization, but becomes increasingly isotropic with increasing intensity. (The angular distributions were measured over a wide range of Xe densities to ensure that they did not depend on space charge.) This survey of AT1 experiments, although incomplete, is adequate for the purposes of this review. Before turning to the theory, we summarize the prominent experimental results. (1) The photoelectron energy spectra show a series of peaks separated by the photon energy Aw (figs. 5-7). ( 2 ) With increasing laser intensity the low-energy peaks are suppressed compared with the higher-energy peaks (figs. 6 and 7). (3) At low intensities the angular distribution ofthe photoelectrons is strongly peaked in the direction of the field polarization, but it becomes increasingly isotropic at higher intensities. (4) Observable differences occur when linearly and circularly polarized fields are used. Some of these have been noted earlier, and others will be discussed in the following section.
0 8.
Above-Threshold Ionization: Theory
The Keldysh theory of strong-field ionization, and its extensions by Reiss, were described in 0 6. The basic approximation there was to treat the binding potentialas a perturbation to the strong external field acting on the electron. The theory leads unambiguously to multiphoton peaks in the photoelectron spectrum; recall eq. (6.44). We will begin our discussion of AT1 theory with a comparison of this predicted peak structure with experiment.
46
ATOMS IN STRONG FIELDS
8.1. PREDICTIONS OF KELDYSH-REISS THEORY: AT1 PEAKS AND
POLARIZATION EFFECTS
From (6.48) and some straightforward algebra one derives the differential form of the transition rate obtained by Reiss [ 19801:
-dR_ - (~onst.)w'/~1 00
dO
( N - z ) ~( N - 2 -
1 GN(p)I2,
N=No
(8.1)
where
z = e2Ai/4mhwc2 = O,/hw is the ponderomotive potential a, photon energy ho,
(8.2) =
e2Ai/4mc2 = e2E:/4m02 divided by the
(8.3)
Io/ho
Eg =
is the ionization potential divided by the photon energy, and NO =
where [ *
*
[ z + EB 1
(8.4)
9
] denotes the greatest integer part, and for linear polarization
cN(p)
=
$i(p)
JN+2n(eAO'P/hwmc)Jn(e2Ai/8homc2).
(8.5)
n
As noted earlier, the algebra is simpler in the case of circular polarization. In this case one obtains (8.1) with
GN(p) = $ii(P)JN([eAOp/homcl
9
(8.6)
where 0 is the angle between p and the field propagation direction, and in both (8.5) and (8.6) p is constrained by the energy conservation condition (6.44):
p2/2m = ( N - z - E , ) ~ w
(8.7)
in the present (Reiss) notation. Reiss [ 1987al compared the prediction (8.1) for circular polarization with the experimental results on the photoelectron energy spectra. For &( p ) he assumes for the ground state of Xe a 5p hydrogenic wave function. Figures 10 and 1 1 show Reiss's comparison with the experimental data of Yergeau, Petite and Agostini [ 19861 and Bucksbaum, Bashkansky, Freeman, McIlrath and DiMauro [ 19861, respectively. The intensities assumed in the calculations for figs. 10 and 1 1 correspond to z = 0.7 and z = 2.1, respectively. In each case the assumed intensity is larger than the stated
47
ABOVE-THRESHOLD IONIZATION. THEORY
1. § 81
-
In-
1
I
1
1
I
1
-
-
2
11
LLI
12
13 14 15 16 Photon o r d e r , n
17
-I
v
al + 0
C
0 .-
.-+-In C
L
b
11
13
15 Photon order, n
17
Fig. 10. Comparison of the predictions of (b) Keldysh-Reiss theory with (a) the experimental results of Yergeau, Petite and Agostini [ 19861 for the photoelectron spectra, for circularly polarized light. Note that the lower part of each bar in figs. 10-16 is for the ionization channel of order (minimum) 11, whereas the upper part is for the channel of order 12. (From Reiss [ 1987al.)
experimental intensity by a factor of about two. (Reiss notes that the experimental intensities were not accurately determined.) Figure 12 shows the experimental results of fig. 1 l a with the theoretical results of fig. 1l b represented by horizontal bars superimposed on the experimental peaks; the peak heights are scaled to give a perfect match at N = 18. The agreement between experiment and the Keldysh theory shown in fig. 12 is clearly quite good, and it is better for the higher peaks. This is to be expected, since the theory ignores the distortion of the photoelectron wave function by the Coulomb potential of the residual ion, and this distortion is less important at the larger electron energies. For the same reason the theory should work better at larger intensities (cf. figs. 10 and 11).
48
ATOMS IN STRONG FIELDS
0
.-
4
$
11 12
13 14 15 16 17 18 19 2 0 21 2 2 Photon o r d e r , n
11
13
15 17 19 Photon o r d e r , n
21
23
Fig. 11. As in fig. 10, but the comparison is with the experimental results of Bucksbaum, Bashkansky, Freeman, McIlrath and DiMauro [ 19861. (From Reiss [ 1987al.)
,-. m
r C
2
h
2 c
.n
b
v
0 0, C
e 2-w
11
13
15 17 Photon o r d e r , n
19
21
Fig. 12. Data shown in fig. 11 with the theoretical predictions represented by horizontal bars superposed on the experimental peaks. The theoretical predictions are scaled to match exactly the experimental peak at N = 18. (From Reiss [1987a].)
ABOVE-THRESHOLD IONIZATION. THEORY
49
200
-
160
I
u
!A Q
120
Q +
0
5
00
.-
.-+ !A
e C
40
I-
’
Ilr, 19 20 21 22 23
l l 12 13 14 15 16 17 I €
n, Photon order
Fig. 13. Theoretical results of Reiss corresponding to fig. 1 1 , but with z = 1.9 and with a Yukawa wave function for the initial bound state. (From Reiss [1987b].)
Figure 13 shows theoretical results of Reiss [ 1987bl corresponding again to the experimental results of fig. 1la, this time assuming z = 1.9 and a Yukawa wave function for the initial bound state. Figure 14 compares the theoretical predictions with z = 0.1 for linear and circular polarizations. Note the suppression of the N = 11 peak in the case of circular polarization, and the dominance of higher-order peaks compared with linear polarization. Figures 15 and 16 show the corresponding results for z = 1 and z = 10, respectively. Important experimental features of the AT1 spectra appear to be well accounted for by the Keldysh theory. These include the suppression of the lowest-order peaks with increasing intensity, the more pronounced suppression of these lowest-order peaks with circular polarization, the greater number of peaks observed with circular polarization, and the bell-shaped envelope of the peaks with circular polarization. With circularly polarized radiation the photoelectron is produced in a state of high angular momentum ( zNA), as required by conservation of total angular momentum. (This is not necessarily so, of course, with linear polarization.) The greater suppression of the lowest-order peaks observed with circular polarization was described by Reiss in terms of a restriction on the available angular momentum channels. For the higher-order peaks coming into play at higher intensities, the greater number of photons involved allows for more possible angular momentum channels, and the ionization rates for circular polarization thus “catch up” with those for linear polarization as the intensity increases (Reiss [1980]). This picture was also used to explain qualitatively the
50
ATOMS
IN STRONG FIELDS
n
11
12
14
13
15
16
n
Fig. 14. Predictions of Keldysh-Reiss theory for z = 0.1 with (a) linear and (b) circular polarization. (From Reiss [1987b].)
bell-shaped envelope of the peaks observed with circular polarization (Reiss [ 1987a1). An explanation of the effect of circular polarization was also given by Bucksbaum, Bashkansky, Freeman, McIlrath and DiMauro [ 19861. For quantitative considerations they base their argument on the Bebb-Gold average frequency approximation ($ 3), in which the N-photon ionization amplitude is proportional to the overlap of r'"IG;.(r)with the final (continuum) state. The basic idea is that with circular polarization the photoelectron is produced in a high angular momentum state, which implies a small overlap and therefore a small N-photon ionization amplitude. If N is large, however, the overlap eventually becomes significant, since r"lC/,(r) peaks at larger values of r as N
51
ABOVE-THRESHOLD IONIZATION: THEORY
1.5 81
n
11
12
13
14
15
16
17
18
19
20
n
Fig. 15. As in fig. 14, but with z
=
1.0. (From Reiss [1987b].)
increases. Thus the low-order peaks with circular polarization may be suppressed, whereas the higher-order ones eventually “turn on”.
8.2. THE PONDEROMOTIVE POTENTIAL
The ponderomotive potential 0,= e2At/4mc2 = e2Ei/4mw2first appeared in our discussion of Volkov states in Q 5. In the Keldysh-Reiss theory it appears in the expression (6.44), or equivalently (8.7), for the kinetic energy of the photoelectron produced in N-photon ionization. In particular, the consequent appearance of the ponderomotive term in (8.4) means that No increases with increasing intensity, i.e., the first peak in the electron energy spectrum should shift to higher energy as the intensity of the radiation increases. This shift
52
ATOMS IN STRONG FIELDS
n
0 *
,o
s
._ t Ill
C
,o
I-
ll
15
19
23
27
31
35
39
43
n
Fig. 16. As in fig. 14, but with z
=
10.0. (From Reiss [1987b].)
of the lowest-orderpeak with increasing intensity evidently accountsfor the observed suppression of the low-order peaks with increasing intensity. However, there is more to the story than this. The expression (6.44) for the photoelectron energy transcends any details of the Keldysh theory. More to the point, (6.44) is simply a statement of energy conservation; the initial electron-plus-field energy is Nhw - I,, the final energy is ( p - ( e / c ) A , cos o t ) * / 2 m , and conservation of energy then requires (6.44) when an average is taken over a field cycle. Thus it is not surprising that several authors (Muller, Tip and van der Wiel [1983], Mittleman [1984], Sz6ke [ 19851, Pan, Armstrong and Eberly [ 19861) discussed the role of the ponderomotive potential in the low-order peak suppression in ways that are completely independent of the Keldysh approach based on the use of Volkov
1, J 81
ABOVE-THRESHOLD IONIZATION: THEORY
53
states. One can say that, as a consequence of energy conservation, the ionization potential is effectively raised from I , to I,, + e2E,2/4m02. If the ionization potential is increased by the field, so that the kinetic energy of the photoelectron is given by (6.44), then evidently the positions of all the AT1 peaks should be shifted downwards by the ponderomotive potential a,. For I z I O l 3 W cm-2, R, x 1 eV, and so the shift should certainly be observable. However, such shifts in the ATIpeaks are not usually observed. (Recall the discussion in 0 7.) The reason is simple: the photoelectron energies are measured outside the focal region of the laser, whereas (6.44) describes an electron in the presence of the field. As the electron moves from a region with E, # 0 to one with E, = 0, its directed kinetic energy increases by the ponderomotive potential R,, much like the increase in kinetic energy of a ball rolling down a hill. This increased kinetic energy then just cancels the decrease associated with the increased ionization potential for an electron in the field, and the net effect is that the ponderomotive potential does not produce any sh$t in the ATIpeaks (Muller, Tip and van der Wiel [ 19831). In other words, the effect of the ponderomotive potential on the AT1 peak structure is observable only indirectly by means of the low-order peak suppression; the observed peaks are not shifted by the ponderomotive potential. A more accurate application of energy conservation takes into account the field-induced level shifts of the initial bound state and the final ionic state. However, the absence of any significant shifts in the observed ATI-peaks (fi 7) can be taken as evidence that these additional bound-state shifts are typically small. In fact, Kibble, Salam and Strathdee [1975] first showed that the A 2 term is cancelled in a low-frequency approximation by the second-order contribution from the A p interaction. This explained the null results of experiments for which Reiss [I9661 had argued that A2 shifts should be observable in bound-bound transitions. Much later, Pan, Armstrong and Eberly [ 19861, and more generally Milonni and Ackerhalt [ 19891, showed in the context of AT1 theory that a second-order contribution from the A * p term to the ground-state level shift has a term that cancels the A* (ponderomotive) contribution in a low-frequency approximation valid for tightly bound states, and the net level shift is approximately just the DC Stark shift ( - 51 aE;), which is small. (This is discussed further later.) Note that the level shift of the initial bound state is ignored in the usual formulation of the Keldysh approximation (recall the comment after eq. (6.27)), but it is not difficult to include in a modified version of the theory (Milonni and Ackerhalt [ 19891). Although the ponderomotive potential does not alter the positions of the AT1
-
54
ATOMS IN STRONG FIELDS
[I. § 8
peaks, it appears to have an observable effect on the angular distribution of the photoelectrons. Specifically, the scattering of the electrons by the ponderomotive potential is believed to be responsible for the intensity dependence of the angular distributions (McIlrath, Bucksbaum, Freeman and Bashkansky [ 19871, Freeman, McIlrath, Bucksbaum and Bashkansky [ 19861). Recall from the discussion in 0 7 that the angular distribution for low intensities is strongly peaked along the field polarization direction, but that it becomes increasingly isotropic with increasing intensity. Freeman, McIlrath, Bucksbaum and Bashkansky [ 19861 accounted for this observation using a model in which it is assumed that (a) the initial angular distribution of the photoelectrons is independent of intensity, and given by the measured low-intensity result; (b) the electrons belonging to a given AT1 peak are produced only along equal-intensity surfaces of the focused laser beam ; (c) the initial kinetic energy of the electrons is the measured kinetic energy minus the ponderomotive potential corresponding to the intensity producing the ionization (recall the preceding discussion); and (d) the radial distribution of intensity at a distance z from the focus is given by I(r, z , t ) = [Io/w2(z)lexp [ - r2/w2(z)lexp { - [ ( t - z / c ) / z I 2 >,
(8.8)
where w ( z ) is the spot size of the Gaussian beam ( w ( z ) = w o , / m ; wo is the spot size at the waist of the beam and zo is the Rayleigh range) and z is the pulse duration. The classical electron trajectories were computed and the angular distributions were determined by averaging over the angles of all the trajectories followed. The only adjustable parameter was the intensity at which the electrons associated with each AT1 peak were stripped from the atom. Freeman, McIlrath, Bucksbaum and Bashkansky [ 19861 report good agreement between such a simulation and their experimental observations, including the effect of using an elliptical rather than a circular focus. The ponderomotive potential in the context of AT1 deserves further discussion here, especially since it has been a source of some controversy in the interpretation of the experiments. We note first that, strictly speaking, the ponderomotive potential appears incorrectly in expressions such as (8.7). In particular, the notion that the ponderomotive potential effectively raises the ionization potential can be misleading because, in the dipole approximation, the A2 term in the Hamiltonian shifts every level, bound and continuum, by exactly the same amount, and therefore, can have no effect on the ionization potentials (Milonni and Ackerhalt [ 19891).
1. § 81
55
ABOVE-THRESHOLD IONIZATION: THEORY
Any shifts of the AT1 peaks can therefore arise only from level shifts associated with the A p term in the Hamiltonian. To second order this shift is Ipim(’[(wim-w)-l+(wim+w)-l]
-(-)c
1 eA,
=
6h
’1 w i i Ixirn12[(aim - m)-’
+ (wim+ w)-’]
(8.9)
m
for the energy level i. The ponderomotive shift of each level can be written, using the dipole sum rule, in the following form for the case of a circularly polarized field :
a,=---”e2A2
2m
2mc’ 3 h
1 om(Ix,,,I’ m
=
3h
omi1
~ ~ ~ 1 ’ . (8.10)
The total (second-order) shift of the energy level i is thus AEi
=
+ 8,=
=
- p1 i ( o ) E , z ,
l x i m l Z[w&/(w& - w’) - mi,]
(8.11)
where E, is the electric field amplitude and cli(w) is the polarizability at frequency w for an atom in state i. Of course (8.11) may be derived from simple classical considerations (Milonni and Eberly [ 19781). As noted earlier, in a low-frequency approximation fii has a zero-frequency part that exactly cancels the ponderomotive shift. One obtains (8.11) with a , ( o ) replaced by the static polarizability. It is worth noting that (8.1 1) holds for all frequencies, and is simply the quadratic Stark shift of level i. Based on these results, let us now reconsider the role of the ponderomotive potential. For further details the reader is referred to Milonni and Ackerhalt [ 19891. Regarding the suppression of low-energy peaks with increasing intensity, consider the implications of energy conservation when n photons are absorbed. The total initial energy is Ei + Di + 8, + nhw, where we include the Stark shift AEi = Sli + 62,. The total final energy is p 2 / 2 m + sZ0, where
56
[I, § 8
ATOMS IN STRONG FIELDS
we include the ponderomotive shift of the freed electron. Thus, p 2 / 2 m = n h o + Ei + = n h o - Ii + Oi,where Ii is the ionization potential of the initial bound state. This result differs from (8.7) in that fii has replaced the ponderomotive potential; i.e. the “threshold shift” is not 62, but 4, and it is SZi rather than the ponderomotive potential 62, that is responsiblefor the suppression of the low-energy peaks. It is easily shown that at low field frequencies E - a,, in which case (8.7)is a good approximation in practice. In principle can be positive at some field frequencies, implying that the ionization potentials can effectively be lowered. In such cases no suppression of lowenergy peaks should occur. It is true, of course, that photoelectrons pick up an additional directed kinetic energy 62, as they leave the field, as was assumed earlier. This means that the detected electrons should have the energy p 2 / 2 m = nh o - Ii + plus the ponderomotive energy no,i.e. nhw + Ei + AEi according to the preceding. In other words, the observedpeaks are shifted simply by the Stark shgt of the initial bound state. These shifts are often small, which is consistent with the experimental observations.
8.3. NUMERICAL EXPERIMENTS ON SIMPLIFIED MODELS
Although it appears at this time that AT1 phenomena are qualitatively well understood, no generally accepted, rigorous theory exists. Conventional perturbation theories become unwieldy at the high intensities of interest, and the major alternative, the Keldysh approximation, becomes suspect for substantial ionization probabilities and, furthermore, suffers from difficulties with the gauge used in its usual (lowest-order) implementation. The problem is to solve the time-dependent Schr6dinger equation with an oscillating applied field. For a one-electron atom this is
V2 + V ( r ) - ere En cos ot $(r, t ) = i h
a -
$(r, t )
(8.12)
at
in the r * E gauge. The most common approach to the problem is to expand $(r, t ) in the complete set of eigenfunctions { $,,(r)} of the time-independent, field-free SchrBdinger equation :
(8.13) Equation (8.12)then leads to a set of first-order, ordinary differential equations
I 9
§ 81
51
ABOVE-THRESHOLD IONIZATION: THEORY
for the probability amplitudes a,(t). For some problems in which only a small number of bound states have significant occupation probabilities, these equations can be solved analytically; for instance, for a two-level atom the equations for the probability amplitudes in the rotating-wave approximation can be written in the form of the optical Bloch equations (Allen and Eberly [ 19751). For ionization problems the set { a ( r ) } must generally include a continuum component, or some approximation to the continuum. One is then typically faced with a large number of coupled differential or algebraic equations to solve on a computer. Recently, several groups have made a direct numerical attack on the timedependent Schrddinger partial differential equation, in effect obtaining an “exact” solution for the time-dependent wave function. A considerable computational simplification in this approach is realized if a one-dimensional model is employed, i.e. if (8.12) is replaced by -
+ V ( x ) - e x E , cos mt
+(x, t ) = ih
a -
at
+(x, t ) .
(8.14)
Numerical solutions of the one-dimensional, time-dependent Schrddinger equation are not difficult, and date back at least as far as the work of Goldberg, Schey and Schwartz [ 19671. Koonin and Meredith [ 19901 provide a listing of a computer program that can be used to solve (8.14). For ATI, of course, the only real justification for (8.14) is the excuse of computational economy, although related problems occur in which a onedimensional model is, at least in part, physically justifiable. An example is a diatomic molecule in a field, for which Goggin and Milonni [ 1988a,b] have studied photodissociation by solving the time-dependent Schrddinger equation with a Morse binding potential appropriate to the H F molecule. Another example is the one-dimensional “hydrogen atom” describing the surface-state electron in an applied field, which we shall find later to be directly relevant to certain experiments. For ATI, one-dimensional models can account for the main features of ATI, such as the multiphoton peaks in the photoelectron spectra, and so they can be used, among other things, to gauge the accuracy of various approximations (e.g.. the Keldysh approximation). We therefore turn our attention to such models. A numerical solution of the time-dependent Schrddinger equation was reported some time ago by Geltman [ 19771,who assumed a simple delta-function binding potential. He compared his results with predictions of the Keldysh approximation as used by Perelomov, Popov and Terent’ev [ 19661 for this
58
ATOMS IN S’I’RONG FIELDS
11, § 8
model with a value of the Keldysh parameter 7 % 1, and he found “gross differences”. In particular, the tunneling theory (Perelomov, Popov and Terent’ev [ 19661 predicted times for 10% ionization that could be too large by several orders of magnitude. Numerical solutions of the SchrBdinger equation (8.11) with applications to AT1 were carried out by Cerjan and Kosloff [1987], Javanainen and Eberly [ 19881, and Collins and Merts [ 19881. These studies included comparisons with the predictions of the Keldysh theory. Cerjan and Kosloff [ 19871 consider a one-dimensional model with a cut-off harmonic oscillator potential. In their numerical experiments on ionization rates versus field intensity and frequency, they found that the Keldysh approximation predicts transition probabilities that are several orders of magnitude too small. Furthermore, it appears from their work that the detailed time dependence of the ionization probabilities cannot be accounted for within the framework of the Keldysh approximation. Javanainen and Eberly [ 19881 assume the binding potential V ( x ) = - 1 / , , f m (atomic units). As in the work of Cerjan and Kosloff [ 19871 the potential is centered at x = 0, so that there is no artificial permanent dipole moment, as would occur for a one-sided potential. Photoelectron spectra were identified with the projection P ( E ) = I ($(x; E)I $(x, t ) ) I *, where $(x; E ) is a field-free wave function of (unperturbed) energy E, and E takes on values both below and above the ionization potential 1, = 0.6698 (atomic units) for the ground state. Figure 17 shows the computed spectra for three different field strengths with a scaled driving o = 0.148, corresponding to five-photon ionization frequency (0.6698/0.148 = 4.53). The AT1 peaks were well resolved after only two field cycles. Note the red-shift of the first peak with increasing intensity, and its eventual disappearance as it is shifted below the ionization threshold, as expected from the discussion earlier of the ponderomotive potential. Comparisons of such numerically computed AT1 spectra were made with the peak structure predicted by the Keldysh theory, and it was concluded that “present versions of the Keldysh model are not accurately representative of an atomic electron”, and that “these simplest models are not supported except in some qualitative features”. Figure 18 shows one such comparison. It should also be noted that Antunes Neto, Davidovich and Marchesin [ 19851 compared the predictions of the lowest-order Keldysh theory for the total ionization probability with numerical computations based on Geltman’s delta-function model, and reported substantial differences. Figure 19b shows a spectrum computed by Eberly and Javanainen [ 19881
1, § 81
ABOVE-THRESHOLD IONIZATION: THEORY
0
2 E/w
59
4
Fig. 17. Photoelectron spectra computed by Javanainen and Eberly for a one-dimensional model system. Assumed field strengths in atomic units are, from top to bottom, E, = 0.05,0.07071, and 0.085 and w = 0.148. (From Javanainen and Eberly [1988].)
1
2
E
Fig. 18. Photoelectron spectra predicted by (a) Keldysh-Reiss theory and (b) numerical computations on the one-dimensional model of Javanainen and Eberly for w = 0.07 and E , = 0.07071. (From Javanainen and Eberly [1988].)
indicating “atom-specific bound-level ‘multiplets’ in AT1 spectra”. In figure 19b the computed spectrum was obtained under the assumption of a smooth-pulse excitation. The “multiplet” structure is associated with the excitation of other bound levels, as indicated in fig. 19a. Note that the lowest-order Keldysh-Reiss
60
ATOMS IN STRONG FIELDS
0
1 0
1
wm o
2
3
Fig. 19. Example of AT1 multiplet structure computed in a one-dimensional model by Eberly and Javanainen, showing the role ofexcited bound states. Part (a) shows the low-lying atomic energy levels, and the arrows indicate laser photons that excite the atomic electron into positive energy states above the ionization threshold. A direct two-photon channel from the ground state is shown, as well as several one-photon channels from excited intermediate states. Part (b) shows the resulting photoelectron spectrum. There are three sets of AT1 peaks, and each one shows structure that can be assigned to the direct two-photon process (highest peak) and to the one-photon ionizations of the excited odd-parity bound states, labelled according to the diagram in (a). (From Eberly and Javanainen [1988].)
approximation does not deal with such excited bound levels. (Recall the approximation (6.27) used in deriving the Keldysh transition amplitude.) The numerical study of Collins and Merts [ 19881, which assumes a square well for V ( x ) , also casts serious doubt on any widespread applicability of the Keldysh approximation. As in the discussion in 6 , these authors regard the (lowest-order) Keldysh theory as a perturbative approximation that assumes a small probability of removing the atom from its initial state. Since they found substantial ionization after only a few field periods for intensities above about lOI3 W cm-2, they conclude that “we must be cautious about applying standard time-dependent perturbation theory or first-order forms as those of Keldysh and Reiss” (Collins and Merts [ 19881). They also found a red-shifting of the AT1 peaks caused by the ponderomotive potential, and a suppression of the lowest peaks with increasing field intensity. Under resonance ionization conditions they observed multiplet structure in the AT1 peaks due to excitedstate ionization, as in the studies of Eberly and Javanainen. Kulander [ 1987a,b, 19881 developed a computer program for numerically solving the fully three-dimensional, time-dependent Schrddinger equation (8.12), and has obtained results in good agreement with some experiments on the multiphoton ionization of Xe. However, the method at present “does not lend itself well to the determination of the final-state electron energy distributions because of the finite size of the grid”, and thus far has been applied
1, I 91
HIGH-ORDER H A R M O N I C G E N E R A T I O N
61
mainly to the computation of total ionization rates and to aspects of the preionization dynamics. Thus, numerical solutions of simple model systems have been useful in assessing the relative efficacy of the analytical approximations used in describing ATI. They also reveal that the qualitative features of AT1 can be recovered by a myriad of techniques and models, which suggests that each model may, in fact, describe only a part of the “true” AT1 mechanism. The idea was explored numerically by Sundaram and Armstrong [ 1988, 19901 in the context of a simple one-dimensional, one-sided potential for a surface-state electron (Jensen [ 19841). The calculation was done using a hydrogenic basis (cf. eq. (8.13)), where the continuum states were normalized within a large box (see, for instance, Cowan [ 19811). As with other models, the principal features of AT1 were recovered. However, the advantage of the basis-set method is the option to turn off the effects of certain couplings or interactions. Using this facility the role of the excited bound states (Sundaram and Armstrong [ 19881) as well as that of interference effects between different ionization channels (Sundaram and Armstrong [1990]) in affecting the features of AT1 were addressed, and a qualitative picture of AT1 was constructed. The essential nonperturbative feature was the inclusion of energy nonconserving steps in the multiphoton excitation mechanism. As in other models, only plausibility arguments, for the basic AT1 mechanism, were presented, although a more quantitative assessment of interference effects (within the context of a different model) has recently appeared (Reed and Burnett [ 19901). It might also be noted that the use of a one-sided potential, for which the reflection symmetry is broken, illustrates that parity considerations are not essential in describing ATI, including the multiplet structure described earlier. Numerical integrations of the time-dependent SchrOdinger equation will be discussed again in relation to harmonic generation and stabilization in strongfield ionization.
6 9. High-Order Harmonic Generation At the intensity levels sufficient for AT1 it was observed that the light scattered by the strongly irradiated atoms contains high-order odd harmonics of the applied field (McPherson, Gibson, Jara, Johann, McIntyre, Boyer and Rhodes [ 19871, Rhodes [ 19871, Ferray, L’Huillier, Li, Lompre, Mainfray and Manus [ 19881). For instance, the latter authors observed up to the thirtythird harmonic of 1.06 pm radiation incident on Ar. They found that the fifth
62
ATOMS IN STRONG FIELDS
lo-’
[I. 5 9
Harnmk ocdpr
Fig. 20. Measured spectrum of scattered radiation from Ar at a Nd : YAG laser intensity of approximately 3 x 10” W c r c 2 . The thirteenth harmonic is absent due to strong absorption of the 81.9 nm radiation accompanying excitation of one of the Ar bound states. (From Ferray, L‘Huillier, Li, Lompre, Mainfray and Manus [1988].)
and higher harmonics are very weak compared with the third, but that the intensities fall off only slowly beyond the fifth, resulting in a “plateau structure” (fig. 20). It is not surprising that only the odd harmonics occur: it follows from basic symmetry considerations that only odd-harmonic generation occurs when the scatterers have inversion symmetry, since the induced polarization in such scatterers must be an odd function of the electric field strength. A rigorous calculation of the spectrum of scattered radiation from a single atom involves the Fourier transform of the autocorrelation function of the induced dipole moment (Sundaram and Milonni [1990]). It is much simpler, however, to work with the expectation value of the dipole moment itself, and to take its Fourier transform to obtain the so-called “coherent” part of the spectrum (Knight and Milonni [19801). This was done by Kulander and Shore [ 19891 and Eberly, Su and Javanainen [ 19891 in their numerical experiments on the time-dependent Schrodinger equations (8.12) and (8.14), respectively, i.e. they identify the spectrum of scattered light as the Fourier transform of
4f) = < W)l er 1 W)> .
(9.1)
Figure 21 shows the results of Kulander and Shore [1989] compared with the experimental data of Ferray, L‘Huillier, Li, Lompr6, Mainfray and
HIGH-ORDER HARMONIC GENERATION
63
-5
-6
-7 N
-9
-1 0
-1 1
-1 2
Photon energy
Fig. 21. Numerical results of Kulander and Shore compared with the experimental data of Ferray, L'Huillier, Li, LomprC, Mainfray and Manus [1988]. The numerical data are scaled to coincide with the relative experimental intensity of the seventh harmonic peak. (From Kulander and Shore [ 19891.)
Manus [ 19881, who reported relative intensities of the harmonics. It is important to note that the numerical data were scaled to coincide exactly with the experimental data for the seventh harmonic. Note the presence of a continuum background in the spectrum, which is much weaker than the strong harmonics and depends on the assumptions made for the pulse duration and shape. A continuum background was also observed in the experiments on Xe at 10 Torr, but this background disappeared at lower pressures (Ferray, L'Huillier, Li, Lompre, Mainfray and Manus [ 19881). The experimentally observed plateau structure, and the fairly sharp cutoff at higher harmonics, also occurred in the numerical experiments of Eberly, Su and Javanainen 119891 for a one-dimensional model atom. They also found a background continuum component in the spectrum. Eberly, Su and Javanainen [1989] argued that the generation of harmonics of the incident field is intimately tied to the generation of the AT1 peaks in the photoelectron energy spectra. In particular, they argued that the
64
ATOMS IN STRONG FIELDS
[I, I 9
generation of the nth harmonic of the field frequency is governed by the same probability amplitude as the corresponding peak in the AT1 electron energy spectrum. Therefore, “AT1 peaks of comparable strength should produce harmonic peaks of comparable strength, and the locations of the cutoffs should be approximately the same in AT1 as in harmonic spectra. Also, there should be no qth harmonic if there is no q-photon AT1 peak but not vice versa” (Eberly, Su and Javanainen [1989]). More recent results are contained in Eberly, Javainen and Rzazewski [ 19911. The plateau structure and the relatively sharp cutoff at high harmonics appear to be fairly generic to strongly driven nonlinear systems. For instance, we found that these characteristics are obtained with a simple two-level model for an atom. Without the rotating-wave approximation the equations for this model are (Allen and Eberly [ 19751) i= -w,y, (9.2a) j = wox +
B sin (wt)z ,
(9.2b)
i = -asin(wt)y,
(9.2~)
where 0, is the transition frequency, x and y involve cross products of upperand lower-state probability amplitudes, and is now twice the Rabi frequency defined in the RWA Bloch equations (4.1). A “harmonic generation spectrum” may be computed as the squared modulus of the Fourier transform of the induced dipole moment e x ( t ) . In our computations we have used a field with a ramp according to the algorithm E
=
E , sin2(wt/4a)sin ( w t ) , 0 < t < 2 n a / w ,
=
E , sin(wt),
t 3 2na/w,
(9.3)
where a, the number of cycles required to “turn on” the field, is typically set equal to five. (In generating the figures a phenomenological damping term was included, which serves to isolate the harmonics.) For fairly weak fields the computed spectrum shows peaks at the first few odd harmonics of the applied frequency w, and the peak amplitudes fall off quickly with increasing harmonic order. However, for larger values of B we observe qualitatively the same plateau and cutoff structure observed experimentally and obtained by numerical integration of the full Schradinger equation. Figure 22, for instance, shows the logarithmic spectrum computed with Q/w, = 4 and w / w o= 0.25. In this simple model, furthermore, the cutoff in the harmonic generation spectrum is linearly proportional to the ratio of the Rabi fequency SZ to the
1, I 91
HIGH-ORDER HARMONIC GENERATION
65
5 4
g
a
2 0
0
10 20 Frequency
30
Fig. 22. Harmonic generation spectrum computed for the simple two-level model with O/o,= 4.0 and w/wo = 0.25. (From Sundaram and Milonni [1990].)
applied field frequency; a linear dependence of the cutoff on the field strength was, in fact, demonstrated experimentally (Rhodes [ 19901). We also noted that in the two-state model the higher-order peaks exhibit the same “effective order of nonlinearity”, as noted by Kulander and Shore [ 19891. This means that the higher harmonics all have approximately the same intensity dependence. Another system that displays the major features found in experiments and in numerical solutions of the Schrodinger equation is a purely classical model of the hydrogen atom. This is not surprising in view of the observation that the principal features of high-order harmonic generation are generic to strongly driven nonlinear oscillators. The prescription for the classical analysis is as follows. Consider again, for instance, the one-dimensional binding potential (again in atomic units) V ( x ) = -e’(x’
+ I)-’’’
(9.4)
used by Eberly, Su and Javanainen [1989]. Take a microcanonical ensemble of classical trajectories (all with energy E equal to the ground-state energy of the quantum system) with initial x-coordinates uniformly distributed between the classical turning points
x l . 2=
(e4,!-’
- 1)-
I/*,
(9.5)
and the corresponding initial momenta determined by energy conservation, i.e. p
=
{ 2 m [ E - V(x)])’/’.
(94
For each of these initial conditions the classical equations of motion dxldt = p / m and dpldt = - V’(x) + F(t) are solved. A time series of the
66
ATOMS IN STRONG FIELDS
I’
I
I
“3
J9
1
Frequency
Fig. 23. Spectrum of higher harmonics computed from a purely classical model (see text). The field parameters were w / o o= 0.2 and the electric field amplitude is equal to 0.0005 times the binding field strength.
ensemble-averaged value of the dipole moment ex can then be constructed, and its Fourier spectrum (fig. 23) displays the gross features of the experimentally observed harmonic generation spectra. Similar results were reported by Bandarage, Maquet and Cooper [ 19901, who performed a three-dimensional classical simulation. The role of shorter-range potentials or the use of approximations that ignore the tail of the Coulombic potential can also be easily studied, given the computational simplicity of the classical model. The principal result here is that more shorter-ranged potentials typically cause a cutoff of the harmonic spectrum at ever-decreasing values of the frequency. It should be noted that the classical dynamics become strongly nonlinear, with the generation of high-order harmonics, at lower values of the driving field than those required for the corresponding quantum dynamics to exhibit high-order harmonic generation spectra. A detailed discussion of similar behavior, although in a different context, is given later. Eberly, Su and Javanainen [ 19891 suggested that classically chaotic trajectories may play a role in producing features seen in the harmonic generation spectrum, most notably the continuum component. Figure 24 shows the classical trajectories (in the phase space of x and p ) corresponding to the spectrum shown in fig. 23. The phase space evolution is entirely regular (not chaotic), but a continuum component is present in the harmonic spectrum. A further increase of the driving field strength leads to a more complicated evolution in phase space, but the corresponding spectra are simply more “noisy” with fewer harmonics present. Still higher values lead to ionization, where only the fundamental and continuum components are seen in the spectra. Thus above-threshold ionization and harmonic generation do not, in general, go hand in hand.
67
HIGH-ORDER HARMONIC GENERATION
I
I
I
I 0.0
I 1 .o
0.6
L
0.0
-0.6
1
-1
.o
X Fig. 24. Regular (nonchaotic) behavior of the phase space trajectories corresponding to the spectrum shown in fig. 23.
We have restricted our discussion so far to isolated atoms. However, a realistic comparison with experiment should account for the propagation of radiation through a medium consisting of atoms. The importance of propagation can be appreciated by recalling the following perturbation-theoretical formula for the intensity at the third harmonic after propagation through a distance L of a medium with third-order susceptibility ~(~'(0) (see, for instance, Milonni and Eberly [ 19881):
where I,(O) is the intensity at the fundamental, which is assumed to be undepleted. Here, Ak = 3k,, - k , , = ( 3 w / c )[n(w)- n(3w)l is the phase mismatch between the nonlinear polarization induced by the fundamental and the third-harmonic field. Although (9.7) is oversimplified when applied to highorder harmonic generation, which cannot be described perturbatively in terms of field-independent nonlinear susceptibilities such as ~ ( ~ ) ( w it )brings , out some important features. The harmonic conversion efficiency can be reduced enormously when there is a large phase mismatch A k . In the case of high-order harmonic generation the question of phase matching is complicated by the fact that the refractive indices n(w)and n ( 3 w ) are intensity-dependent in intense fields. Furthermore,
68
ATOMS IN STRONG FIELDS
[I, § 9
the effects of photoelectrons on the refractive indices may also play a role at very high intensity levels. Equation (9.7) shows, however, that even if perfect phase matching is realized, the frequency dependence of the harmonic generation spectrum is not simply that contained in 1 x ( ~ ’ ( ~ ) I 2 : there is also the factor w2. The number of then varies with photons at frequency 3w (which is proportional to w frequency as w I ~ ( ~ ’ I (in0the ) perturbation-theoretic approximation and when perfect phase matching occurs. The factor w is a consequence of the approximately one-dimensional nature of the problem when the harmonic field propagates as a beam of radiation. For the case of a single atom, on the other hand, the harmonic radiation is scattered in three dimensions, in which case the factor w multiplying 1 x ( ~ ’ ( ois )replaced /~ by w3. These frequency prefactors arise also in the nonperturbative case appropriate to high-order harmonic generation (Sundaram and Milonni [ 19901). The single-atom prefactor w 3 has been included in the harmonic generation computations by Potvliege and Shakeshaft [ 19891 but not by Kulander and Shore [ 19891 or Eberly, Su and Javanainen [ 19891. However, for modeling the experiments in which the harmonic fields are “coherent” beams generated in a medium, the appropriate frequency prefactor, neglecting phase mismatch, is not w3 but w. When a factor of w is included in the spectra computed by Kulander and Shore [ 19891, one finds, in fact, that the computed lower-order peaks are brought into better agreement with experiment. However, the higher-order peaks in the plateau region are then too strong compared with the relative experimental peak strengths. A plausible resolution is that, since the differences in refractive indices between the fundamental and the harmonics increase with increasing harmonic order, phase mismatch becomes increasingly important as the harmonic order increases. The good agreement between the numerical computations and the experimental results would then be preserved at the lowest harmonic orders, whereas at the higher orders the phase mismatch would weaken the peak intensities and mitigate the effect of the prefactor w. For studies of high-order harmonic generation with approximate treatments of propagation effects we refer the reader to Shore and Kulander [I9891 and L‘Huillier, Li and Lompre [ 19901. The reader is also referred to the recent experimental study of Sarukura, Hata, Adachi, Nodomi, Watanabe and Watanabe [ 1991 1. Using 280 fs KrF laser pulses of intensity from 1015-1018W ~ m - ~they , observed the twenty-fifth harmonic in Ne. The 9.9 nm radiation at this harmonic is evidently the shortest-wavelength coherent radiation obtained thus far in the soft X-ray
1, § 101
DISCUSSION
69
region. Results were also reported with He and Ar as the nonlinear media. Anomalous harmonic peaks were found in all these spectra, and in addition, the plateau structure of previous experiments was not found. Possible reasons for these discrepancies were briefly discussed.
8
10. Discussion
The interactions of atoms with super-intense fields are currently under intense experimental and theoretical investigation, and it should be clear that we had to omit not only many references to the literature but also important details of both experiments and theory. Nevertheless, in qualitative terms, at least, the main features of the electron energy spectra in ATI, and of the high-order harmonic generation spectra, seem to be fairly well understood (albeit within a wide range of schemes and with many outstanding questions). On the theoretical side, what is needed (as usual!) is something better than a lowest-order perturbation theory. The Keldysh-Reiss theory evidently contains important elements of the (experimental) truth in that it predicts AT1 peaks, the suppression of low-order peaks in strong fields, and qualitative differences between linear and circular polarization. At the same time, this approximation is not reliable under conditions of strong ionization, or when excited-state populations are substantial, and there are some formal difficulties connected with gauge invariance. We note that other nonperturbative models that address successfully, in a more quantitative sense, specific features seen in AT1 do exist (see, for instance, Edwards, Pan and Armstrong [ 1984, 1985]), but a model accounting for all aspects of AT1 remains elusive. Bucksbaum, Bashkansky and Schumacher [ 19881 have made what can be considered the first detailed comparison of the Keldysh approximation to the experimental AT1 results. They ionized helium with linearly or circularly polarized 532 nm pulses (FWHM z 80 ps) with an intensity of about l O I 4 W cm - ’. In their computations with the Keldysh-Reiss theory they included averages over the field temporal and spatial variations appropriate to the experiments, and used a Hartree 1 s variational wave function for the initial state of He. Figures 25 and 26 compare the experimental data and KeldyshReiss predictions for circular and linear polarization, respectively. In fact, Bucksbaum, Bashkansky and Schumacher [ 19881 noted that good agreement could be obtained if a 30% higher value of the peak intensity was assumed than the 2 x loi4W cm used in obtaining the theoretical predictions shown. (Recall also the discussion related to figs. 10 and 11, where a ~
70
ATOMS IN STRONG FIELDS
PP,olops obsorbed
0008.. .
.
15
.
. .
20 ,
,
.
.
..
..
,
.
. .
,
25 .
>
',
p0.006 In
0.004 c
-:0.002
W
0.000
0.08, .
,
1
,
. .
, .
,
Electron energy (eV)
Fig. 25. Comparison of the predictions of Keldysh-Reiss theory with experimental results, for circular polarization. (From Bucksbaum, Bashkansky and Schumacher [ 19881.)
Photons obsorbed 12
14
16
18
20
22
< 0.3
3'0 0.2 .L u
-y o . 1
W
0.0
< 0.3
Theory
c
$0.1
W
O"0
5 10 15 20 25 30 Electron energy (eV)
Fig. 26. Same as in fig. 25 but for linearly polarized light. (From Bucksbaum, Bashkansky and Schumacher [1988].)
similar increase in the intensity was needed to give good agreement with the experimental data.) Note, however, that the theory predicts electron yields that are about an order of magnitude too large. It should be realized, of course, that the ionization in this case scales roughly as the tenth power of intensity in a perturbation-theoretic rate description, so that this order-of-magnitude disparity should, in fairness, not be considered as bad as it might at first appear. In fig. 26 the predicted yields are about right, but the predicted spectrum is in poor agreement with the experimental data. It was also found that the overall
1, § 101
71
DlSCLlSSlON
agreement between theory and experiment for the angular distributions of the photoelectrons with linear polarization was “quite poor”. Faisal [ 19731 obtained what is now called the Keldysh-Reiss amplitude from a space-translation approach, which involves a unitary transformation of the time-dependent SchrOdinger equation using the operator
S(t) = exp (i
j,‘
-
d t ’ [ ( e / m c ) A ( t ’ )p - (e’/Zmc’)A”(t’)]/h)
Defining a transformed state vector SchrOdinger equation for I+) is i h ( a / a t ) I+)
=
I +) such that 1 t+h)
=
.
(10.1)
S 1 +), the
[ S + H S- ihS+(aS/at)] I+)
where we use the space-translation property e i p - R / h V(IrI)e-’P*R’h =
V(lr - R I ) .
(10.3)
The Keldysh-Reiss results are obtained when (10.4) i.e., for r b I ( e / m c )(A,/w)l = I eE,/mw’ I in the case of a monochromatic field. This condition basically says that the applied field strength must be small compared with (hw/I,)’ times the atomic binding field, where I,, is again the ionization potential. For hydrogen we can write this condition as lerE,I (w/w,)210,where I , = mwir’ is the ionization potential. For r = u0, then, the condition is that the intensity I Q 5 x 10” W c m - 2 when w corresponds to 1.06 pm. This is another indication that the Keldysh theory is not ‘bexactOin the applied field and, in fact, becomes suspect at very high intensities. Guo, Aberg and Crasemann [ 19891 formulated the ionization problem in terms of scattering theory and quantized-field Volkov states. They find that a final scattering state exists only when the ponderomotive potential is an integral multiple of the photon energy in the nonrelativistic and high-photonnumber limits. We have focused our attention on Keldysh-type approaches because these provided the most direct and qualitatively appealing interpretations of ATI. Another approach is that of Chu and Cooper [ 19851, which is based on an ab initio Floquet formulation of multiphoton processes (Chu and Reinhardt
+
12
ATOMS IN STRONG FIELDS
[I, 8 10
[ 19831). This approach yielded additional insights into shifts of the photoelectron energies and the role of continuum-continuum transitions. Almost any theory that includes multiphoton transitions can be expected to account for the existence of the AT1 peaks. At this time the Keldysh-Reiss theory provides the simplest theoretical guide to the interpretation of the experimental data, although, as we have indicated, it has some serious shortcomings. Milonni and Ackerhalt [1989] showed how the theory can be modified to maintain gauge invariance and account for the population of intermediate states. Parker and Stroud [ 19891 found good agreement between a modified Keldysh theory and the predictions of a numerical solution of the SchrBdinger equation for a model problem. (Their modification is exactly the same as that proposed by Milonni and Ackerhalt [ 19891.) A further recent development in the theory of atoms in intense fields should be mentioned, namely, the “stabilization” that occurs at high frequencies and very large field strengths. This effect was described using the so-called “Kramers-Henneberger” transformation (see Henneberger [ 1968]), described by eqs. (10.1) and (10.2) (Gersten and Mittleman [ 19761, Gavrila and Kaminski [ 19841, Pont, Walet and Gavrila [ 19901) and references therein, Su, Eberly and Javanainen [ 19901). For a monochromatic field A(?)= A , cos at, the shifted potential appearing in (10.2) in the case of the Coulomb potential is - e2
V ( J r+ a ( t ) l ) = JY
+ a.
sinotl ’
(10.5)
where a. = - e A , / m o c . Suppose the field frequency o is sufficiently large that the potential (10.5) may effectively be replaced by its time average, which for the Coulomb potential is shown in fig. 27. In such a time-averaged approximation the energy eigenstates of the driven atom (i.e. the eigenstates of the potential shown in fig. 27) will obviously all be stable, since no time-dependent perturbation occurs after time-averaging. Thus no transitions will occur among these eigenstates (referred to as “Gavrila states” in the recent literature), and in particular, no photoionization ; the atom is stabilized against photoionization. Pont, Walet and Gavrila [ 19901 found in numerical computations for the time-averaged potential that, for field intensities large enough so that I a. I z 10 Bohr radii, the wave functions have a two-peaked (dichotomous) structure, with the peaks occurring near the classical turning points. Using their one-dimensional model described earlier, Su, Eberly and Javanainen
I , § 101
DISCUSSION
73
Fig. 27. Effective time-averaged Coulomb potential, in the Kramers-Henneberger gauge. The plane passes through the symmetry axis defined by 01" and the peaks lie along the polarization direction ofthe field. Distances are in units of 1 01" 1, and the vertical axis in atomic units represents - I 0 1 ~ 1 times the time-averaged potential. (From Gavrila and Kaminski [ 19841.)
[ 19901 found such dichotomous wave functions in numerical solutions of the time-dependent Schrodinger equation. In their model the field frequency corresponds to a photon energy of 14.12 eV compared with the ionization threshold of 18.21 eV. They find that the ionization probability saturates as a function of time for sufficiently large intensities ( % 10'6-10'8 W cmP2), and that the degree of this stabilization increases with increasing field intensity. The reader is referred to the cited literature for details of the stabilization effect, which for a number of reasons may be difficult to observe experimentally. We also note that a dynamical interpretation of stabilization, using notions from nonlinear dynamics, is contained in Jensen and Sundaram [1990] and Sundaram and Jensen [ 19911. Other aspects ofthis phenomenon are contained in Burnett, Reed and Knight [ 19921. Finally we mention one more contribution to the theory of atoms in intense fields (Wasson and Koonin [ 19891). Kirschbaum and Wilets [ 19801 employed modified classical equations of motion, together with energy minimization, to determine the ground-state energies and r.m.s. radii of multielectron atoms. Their predicted single-electron ionization energies are in reasonably good agreement, to within a factor of two and often much better, with experimental
14
A T O M S I N STRONG FIELDS
[I, § 11
values. The modification of the ordinary classical equations of motion consists of the addition of two potentials, one simulating the Pauli exclusion principle and the other the Heisenberg uncertainty principle. The latter potential is a Gaussian function that prevents the separations I ri - rjl and I pi - p,I from becoming too small. Wasson and Koonin [ 19891 used this “molecular dynamics” method to treat the strong-field ionization of He and Be atoms and, in particular, to study the possibility of collective ionization. It is hoped that this survey will motivate some readers to look more deeply into the theory of strong-field ionization, and perhaps develop new and better models. Theoretical breakthroughs in this area might also be brought to bear on some deeper and more general questions about quantum theory itself, some of which we shall now consider.
8 11. What is Chaos? The term chaos implies unstable, erratic, seemingly random behavior. In physics chaos refers specifically to an effectively random type of behavior that is nevertheless described by a deterministic set of equations; the chaos is not due to any stochastic or random input to the system, but is intrinsic to the system itself. Thus a chaotic system will evolve in a perfectly deterministic manner, so that the future (and the past) is in principle perfectly predictable from the present state of the system. However, the sensitivity to initial conditions is so severe that, as a practical matter, any detailed long-term predictability about the system is lost. The system exhibits deterministic chaos. During the past decade, and especially in recent years, chaotic behavior has attracted widespread attention, and it is easy to see why. First, many working scientists have finally realized that simple systems described by just a few seemingly innocent equations can exhibit chaos; one need not have a huge number of complicated-looking equations to have chaos. (“Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties” (May [ 19761.) Second, a few universal routes to chaos have been discovered, and systems as diverse as lasers, bouncing balls, chemical reactions, and fluid flows, among others, have exhibited these same routes from order to chaos as some parameter or “knob” is varied. An underlying theme in all this work is the hope that one day we can claim an understanding of turbulence in terms of deterministic rules of evolution. “Unaware of the scope of simple equations, man has often
I , § 111
WHAT IS CHAOS?
75
concluded that nothing short of God, not mere equations, is required to explain the complexities of the world (Feynman, Leighton and Sands [ 19641).
I 1.1. PRELIMINARY NOTIONS
We shall begin by considering a simple example, the so-called logistic mupping. This famous example might understandably appear highly artificial to the uninitiated, but after discussing it briefly, however, we shall argue that it is not very artificial after all, and that real physical systems can, and often do, show much the same type of chaotic behavior. The logistic map is defined by the iteration
x,,
I
=
41X,A1 - x,)
9
(1 1.1)
where both the “knob” 1and the initial value xo of x lie between zero and one. In the case 1= 1 it is convenient to let x, = sin’n0,. Then (1 1.1) implies sin’nO,+ I = (2 sinno, c0sn8,)~ = sin22n0,, or On+ = 28,, or
0,
=
2“00 (mod 1 ) .
(11.2)
It is easy to see that this system with 1 = 1 has the property of “very sensitive dependence on initial conditions”: ifwe change the initial seed 0, to 0, + E , then O,, changes by 2 “ =~ E e“ log’. Thus, there is an exponential separation with “time” n of initially close “trajectories”. The rate of exponential separation, namely log2, is called the Lyupunov exponent, and its positive nature in this example means that we have a highly sensitive dependence on the initial conditions. Such an exponential sensitivity to initial conditions is what is meant by the phrase “very sensitive dependence on initial conditions” and may be taken as the definition of chaos. To show more clearly the sensitive dependence on initial conditions implied by a positive Lyapunov exponent, let us write 0, in base-2 notation. For instance, we can write the number 1/2 + (112)’ + (1/2)4 + (1/2)7 + . * as 0.1 101001. .. In base-2 the algorithm (1 1.2) amounts to just shifting the “decimal” point to the right. Thus, if 0, = 0.1 101001.. . then 0, = 0.101001.. . , 0, = 0.01001.. ., 0, = 0.1001.. ., etc. Obviously, 0, will depend on the nth and higher digits of $, and when n is very large, the value of 6, depends extremely sensitively on the precise value of $. Such a “Bernoulli shift” is well known in various contexts. If, for instance, we iterate the map (1 1.2) on a digital computer, then after a relatively small number of iterations (typically about 50 on a 16-digit machine), we generate
16
[I, § 1 1
ATOMS IN STRONG FIELDS
numerical “noise” simply because in the digit shifting we eventually pick up machine round-off errors. In this sense it may be said that chaos will beat any computer! This example illustrates that detailed, long-term predictions about a chaotic system are impossible as a practical matter, since (1) we can never know initial conditions with infinite precision, and (2) even if we could, no conceivable computer can handle an infinite string of digits. One might ask how random is the chaos generated by the iteration (1 1.2). How much is it like the randomness in a game of coin tossing? We can, in fact, compare the iterates 8, generated by (11.2)to the results of coin flipping by associating 0 < 8, < 1/2with heads (H) and 1/2< 0, < 1 with tails (T). Write 8, in base 2 as 8, = O.d,d, d,, = 2 . . . with each di equal to either zero or one, and call 0, “heads” if it lies between 0 and 1/2,in which cased, = 0, and “tails” if it lies between l/2and I , in which case d,, = 1. Now suppose in tossing a coin we come up with some sequence of heads and tails like HTHHTHTTH . . . . It is easy to see that this same sequence can be reproduced with (1 1.2)simply by choosing 0,, appropriately, because 8, = O.d,d, ,d, 2 . . . corresponds to heads or tails depending on whether d, = 0 or 1, and the choice of 8,) = O.d,d, d2 * * * d,d, d, 2 . . . determines a unique sequence of heads and tails. Thus, any arbitrary sequence of heads and tails corresponds to some choice of 8,, in the iteration ( 1 1.2). We can well imagine that our coin flipping game involves some very complicated equations that reduce to some type of chaotic mapping. This raises the question of whether things we usually think of as random are really just examples of deterministic chaos, with underlying deterministic rules of evolution. Is the unpredictahility of random processes just a consequence of extreme sensitivity to initial conditions? Such questions are part of the reason for the interest in chaos. Some large and complicated systems of equations required to describe “real-world’’ phenomena can have erratic, chaotic solutions. It is perhaps surprising to many people, however, that relatively simple-looking equations can have chaotic behavior, with solutions behaving as randomly as coin tossing. This idea was first emphasized in modern times by Lorenz [ 1963, 19641, although Maxwell, Born and others expressed related views earlier. Before leaving the example of the logistic mapping, we should emphasize that much more can be said about it. In particular, we have focused our attention on A = I . If we choose A between 0 and 1/4,however, after a few initial transient x,, we find the fixed point x = 0; in a physical system this attractor would represent a stable equilibrium. For A between 1/4and 3/4,we find similarly the +
+
+
+
+
I,§ 111
I1
WHAT IS CHAOS?
fixed point x = 1 - (1141). For 3/4 < 1< 1, however, the logistic map has no fixed points. For 1= 0.76, for instance, after some initial transients we find that the sequence {x,} settles into a two-cycle oscillation: {0.7306,0.5984,0.7306, 0.5984, . . .}. This period-two attractor is independent of the initial seed xo. It is said that a period-doubling bifurcation occurs at 1= 314. As 1is raised further, we find increasing period doublings (four cycles, eight cycles, etc.), until eventually the period has doubled ad infinitum and we generate a chaotic sequence of x,, a so-called strange attractor. The case 1= 1 is an example of the chaotic regime of the logistic map. Remarkably, a quantitative universality is present in this period-doubling route to chaos: a wide class of systems shows the period-doubling route to chaos, the knob parameter (e.g., 1in the example of the logistic map) converges geometrically to the value required to reach the chaotic regime, and the rate of convergence is universal to all these systems. This is the essence of the Feigenbaum universality in the period-doubling route to chaos (see, for instance, the papers in Cvitanovic [ 19841). This period-doubling route to chaos is found in systems like x, = f ( x , ) , in which f ( x ) is continuously differentiable and has one hump, as in the example of the logistic map. (It is also required that the Schwarzian derivative be negative over the whole mapping interval .) In physics we are nearly always concerned with continuous flows like differential equations (e.g., F = ma or the Schrsdinger equation) rather than mappings, although various ways exist, at least in principle, to relate continuous flows to mappings. One technique is the so-called PoincarP map, or suvface of section. To illustrate this method, consider the system +
x=
f(X1
y, z)
,
(11.3a)
7
( 1 1.3b)
(11.3~) Suppose we plot points ( x n ,y,) = (x(t,), y(t,)) for times t, for which z(t,) = 0 and i(t,) < 0 (fig. 28). Then the evolution (x,~,Y , ~+ ) (x, ,y , ,) defines a Poincare map. Although the only known general way of constructing this map is by numerical integration of the differential equations, it is useful to know that such a discrete mapping exists in correspondence to a continuous flow. For instance, the extreme sensitivity to initial conditions (chaos) found in mappings like (1 1.1) is also found in systems of differential equations. In fact, it is now well established that real physical systems exhibit deterministic chaos and the identical routes to chaos (e.g., period doubling) that are found in discrete +
,
+
78
ATOMS I N STRONG FIELDS
Y
Z Fig. 28. Construction of a Poincare map with the xy plane as the surface of section. In this example point 1 is mapped into point 2, and point 2 is mapped into point 3.
mappings like (1 1.1). Thus, certain lasers go from stable operation to chaotic output as the discharge current, or some other knob analogous to 1,is varied, and the observation of a well-characterized route to chaos indicates that deterministic chaos is being observed. For details the reader is referred to some introductory books on deterministic chaos (Lichtenberg and Lieberman [ 19831, Schuster [ 19841, Milonni, Shih and Ackerhalt [ 1987]),
11.2. HAMILTONIAN SYSTEMS
The logistic map is an example of a dissipative system. Such systems have attractors, including strange attractors, and correspond to physical systems in which some sort of friction or damping is present. Here we are interested in Hamiltonian systems, which we may define by the existence of a function H ( q , , q 2 , . . . ,q N ;p I , p 2 , . . . ,p N ) of generalized coordinates qi and momenta p i such that the equations of motion take the form
4i = a H / a p i ,
(1 1.4a)
pi =
(11.4b)
- aH/aqi,
where qi and pi are said to be conjugate variables, and any set of q’s and p’s whose time evolution is governed by (1 1.4) is called a set of conjugate variables. A Hamiltonian system with N degrees of freedom has a 2N-dimensional phase space o f p and q. Equations (1 1.4) define trajectories in this phase space, and the uniqueness theorem for differential equations ensures that trajectories
I.§ Ill
19
W H A T IS CHAOS?
in this phase space do not intersect each other. At least one constant of the motion is associated with each trajectory, namely the Hamiltonian function H . In Hamiltonian systems phase-space volumes are conserved (Liouville’s theorem) and there are no attractors. Chaos can occur in Hamiltonian systems, however, and it is defined in the same way as in dissipative systems: there is a positive Lyapunov exponent, implying the property of very sensitive dependence on initial conditions. Transformations from variables (q, p) to ( Q , P) can often simplify a problem. Of particular interest are the canonical transformations, i.e. transformations that preserve the Hamiltonian form of the equations of motion: =
Qi
P,
=
aHIaP,, -
( I 1.5a)
aHjaQ,,
(11.5b)
where H is the Hamiltonian expressed in terms of the coordinates ( Q , P). The canonical transformation theory of classical Hamiltonian systems is, of course, treated in detail in standard textbooks such as Goldstein [ 19801. Nevertheless, it may be useful to review some pertinent points in perhaps a slightly different fashion in order to introduce the concept of integrability, which is usually not discussed in the standard texts. For simplicity we shall consider the example of N = 1, and assume that the phase-space coordinates (q , p) and (Q, P) are Cartesian. Consider the area S enclosed by a closed curve C in the (q, p ) phase spacing. According to Stokes’ theorem we can write this area as dq dp = s c p ( q ) dq. In the ( Q , P) representation, similarly, this same area is dQ dP = P(Q) dQ, and therefore we have ( p dq - P dQ) = 0. (Note: (q, p ) and (Q, P) here are viewed simply as different ways of labeling the same phase-space point, and so we refer to the same S and C in writing these equations in the different coordinate systems.) Using
ss
sc
ss
sc
Icd(Qp)= I c ( Q d P + P d Q ) = O ,
sc
( p dq + Q dP) = 0. we can also write Since the closed curve C is arbitrary, it follows that p dq + Q d P must be the exact differential of some function F(q, P):p dq + Q d P = dF(q, P), or P
=
a m , p)iaq,
Q = a m , p)iap,
(11.6a) (1 1.6b)
which determines (Q, P) in terms of (4, p). Recall that F(q, P) is called the
80
ATOMS IN STRONG FIELDS
[I,! 11
generating function for the transformation (q, p ) -+ (Q, P). An example is F = qP, the identity transformation (Q = q, P = p ) . The generating function F(q, P) here is of the class that Goldstein [ 19801 denotes F,. A simplification is achieved if the generating function can be chosen in such a way that H depends only on Q or P. Thus if R = H ( P ) , then P = - aH/aQ = 0, Q = alT/aP = constant = V, or
P = constant, Q
= vt
(11.7a)
+ 6.
( 1 1.7b)
In the case of N degrees of freedom this generalizes to Pi = constant and Qi= vit + Si. The vi and Si are the 2N constants of integration. The problem of reducing H to the form R(P),for instance, amounts to finding the correct generating function. From ( 1 1.6) and H(q, p ) = H(4, aF/aq) = H ( P ) = constant = E(P), we see that Fmust satisfy the Hamilton-Jacobi equation H(q, w a q )
=
( 1 1.8)
E(P)
with an easy generalization to the case N > 1. Once the solution to the Hamilton-Jacobi equation is found, the solution of the problem in terms of the Q s and P's is trivial (eq. ( 1 1.7)), and we can perform the inverse transformation (Q, P) -+ (q, p ) to find the original variables (q, p ) as functions of time. 1 1.3. INTEGRABILITY, TORI, AND QUASIPERIODICITY
The solution of the Hamilton-Jacobi equation is usually far from trivial. In some problems, however, coordinates may be chosen in such a way that the Hamilton-Jacobi equation is separable, with one independent equation for each of the N degrees of freedom. As a trivial example, consider two coupled harmonic oscillators, for which the Hamiltonian in normal-mode coordinates may, for our purposes, be taken to be H = i ( p : + p i + w:q: + w i q i ) . The Hamilton-Jacobi equation is (aF/aq,), + (
Writing F
=
a ~ / a ~ +, )4:~ + 4;
=
2 ~ .
(11.9)
Fl(ql,P,)+ F2(q2,P2), we obtain
(aFi;./aqi)2 + q'
=
ai, i
=
1,2,
(11.10)
where the separation constants aisatisfy a, + a2 = E. The separation constants of such an integrable system are called isolating integrals; they are invariants of the motion.
I,$ 111
81
W HAT IS CHAOS?
More generally, a system with N degrees of freedom is said to be integrable if and only if N independent isolating integrals exist. The isolating integrals must be “in involution”, i.e. the Poisson brackets of all pairs vanish. Loosely speaking, then, a system with N degrees offreedom is integrable if and on& i f there exist N independent constants of the motion, or N - 1 constants of the motion in addition to the Hamiltonian itself. Thus all systems with a single degree of freedom are trivially integrable. It is convenient for integrable systems to introduce action-angle variables. Consider again a system with a single degree of freedom, and assume the motion in phase space is bounded. Then, since trajectories in the two-dimensional phase space do not cross, it follows that they must be closed loops. (If this is not obvious geometrically, it may be proved as a consequence of the Poincare-Bendixson theorem (see, e.g., Milonni, Shih and Ackerhalt [ 19871). In other words, the motion must be periodic. Denote (Q, P) by (0, J). If F is the generating function for the transformation (4, p ) -+ (0, J ) , then p = aF/aq, e = aF/aJ, and ae/aq = a(aF/aJ)/aq = a(aF/aq)/aJ = ap/aJ. Letting denote an integration over a period of the motion, therefore, we have
s
aaJ s p d q = s g d q = I d O = l , where we choose 0 such that the integral J=
s
pdq,
(1 1.11)
s d 8 is unity. Thus we may take (11.12)
where J and 8 are the action-angle variables. From (11.7) it follows that J = constant and 8 = v(t - to), and from (11.1 1) that v = 1/T, where T is the period of the motion. In other words, v = aH/aJ = aE/aJ is the frequency, and once we have the Hamiltonian as a function of the action variable, we can calculate the frequency without having to solve the equations of motion. This is one of the major advantages of action-angle variables as proclaimed in the textbooks. The generalization to integrable systems with N > 1 is straightforward. If the Hamilton-Jacobi equation is separable, we obtain the N ordinary differential equations H,(qi, aF,/aq,) = ai, where a1 + a2 + . . . + a,, = E , and each of these equations can be solved simply by quadrature, i.e. by solving for aFi/aqi and integrating over qi. The action-angle variables (Oi, Ji) form a set of canonical variables defined by j i = - aH/at$ = 0 and ei = aH/aJiE v,(J), or
J
constant,
(1 1.13a)
e=v(J)+S,
(1 I. 13b)
=
82
[ I , § 11
ATOMS IN STRONG FIELDS
in the notation J = ( J , , J2, . . . ,J N ) . These are obvious generalizations of the case N = 1. For an integrable system N independent constants of the motion exist, and so the motion of trajectories in phase space is confined to an N-dimensional surface in the 2N-dimensional phase space. Now, when the angle variables Oi each change by one, the (qi, pi)return to their original values before the change. This means that for integrable systems the trajectories are, in fact, confined to N-dimensional tori in phase space. A harmonic oscillator, for instance, describes a one-dimensional torus or loop (e.g., p 2 + q 2 = constant) in the two-dimensional phase space. Two coupled oscillators with two incommensurate frequencies likewise describe a two-torus or doughnut in phase space (fig. 29), and so on. Since the 8;s are periodic with period one, any trajectory on the N-torus can be expressed as a discrete Fourier series: q(t) =
1 A n ( J )e2nin.O = 1 A n ( J ) eznin.(vr+ I1
p(t) =
1 Bn(J)e2nin.e= n
8)
(11.14a)
n
B,,(J)e*ni"*("f+a),
(1 1.14b)
I1
where n represents an N-dimensional set of integers. Thus the power spectra (modulus squared of Fourier transform) associated with integrable motion consist of sharp spikes (fig. 30), i.e. integrable systems are quasiperiodic in their time evolution. Now an important feature of quasiperiodic functions of time (e.g., (1 1.14)) is their recurrence property: let y ( t ) be any of the p's and q's in (1 1.14). Then for any E > 0, there exists a T ( E such ) that any interval of length T ( E of ) the real line contains at least one point t' such that Iy(t) - y ( t ' ) l < E for any t (Corduneanu [ 19681). In other words, given y ( t ) , we can always find a t' such that y ( t ' )is as close to y ( t ) as we wish, and there are an infinite number of such times t ' . Periodic functions are quasiperiodic, but of course quasiperiodicity does not imply periodicity. Thus the function y ( t ) = a cos wIt + b cos o , t is quasiperiodic, but not periodic unless wI and w 2 are commensurate frequencies, i.e. unless w 1 / 0 2 is a rational number.
Fig. 29. A two-torus for two-frequency quasiperiodic motion.
1,s 111
WHAT IS CHAOS?
83
U Fig. 30. Typical power spectra for (a) quasiperiodic motion and (b) chaotic motion.
Quasiperiodic motion is regular, or nonchaotic, i.e. quasiperiodic systems do not have positive Lyapunov exponents. Thus, although quasiperiodic motion can certainly look highly complicated and seemingly irregular, it cannot be truly chaotic in the sense of exponential sensitivity to initial conditions. The sensitivity is at most linear. Therefore integrable systems cannot be chaotic: integrability and chaos represent two opposite extremes of behavior, one being regular and predictable, the other irregular and without detailed long-term predictability. Since quasiperiodicity implies predictability and order, it follows that chaos implies nonquasiperiodic motion. Thus chaotic trajectories cannot have a purely discrete spectrum as in ( 1 1.14), but must have a broad-band, continuous component to their power spectra (fig. 30). The computation of power spectra is therefore a relatively simple test for chaos. Although it is usually true that a continuous spectrum of bounded motion implies chaos, it is not always true; the certain test for chaos is the existence of at least one positive-definite Lyapunov exponent. Unfortunately, the computation of the spectrum of Lyapunov exponents is usually a fairly laborious numerical task compared with the computation of spectra using the fast Fourier transform algorithm. This subsection concludes with two small peripheral points. First, we note
84
ATOMS IN STRONG FIELDS
[ I , § 11
that all of the standard textbook systems for which we have analytical solutions (e.g., coupled oscillators or the Kepler problem) are integrable. Indeed, it is sometimes implied in textbooks that all systems are integrable! Second, it took a long time before it was realized that not all bounded motions have purely discrete Fourier spectra. In a sense the hypothesis that all bounded motions have discrete Fourier spectra goes back to the time of Ptolemy, when it was believed that all motion could be decomposed into perfect circular motions. Poincare, near the turn of the century, was perhaps the first person to clearly understand that bounded systems exist whose spectra are not purely discrete, and that there are nonintegrable systems.
11.4. THE KAM THEOREM
Consider an integrable system with Hamiltonian H , ( J ) and perturb it with a nonintegrable perturbation, so that the total Hamiltonian becomes H
=
H ( J ) + EH,(8, J ) ,
(1 1.15)
where E is a parameter characterizing the strength of the perturbation. When E = 0, we have Ji = constant and Oi = v,t + hi,with vi = BH,,/i3Ji, and the Oiare angles on an N-torus. When E # 0, the (O,, Ji) are no longer action-angle variables of an integrable system, because then H depends on 0 as well as J . What happens to the system if E is very small, so that the system is “nearintegrable”? In particular, what happens to the N-tori to which all trajectories were confined when E = O? The latter question is addressed by the KAM (Kolmogorov-Arnold-Moser) theorem (Arnold [ 19631). Suppose that the perturbation is sufficiently smooth and small, and also that the frequencies v i ( J )associated with the unperturbed Hamiltonian are linearly independent, i.e. for any set (n,,n2, . . . , nN) of integers that are not all zero, N
C
nj v j ( J ) # 0 .
(11.16)
.j = 1
The frequencies in this case are said to be noncommensurate. Under these conditions the KAM theorem asserts that most of the N-tori (KAM tori) of the unperturbed system are not destroyed but only deformed. (We say most tori because tori with commensurate frequencies, or nearly commensurate frequencies, may be destroyed.) The proof of the KAM theorem is long and complicated. The basic idea of the proof is to find a canonical transformation to new action-angle variables,
1,s I l l
85
WHAT IS CHAOS?
such that the transformed Hamiltonian depends only on the new action variables. This leads to a complicated equation for the required generating function that is solved in terms of a perturbation expansion in E. Here, a small-denominators problem arises with perturbation theory, which in the proof is circumvented by a so-called accelerated convergence procedure. In summary, the KAM theorem says that for small nonintegrable perturbations, most of the tori associated with the unperturbed, integrable Hamiltonian will be preserved. However, the theorem does not tell us exactly what “small” means. Furthermore, it leaves open the possibility of overlapping resonances, which can, in fact, result in the destruction of KAM ton, the breakdown of integrability, and the onset of chaos.
11.5. RESONANCE OVERLAP
We follow here the discussion of Walker and Ford [1969] and consider the case N = 2. Suppose, first, that the Hamiltonian is of the form H = H,(J,, J,)
+ f ( J , , 5,) cos(m8, + no,).
(1 1.17)
For f = 0 the trajectories are confined to two-tori. If we can find a canonical transformation to new action-angle variables (O;, 8; J ; ,J ; ) , so that the transformed Hamiltonian is of the form = H ( J ; , &), then the motion of the perturbed system is also confined to two-tori. According to the KAM theorem, this should be possible for most tori if the perturbation is sufficiently small. Now the generating function
F ( 8 , , 8 , ; J ; , J ; ) = 8,J; +8,J;+B(J;,J;)sin(m8,
+no,)
(11.18)
transforms (1 1.17) to
H = HdJ;, where
J;)
+ {[mv,(J;,
J;)
x cos(m8,
+ nv,(J;, J ; ) I W ; , J ; ) + f(J;,J;)}
+ no,),
(11.19a)
v~(J;,J ; ) = aH,(J;, J;)/aJ;
(1 1.19b)
if we retain only the lowest-order terms, assuming the generating function (11.18) is close to that for the identity transformation ( F = 8 , J ; + 8,J;) for which B ( J ; , J ; ) = 0. The Hamiltonian (1 1.19a) is a function only of the action variables J ; J ; if
N J ; ,J ; ) =
- f ( J ; , J;)/[mv,(J;, J ; )
+ nv2(J;, J;)] .
(1 1.20)
86
ATOMS I N STRONG FIELDS
[I,
B 11
With this choice for B , we can expect the perturbed tori to be close to those for the unperturbed system, since the transformation was chosen to be close to the identity transformation. This argument breaks down if (1 1.20) is not small, since then the generating function (1 1.18) is not close to that for the identity transformation. In particular, if there are frequencies vi for the system (1 1.17) satisfying the resonance condition Imv,(J,, 52) + nvz(J1, 5211 Q If(J1,
J2)I
1
(11.21)
the angle-dependent perturbation in (1 1.17) might be expected to greatly distort the unperturbed tori. In other words, we can expect a resonance such as mv, + nv, r 0 to substantially distort the unperturbed tori. More generally we can suppose the perturbation to have the Fourier expansion (1 1.22) and if there is a resonance associated with m0, + no,, we can anticipate that the tori distorted by this resonance are strongly affected also by terms m' 0; + n' O;, such that m' /n' is sufficiently close to m / n . Because ofthe (J1,J,) dependence of v I and v,, and because J , and J, can be thought of loosely as the "radii" of the unperturbed tori, a relation like (11.21) determines the distortion of a zone in the action-angle phase space of the unperturbed tori. If such resonance zones overlap, so that in the region of overlap the unperturbed tori are distorted by a large number of terms like cos(m0, + no,), we may anticipate the complete destruction of the unperturbed tori. This possibility becomes stronger as the perturbation is increased, since this makes (1 1.20) larger and takes the transformation further from the identity. To see what happens when multiple resonances and resonance overlap are present, we consider the example of Walker and Ford [1969]. Suppose that the Hamiltonian is given by (1 1.17), vi = aHo/aJi, and that (1 1.21) can be satisfied. There are two constants of the motion, namely, Hand I = nJ, + mJ,; the constancy of the latter may be verified by taking its Poisson bracket with H. Since these two constants of the motion occur in involution, the system (1 1.17) is still integrable. Walker and Ford consider the unperturbed Hamiltonian H,
=
J, + J , - J: - 3J1J2t 522,
(11.23)
where the actions J,, 5, are related to the Cartesian coordinates (4,p ) by the
87
I , § 111
formulas qi = &.
pi = -
c~~e, ,
m.sin a,,
(1 1.24a) (11.24b)
which means that J, = $(p,' + 4.): Therefore, the unperturbed motion in the (q,, p i ) planes is confined to concentric circles. The condition that the frequencies VI =
1 - 25, - 35,
( 1 1.25a)
and
are positive implies 0 < E < 3/ 13. In the case of a "2-2" resonance with
H
=
H,(J,, J,)
+ c c l , ~ ,C O S ( ~-~2e2), ,
(11.26)
there is the additional constant of motion I = J 1+ J 2 ,
(11.27)
which makes it possible to study the system in considerable detail with only simple analytical methods. From (1 1.23), (1 1.26), and (1 1.27) it is easily verified, for instance, that , I2 =E , (3 + ~ t c o s 2 8 , ) ~-: (51 + I ~ c o s ~ ~ , +) IJ +
(11.28)
where O2 has been set equal to 3n/2. Equations (1 1.28) and (1 1.24) can be used to algebraically determine the (q, p ) curves in the planar surface of section defined by q2 = 0, p 2 = q2 2 0. A typical result is shown in fig. 31. Note that the concentric circles associated with the unperturbed motion are only slightly distorted, except in the 2-2 resonance zone corresponding to the crescentshaped loops. We can relate the more strongly distorted (crescent-shaped) curves to the resonance condition (1 1.21), which in this case becomes 2v1 E 2v2, or from (1 1.29, J, E 5J2.
(11.29)
Consider the fixed points at the center of each crescent. These points correspond to stable periodic trajectories, with J , , J2 constant and 8, = 8,.
88
ATOMS IN STRONG FIELDS
P1
I
Fig. 3 1. A typical surface of section obtained algebraically from eqs. ( I 1.24) and (1 1.28),showing the distortion of the concentric circles of the unperturbed motion. (From Walker and Ford [1969].)
From the equations of motion
jl =
j,
- aH/ae, =
= -
- 2 ~ 55,, sin(28,
-
28,),
a ~ / a e =, 2 ~ 55,, sin(28, - 28,),
e,
=
aH/aJ,
9,
=
a ~ / a =~ 1, - 35, + 25, + aJ, cos(28, - 28,),
=
1 - 25, - 35,
we then infer that 28, - 28,
J1/J2= ( 5
=
+ aJ2 C O S ( ~- ~28,), ,
( 11.30a)
(1 1.30b) ( 11 . 3 0 ~ )
(1 1.30d)
n or 371 and
+ a)/(l + a ) .
(11.31)
The solutions 28, - 28, = n and 3n are chosen instead of 0 or 211 in order to investigate the points at the centers of the crescents in fig. 3 1, for which p , = 0. (Recall (1 1.24) and the fact that 8, is held fixed at 3n/2 in fig. 3 1.) For small a, (1 1.31) is equivalent to (1 1.29). This confirms the claim that the tori are distorted most strongly in the 2-2 resonance zone. A 3-2 resonance associated with a Hamiltonian like H = H,(J,, 5,)
+ U J : ’ ~ J C, O S ( ~- ~28,) ,
(1 1.32)
can be similarly studied algebraically due to the additional constant of motion
WHAT IS CHAOS?
89
Fig. 32. Distortion of the concentric circles of the unperturbed motion in the case of a 3-2 resonance. (From Walker and Ford [1969].)
I = 2J, + 3J2 (Walker and Ford [ 19691). Figure32 shows how the preceding results are modified. In this case the points at the center of each of the crescents shown correspond to a single periodic solution, so that the 3-2 resonance is said to consist of a chain of three islands. What happens if two resonances are possible? Walker and Ford consider the Hamiltonian
H = H , ( J , , J ~ ) + ~ J , J ~ c o- 2s (e ~, )~+,a . ~ , ~ , 3 / ~ ~ 0 ~- (328 8, ),,
(11.33)
in which the unperturbed tori can now be distorted by both the 2-2 and 2-3 resonances. In this case the surfaces of section had to be studied numerically by the integration of the equations of motion (with o! = 0.02). It can be shown that the unperturbed 2-3 torus does not exist for E 6 0.16. Figure 33 shows the numerical results for (q2, p 2 ) curves of the Hamiltonian (1 1.33) with E = 0.056; note the similarity to fig. 3 1. For E slightly greater than 0.16, the 2-2 and 2-3 resonance zones are widely separated. (The 2-3 zone lies inside the 2-2 zone.) Figure 34, for E = 0.18, resembles a superposition of figs. 31 and 32, since the two resonances are, in effect, acting separately to perturb tori in different regions of phase space. Walker and Ford estimated that the 2-2 and 2-3 resonance zones should first begin to overlap when E = 0.2095. Figure 35 shows the numerically generated (q2, p 2 ) curves for this energy. Evidently a small zone of unstable, chaotic
90
ATOMS IN STRONG FIELDS
Fig. 33. Surface of section
motion has appeared in the region of resonance overlap, i.e. resonance overlap has destroyed the KAM tori. Much more can be said about resonance overlap, the destruction of KAM tori, and the onset of nonintegrable, chaotic motion. For our purposes, however, the foregoing discussion and example will suffice, and the reader inter-
Fig. 34. Surface of section for the Hamiltonian (1 1.33) with E [ 19691.)
=
0.18. (From Walker and Ford
I,§ Ill
WHAT IS CHAOS?
91
i'
Fig. 35. Numerically generated surface ofsection for Hamiltonian (1 1.33) with E = 0.2095, where the 2-2 and 2-3 resonance zones are predicted to begin overlapping. (From Walker and Ford [1969].)
ested in more detail can find a large literature to assist him. (The collection of reprints compiled and introduced by MacKay and Meiss [ 19871 would be a good place to begin.) We shall now examine resonance overlap and the onset ofchaos in driven systems, which include atoms and molecules driven by a laser field, and provide the principal testing ground for the study of chaos in atoms and molecules.
11.6. RESONANCE OVERLAP IN DRIVEN SYSTEMS
We shall extend the idea of resonance overlap to driven systems by means of a simple but important example, the kicked pendulum (or rotor). In § 14 we shall be more systematic in applying the concept of resonance overlap to the problem of a hydrogen atom in a monochromatic field. Kicked systems, i.e. systems driven by a series of delta-function impulses, are convenient computationally because their dynamics reduce to discrete mappings rather than continuous flows in phase space. The Hamiltonian for the kicked pendulum is co
H=-
2m12
-
(mi2w,')c o d
1 n = --oo
6 ( t / T - n) ,
(11.34)
92
ATOMS
IN STRONG FIELDS
and the equations of motion are 00
6(t/T - n) ,
Po = - (ml’wi) sin8 n=
8
=
( 11.35 a)
-00
( 11.35b)
pe/m12,
where m and I are the pendulum mass and length, wo is the natural oscillation frequency for small displacements, and T is the period of the delta-function kicking. If T + 0, the force term is on continuously. For T # 0 we can imagine that the gravitational force acting on the pendulum is being switched on and off periodically to provide the kicks. Although the model is artificial, it plays an important illustrative role not only classically but also in quantum chaos (9 12). From (1 1.35) we obtain by integration the mapping pn
+
I
=
p , - (ml’wi T ) sin 8, ,
e n + 1 = en
+ P n + LTIml’
9
( 11.36a)
(1 1.36b)
where pn, 8, are the values of p e and Ojust before the nth delta-function kick. Writing p , = (mI’/T)P,, and replacing 8, by 8, + n, we have the standard map (or Chirikov map) Pn+I
=
Pn + K sin$,, ,
o n + 1 = on
-t pn+ 1
9
( 11.37a)
(11.37b)
where K = (o,T)’.For K + 0 the gravitational potential is on continuously and of course (1 1.37) is integrable. For small K we expect from the KAM theorem that most trajectories lie on invariant curves in the phase space of the near-integrable system. Numerical experiments indicate that some of these KAM curves are broken as K 4 1, but for K S 1 essentially all of them are broken and most trajectories are chaotic. As its name implies, the standard map has been studied in considerable detail (Lichtenberg and Lieberman [ 19831). To summarize, the fixed points are easily found from (1 1.37) to be P
=
2nn, n
8=0,7t,
=
integer,
(11.38a) (1 1.38b)
where it is understood that 8 is given modulo 2n. The fixed points (P, 0) = (2nn,x) for K < 4 are stable against small perturbations. There are also n-cycles with n 2 2. For instance, it is not difficult to show that there are
WHAT IS CHAOS?
93
Fig. 36. Phase curves for a kicked pendulum for small K.
stable two-cycles with P = 2x(n + 1/2) and 8 = 0, x for K < 2. Thus, for small K we can expect the type of phase curves illustrated in fig. 36, which indicates the stable motion about the stable period-1 and period-2 fixed points, as well as the “separatrices” dividing these different regions of phase space (Lichtenberg and Lieberman [ 19831, Milonni, Shih and Ackerhalt [ 1987I). Figure 37 shows the numerical results for the standard map with K = 0.5 and 1.0. For K = 0.5 we see the type of behavior shown in fig. 36. At K = 1 trajectories that are possibly chaotic can be seen, and a computation of the Lyapunov exponents confirms that these trajectories are indeed chaotic. The destruction of KAM tori, and the onset of widespread chaos, can be understood as a consequence of resonance overlap as follows. As a first approximation, we consider the unperturbed Hamiltonian H
=
pi/2m12 - (ml’o;) cos 0 = (mlT2) (Pi- K cos 8),
( 1 1.39)
where again po = (ml*/T)P,. Since the regions with P , r 2xn in fig. 36 are separated by APo = 2x, and the maximum variation of P, about each region is 2@, we can expect a resonance overlap when 2(2& = 2x, or K
=
(in)’ z 2.47 .
(11.40)
This roughly agrees with what is found in numerical experiments, i.e. that widespread chaos sets in for K z 1. Better agreement between the prediction based on resonance overlap and experiment can be achieved by including higher-order resonances (Lichtenberg and Lieberman [ 19831, Chirikov [ 19791).
94
ATOhlS I N STRONG FIELDS
0
2
4
6
6
Fig. 37. Computed phase curves for the kicked pendulum with ( a ) K
=
0.5 and (b)K = 1.0
In 13 we shall consider more systematically the idea of resonance overlap in one-dimensional nonlinear systems driven by a periodic force - systems sometimes said to have a dimension of 1.5. Following Chirikov, we shall show that such systems may be related approximately to the pendulum system. In the remainder of this section, however, we discuss the energy gained by the pendulum as a result of the kicking and the difluse energy growth associated with the onset of chaos in the kicked pendulum. Figure 38 shows computed results for the energy (obtained from (P,')) for different values of K , where the average in each case is taken over a set for 40 different values of 19,. In the chaotic regime of large K the average of the square of the angular momentum, and therefore the average energy, grows in a random-walk, diffusive manner:
(P2)ziK2t.
(11.41)
Such results were first discussed by Casati, Chirikov, Izrailev and Ford [ 19791, who gave the following intuitive explanation for the diffusive energy growth. We note that from (11.39a) it follows that n- . I ..
sinq.,
P,, - PO = K j=O
( 1 1.42)
WHAT IS CHAOS?
95
Fig. 38. Energy in the kicked pendulum for (a) K = 0.5, (b) K = 4.0, and (c) K = 10.0. In each case the plotted energy is an average over 40 uniformly distributed values of the initial angle 0,.
96
ATOMS IN STRONG FIELDS
[I, § 12
and therefore that n- I n- 1
(p,, - Po)2 = K 2
1
sin 6, sin 9 .
(11.43)
i=o j = o
Now, given the chaotic evolution of the system for large K , we might regard the 0, as uniformly distributed random variables, in which case the average of (1 1.43) over the 0, is proportional to n. Thus we can see how chaos gives rise to a linear dependence of the average energy on time n. Furthermore, if we regard P,, in (1 1.42) as a sum of independent random variables, it follows from the central-limit theorem that P has a Gaussian distribution function :
f(P)= ( K f i ) - exp ( - P 2 / K 2 t ).
(11.44)
Indeed, such a distribution function was found experimentally by Casati, Chirikov, Izrailev and Ford [ 19791. In classical models of laser-driven atoms and molecules, the onset of chaos and diffusive energy growth in less artificial models has been associated with dissociation or ionization: loosely speaking, the trajectory of a particle in a potential well, driven to chaos by an external force, will meander about haphazardly in phase space and eventually “find” the continuum and escape from the well, corresponding to dissociation or ionization. This suggests the attractive possibility that chaos might play a role in atomic and molecular physics. However, it turns out that quantum mechanics brings a skunk to the garden party. We turn our attention now to the problem of quantum chaos.
6 12. Questions of Chaos in Atomic Physics The issue of quantum chaos raises several questions, the most important of which is whether chaos is even possible in quantum systems. We first consider this issue. 12.1. IS THERE ANY QUANTUM CHAOS?
For classical systems chaos means a highly sensitive dependence on initial conditions - the existence of at least one positive-definite Lyapunov exponent. In quantum mechanics there does not appear to be any generally accepted “hard number” that can be computed to determine unambiguously whether a system is chaotic. Indeed, the limits imposed by the uncertainty principle on simultaneous measurements of 4’s and p’s make it impossible to think of
I , $ 121
QUESTIONS OF CHAOS IN ATOMIC PHYSICS
91
exponential separation of initially close trajectories in the same way as in classical dynamics. Thus a characterization of quantum chaos must evidently rely on something other than exponentially separating trajectories. Several arguments have been advanced against the possibility of quantum chaos. One such argument is that, since chaos only arises in nonlinear dynamical systems, and the SchrBdinger equation is linear, the wave function can never evolve chaotically. This is a weak argument, however, because a linear but inznite-dimensional dynamical system may be equivalent to a nonlinear system. To see this, consider a nonlinear system of first-order differential equations. Define each nonlinear term to be a new dependent variable, and write differential equations for these new dependent variables. This leads to new nonlinear terms, which we define to be new dependent variables, and for which we write new differential equations. Continuing in this fashion, we obtain an increasing number of linear equations. It is easy to see that a finite-dimensional nonlinear system may be “embedded” in this fashion in an infinite-dimensional linear system. Sometimes the Schrodinger equation is equivalent to a finite-dimensional system (e.g., the optical Bloch equations for a two-level atom in an applied field), but in general, it is not. The linearity of the SchrOdinger equation then cannot be used by itself to argue that quantum chaos is impossible. One can, however, make a compelling argument against the possibility of quantum chaos in systems with purely discrete energy eigenvalues. Specifically, one can easily prove the quantum recurrence theorem for such systems. Bocchieri and Loinger [1957] stated this theorem as follows: “Let us consider a system with discrete energy eigenvalues En;if $(to) is its state vector at the time to and E is any positive number, at least one T will exist such that the norm 11 $ ( T ) - $(to) )I of the vector $ ( T ) - $(to) is smaller than 6.’’ The proof is not difficult, and the result can be understood simply by noting that the state vector at time t can be written as (12.1)
where the En are the discrete energy eigenvalues and the t,bn are the corresponding eigenvectors. Thus the norm of t,b(t) is a quasiperiodic function of time, and so has the recurrence property noted in 8 1 1 . 3 . (Note that the mean recurrence times with which a quasiperiodic function takes on particular values can, in fact, be predicted (see Kac [1943] and Mazur and Montroll [ 19601)). This quantum recurrence theorem is the analogue of the classical Poincare recurrence theorem, which says that any initial point (q, p ) in the phase space
98
ATOMS IN STRONG FIELDS
[I, § 12
of a system of finite volume will be revisited as closely and as often as desired if one waits long enough. In this sense the quantum analogue of a classical system of finite volume is a system with discrete energy eigenvalues. However, it should be noted that the quantum recurrence theorem is more far-reaching than the classical one, because whereas nearby points in classical phase space may have quite different recurrence times, many similar quantum states can exist with similar recurrence times (Hobson [ 19711). It is also worth noting that recurrence per se is not enough to rule out the possibility of chaos, since the Poincare recurrence theorem certainly does not prohibit classical chaos. The important feature about the quantum recurrence theorem in this connection is the assumed quasiperiodicity, which is what rules out the possibility of chaos. In the case of the classical Poincark theorem the recurrence is proved without the requirement of quasiperiodicity ; recurrence and chaos are not incompatible. We are primarily interested in driven systems, specifically atoms in applied fields. For periodically driven systems, e.g., an atom in a monochromatic field, a recurrence theorem similar to that given earlier for nondriven systems can be proved. Hogg and Huberman [ 19821 have proved, using arguments similar to those of Bocchieri and Loinger [ 19571, that “under any time-periodic Hamiltonian, a nonresonant, bounded quantum system will reassemble itself infinitely often in the course of time”. According to them, “this in turn implies that no strict [quantum chaos] is possible.. .”. However, the recurrence time may be exceedingly large. Peres [ 19821, for instance, considered an example where the recurrence time may exceed the age of the universe. Furthermore, as noted earlier, recurrence per se does not necessarily rule out the possibility of chaos. What does rule out chaos is the assumption made by Hogg and Huberman that the quasi-energy spectrum is purely discrete. Note that Casati and Guarneri [ 19841 proved that driving frequencies exist for which the quasi-energy spectrum of the kicked pendulum may be continuous, although such cases have not been found in numerical experiments. (Note: for a time-periodic Hamiltonian we can apply Floquet’s theorem and write the state vector as I Y ( t ) ) = e-’”‘ I $ ( t ) ) , with I $ ( t ) ) periodic with the period of the Hamiltonian. The o,which are analogous to the k vectors in the energy-band theory of solids, define the quasi-energy spectrum (referred to earlier)). We saw earlier how the chaos in the classical theory of the kicked pendulum leads to diffusive-like behavior and, in particular, to an energy growth proportional on average to the time. It is interesting, therefore, to consider now the quunturn theory of the kicked pendulum. The eigenstates of the unperturbed
99
QUESTIONS OF CHAOS IN ATOMIC PHYSICS
1, § 121
pendulum in the quantum theory are
$#)
=
(2n) - eino,
(12.2)
and the corresponding energy eigenvalues are En = n2h2/2m12, n
=
0,
1, + 2 , , . . .
(12.3)
Consider the general problem of a quantum system described by the Hamiltonian H
=
Ha + A(x)F(t)
00
1
6 ( t / T - n)
(12.4)
n = -cc
Let I Y ( k ) ) be the state vector just prior to the kth kick. Just after the kth kick the state vector is exp [ - iA(x)F(k T )T / hI I W k )) , and between kicks the evolution of the state vector is governed by the time evolution operator exp ( - iHat/A). Thus,
I Y(k + 1))
=
exp(-iH,T/h)exp[ -iA(x)F(kT)T/h]
Writing I Y ( k ) ) = Zn a,@) 1 t,hn), where H , I I)n) (12.5) the quantum map
=
I Y(k)) . (12.5)
En I I)n), we obtain from
(12.6)
In the case of periodic kicking (F(t) = constant), Vnm(k)is independent of k. In the example of a periodically kicked pendulum the quantum map has the form (12.6), with V,,,
exp(im12co,2TcosB/h) I I),,,) exp( - in2hT/2m12)
=
($,,I
=
(27r-I exp(-in2z/2) jaZRdBexp[i(m- n)B] exp[i(K/z)cosB], (12.8)
where z = AT/m12 and K = (w,T)2.Note that K is identical to the parameter appearing in the classical standard map (1 1.37). In the quantum map, however, the additional parameter z appears, which vanishes in the classical limit A -,0. Using the well-known representation of the Bessel function J, of the first
100
ATOMS I N STRONG FIELDS
kind, we can write (12.8) in the form V,,,
=
(2n)-’ exp(-inzz/2)
f
x
jOZff
dOexp[i(rn - n)O]
b s ( K / z )exp(isO),
(12.9)
s = --co
with bs(x) 3 isJs(x).Thus, 00
V,,,
=
exp( - inzz/2)
1 b,(K/z) (1/2x) -a
jo Zn
dOexp[i(rn - n t s)O] , (12.10)
and the quantum map (12.6) for the kicked pendulum becomes
c,(k + I) =
C bn-,(K/z)exp(
-inzz/2)c,(k),
(12.11)
n
where we have let c,(k) = a,(k) exp (inz2/2). The energy expectation value for the kicked pendulum is obtained simply by iterating the map (12.11) and using the expression (12.12) for the energy measured in units of rnw,2I2/2.Here N is an integer, typically about 400 in the computations reported below, chosen to be sufficiently large that the total probability is conserved at each iteration. Results for ( E ( k ) ) were first reported by Casati, Chirikov, Izrailev and Ford [ 19791. In fig. 39 we show ( E ( k ) ) for z = 1 and K = 10. Comparing with the classical results shown in fig. 38, we see that a substantial quantum suppression of the classical diffusive energy growth occurs: eventually the energy expectation value stops growing approximately linearly with time, and instead there is a “saturation” of the pendulum energy, or at least a much slower energy growth than at short times. (Note that, based on the energy-time uncertainty relation, we expect the best agreement between the classical and quantum theories for short times, since a small At implies a large AE, making the discrete quantum energies of the pendulum unresolved.) This quantum suppression effect, which was first found in the numerical experiments of Casati, Chirikov, Izrailev and Ford [ 19791, does not appear to depend on the particular choice for z ( # 0). This quantum suppression of the classically predicted diffusive energy
101
QUESTIONS OF CHAOS IN ATOMIC PHYSICS
90
x
60
P C
w 30
0 0
200
100
300
k
Fig. 39. ( E ( k ) ) for K
=
10.0 and T = 1, showing the quanrum suppression of the classically predicted diffuse energy growth.
growth was nicely explained by analogy with Anderson localization (Grempel, Prange and Fishman [ 19841). Consider the tight-binding model of an electron on a one-dimensional lattice, for which the Schrtidinger equation for stationary states can be written in the familiar form (12.13)
where a, is the probability amplitude and En the energy for an electron at the nth lattice site. If the site energies E, are assumed to be independent random variables, the stationary states are found to be exponentially localized in space, i.e., there is no quantum diffusion. Grempel, Prange and Fishman [ 19841 related this quantum suppression of diffusion in the kicked pendulum to Anderson localization in the tight-binding model. From the Schrodinger equation for the kicked pendulum they obtain an equation of the same form as (12.13). The lattice sites of the tight-binding model correspond to the integer values of quantized angular momentum in the kicked pendulum. In place of the random diagonal terms E, of the tight-binding model is a “pseudorandom” sequence { T,} in the case of the kicked pendulum. Specifically,
T,
=
t a n [ i ( o T - n’z)],
(12.14)
where o belongs to the quasi-energy spectrum. Provided z is not a rational multiple of 4n, { T,} is effectively a random sequence, having decaying correla-
102
ATOMS IN STRONG FIELDS
[I, 8 12
tions and a broadband power spectrum, much like the sequence {sin’nfl,}, O n + , = 20,, discussed earlier in connection with the logistic map. Thus, the analogy to Anderson localization, and hence the quantum suppression of the classically predicted diffusion, can be traced directly to the discrete energy-level spectrum of the kicked quantum pendulum. Note that when z is an integral multiple of 4n, the factor exp( - in2z/2) in (12.11) is unity. In this case, and whenever z is a rational multiple of 4n, we have a so-called “quantum resonance” in which the energy grows monotonically as t 2 (or actually n 2 in this kicked system). Since these quantum resonances are very special cases, we shall not discuss them further. For driven systems the quantum suppression of the classically predicted diffusive energy growth is seen in models other than the kicked pendulum, and would appear to cast a long shadow over all classical models of laser-driven atoms and molecules. For such systems this may be the strongest evidence against the possibility of quantum chaos. Classical theories of the microwave ionization of hydrogen have been remarkably successful, however, and it is well known that classical trajectory analyses have been very useful in other problems as well. Before further discussion of these issues, a different aspect of quantum chaos concerning energy eigenvalue distributions will be mentioned.
12.2. REGULAR AND IRREGULAR SPECTRA
Much work has focused on the energy eigenvalue distributions of systems that are chaotic in the classical limit. These studies may be traced in part to a suggestion of Percival [ 19731. Arguing from the correspondence principle, Percival suggested that in the semiclassical limit the spectrum of a quantum system consists of a regular part and an irregular part. Regular regions of classical phase space give rise to the regular spectrum, where the actions .Ii are given by the Einstein-Brillouin-Keller (EBK) quantization conditions. Irregular regions of phase space, where KAM tori have been destroyed, give rise to the irregular part of the spectrum. Percival suggested that in the latter case the energy level distribution depends sensitively on small changes in the nonintegrable perturbation. This suggestion was supported by numerical experiments on the Henon-Heiles potential by Pomphrey [ 19741. It should be noted that the energy levels and spacings may be more orderly when the classical motion is chaotic than when it is regular (Berry [1984], Pechukas [ 19831, Bohigas, Giannoni and Schmit [ 19841) in the sense that adjacent levels repel each other and the spectrum has a rigid character. It has
I , § 121
103
QUESTIONS OF CHAOS I N ATOMIC PtIYSICS
been predicted that in the chaotic regime the distribution of successive levels peaks at a nonzero value, whereas in integrable cases a clustering of levels and a maximum of the distribution occurs at zero separation (Berry and Tabor [ 19771, Berry [ 1977a,b], Zaslavskii [ 19771). This prediction was supported by McDonald and Kaufman [ 19791 in numerical experiments on a “stadium” problem. Based on these and related studies, it appears fair to say that there is quantum chaos in the sense that the wave functions, matrix elements, and energy spectra can all reflect the chaotic character of the corresponding classical motion. (A recent review by Izrailev [ 19901 addresses questions about the spectral statistics and the structure of the eigenfunctions in the regime of strong classical chaos.) However, no current evidence indicates that the time evolution of the wave function can exhibit the hallmark of classical chaos, namely the “very sensitive dependence on initial conditions”, corresponding to a posirive Lyapunov exponent. 12.3. QUANTUM SYSTEMS CAN MIMIC CLASSICAL CHAOS
Although at present no known examples of the chaotic time evolution of a wave function exist, it can be said that certain consequences of chaos in classical systems can be exhibited by quantum systems. One aspect of chaos is the decay of correlation functions of chaotically evolving quantities. To see that quantum systems can exhibit a similar effect, consider the example of a kicked two-level atom. In this case H , = (ho,/2)a,, where a, is the Pauli spin-1/2 matrix in the standard representation and o, is the transition frequency. For the perturbation we take A ( x ) = - d E o, = Asla,. Then, exp[ - iA(x)F(kT)T/A] = exp[iQF(kT)Tax] =
cos[Q(k)T] + ia,sin[Q(k)T]
with SZ(k) 5 QF(kT). Defining c,(k) = a,(k) exp(ikE,T/A), n write the quantum map (12.6) in the form
(12.15) =
1,2, we can
c,(k+ l)=cos[Q(k)T]c,(k)+isin[R(k)T]exp(-ikw,T)c,(k),
(12.16a)
c,(k+ l ) = i s i n [ Q ( k ) T ] exp(ikw,T)c,(k)+cos[Q(k)T]c,(k). (12.16b) Consider first the case of periodic kicking, i.e. Q(k) + SZ, independent of k. We define the autocorrelation function of the state vector as C(z)
=
Jimz
l
j,, T
dt (Y(t)I Y(t +
7))
.
(12.17)
104
ATOMS IN STRONG FIELDS
[I, J 12
For the kicked two-level atom we consider
Numerical experiments indicate that in the case of periodic kicking this autocorrelation function is quasiperiodic rather than decaying. And, of course, since the quantum map (12.16) is linear, its iterates evolve in an orderly, nonchaotic way. In the case of quasiperiodic impulses, however, I C(k)I can be a rapidly decaying function of k, at least over long intervals. Consider, for instance, the kicking with F(t) = c o s d t , in which case (12.19) Q(k) = n cos (a’ k T ) = 62 cos (2nko’/w) = n cos (2nkx), where x = w ‘ / w is the ratio of the two driving frequencies. A rational value of x means that the two driving frequencies are commensurate; otherwise they are incommensurate. For rational values of x we find nondecaying, quasiperiodic autocorrelation functions. For irrational values of x , however, the autocorrelations decay rapidly for large values of QT. The correlations do not actually go to zero: small but finite correlations occur even for very large values of k, as well as occasional peaks as high as 0.4 in 1 C(k)(. However, the behavior of the autocorrelation function of the wave function is dramatically different from the case of rational x . Furthermore, the time evolution of the wave function has a broadband power spectrum. For quasiperiodic kicking the motion of the wave function on the Bloch sphere associated with a two-level atom can also be “ergodic”, in the sense that the Bloch sphere appears to be covered uniformly. These results are discussed by Milonni, Ackerhalt and Goggin [ 19871. This simple example illustrates several important points as follows : (1) Ergodicity and chaos are not the same. In this example the motion on the Bloch sphere can be ergodic, but not chaotic in the sense of a positive Lyapunov exponent. (It is found in all cases that the map is not chaotic in the rigorous sense of “very sensitive dependence on initial conditions.”) (2) Quantum systems can effectively show features like broadband power spectra and decaying correlations, which are consequences of chaos in classical systems, without being chaotic in the classical sense. These features may be among the strongest possible manifestations of any sort of quantum chaos. (3) Quasiperiodically driven quantum systems can display a qualitatively different type of behavior than periodically driven systems. For an intuitive understanding of this “chaotic” behavior under quasiperiodic driving, consider the factors cos [ n(k)T] = cos [ S1T cos(2nkx)] and
I,! 121
QUESTIONS OF CHAOS
IN ATOMIC PHYSICS
105
sin [ QT cos (2nkx)l appearing in the SchrBdinger equation (12.16). For large values of QT and irrational values of x, these functions vary erratically (but, of course, not chaotically!) with k . In particular, the autocorrelation functions are predominantly decaying over large intervals. This does not occur at small values of QT, nor for rational values of x. For large QT and irrational x, therefore, the probability amplitudes are being driven “chaotically” and therefore evolve “chaotically” in a loose sense of the term. It is also worth noting that the angles 0, = 2nkx satisfy the circle map 0, = 0, + x, and for irrational x it can be shown that the circle is filled in densely (ergodically). This simple example suggests a re-examination of the quantum kicked pendulum for the case of two incommensurate driving frequencies (Shepelyansky [ 19831, Milonni, Ackerhalt and Goggin [ 19871). In the case F(t) = cos d t the parameter K appearing in eq. (12.1 1 ) for the kicked pendulum is replaced by K ( k ) = K cos(2nkx), where again x = o ‘ / w is the ratio of the two driving frequencies. For rational x numerical experiments reveal the localization behavior characteristic of periodic kicking. For irrational x, however, there is evidently a dfusive energy growth, as in the classical kicked pendulum. Figure 40, for instance, shows the energy expectation value ( E ( k ) ) for K = 10, 7 = 1 , and an irrational value of x. No evidence exists of any quantum suppression (localization) of the classically predicted diffusive energy growth, even after a large number of impulses. Shepelyansky [ 19831 suggested that the diffusive time scale in this case increases exponentially with K . Based on these results +
,
k
Fig. 40. Energy expectation value ( E ( k ) ) for the kicked quantum pendulum for K = 10.0, z = 1 , and x irrational, showing energy growth proportional to r in the case of quasiperiodic impulses with incommensurate frequencies.
I06
A r O M S I N STRONG FIELDS
[ I . § 12
it does not appear that diffusive energy growth is generally ruled out in driven
quantum systems. Let us emphasize again, however, that the quantum systems subjected to quasiperiodic impulses considered here are not chaotic in the classical sense of a positive Lyapunov exponent. Regarding the periodically kicked pendulum, it is perhaps not terribly surprising that the classical and quantum dynamics are, for sufficiently long times, completely different: as we move up the energy scale, the energy levels get farther and farther apart and, the higher we go, the more the distinctly quantum features will manifest themselves. In fact, the monotonic growth of energy with quantum number plays a crucial role in the analogy to Anderson localization mentioned earlier. The situation is quite different, of course, for an atom or molecule, where the energy spectrum is discrete only up to some ionization or dissociation limit. Detailed comparisons between classical and quantum dynamics have been made only for a few of these systems, but such comparisons have shown, by and large, that the two theories agree well in some of their predictions for ionization or dissociation probabilities. We shall see an example of such good agreement in the following sections. We shall now summarize some results for the Morse oscillator (Goggin and Milonni [ 1988a,b]). The Hamiltonian for the driven Morse oscillator is
H
=
p2/2m
+ D(l
-
e-ar)2 - d x ELcos(oLt),
(12.20)
where D and c1 are the dissociation energy and range parameter, respectively, of the Morse potential, and d is the dipole moment gradient. The classical Newton’s equation of motion can be written in the scaled form d2X/dz2 = -(4/B2) ( e - X - e - 2 x ) + 2K cos(pz),
(12.21)
where z = (DB2/h)t, X = ax, p = ho/DB2, K = dEL/aDB2, and the dimensionless parameter B = h a / m . In the computations reported by Goggin and Milonni, parameters corresponding to the H F molecule were used (Walker and Preston [ 19771). In this case there are 24 bound states of the Morse potential. In terms of the same scaled variables the Schrddinger equation is
ia$/az
-a2$/aX2
=
+ r 2 ( i - e-x ) $ -
KXcos(p~)l(/.
(12.22)
The classical and quantum predictions for the driven Morse oscillator were compared as follows. Equation (12.22) was solved numerically, and the probability P,(O = $,*I I (12.23)
1 < w> n
3
1 9 8 1-21
QUESTIONS OF CHAOS
IN ATOMIC PHYSICS
107
where the summation is over the discrete eigenfunctions I II.,),was computed. The dissociation probability at time t is then 1 - P,(t). In the classical theory eq. (12.21) is solved numerically for an appropriate ensemble of classical trajectories, and the dissociation probability is defined as the fraction of trajectories escaping the Morse well (Goggin and Milonni [ 1988a,b]). The classical resonance overlap criterion makes a fairly accurate prediction for the critical field strength K , necessary for dissociation. Figure 41 compares K , predicted by the classical resonance overlap criterion (solid curves), classical dynamics (o), and quantum theory ( + ) as a function of the initial energy E of the unperturbed Morse oscillator. Figure 41a is for the case of an N = 1 classical nonlinear resonance, whereas fig. 41b is for an N = 4 resonance. For the Morse oscillator such resonances occur when the laser frequency wL = Nw,,,/-, where w,, = is the natural oscillation fre-
d w
0
0.6
0.3
0.9
E /D
0.80
0.92
0.86
0.98
E /D
Fig. 41. Critical field strength K , for dissociation in the driven Morse oscillator, versus E / D (see text), predicted by the resonance overlap condition for (a) N = 1, and (b) N = 4 nonlinear resonance. (From Goggin and Milonni [1988a].)
108
ATOMS
IN STRONG FIELDS
[I, § 12
quency for the nearly harmonic motion near the bottom of the well. Note that the three predictors for K , plotted in fig. 4 1 come into better agreement as E / D increases, as might be expected. The differences between the classical and quantum predictions are most pronounced near higher-order classical resonances ( N > 1) (Goggin and Milonni [ 1988a1) and quantum multiphoton resonances (Walker and Preston [ 19771). The resonance overlap criterion was used to predict the amplitude of the driving force necessary for the onset of global chaos. The chaotic meandering of trajectories leads to dissociation as the system eventually “finds” the continuum. In other words, the classical picture of dissociation (ionization) appears to be closely tied to the onset of chaos. For dissociating (or ionizing) systems, however, it is not clear how to define chaos rigorously, because the computation of Lyapunov exponents requires, in principle, a t + 00 limit. For such systems it has been suggested that the issue of quantum chaos might be phrased as follows: How, if at all, does resonance overlap manifest itself quantum mechanically? (Goggin and Milonni [ 1988a,b]). The width of a classical resonance turns out to be proportional to the square root of the applied field amplitude. Based on the simplest semiclassical quantization procedure, therefore, one might surmise that the number of quantum levels coupled by the field should be proportional to the square root of the field amplitude under conditions of classical resonance overlap, which is corroborated rather well in numerical experiments on the Morse oscillator (Goggin and Milonni [ 1988a,b]). Since the width of a classical resonance corresponds to a spread An of the number of quantum levels mixed by the field, an overlapping of the classical resonances is associated with the mixing of a large number of quantum levels. When this happens, the quantum dynamics is complicated and can mimic the chaotic classical dynamics, but the quantum dynamics are quasiperiodic, not chaotic (Goggin and Milonni [ 1988a,b]). Classically, resonance overlap leads to diffusive motion in phase space, leading to dissociation. Quantum mechanically, it is the spread of population with increasing field strength that gives rise to dissociation. We have already discussed the fact that quantum systems with even a relatively small number of incommensurate energy levels or frequencies can mimic classical chaos in their randomness. In particular, the wave function can exhibit properties that are consequences of chaos, such as broadband spectra, decaying correlations, and certain ergodic properties, without evolving chaotically in time (Milonni, Ackerhalt and Goggin [ 19871, Goggin and Milonni [ 1988a,b]). Of course, the view that quantum chaos is generally not
I , § 131
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
109
possible, and that quantum dynamics can only mimic chaotic behavior in some circumstances, will have to be modified if any examples of true quantum chaos are discovered. The next section discusses these issues in greater detail and examines some laboratory experiments.
5 13. Microwave Ionization of Hydrogen: Experiments and Classical Theory Bayfield and Koch [ 19741 reported the first experimental studies of the microwave ionization of highly excited atoms. Hydrogen atoms with principal quantum numbers n x 66 were produced by charge transfer from Xe using an 11 keV proton beam, and then passed through a microwave cavity. For the microwave frequency of 9.9 GHz, Bayfield and Koch found substantial ionization above a field strength of about 20 V cm- I . The ionization depended strongly on the intensity but not the frequency of the microwaves. Note that since the n = 66 --t 67 transition has a Bohr frequency of 22 GHz, the applied field frequency is only about 40% of the resonance frequency. Furthermore, the ionization energy 13.6 eV/662 is about 76 times as large as the photon energy. Thus, unlike the case of the AT1 experiments so far, the ionization process involves a very large number of photons. An interesting aspect of this microwave ionization is that classical theory is remarkably successful, and also predicts chaotic trajectories for the electrons. Since in the experiments the electrons are presumably described perfectly by quantum mechanics, the possibility exists of direct experimental studies of quantum chaos. In fact, largely for this reason the original experiments were extended in recent years by Bayfield and Koch and their collaborators, and at the time of writing further important experiments are in progress. Note that, because the highly excited electrons are weakly bound, and the applied field strength is typically about 10% of the Coulomb field, the Bayfield-Koch experiments are indeed strong-jield experiments. In other words, since the experiments involve weakly bound electrons, they can explore strong-field regimes with ordinary weak fields. 13.1. IONIZATION EXPERIMENTS
The recent experiments of Bayfield and collaborators have focused on the distribution over n states of the highly excited electrons, whereas those of Koch
110
[I, § 13
ATOMS IN STRONG FIELDS
and collaborators were concerned mainly with ionization. (Note that ionization here always refers to the sum of “true” ionization and excitation to bound states above some “cut off’ value n,. In fact, n, can be used as an independent diagnostic parameter in the experiments, as discussed in Koch, Moorman and Sauer [ 19901). Since this review is concerned with ionization, we shall focus on the latter experiments. First, however, we shall write out the classical equations of motion to see the scaling factors involved. We shall find that the Keldysh adiabatic tunneling parameter y considered in 0 6 also arises naturally in classical theory. The classical equations of motion for an electron in the Coulomb field of the proton plus a linearly polarized monochromatic field Eol cos cut are (13. la)
drldt
=
dpldt
= - (e2/r2)i
p/m,
+ F,,,i
cos W t .
The caret ( ” ) labels the unit vectors. We define F,,, force, and frequency parameters as
(13.lb) =
eEo and the energy,
(13.2a) (13.2b) (13.2~) where a,, and vat are, respectively, the semimajor axis and the root-meansquare velocity of an orbiting electron. Using these definitions in (13.1), we obtain the following equation of motion for R = r/a,,: d2R/dt2= - R P 2 R+ ( F m a , / F a , ) E c o s ( o z / o a , ) ,
(13.3)
where now the time variable is z = Watt. Thus the classical dynamics of the hydrogen atom in a monochromatic field depend only on the dimensionless ratios o/w,, and Fmax/Fat.The Keldysh parameter y is just their ratio: y = (m/ma,)/(Fmax/Fa,), which is eq. (6.51). It is also possible to write y in the form
Y = ho/(Frnaxna,)
(13.4)
for an electron with principal quantum number n, where a,, is the Bohr radius. This is the form used by Bayfield and Koch [1974] in their discussion. They use units in which m = e = h = 1, and so write y = o/nF,,,. For an electron with principal quantum number n, we have Fmax/Fa,= n4Fma,/(e2/ai)and w/wat = (maZ/h)n’w, so that n4FmaXand n 3 0
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
111
in atomic units are the natural parameters suggested by classical theory for plotting experimental data as F,,,, w, and n vary. Note that for n3w = 1, the applied field is resonant with the classical orbital frequency, and nearly resonant with the quantum n -+ n f 1 transition frequency. For n x 66 the electric field due to the nucleus is about 275 V cm- I , and we estimate y z 5.9 for a 9.9 GHz applied field of amplitude 20 V cm- I . The first indication of the success of classical theory in modeling the microwave ionization of hydrogen came from the numerical experiments of Leopold and Percival [ 19781. They integrated the classical equations of motion (13.1) for a range of initial conditions chosen from a (classical) microcanonical distribution corresponding quantum mechanically to an equal distribution of population over degenerate (1, rn) states. They defined and computed the “compensated energy”
E c =’(2 px + P.:) + i
[ ~ -z (Fmax/w)sinmt12 - r - ’
(13.5)
in scaled units. Ec was found to be useful because it removes the rapid oscillations of the energy due to the oscillating applied field; in the absence of the Couloinb field it is constant. A compensated energy E, > 0 was taken as an indication of ionization. Leopold and Percival identified four types of trajectories: (1) trajectories confined to tori; (2) trajectories that rapidly ionize; (3) trajectories excited to very high levels with subsequent ionization; and (4) trajectories excited to very high levels without subsequent ionization. The ionization probability as a function of time was fitted well by the formula Pi,,(t)
=
(1 - QT) [ 1 - e - B ( r ) r ] ,
(13.6)
where QT is an estimated probability that the motion is confined to a KAM torus and jl is nearly constant until ionization is nearly complete. Leopold and Percival reported computations with y = 6 and 7, corresponding to a field frequency of 9.9 GHz and F,,,/F,, = 0.072 and 0.061, respectively. Ionization probabilities for y = 6 and 7 were computed to be 6 2 4 0 % and 40-50%, respectively, compared with the values of 62 and 50% inferred by Bayfield and Koch in their experiments. This good agreement between classical theory and experiment has been even more impressive (although restricted to the regime n3w < 1) in more recent computations and experiments, as we shall see later. The ionization experiments we discuss used microwave frequencies in the 6 to 12 GHz range (Koch [ 1986, 19881). (More recent experiments considered frequencies as high as 36 GHz, at which scaled frequencies in excess of one can be explored; see, for instance, Koch, Moorman and Sauer
112
ATOMS IN STRONG FIELDS
s
ti, 13
[1990].) For fixed frequency w the scaled frequency n3w was varied by varying the principal quantum number n. The hydrogen beam entering the microwave cavity (fig. 42) is produced by charge transfer of a proton beam ( x 14 keV) in Xe. The H(n) atoms are distributed approximately as n-,. The highly excited states were pumped by a double-resonance method employing two CO, laser beams and the static field regions labeled F, , F,, and F, in fig. 42 (Koch [1983]). The Rydberg atoms entering the microwave cavity see a field requiring about 60 periods to ramp up to a steady wave lasting about 300 periods, and then turning off in about 60 periods. Two different methods were used to record the ionization data. One method employs a static field VIabelE 200 V applied to the cavity, enabling an “energylabeled” detection of the protons produced inside the cavity (Koch [ 19831). This method has the advantage of greatly improved signal-to-noise ratio, but the disadvantage that the static field outside the cavity can itself ionize atoms with n x 75. The other method does not measure ionization directly but involves a longitudinal static field downstream from the cavity, and measures how the signal of detected atoms is quenched by the microwave field. Except for n-values above about 75, the two methods give almost the same ionization data (Koch [ 19881). For n = 71-90 the second method is used. Figure 43 shows experimental results for initial states n = 65-74 for the fixed microwave frequency w/2n = 9.923 GHz and varying microwave field amplitude (Koch [1988]). Note that the four curves for n = 67-70 are nearly identical, but that a change in n by just one or two (n = 65,66,7 1, and 72) brings the ionization curve away from these four clustered curves. An additional change in n by one or two (n = 73 and 74) gives completely different ionization curves; note in particular the differences at the lower microwave field strengths. Figure 44 shows a semilogarithmic plot of the threshold field srrengths co2 loser beams
I
I I
I
r_- i-c
mirror
c
Ion detection
L
-
w
-
Fig. 42. Schematic illustration of the experimental arrangement of Koch et al. (From Koch [1988].)
1.8 131
113
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
-
-
Wd
0.0 6
14
10
18
22
26
Microwave electric f i e l d amp1 ( V / c m
)
Fig. 43. Ionization curves for n = 65-74 for a fixed microwave frequency of 9.923 GHz and varying field strengths. (From Koch [1988].)
3.0
2.8
E \
>
.-C
E .-
2.6 2.4 2.2
2.0
L
I
1.8
1
1.6
I
I
I
I
1
I
1.4 c L
0
1.2 1 .o
0.8 0.6
0.4 0.2 0 50
30
70
90
Principal quantum number
0 10% Threshold
X 90%Threshold
Fig. 44. Semilogarithmicplot ofthe threshold field strengths for 10 and 90% ionizationsfor initial states n = 32-90, for a fixed microwave frequency of 9.923 GHz. (From Koch [1988].)
114
[I,
ATOMS IN STRONG FIELDS
S 13
necessary for 10 and 90% ionization for the same frequency, with n ranging from 32 to 90. From n = 32 to n = 90 the threshold fields fall from around 1 kV cm - to only a few V cm - beyond n z 80. Another remarkable thing is that the curve is not monotonic; for instance, the threshold field strength is seen to increase as n increases from around 83 to 88, although, of course, the electron is less tightly bound as n increases. Note, furthermore, the staircase structure. In particular, the plateau around n = 70 reflects the clustering of the n = 67-70 curves of fig. 43.
'
'
13.2. RESONANCE OVERLAP FOR THE CLASSICAL, ONE-DIMENSIONAL HYDROGEN ATOM
Given these strange features, the accuracy of classical theory is all the more remarkable. To begin the discussion of the classical theory and its comparison with experiments on the microwave ionization of hydrogen, we shall apply the Chirikov resonance overlap analysis to the one-dimensional hydrogen atom. For the unperturbed Hamiltonian
H,
=
p2/2m - e 2 / x , x > 0
=a,
(13.7)
x
we determine the action variable from ( 1 1.12):
where x I and x2 are the classical turning points: x, = 0, x2 = e2/1EI = a. After a simple integration we obtain J = (me4/2lE1)'l2 or lEl = me4/2J2 and
H,(J)
= - me4/2J2
(13.9)
for the unperturbed Hamiltonian in terms of the action. Note that the simple semiclassical quantization prescription J --t n h gives the correct Bohr levels E,, = - me4/2n2h2. From (13.9) and = aH,/aJ = me4/J3 = constant we have the angle variable 8 = 8, + ( m e 4 / J 3 )( t - to). For the resonance overlap analysis that follows we require the electron coordinate x in terms of 0. This functional dependence can be determined from x = p / m , or
e
I O x d x ( - IEl
1'"
+ e2/x)-1/2 = (2/m)'/2
'dt.
(13.10)
1, § 131
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
115
Again, a straightforward integration yields
e = 2[sin-'Jxla =
-
J-1,
271 - 2[sin-'Jx/a
-
J1-,
P>O
(13.1 la)
p
(13.1 lb)
if we choose Oo = to = 0. With these preliminaries out of the way, we shall present a more systematic version of the Chirikov resonance overlap analysis than we used in 0 11.6. The perturbed Hamiltonian of interest at present is H = H,(J) - exE, cos wt, or
H
=
H,(I) - eEo cos wt x(I, 0).
(13.12)
We use I instead o f J now to denote action so as not to confuse with the Bessel functions arising below. Now let us write (13.13a) dOx(I,O)cosNO= 271
dOx(I,O)eiNU, (13.13b) 271
so that oci
H
=
H,(I) - 2eE0 c o s m
V N ( I )cosNO N=0
00
=
H,(I) - eEo
1
V N ( I ) [cos(NO + wt) + cos(NO - a t ) ] . (13.14)
N=O
In the spirit of a rotating-wave approximation in which rapidly oscillating terms are ignored, we replace (13.14) by HRwA
=
Ho(I) - e E o V N ( I )cos(N0 - wt)
(13.15)
near an N-resonance defined by w = NO = Nwo(I) = NaHo/aI. We also write I = I N + NP, where I N is the value of the action that gives exactly the N-resonance condition ( w = NwO(IN)), and assume that NP is a small correction to I,, so that V N ( I )E VN(I,) in (13.15). Thus, (13.16a) P = ( l / N ) j = - (1/N)aHRw,/aOG eE, V N ( 1 N ) sin Q , where Q 3 ot - NO. Furthermore, Q = w - NO = w - NaHo/aI, and after some straightforward algebra and the approximation that NPII, is small, we obtain
Q = P / M N , M N = N-2(dwo/dI),'
=
I$/3N2me4.
(13.16b)
116
ATOMS IN STRONG FIELDS
Equations (13.16) follow from the Hamiltonian
H,,,,
=
P2/2M,
+ eE, VN(IN)cos Q
(13.17)
for a pendulum. Obviously, the approximations leading from (13.12) to (13.15) to the pendulum Hamiltonian (13.17) are independent of the form of Ho(I). In other words, any sinusoidally driven, one-dimensionalnonlinear system can be approximated by apendulum system near a weaklyperturbed N-resonance defined by o = Noo(IN). The specific forms of MN and V N ( I N ) will, of course, depend on the system under consideration. Conservation of energy for the pendulum system (13.17) implies that the maximum excursion of P is A P = I 2eM,E0V,(I,)I ‘ I 2 , so that the corresponding range or “width” in the action variable is given by
111,
=
N A P = N/2eM,EoV,(I,)I
(13.18)
The Chirikov resonance overlap criterion (Chirikov [ 19791) states that the onset of chaos should occur when neighboring resonances overlap:
(Recall the discussion in 8 11.5, where the overlap of resonance zones led to the breakup of KAM tori and the appearance of chaotic trajectories.) Now, for an N-resonance in hydrogen we have w = Noo(I,) = Nme4/I,& or
I,
=
(Nme4/w)’I3
(13.20)
for the action resonant with the Nth subharmonic of the applied field frequency. To evaluate AI,,,, we now require V,(I,). From (13.13) and 8 = wot we have, with T denoting the period of the unperturbed notion,
jo T
VN(IN) = O0 271
d t x exp(iNwot)
-i joTdtxexp(iNw0t)= -- 1 -
2nN
nN
J: -
d x sin NO
Jxlaol)(13.21)
1 , s 131
117
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
from (13.11). The change of variable x
a sin’u puts this integral into the form
=
dy siny sin(N siny - Ny)
= - (a/N)J&(N),
where J b is the derivative of the Bessel V N ( I N )= (21$/Nme2)Jh(N) and, from (13.18), AIN = (212/rne3) IeEoJb(N)/3NI =
function
(13.22)
J,,,. Thus,
’/’
(2Ne/o) leE0Jh(N)/3NI ‘ I 2
(13.23)
With this result and (13.20) we can now use the Chirikov criterion (13.18) to determine a threshold field strength E , for resonance overlap and the onset of chaos. Before comparing classical predictions with experiment, we should mention some shortcomings of the lowest-order resonance overlap analysis just presented. It ignores any interactions among the “primary” N-resonances ;coupling of these primary resonances produces “secondary” resonances, which further facilitate the breakup of KAM tori. The lowest-order resonance overlap analysis can therefore be expected to overestimate the field strength necessary for ionization; typically it overestimates by a factor of about two to three. (Recall the remark following the resonance overlap estimate (1 1.40) for the kicked pendulum.) Any additional perturbation (such as a static field) would also affect the threshold field value, and the application of lowest-order resonance overlap analysis in that case is considerably more subtle (Stevens and Sundaram [ 19871). The resonance overlap analysis can also be misleading if the field frequency o is small compared with the nonlinear oscillator frequency wo(Io),where Zo is the initial action. Since wo(Io)= me4/I: and w me4/If near an N = 1 resonance, this occurs when I , is small compared with I , . In this case the onset of chaos would presumably require the width in I , not only to reach up to I,, but also to reach out to I,. Jensen [1984] gave analytical estimates accounting for the effect of electrons with I , < I , in two cases: (1) w 4 oo(Io)and (2) w E wo(Zo).In the second case the onset of ionization for the near-resonance electrons is estimated by calculating the field strength required for the N = 1 resonance island centered at I,to trap an electron with an action Zo < I , . This analytical estimate is labeled as the “m = 1 trapping threshold” in fig. 45, which compares analytical estimates for ionization threshold field strengths with the results of numerical experiments based on the classical equations of motion. The prediction
118
ATOMS IN STRONG FIELDS
f
0.05 -
NUMERICAL IONIZATION THRESHOLDS
+
0.075
0.1 0.2
43
m=l TRAPPING THRESHOLD
0.4 0.5 0.6 0.7 0.8
0.9
1.0
m: Fig. 45. Comparison of analytical approximations for ionization thresholds of a one-dimensional, classical hydrogen atom with the results of numerical experiments (see text). (From Jensen [ 19841.)
based on (lowest-order) resonance overlap shown in fig. 45 assumes that N is large in order to use an asymptotic expression for J&(N). It should be mentioned that when the lowest order resonance overlap analysis is carried further, resonances occur not only when the field is an integral multiple of the oscillator frequency o&), but also at rational multiples of a,,(]).Recalling that coo(]) = me4/13, we see that nonlinear resonances should occur at rational values of the scaled field frequency n 3 0 (atomic units), i e . at n3w = 1, 2/3, 1/2, 2/5, 1/3, .., (Jensen [1987]). This is un important considerationfor interpreting the early experiments, whichfocused on the regime o < o,(l)or, in scaled units, n 3 0 < 1. For n = 70, for instance, and o / 2 n = 9.9 GHz, we have n 3 0 E 1/2.
13.3. COMPARISON OF CLASSICAL THEORY WITH IONIZATION
EXPERIMENTS
The laser excitation of highly excited excited hydrogen atoms in the recent experiments of Koch’s group (fig. 42) produces a unique (n, 0,O) state. However, the distribution over degenerate substates is altered by stray fields before
119
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
the atoms enter the microwave cavity, and consequently the microwave field interacts with atoms distributed approximately uniformly over m-substates of fixed principal quantum number n (van Leeuwen, v. Oppen, Renwick, Bowlin, Koch, Jensen, Rath, Richards and Leopold [ 19851). Therefore, the approximation of a one-dimensional hydrogen atom is not strictly applicable. Nevertheless, the one-dimensional approximation is reasonable, since the extremal states in which the electron and nucleus tend to be stretched apart along the direction of the linearly polarized field are expected to ionize most strongly. The one-dimensional model, for instance, was used to interpret the staircase structure noted earlier in connection with fig. 44 (Jensen [ 19871). Figure 46 shows data plotted in terms of the scaled variables n4F and n 3 0 , as discussed in Q 13.1. In such a plot the steps in fig. 44 translate into peaks centered at rational values of the scaled frequency n 3 0 ( n 3 0 = 1, f , $, . . .), corresponding to classical nonlinear resonances. Evidently the overlap of nonlinear resonances causes ranges of different initial actions - corresponding quantum mechanically to different initial principal quantum numbers - to respond to the field in a similar way. This appears to gives rise to the staircase structure in fig. 44. However, this interpretation does not account for the
4,
0.0
0.5
1.0
1.5
n3w
Fig. 46. Scaled threshold field strength n4F for 10% ionization plotted as a function of scaled frequency n 3 0 . Solid curve shows results of classical one-dimensional theory, and the dashed line is the estimate for the quantum delocalization border, for n = 66. (After Jensen [ 19871.)
120
ATOMS IN STRONG FIELDS
11, I 13
increase with n of the threshold field strength for ionization near n = 67-70 and n = 82-90. Such details evidently require numerical simulations based on the classical equations of motion. These classical Monte Carlo simulations are performed by averaging over classical trajectories, as in the earlier work of Leopold and Percival [ 19781. Figure 47 shows the remarkable agreement between classical theory and experiment obtained in this way by van Leeuwen, v. Oppen, Renwick, Bowlin, Koch, Jensen, Rath, Richards and Leopold [1985]. Both one- and two-dimensional classical models show excellent agreement with experiment, with the two-dimensional treatment showing generally better agreement (see fig. 48). Notice, in particular, that the classical theory can accountfor the observed increase with n of the ionization threshold near n = 67 and n = 82. The only three-dimensional simulations to date have been classical (Rath and Richards [ 19921). Figure 48 compares such simulations with experimental data on the ionization probability versus scaled field strength for the n = 58 level (Koch, van Leeuwen, Rath, Richards and Jensen [1987]).
E
.-c
E 8
Fig. 47. Comparison of experimental data for the field strength for 10 and 90% ionization with one-dimensional classical theory for 10% ionization, including effects of field turn-on. (From Koch [ 19881.)
121
MICROWAVE IONIZATION OF HYDROGEN: CLASSICAL THEORY
C
.-
I
0.04
0.06
0.08
0.10
0.12
n - 4 ' F O (closs. scoled microwove field E x p t ( 3 D ) + Class. 2 D 0 Class. 3 D
-
Fig. 48. Comparison of experimental data for ionization of hydrogen (n = 58) atoms with the predictions of two- and three-dimensional theories. (From Koch, van Leeuwen, Rath, Richards and Jensen [1987].)
(The simulations included the turning on and off of the field, which is important for an accurate comparison of theory and experiment.) Also shown are results of classical two-dimensional simulations, which are less satisfactory than the fully three-dimensional ones. Some experiments were also performed to test the validity of the classical scaling relations discussed in § 13.1 (Koch [1988]). In these experiments ionization was studied using different microwave frequencies o and different ranges of the principal quantum number n giving the same scaled frequency n30. The data shown in fig. 49 for w/2n = 9.92 GHz and o / 2 x = 11.89 GHz indicate that the classical scaling prediction works well. Despite this impressive agreement with experiment, the data shown contain evidence of at least two exceptions to the success of classical theory (Koch [ 19881). One exception is near the peaks in fig. 47, which correspond to resonance islands. For n = 69, for instance, the three-dimensional classical theory predicts a much slower rise with the field strength than that shown in fig. 43. For w/2n = 9.92 GHz, n = 69 implies n 3 0 = 0.496 z 1/2. Quantum mechanically, n 3 0 = 1/2 for n = 69 means that w is close to a resonant two-
122
ATOMS IN STRONG FIELDS
0.15
I 43
-
a
I 44
I 46
I 48
I
47
1 48
1
I
I
+
60 E n (11.89GHz)
-
I 0.02cHz 11.8OCHs
‘d 0.07 0.14
0.18
0.18
0.20
0.22
0.24
n3u (CLASS. SCALED FREQUENCY) Fig. 49. Experimental evidence supporting classical scaling for 10 and 90% ionization thresholds. The label “50” at the top right should actually be “49” (P. M. Koch, private communication). (From Koch [1988].)
photon transition to either n = 68 or n = 70. This failure of classical theory near such a resonance is not surprising: it is well known that classical lheory generally ,fails near multiphoton resonances. Another failure of classical theory was observed when nonmonotonic “bumps” occurred near the ionization threshold for certain n-values, such as in fig. 49 for n = 48. Such features are accounted for by one-dimensional quantum simulations.
13.4. REMARKS
Classical theory offers the following picture of the ionization process: resonance overlap leads to chaotic electron trajectories, and the energy of a chaotically meandering electron grows in a diffusive-like fashion, resulting in ionization. We have seen that the classical theory is, for the parameters considered, primarily in excellent agreement with experiments on the microwave ionization of highly excited hydrogen atoms. One argument explaining the accuracy of classical theory is that the quantity AJN representing the width in the action variable for a classical N-resonance is large compared with the quantum unit of action, h ; the condition A J N / h9 1 can be shown to take the form
1 . 8 131
MICROWAVE IONIZATION OF HYDROGEN CLASSICAL THEORY
123
y Q fJh(N)n,2,where no is the principal quantum number corresponding to the classical orbit and y is the Keldysh parameter introduced earlier (9 6). This condition would appear to be well satisfied in the experiments thus far, except that the microwave frequencies in the experiments we have considered were actually too small to realize any of the classical nonlinear resonances with integer N . (Recall that n 3 0 < 1 in the experiments thus far.) But how is it possible, given the apparent inability of quantum mechanics to produce chaos in the strict sense of a “very sensitive dependence on initial conditions”, for the chaotic classical dynamics to work so well? Although highly excited hydrogen atoms in microwave fields provided the principal experimental testing ground for this question of quantum chaos, the question can be studied independently of any laboratory experiments by comparing the classical and quantum theories for various model problems. In a sense the experiments in the laboratory can be regarded as furnishing analog computers for the quantum calculations! Various studies have focused precisely on the comparison of classical and quantum theories of driven nonlinear oscillators. We have already mentioned, for instance, the work of Walker and Preston [1977] comparing the two theories for a sinusoidally driven Morse oscillator under conditions of negligible dissociation. The most important conclusion drawn from their work, perhaps, is that the classical theory is in reasonably good qualitative agreement with quantum theory except near (quantum) multiphoton resonances. The extension by Goggin and Milonni [1988a,b] to include the chaotic regime of the classical theory, and therefore the possibility of dissociation, confirmed Walker and Preston’s result that classical theory fails most substantially near multiphoton resonances, and showed that it also fails near higher-order classical resonances. As noted earlier, experimental evidence indicates that classical theory is inadequate near multiphoton resonances in the microwave ionization of hydrogen. Computations for the driven Morse oscillator were also extended to the case of two-frequency driving (Goggin and Milonni [ 1988bl. It was reported that in both the classical and quantum theories the presence of a second driving frequency generally acts to reduce the threshold field strength required for dissociation. Analogously, both numerical and laboratory experiments on the microwave ionization of hydrogen showed that two-frequency driving tends to increase the degree of ionization compared with the case ofperiodic driving (Delone, Krainov and Shepelyansky [ 19831, Moorman, Galvez, Sauer, Montazawi-M, van Leeuwen, v. Oppen and Koch [ 19881). These results are consistent with the idea discussed earlier (0 12.3) that quasiperiodically driven systems can more easily mimic aspects of chaos such as diffusive energy growth.
124
ATOMS IN STRONG FIELDS
[I, I 14
It is also worth recalling the argument that classical resonance overlap (leading to classical chaos) translates quantum mechanically into a strong level mixing by the field (Bliimel and Smilansky [1985, 19871, Bardsley, Sundaram, Pinnaduwage and Bayfield [ 19861, Goggin and Milonni [ 1988a,b]). In particular, simple semiclassical quantization ( J + nh) and the fact that AJN is proportional to (cf. eq. (13.18)) suggests that, quantum mechanically, the number of levels mixed by the strong field should be proporThis trend was found to hold in quantum simulations of the tional to driven Morse oscillator (Goggin and Milonni [ 1988a,b]), and suggests that one quantum manifestation of classical chaos might be a spreading of the wave function to cover a considerable number of eigenstates of the unperturbed system. This, in turn, implies that a large number of incommensurate frequencies are important in the time evolution of the state vector. Then the quantum system, although quasiperiodic, can evolve “erratically” enough that the dynamics of dissociation (or ionization) resembles qualitatively what is predicted by classical chaotic dynamics. In particular, if the number An of levels mixed by the field becomes large enough, the state vector can spread to include the continuum (for a variant of this idea, see BlUmel and Smilansky [ 19871). This presumably would be analogous to classical chaotic trajectories meandering into the continuum.
&
A.
Q 14. Microwave Ionization of Hydrogen: Quantum Theory The fact that the experiments involve the absorption of a large number ( x lo2) of photons from a field comparable in strength to the Coulomb field
has made accurate quantum mechanical calculations difficult. However, the success of one-dimensional classical models suggests the same simplification for the quantum calculations, and indeed fairly extensive quantum calculations were carried out for the system described by the SchrBdinger equation (8.14) with V ( x ) = - e 2 / x . (A detailed analysis of the conditions under which the “real” three-dimensional system behaves as a pseudo-one-dimensional one is contained in Sundaram [ 19861.) In scaled units this equation is i
a + -i1 ax2-x a2$ 1 $ - (XF cos w t ) $
-=
(14.1)
at
As in the case of AT1 ( 0 3), we are thus led to a simple equation allowing us to address interesting and fundamental quastions about the interaction of atoms with strong fields. Also, as in ATI, the simplicity of eq. (14.1) is decep-
I,
I 141
MICROWAVE IONIZATION OF HYDROGEN: QUANTUM THEORY
125
tive, since it has been difficult to solve this equation under conditions involving a large number of bound states in addition to the continuum states. One approach to the solution of (14.1) is to use the usual type of basis-state expansion for $(x, t ) :
$(x, t ) =
C an(t>+n(x)+ n
s
dk a,(t>+k(x)
(14.2)
9
where +,,(x) and +,(x) are bound- and continuum-state eigenfunctions, respectively, of the unperturbed system. For the one-dimensional hydrogen atom discussed earlier ($ 13.2), the $,,(x) satisfy the eigenvalue equation
dx2
+ ( E + 2/x)+ = 0
(14.3)
The eigenfunctions +,,(x) associated with the energy eigenvalues En = - l/n2 are given, except for a normalization constant, by
$(x)
=
e-"/"L',- ')(2x/n),
(14.4)
where L',-') is an associated Laguerre polynomial (see Susskind and Jensen [1988] for a detailed quantum mechanical analysis of the onedimensional, one-sided Coulomb potential). In fig. 50, I &(x) I is plotted for the lowest three bound states. Some of the matrix elements (+,,I x I + m ) are plotted in fig. 51. For n = m the (+,I x 1 + m ) are given exactly by 1.5n2. The fact that (+,,I x I +,,) is nonvanishing is a consequence of the one-sided nature of the assumed one-dimensional Coulomb potential. In the first extensive computations on the system (14.l), reported by Casati, Chirikov and Shepelyansky [ 19841, the initially excited level had principal quantum number no = 45, 56, or 66, and a total of 200 levels was included in the range roughly from n = 20 to n = 200. Continuum states were not included in the computations. Comparisons were made with classical-trajectory results corresponding to the same initial n. To gauge the degree of diffusion over the unperturbed eigenstates, the second moment of the distribution function over the n's was computed:
'
M,
= ((n -
(n))')/nt
=
(An')/nt.
(14.5)
The most interesting result of the computations, perhaps, was a quantum suppression for n ; o > 1 of the classically predicted chaotic diffusion. This result is consistent with expectations based on simple models like the periodically kicked pendulum, and confirms again the fact that quantum systems
126
ATOMS IN STRONG FIELDS
X
Fig. 50. I "$I
1
for one-dimensional hydrogen for n = 1,2, and 3; x is measured in units of the Bohr radius a,.
appear to enjoy a greater degree of dynamical stability than their classical counterparts. For short times, M , was close to the classical value, as expected, but then typically oscillated about a constant value, whereas the classical M2 continued to grow with time in the chaotic regime. Differences between the classical and
127 1
I
I
-
r n z l
-
I
I
I
I
I
-
-
m:3
-
-
I
Fig. 5 1. Some matrix elements ($,I
I
x 1 $") for one-dimensional hydrogen; x is specified in units of the Bohr radius a,.
quantum dynamics are illustrated in fig. 52 for the case no = 66, n,)o = 1.2, and n:F = 0.03 and 0.04. Note that one of the two quantum cases shown demonstrates continued growth of M , with time (curve 3), but at a much slower rate than in the corresponding classical case (curve 4). Figure 53 shows an averaged distribution function computed by
128
[I, § 14
ATOMS IN STRONG FIELDS
f
I
0.03. r4
I 0.02-
0.01. 1
I
0
20
60
40
80
T
Fig. 52. Dependence of M 2on time T = mr/2n. Curves 1 and 3 are obtained by quantum computations for no = 66 and n t F = 0.03 and 0.04, respectively. Curves 2 and 4 are obtained by corresponding classical computations. (From Casati, Chirikov and Shepelyansky [ 19841.)
x,
n
Fig. 53. Distribution function obtained by averaging f,(t) over 40 values of 7 for the case of curves 1 and 2 of fig. 52. The quantum and classical results are given by the solid and dashed curves, respectively.The transition point for classical chaos is n,, and the arrows are drawn with equal (on the energy scale) spacing A E = hw. (From Casati, Chirikov and Shepelyansky [ 19841.)
1,
s 141
MICROWAVE IONIZATION OF HYDROGEN: QUANTUM THEORY
129
Casati, Chirikov and Shepelyansky [ 19841 for the cases 1 and 2 of fig. 52. The average is over 40 values of the scaled time t = ot/2n:in the interval 80 < t < 120. The classical distribution (dashed line) is fairly broad, whereas the quantum distribution (solid line) peaks at the initial value no = 66. The peaks in the plateau in such quantum distributions were associated with multiphoton resonances. Casati, Chirikov, Shepelyansky and Guarneri [ 19861 argued that, actually, two critical field strengths, one classical and one quantum, need to be considered. The resonance overlap analysis of Q 13.2 for n i w > 1 leads to the estimate Fc E 1/50r1,5wI/~
(14.6a)
n$F,
(14.6b)
or %
1/50(niw)1/3,
where F, is the critical field strength for classical resonance overlap, the onset of chaos for nearly all initial conditions, and diffusive energy growth leading to ionization. However, we have seen that numerical experiments indicate a quantum suppression of the classical diffusive energy growth. Numerical data and semiclassical arguments (Casati, Chirikov and Shepelyansky [ 19841) indicate, furthermore, that the distribution over n levels reaches a steady-state form with a localization length in n given by 1% 10F2/3w10/3
(14.7)
about the initial value no. Note that the classical diffusion rate may be related to the localization length (Shepelyansky [ 19861). (Recall the discussion in Q 12.1 on the analogy between quantum suppression and Anderson localization. A recent article by Jensen, Susskind and Sanders [ 19911contains a simple discussion of the localization picture in terms of a succession of two-level transitions.) If this “photonic localization” length is small, it is argued that the classically predicted diffusive energy growth and ionization are suppressed by quantum mechanics. However, if the field strength is sufficiently large that I becomes comparable to the number of photons required for ionization (Nl = 1/2n;w, atomic units), then presumably it becomes possible for ionization to occur. The condition I = NI gives (14.8a) F,
= O. 4(n:
0)7/6/&
(14.8b)
I30
ATOMS IN STRONG FIELDS
[I, § 14
for the critical field strength necessary to overcome the quantum localization and achieve ionization. This critical field strength defines what Casati, Chirikov and Shepelyansky [ 19841 call the quantum delocalization border, which is indicated by the dashed line in fig. 46 (see also Casati and Guarneri [ 19881). Casati, Chirikov, Shepelyansky and Guarneri [ 19861 reported that the quantum delocalization theory was corroborated well by numerical experiments on the one-dimensional hydrogen atom. Some of their numerical studies included a coupling to the continuum by employing a so-called Sturmian basis, with each Sturm function being a superposition of bound and continuum eigenfunctions. Up to 600 Sturm functions were included. We refer the reader to Casati, Chirikov, Shepelyansky and Guarneri [ 19871 for a review of their numerical techniques and many numerical results on the sinusoidally driven, one-dimensional hydrogen atom. As noted earlier, the experiments considered thus far were in the regime n,30 < 1. More recent experimental work, however, has explored the regime n,30 > 1. Although some experimental evidence exists to support the dynamical localization picture (Bayfield, Casati, Guarneri and Sokol [ 1989]), an active debate (based on experimental data) is continuing on the extent of the confirmation of the theory. (Koch, Moorman and Sauer [1990], and references therein, discuss this issue in detail.) It might be noted in this connection that the delocalization border is lower in the two- and three-dimensional cases, so that the experiments may have to employ atoms prepared in effectively onedimensional states to probe the differences between the classical and quantum critical field strengths for ionization. The work of Bltimel and Smilansky [ 1985,19871 handled the continuum by (a) including bound-continuum coupling but ignoring continuumcontinuum coupling, or (b) using a Sturmian basis that discretizes the continuum (Casati, Chirikov, Shepelyansky and Guarneri [ 19861). The assumption of periodic driving allows one to employ Floquet theory and quasi-energy states (Q 12.1). This eliminates any need to integrate equations over many periods of oscillation of the applied field. Bltimel and Smiiansky describe an approximate division of the quasi-energy states into two classes: (a) those overlapping mainly low-n states and (b) those overlapping mainly high-n states. The latter have expansion amplitudes in the bound states that decay with n according to a power law. The transition between the two classes, and the breakdown of localization, occurs at some critical value of n that Blumel and Smilansky estimate using the Hose-Taylor criterion (Hose and Taylor [ 19831). Basically this criterion is the assertion that delocalization
1, § 141
MICROWAVE IONIZATION OF HYDROGEN: QUANTUM THEORY
131
10-
LO
60
no
80
Fig. 54. Critical field strength for ionization as a function ofinitial quantum number no calculated by BlUmel and Smilansky for w = 1.5 x 10- (atomic units). The squares label the experimental points from Koch et al., the solid points are calculated points, and the solid curve is based on a perturbative approach using the Hose-Taylor criterion. (From Blilmel and Smilansky [ 19871.)
(“quantum chaos”) occurs when the eigenstates of the perturbed, nonintegrable Hamiltonian overlap the eigenstates of the unperturbed system with a probability exceeding a half. From this criterion, and a more accurate numerical calculation, a critical field strength for delocalization and ionization is determined in terms of w and the initial level no. Figure 54 shows the excellent agreement with experimental critical field strengths determined in this way by Bliimel and Smilansky. Quasi-energy states were also used by Bardsley, Sundaram, Pinnaduwage and Bayfield [ 19861, although without considering the effects of continuum states. They observed a transition from localized to delocalized quasi-energy states as the scaled field strength was increased, and showed good agreement with the distributions over n-states seen in the Bayfield experiments. The effects of continuum states were later included, using a complex-coordinate method (Bardsley and Comella [ 1986]), and good agreement with the experiments of Koch’s group was reported for the threshold field strength for 10% ionization from the initial state no = 63 and with n ; o ranging from 0.4 to 1.0. Unfortunately, Floquet theory cannot be used in any straightforward way to handle nonperiodic driving, such as occurs when the field is turned on and off adiabatically. Such effects are known to be important for a detailed comparison with experiment. At the time of writing, many aspects of the quantum theory for the microwave ionization of Rydberg hydrogen atoms are under intensive
132
ATOMS IN STRONG FIELDS
[I, 8 15
investigation, and our discussion here is intended only as an introduction to this important aspect of quantum chaos.
6 15. Summary and Open Questions The theory of the interaction of atoms with arbitrarily strong laser fields has yet to be fully worked out. This review has considered the simplest conceivable problem, namely, a one-electron atom in a perfectly monochromatic field. In fact we have focused considerable attention on the even simpler problem of a one-dimensional atom in a monochromatic field. Such problems are still difficult, still hold surprises, and are becoming increasingly important as the level of readily achievable laser powers continues to grow. We introduced and surveyed two photoionization problems of current interest: above-threshold ionization and the ionization of highly excited hydrogen atoms by microwave fields. Both are strong-field problems, although in the latter the strong-field regime is realized by working with weakly bound electrons and modest field strengths rather than tightly bound electrons and very large field strengths. In the latter case, the phenomenon of high-order harmonic generation, which occurs in the weak-ionization regime, was also discussed. Although the main features of above-threshold ionization seem fairly well understood, the theory at this time is lagging well behind the experiments. The leading theoretical benchmark is perhaps still the Keldysh approximation, but this approach has serious limitations. Furthermore, it is not clear whether approaches of this type can ever be meaningfully extended to include aspects like two-electron excitations. Thus far the theory of the microwave ionization of hydrogen has evolved largely independently of the theory of above-threshold ionization. The main interest at present lies in its relevance to the problem of quantum chaos. However, there appears to be no fundamental reason why the same methods and ideas cannot be applied more generally to other problems of atoms in strong fields. Are the concepts of quantum localization and delocalization important more generally? Are questions of chaos more prevalent in strongfield interactions, or do they arise only in limiting situations where classical theory is a good approximation? In this context it is worth mentioning another mechanism seen in the quantum suppression of classically predicted chaotic behavior. This effect is referred to as “scarring” (Heller [ 19841) and, unlike dynamical localization, depends on local structures in classical phase space. “Scarring” dispels the
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expectation that highly excited quantum states of chaotic systems would appear to be locally random; instead, typical wave functions were found to peak strongly along the unstable, periodic orbits of the system (Heller [1984]). In recent years, “scars” have been identified in a wide variety of systems, including the interaction of excited hydrogen with microwaves (Jensen, Sanders, Saracen0 and Sundaram [ 19891). Surprisingly, the effect also seems pertinent to the most recent strong-field phenomenon of stabilization (Jensen and Sundaram [ 19901). This appears to support the suggestion that perhaps a common approach can be used to deal with seemingly diverse strong-field phenomena.
Acknowledgements We are pleased to acknowledge helpful discussions with J. R. Ackerhalt, L. Armstrong Jr, J. S. Cohen, L. A. Collins, G. Csanak, J. H. Eberly, R. V. Jensen, G. A. Kyrala, A. L. Merts, M. H. Mittleman, L. Pan, G. T. Schappert and B. W. Shore.
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LIGHT DIFFRACTION BY RELIEF GRATINGS: A MACROSCOPIC AND MICROSCOPIC VIEW BY
E. POPOV Institute of Solid State Physics Bulgarian Academy of Sciences 72 Trakia blvd. Sofia 1784, Bulgaria
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CONTENTS PAGE
$ 1. INTRODUCTION
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$ 2 . QUASIPERIODICITY: A FUNDAMENTAL PROPERTY OF GRATINGS . . . . . . . . . . . . . . . . . . . . . .
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$ 3. PHENOMENOLOGICAL APPROACH: A STEP TOWARD THE PHYSICAL INTERPRETATION OF GRATING 158 PROPERTIES . . . . . . . . . . . . . . . . . . . . . $ 4 . MICROSCOPIC PROPERTIES OF LIGHT DIFFRACTED
BY RELIEF GRATINGS . . . . . . . . . . . . . . . . 168 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 185 REFERENCES
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6 1. Introduction Diffraction gratings have attracted the attention of scientists for centuries, but they continue to exhibit such fascinating properties that the interest in them is still increasing. There are two primary reasons for gratings being used in several scientific disciplines. First, since they are used as a fundamental dispersive element, both manufacturers and users are vitally interested in the investigation and, if possible, optimization of grating properties. Moreover, the manufacturing process itself generates many problems for research into mechanics, lasers, optics, and materials (see, e.g., Hutley [ 19821, Loewen [ 19831). Second, although simple in structure, gratings have proved to be so complex in behavior that the investigation of light diffraction by a periodically corrugated surface continues to be one of the most fascinating problems of the electromagnetic theory of light scattering. Throughout its development the theory has never been able to predict the entire spectrum of grating peculiarities, and as new phenomena (i.e. anomalies) appear that need further investigation, new theories arise. In addition, because of their structural simplicity, gratings are playing an important role as a model for other configurations, such as quasiperiodical and random rough surfaces (e.g., Maystre [ 1984b1).
1 . 1 . GRATING ANOMALIES
Contrary to the nonscientist, the scientist is usually thrilled with the word “anomaly” - a paradox that is explained by the implication of something abnormal or unexpected, i.e., a new object for investigation. When Wood [ 19021 observed an unexpected property of diffraction gratings (i.e. a change in diffraction efficiency by a factor of more than 10 occurs in a spectral region not larger than the distance between sodium lines), he called that phenomenon “anomalous”. Each more or less rapid change in the diffraction efficiency of gratings is called an anomaly, even if it becomes a theoretically normal phenomenon. The great interest in the almost century-long investigation of anomalies has several explanations: 141
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(1) Their appearance is connected to some physical phenomena that attract attention by themselves. (2) Many anomalies result from surface wave excitations and could provide information on surface properties. (3) Since it is more important to have a uniform rather than a high diffraction efficiency for most grating applications, anomalies must be avoided. (4) In some cases high efficiency values could also be anomalous, and detailed investigations of anomalies could result in interesting applications. ( 5 ) The theory of anomalies provides a valuable stimulus for the development of recent numerical methods for the analysis of the diffraction of light by relief gratings. The first reviews on grating anomalies were published by Twersky [ 1956, 19601 and Millar [ 1961a,b]. More recently, Neviere [ 19801 and Maystre [ 19821 have contributed significantly to the physical understanding of anomalies and their classification. Although Q 3 discusses in detail the present-day interpretation of different types of grating anomalies, some preliminary observations can be made. Three main types of anomalies can be distinguished: ( 1 ) threshold phenomena that are connected with the energy distribution in propagating orders when some orders become evanescent; (2) resonance anomalies that are due to the excitation of a guided wave along the corrugated surface (phase matching between the incident wave and a solution of the homogeneous problem are ensured by the grating); and (3) nonresonance anomalies (sometimes called “broad” anomalies). It is surprising that of the entire range of grating properties, almost nothing exists that has not been classified as an anomaly; as shown in Q 2.2.1, even a completely regular type of behavior can be considered to be anomalous.
1.2. GRATING PROPERTIES AND PHYSICAL INTUITION
Over the past decade, many reviews and monographs have described the various theoretical, experimental, and technological problems and achievements of gratings (e.g., Petit [ 19801, Hutley [ 19821, Loewen [ 19831). Today scientists believe and have proved that the electromagnetic theory of gratings based on the numerical implementation of the fundamental Maxwell’s equations with appropriate boundary conditions can solve any problem involving light diffraction by periodic structures. Two “minor” difficulties, however, make this conclusion hypothetical rather than real, namely, numerical problems and cognition. The latter difficulty is typical for an understanding of
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complex objects. Physical intuition is never satisfied simply by a knowledge of the basic facts; rather, understanding needs the determination of connections and analogies with something previously recognized and familiar. This idea was illustrated by Mermit [ 19901 using the example of recent quark theory: the knowledge of something being just an allowed combination of quarks contributes little to a real understanding of this entity. Similarly, sometimes a rigorous numerical modeling of the grating properties is insufficient to draw connections and analogies; calculations can stimulate, but not substitute for, thinking. A good example is the well-known property of metallic gratings to support two (the zeroth and first) diffraction orders in a Littrow mount, where the dispersive order propagates in a direction opposite to the incident wave: a quasiperiodicity in the diffraction efficiencies is observed as a function of the depth of the grooves. The theoretical explanation for this curious phenomenon, which is the focus of our interest here, could be found by starting from the simplest approaches based on Fresnel's principles, followed by scalar theories, and concluding successfully with recent electromagnetic methods. However, the further we go into more rigorous descriptions, the less helpful is physical intuition (see discussion by Petit and Cadilhac [ 19871). For example, the simplest possible explanation is based on the following considerations: let us investigate a blazed (echelon) grating with a 90" apex angle (fig. 1) constructed from a perfectly conducting material, with light incident perpendicular to the larger facets. If the ratio of wavelength A to the period d is properly chosen: 2 sin$
=
m(A/d), m
=
0,
1, & 2,. . . ,
(1)
the mth diffracted order propagates backwards towards the incident wave (fig. 1). If the angle of the grooves is equal to the angle of incidence $, then a maximum efficiency could intuitively be expected for this mth order: light is incident normally on the large facet and is reflected backwards as if by a flat
Fig. I. Schematic representation of a blazed grating.
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LIGHT DIFFRACTION BY RELIEF GRATINGS
[I4 I 1
mirror. Equation (1) shows that the phase difference between rays reflected by two consecutive facets is equal to an integer times 2 ~i.e., ; interference maxima appear in the far-field zone. According to eq. (l), such a maximum in the diffracted efficiency is repeated quasiperiodically with increasing m. repeated quasiperiodically with increasing m. Such a simple intuitive approach, however, could not explain why this phenomenon is independent of the sign of the angle of incidence (the dashed lines in fig. 1). If two diffraction orders are propagating, a perfect blazing in the Littrow mount can be observed when the incident wave is symmetrical with respect to the normal of the grating and is not perpendicular to any of the facets. Intuition fails completely when one realizes that this behavior in a Littrow mount is typical of the triangular and any other form of profile. Moreover, this quasiperiodicity (consecutive peaks and dips in the dependence of the efficiency on the groove depth) is common for other types of grating materials and incident conditions. This property is described in detail in 0 2 for a variety of gratings. The results are obtained using rigorous electromagnetic methods. Most of these results have been confirmed experimentally. Section 3 explains in phenomenological terms the phenomena that were introduced in 0 2. Peculiarities in the efficiency are expressed using a small number of parameters (usually the poles and zeros of scattering matrix components), and their trajectories are followed when the groove depth is varied. These trajectories enable us to draw some unexpected new connections between different phenomena, such as surface wave excitation, Littrow and non-Littrow perfect blazing, and resonant and nonresonant total absorption of light. Section 3 also proves that the phenomenological parameters exhibit a quasiperiodical type of behavior. Thus it becomes obvious that the generality and importance of such behavior can be compared with the grating equation: whereas the latter guides the directions of the diffracted light, the quasiperiodicity determines the amount that is diffracted. In 5 4 a microscopic near-field picture is presented that enables us to visualize rigorous numerical results and to see what happens, to find out why it happens, and to determine why it is so general. In summary, it is shown that the behavior of reflection gratings is determined by the formation of regions with closed energy flow lines. Two main reasons explain why this physical interpretation (simple in its essence and illustrative in its presentation) of the numerical results was found only recently. First, it has not been clear where to look for a simple interpretation, since when dealing with the near-field zone, most scientists are interested in the energy power density and not in the energy flow direction. Second, theoretical and numerical difficulties prevent the possi-
1 1 7 8
11
INTRODUCTION
145
bility of obtaining the behavior of the electromagnetic field very close to the grating surface with sufficient precision, particularly for deep grooves, which is necessary because most phenomena are manifested only in deep gratings.
1.3. THEORETICAL APPROACHES TO GRATING PROPERTIES
As discussed in Q 1.2, intuitive approaches not only failed to work even for the simplest configurations, but they also could not explain satisfactorily any of the grating anomalies, a topic of great interest since Wood’s [ 19021 astonishing observation of the anomalous diffraction of light by a metallic grating. Lord Rayleigh [1907] was the first to realize that Fresnel’s theory “tends to fail altogether when the (grating) interval is reduced so as to be comparable with the wave-length”. His proposal for decomposition of the field into propagating and evanescent waves has provided a basis for the more recent approaches, although it includes diffracted orders that propagate only “upwards”. By 1941 the Rayleigh method, as a precursor of recent electromagnetic theories, became commonly accepted (see Fano [ 19411) as “following, in particular, work by Madden and Strong [ 19581 it becomes apparent that physical optics failed to give a satisfactory solution” (McClellan and Stroke [ 19661). Unfortunately, approaches based on the Rayleigh method cannot, in general, describe the field inside the grooves (except for shallow gratings, see Petit and Cadilhac [ 19661, Neviere and Cadilhac [ 19701, Millar [ 197 la,b], Van den Berg and Fokkema [ 1979]), although they could sometimes predict the far-field behavior (diffraction efficiencies) (Wirgin [ 1979a-c, 1980, 19811). Van den Berg [ 19811 made an illustrative analysis investigating the convergence of the diffraction efficiencies and the satisfaction of the boundary conditions when the number of diffraction orders used in the calculations was increased; the convergence of the far-field results is not always accompanied by a fulfillment of the boundary conditions on the grating surface. Differential methods are relatively simple to implement with computers (Petit [ 1966b], Neviere, Vincent and Petit [ 19741, Moaveni, Kalhor and Afrashteh [ 19751, Chang, Shan and Tamir [ 19801, Vincent [ 19801, Moharam and Gaylord [ 1982]), but they also exhibit convergence difficulties for highly conducting gratings when the magnetic field vector is parallel to the grooves (see Vincent [ 19801).Thus, approaches must be used that are much more complicated and thus less attractive for programming, e.g., the integral method (Petit and Cadilhac [ 19641, Wirgin [ 1964, 19681, Uretsky [ 19651, Petit [ 1966a1, Pavageau, Eido and Kobeisse [ 19671, Neureuther and Zaki [ 19691, Dumery
I46
[II, § 1
LIGHT DIFFRACTION BY RELIEF GRATINGS
and Filippi [ 19701, Van den Berg [ 19711, Maystre [ 1972, 1973, 1978a,b, 19801) or the conformal mapping approach (Neviere, Cadilhac and Petit [ 1972, 19731). It is not our purpose to discuss in detail the nature, efficiency, and difficulties of different methods of light diffraction by periodic structures, but only to outline the main problems in the near-field investigations. Extended reviews on electromagnetic theory and its computer implementations can be found elsewhere (Petit [ 19801, Maystre [ 1984a1). The importance of near-field results by themselves, and not only as a basis for the correct determination of efficiencies, concludes with a description of a brilliant example from a paper by Maystre [ 1984133. A theory for electromagnetic scattering by a finite corrugated surface (figs. 2 and 3) is developed using the rigorous integral method of light diffraction by periodic structures. These impressive results warrant special attention: the distribution of electromagnetic field energy density in the near vicinity of a groove is almost equivalent to the corresponding distribution for an infinitely extending corrugation, provided the groove under investigation is separated from the surrounding plane region by several grooves. This phenomenon, called "short-coupling range", seems to have far-reaching, although not fully determined consequences. The first practical uses are in the theory of light diffracted by random rough surfaces that make it possible to reduce the computer memory storage and computation time 0.20 0.10 0. 00 - 0.10 0.20 -5.
I
t
-
20.
0.
n, v v v, ri
5.
x
t
f
10.
,
4. 0,
TM
1.0 -5.
1" A'" 0.
5.
1
10.
Fig. 2. The top part of the figure shows the shape of a finite grating with three grooves. The two other curves represent the difference between the actual surface current density and that corresponding to a perfect mirror: 0, = 30", I / d = 0.9, d = I , h/d = 0.4. (Afler Maystre [1984b].)
147
QUASIPERIODICITY
0.20 0.10 0.00 -0.10 -0.20 0.
5.
10.
-5.
0.
5.
10.
-5.
0.
5.
10.
-5.
20.
,
15.
10.
5. 0.
Fig. 3. The same as fig. 2, but with eight grooves. (After Maystre [1984b].)
significantly. Applied directly to the results given in 5 4, the short-coupling range phenomenon permits us to conclude that, in general, these results are valid for structures with a limited number of grooves and, perhaps also, for rough surfaces.
0 2. Quasiperiodicity: A Fundamental Property of Gratings 2.1. STATEMENT OF THE PROBLEM
Let us investigate a plane wave scattered by a periodic structure with a period d. The electromagnetic field F(x, y ) that is a solution of this problem is periodic to within a phase term, F(x + d, Y ) = F(x, 14exp [ i ( 2 d d ) x l ,
(2)
where Fis any field characteristic and, in particular, its vector component along the grooves. We assume a classical diffraction case when an incident (and diffracted) wave vector is perpendicular to the grooves. We deal with a periodic corrugation (fig. 4) of the interface between two media that are characterized by their (complex) refractive indexes n,, j = 1,2. The upper medium is assumed to be lossless. The x-axis of the coordinate system introduced lies in the grating
148
LIGHT DIFFRACTION B Y RELIEF GRATINGS
I
-
X
Fig. 4. Schematic representation of light diffraction by a grating.
plane and is perpendicular to the grooves. The z-axis is parallel to the grooves. Light is incident at an angle of Oo with respect to the y-axis. Two fundamental cases of polarization are identified that are mutually independent in the classical diffraction case (e.g., Petit [ 19801). They are represented by the z-component of their electric ( E ) or magnetic ( H ) field vectors: E , , TEcase, If,, TMcase.
F(x, Y ) =
(3)
Due to the periodicity of the corrugations, incident light is diffracted into several orders with directions determined in the upper medium by the grating equation sine,
=
sine,
+ m ( A / d ),
m
=
1, & 2, . . . .
0,
(4)
If the second medium is also lossless, transmitted diffraction orders exist in directions determined by n 2 sin 8::)
=
+ m(A/d),
n , sin 0,
Diffracted wave vectors are equal to k , k
=
=
0,
1,
2, . . . .
k(a,,
x,,
0), where
=
(5)
27t/A,
a,,, = n, sin 6:)
x;
m
=
($
, j
=
1, 2 ,
- a y 2 .
If x,, is real (for m E U), the corresponding order propagates away from the corrugated region, and if 1 , is imaginary, the order is evanescent. In the far-field zone defined by I + co,the total field is a sum of the incident and jjl
149
QUASI PERIODICITY
diffracted terms
F(x, Y ) = a0 ~ X [ik(aox P - xOY)I +
1 6,
exp [ik(amx + x ~ Y ) I
(7)
moU
with amplitudes a, and {brn}. In general, the wavelength-to-period ratio determines the direction in which the light is diffracted (eqs. (4)and (5)). Another set of parameters, namely the groove profile f(x), determines how much of the incident light is diffracted into those directions. For a fixed profile form the most general characteristic usually becomes the groove depth h
=
max [ f(x)] - min [ f(x)] .
(8)
According to eq. (2), the total electromagnetic field is periodic along the x-axis parallel to the grating vector. Much more surprising is the existence of a quasiperiodicity in diffraction order amplitudes (their modulus and phases) as a function of groove depth h at other fixed parameters. It is impossible to predetermine this fundamental property from general considerations. It is investigated in this section for different grating periods, mountings, and materials.
2.2. REFLECTION GRATING SUPPORTING TWO DIFFRACTION ORDERS
2.2.1. Littrow mount Two diffraction orders propagate in the first-order Littrow mount (when eq. (1) is satisfied for m = - 1) when the wavelength-to-period ratio A/d lies in the interval ($, 2). Consecutive rises and falls of the diffraction efficiency with increasing groove depth are observed (fig. 5). This well-known type of behavior has been widely discussed. Although the behavior is regular, sometimes definite points in the groove depth dependence of the efficiency are considered to be anomalous: for decades the most unusual feature was thought to be the first specific region in fig. 5 with a zero efficiency in the specular order, accompanied by a maximum energy amount diffracted in the first order. This phenomenon is called “perfect blazing in a Littrow mount” (Roumiguieres, Maystre and Petit [ 19761, Breidne and Maystre [ 19801, Maystre, Cadilhac and Chandezon [ 19811, Maystre and Cadilhac [ 1981]), or “Bragg-type anomaly” (Tseng, Hessel and Oliner [ 19691, Hessel, Schmoys and Tseng [ 19751). The primary reason for this interest is the practical importance of the maximum efficiency
150
LIGHT DIFFRACTION BY RELIEF GRATINGS
0
0.75
1.5
h/d
Fig. 5 . Littrow mount efficiency as a function ofgroove depth of a sinusoidal perfectly conducting grating: d = 0.5 pm,A = 0.6238 pm, TM polarization. The solid line represents the zeroth order and the dotted line represent the first order. (After Popov, Tsonev and Maystre [1990a].)
for the use of gratings. For metallic gratings the ratio of groove depth h, to period corresponding to perfect blazing is approximately equal to h,/d x 0.40 for TM polarization, being greater and strongly dependent on the wavelength for TE polarization (Petit [ 19801). For deeper grooves the first-order efficiency q - gradually decreases, and at some value of h, almost equal to two times h,, q - becomes zero. The grating acts like a flat mirror throughout a large interval of angles of incidence (fig. 6): provided q - is zero in the Littrow mount, it is almost zero everywhere (independent on the mounting). This phenomenon is called the “antiblazing” of gratings, which is explained mathematically by energy conservation and symmetry considerations (Mashev, Popov and Maystre [ 19881).
0.2
0.6
1.0
a0
Fig. 6. Angular dependence of the first-order efficiency of a perfectly conducting sinusoidal grating, d = 0.5 pm, 1 = 0.6328 pm, TM polarization, a, = sin 0,. (After Mashev, Popov and Maystre [1988].)
11, § 21
151
QUASIPERIODICITY
A further increase of the groove depth leads to a repetition of the behaviour of the efficiency; i.e., blazing is followed by antiblazing. A perfectly conducting substrate makes the blazing perfect, and all the incident energy is reflected back partially or totally into the zeroth or first order, their sum remaining equal to unity. For real metallic gratings in the visible region a similar quasiperiodic type of behavior is observed, the only difference being a slow reduction of the maximum efficiency values (Chandezon, Dupuis, Cornet and Maystre [ 19811).
2.2.2. Non-Littrow mount The quasiperiodicity of the efficiencies of highly reflecting gratings in the Littrow mount also appears for other incident angles. This can be expected in the light of the previously discussed antiblazing effect, since the first-order efficiency over a large angular interval becomes almost zero provided that it is zero in the Littrow mount (fig. 6). A typical example is presented in fig. 7; the efficiencies in the Littrow mount and in the grazing incidence exhibit similar quasiperiodic behavior. The maximum values in the grazing incidence are significantly lower than in the Littrow mount, but the positions of the minima coincide. In addition, fig. 7 presents results that contribute greatly to the generality of this type of behavior: the diffraction losses of a surface wave propagating along the same corrugated surface oscillate with the same period.
0
0.65
1.3
h /d
Fig. 7. Groove depth dependence of the efficiencies q - , (the solid line represents the Littrow mount and the dashed line represents grazing incidence, with ci0 = 0.99) and of the diffraction losses (dotted line) of a plasmon surface wave: d = 0.5 pm, I = 0.6328 pm, TM polarization. (After Popov, Tsonev and Maystre [1990b].)
152
LIGHT DIFFRACTION B Y RELIEF GRATINGS
[II, § 2
2.2.3. Surface waves on corrugated metallic surfaces As mentioned in 0 1.1, surface waves play an important role in grating anomalies. Their excitation leads to the formation of narrow dips and peaks in the diffraction efficiency (discussed in detail in 0 2.3.2 and 0 3.1). In this section the properties of the surface wave itself are analyzed. Two diffraction orders propagate in the upper medium (air, with n , = 1) when
I/d- l
(9)
In the case presented in fig. 7, I / d - 1 = 0.2656. For angles of incidence characterized by I sin Oo I < 0.2656, only the zeroth-order wave propagates; this case is examined in detail in the next subsection. The other limiting case, 1 sin O0 1 > 1, corresponds to surface wave propagation along the corrugated boundary. This surface wave has a propagation constant ku, (=I?sin $) and is characterized by an exponentially decreasing field with increasing distance from the interface. For a highly conducting metal-air boundary these waves are usually called plasmon-polariton surface waves (PSWs), which exist when the real part of dielectric permittivity of the metal is less than - 1 (Neviere [ 19801, Maystre [ 19821). The homogeneous problem (the existence of a nonzero scattered field without an incident field) on the plane surface has a solution with a propagation constant equal to 1 us
=
d m
and exists only for the TM polarization on a bare metallic surface. When the media are lossy, even on a flat boundary the surface wave propagation constant becomes complex (eq. (10) with complex values of n2). Its imaginary part Ima, corresponds to the absorption losses in the metal. For highly conducting metals the real part of a, is slightly greater than unity. The surface corrugation has two simultaneous effects, resulting in the growth of Imu,. First, it modifies slightly the absorption by the surface (fig. 8). By increasing the groove depth, the absorption losses grow proportionally to the average increase in the local-field density (Popov, Tsonev and Maystre [ 1990bl) rather than to the total area of the absorbing surface. This conclusion is further confirmed for deeper grooves: absorption losses exhibit consecutive rises and falls as a function of the groove depth (fig. 8b). Second, provided that the grating period is appropriately chosen to couple the surface wave to a
11, I 2 1
153
QUASIPERIODICITY
5.0
0.016 m m
losses
x
c
m
-0
C
2.5
--.
D
0.1 h /d
0.2
+d
n
n
. . middle . ..
0 0
.s b
L
0.8
0.08
0
0 0.7
1.o h/d
1.3
Fig. 8. Absorption losses of a plasmon surface wave as a function of groove depth (solid line) compared with the field intensity at the top, bottom, and in the middle ofthe grooves: d = 0.5 pm, 1 = 0.6328 pm, TM polarization. (a) For a shallow grating, where the losses and intensities are given compared with their values on a flat surface. (b) deep grating, where the absolute values of the field intensity and absorption losses are given. (After Popov, Tsonev and Maystre [ 1990bl.)
propagating order -l
2.3. GRATING SUPPORTING A SINGLE DIFFRACTION ORDER
2.3.1. Perfectly conducting grating We shall now examine the other case in eq. (9): if a plane wave is incident on the grating at an angle such that I sin 6, I c I / d - 1, only a single order propagates in the upper medium. In contrast to the case where a surface wave propagates (eq. (1 1)) this is the specular order that has a number m = 0.
I54
LIGHT DIFFRACTION B Y RELIEF GRATINGS
0
1.4 h /d
2.0
Fig. 9. Groove depth dependence of the phase of a wave reflected by a perfectly conducting sinusoidal grating: d = 0.3 pm, I = 0.6328 pm, TM polarization, a. = 0.6328. (After Popov, Tsonev and Maystre [1990a].)
Assuming perfect conductivity of the substrate, n o variation of reflectivity can be expected due to the conservation of energy. Peculiarities can be found only in the phase of the reflected wave (fig. 9). It varies almost periodically when the groove depth increases by a large amount.
2.3.2. Total absorption of light by metallic gratings In 1976 an unexpected theoretical discovery was made by Maystre and Petit [ 19761 that was verified experimentally by Hutley and Maystre [ 19761. Whereas a flat mirror reflects almost all incident light, an only slightly corrugated surface under certain conditions can absorb it totally (fig. 10). These conditions include a set of angles of incidence, and wavelength and groove depth values, that are peculiar to a given substrate material and form of groove (fig. 10). During the following 15 years, this phenomenon (sometimes called Brewster’s effect in metallic gratings) was investigated thoroughly. Its importance in grating studies can be compared with Wood’s discovery of anomalous grating behavior. Its connection with the plasmon surface wave excitation was revealed and its resonant nature was proved (see § 3). We want to emphasize here that this phenomenon (total absorption of light, or TAL) is not an isolated case. It represents a manifestation for shallow gratings of a more general phenomenon (fig. 11). Investigations in a large groove depth interval indicate that repeating minima of reflectivity appear. A suitable choice of angle of incidence can diminish any reflection minima to a value of zero. It is remarkable that for an aluminum sinusoidal grating with red light (A = 0.6328 pm), the three cases of total light absorption (for shallow, deep, and very deep grooves) almost coincide in position (fig. 1 1 ) .
11, § 21
155
QUASIPERIODICITY
*
1 ..
0.5
.~ h;
0.01
*
h.O.021
1%
0
Fig. 10. Zeroth-order efficiency ofa silver grating versus sin 0,: TM polarization, d = 1/2400 mm, = 0.5 prn, given for different groove depth values, shown in the figure in microns. (After Maystre and Neviere [1977].)
I
0
0.7 h/d
1.4
Fig. 1 I . Reflectivity (q,,) of an aluminum sinusoidal grating as a function of the groove depth: d = 0.5 prn, I = 0.6328 prn, TM polarization, a, = 0.25762. (After Mashev, Popov and Loewen [ 19891.)
The behavior of these anomalies when varying the entire set of system parameters will not be discussed in detail, but a few properties are worth noting. First, when a grating supports a higher number of orders, zeros in their amplitudes also exist, but these zeros are usually separated; they are manifested for different values of groove depth and angle of incidence (Maystre [ 19821). The zeros of different orders can coincide only for highly specific sets of system parameters ($ 3.2, fig. 18). Second, resonance anomalies exist throughout the entire spectral region where the wavelength-to-period ratio is
156
LIGHT DIFFRACTION BY RELIEF GRATINGS
Fig. 12. Reflectivity of a sinusoidal aluminum grating as a function of the groove depth and wavelength: d = 0.5 pm, TM polarization. The angular deviation from the first-order cutoff is kept constant equal to 0.052”. (After Popov [1989].)
suitable for the excitation of surface waves. Because of the stronger coupling between surface waves propagating in opposite directions, the spectral behavior of anomalies in deep gratings is more complex than that for shallow grooves (fig. 12).
2.4. DIELECTRIC GRATINGS
The great interest devoted to perfect blazing in a Littrow mount derives primarily from its “blazing” properties, i.e., maximum diffraction efficiency in the dispersive order. It is usually assumed that the metallic substrate is a natural tool for obtaining high efficiency values. For some applications, however, even a small amount of energy absorbed by the substrate can be critical. These include the two limiting cases: either when very high light intensities can damage the grating material if absorbed even partially, or for very low intensities that are below the threshold of detection or laser generation.
I57
QUASIPERIODICITY
11, § 21
It was proposed that a stack of layers on the corrugated metallic surface would increase the total reflectivity of the system (Maystre, Laude, Gacoin, Lepere and Priou [ 19801). However, in contrast to dielectric multilayered fiat mirrors, waveguide modes can be excited in grating structures that lead to multiple anomalies (Mashev and Popov [ 19841, Mashev and Loewen [ 1988]), thus limiting the application of these gratings. In the 1980s two different mountings were proposed that make possible an almost perfect blazing by dielectric gratings. In the first mounting the idea of metallic gratings was suggested to reduce to a minimum the number of propagating orders using total internal reflection (TIR) (Popov, Mashev and Maystre [ 19881). With light incident from the dielectric substrate side at an angle larger than the critical one for TIR, no diffracted waves are possible in air provided the period is small enough:
32, > 2n,d
(12)
in the first-order Littrow mounting. The groove depth dependence of the first-order efficiency is shown in fig. 13. Rises and falls similar to those of the efficiencies in metallic gratings are observed. An increase leads to a maximum of almost 100% followed by a minimum value of zero, etc. The two main differences in comparison with fig. 5 are that in the case of dielectric gratings, first, blazing occurs for a much greater groove depth-to-period ratio, and second, the maximum for TE polarization precedes the TM peak. In the second case, when very high diffraction efficiency values are obtained. 1.0
0.5
0.0 0
200
400
h[nrn]
Fig. 13. Groove depth dependence of the efficiency of the first reflected order of a sinusoidal dielectric grating used from the substrate side in the Littrow mount: n, = 1.5, n2 = 1, d = 0.26 pm. The solid line represents the TE polarization (1 = 0.55 pm),the dotted line represents the TM polarization (A = 0.55 km),the dashed line represents the TE polarization (A = 0.65 pm), and the chain line represents the TM polarization (1 = 0.65 pm), (After Popov, Mashev and Maystre [ 19881.)
158
LIGHT DIFFRACTION BY RELIEF GRATINGS
Fig. 14. Groove depth dependence of the efficiency of the first transmitted order for three different profiles. The light is incident from the substrate side: n , = 1.66, n2 = I, d = 0.4476 pm, I = 0.6328 km, TE polarization, 8, = 45". (After Yokomori [1984].)
a dielectric grating is used in a transmission regime in the so-called Bragg mounting, which requires the dispersion order to propagate symmetrically to the transmitted zeroth order with respect to the normal to the grating (Moharam and Gaylord [ 19821, Enger and Case [ 19831, Moharam, Gaylord, Sincerbox, Werlich and Yung [ 19841, Yokomori [ 19841). The groove depth dependence of the efficiency is shown in fig. 14, and is similar to that presented in figs. 5 and 13. A quasiperiodicity is again observed, although its manifestation is now less obvious due to the larger groove depth values required for blazing and antiblazing.
4 3. Phenomenological Approach: A Step Towards the Physical Interpretation of Grating Properties By using a phenomenological approach to grating behavior, scientists expect to identify most variations and regularities in a small number of parameters. It is generally unnecessary, but strongly desirable for these parameters to have some physically obvious meaning. Nevertheless, it is satisfactory that the phenomenological approach may simplify some complex irregularity so that both the process and the results of simplification will bring us closer to an understanding of the investigated phenomenon. Three investigative directions have been applied to the properties of the diffraction of light by periodically corrugated surfaces using a simplification (regularization) approach, each of which is a solution for a generally stated problem : (1) What is the physical reason and interpretation of the narrow anomalies, best illustrated by the examples shown in figs. 10 to 12?
11, § 31
PHENOMENOLOGICAL APPROACH OF GRATING PROPERTIES
159
( 2 ) Why do broad anomalies (i.e. regions of perfect blazing and antiblazing) exist, and what are their relationships with narrow anomalies? (3) What is the physical background to the quasiperiodicity of the grating properties ? These three topics are discussed later in detail, but it is worth mentioning that in contrast with other physical problems, the phenomenological approach in grating theory is as rigorous as the recent numerical methods based on electromagnetic theory. This approach consists only of the proper choice of a larger number of regular parameters (believed to have a deeper physical meaning) rather than any simplification (approximation) of the theoretical methods. These phenomenological parameters can be determined with the same accuracy as any other quantity (e.g., phase, amplitude, efficiency). In fact, both sets are calculated using only slightly modified computer codes. In this section we distinguish between resonance and nonresonance phenomena, calling them anomalies in the traditional way, although some demonstrate regular behavior. Generally, resonance anomalies are due to a surface wave excitation; a resonance between the incident wave and a solution of the homogeneous problem is ensured by the periodicity of the grating. Mathematically, this solution is expressed (discussed in detail in 8 3.1) by a pole of the scattering matrix. Thus, as a working definition, it can be assumed that resonance anomalies are accompanied by a nearby pole, in contrast to the nonresonance ones. A preliminary observation should be made for clarification : resonance anomalies are usually narrow and nonresonance anomalies are broad, but there are some exceptions. Narrow dips in the angular and wavelength dependences of the efficiencies exist in the case of the almost total absorption of light by metallic gratings at grazing incidence (Mashev, Popov and Loewen [ 1988]), although this is a nonresonance anomaly. The half-width of minima and maxima in the reflectivity of corrugated waveguides due to waveguide mode excitation (resonance anomaly) depends strongly on the depth of the grooves (Popov, Mashev and Maystre [ 19861) and can become sufficiently large. From a historical point of view, however, resonance anomalies sometimes are called narrow and nonresonance anomalies broad. That convention can safely be used only if the background reasons for specific anomalies are known. 3.1. RESONANCE ANOMALIES
Fano [ 19411 was the first researcher to draw a clear connection between narrow anomalies and surface wave excitation, distinguishing between “sharp”
160
LIGHT DIFFRACTION RY RELIEF GRATINGS
[IL I 3
and “broad” anomalies. Hessel and Oliner [ 19651 provided the following recently accepted formulation of a phenomenological approach to resonance (sharp) anomalies. The surface wave propagation along a flat or corrugated surface is expressed mathematically by a solution of the homogeneous problem for light scattered by the surface without an incident wave. This solution requires the existence of a pole a p of the scattering matrix S of the system, a pole that is equal to the surface wave complex propagation constant a s . The scattering matrix determines the connection between the incident a and diffracted b wave amplitudes
b=Sa.
(13)
In fact, to have a nonzero solution for b without incident waves (a = 0), a zero of the determinant of the inverse of S is required. In the first-order approximation, det I S - I I a. - a p ; i.e., each component of S is inversely proportional to a, - ap. Periodic corrugation multiplies the pole by the grating period (due to the coupling, the pole of one component of S becomes a pole of all the components):
-
a,” =
Org
+ nlld.
(14)
In particular, if a plane incident wave vector is phase matched to any homogeneous solution through eq. (14), the amplitude of any diffraction order (namely, the mth) should be proportional to b,,,w-.
1 a. - a,“
For propagating orders (and, in particular, for the specular one) no anomalies occur without a corrugation. Thus, the pole is not manifested and has to be compensated by a zero a; in the numerator such that a‘ = a p when h = 0. Corrugation removes the requirement for their equivalence, leading to a splitting, and the pole and the zero could be manifested in the amplitudes:
The different indexes in eqs. (15) and (16) have a different physical meaning. The index n of the pole corresponds to a surface wave excitation through the nth order. The amplitude (and the zero) index m represents the number of the order of interest. Equation (16) represents the noted phenomenological formula that enables
11, § 31
PHENOMENOLOGICAL APPROACH OF GRATING PROPERTIES
161
us to determine the behavior of the efficiencies in the region of a resonance anomaly (it is referred to as “resonance” due to the pole to which it has to be matched). Numerical results fully confirm the foregoing speculations that resulted in eq. (16). In general, pole and zero can be found for complex values of ao, and their influence on the diffraction efficiency and the phase behavior can be satisfactorily traced for real angles of incidence following eq. (16). Its validity has been proved by a comparison of theoretical and experimental results (McPhedran and Maystre [ 19741, Maystre and Neviere [ 19771, Maystre, Neviere and Vincent [ 19781). The coefficient of proportionality is a slowly varying function of the grating parameters. Pole and zero are independent of the angle of incidence and are slowly varying functions of groove depth and wavelength for a fixed form of groove profile. This characteristic is valid in the regions where the pole and zero are simple. When an anomaly interaction appears (e.g., when two surface waves are excited simultaneously), it can be expressed in the phenomenological formula by two (or more) terms with different poles and zeros corresponding to the excitation of separate surface waves. For shallower gratings the splitting between the pole and the zero is smaller, in general, and the anomaly in the diffraction efficiency is sharper. For a perfectly conducting grating with a single propagating order, the energy balance criterion requires that the pole and the zero remain mutually complex conjugated. Let us remember that corrugation initiates diffraction losses of the surface wave, expressed in the growth of the imaginary part of the pole. Thus, the zero could also become complex. For real metallic substrates both the imaginary and real parts of the pole and the zero are slightly modified. Figure 15 presents their trajectories when the groove depth is varied, corresponding to the efficiency behavior seen in fig. 10. The position of $ is almost symmetrical to ap with respect to their initial position at h = 0. On increasing the corrugation depth, the trajectory of a; crosses the real a. axis when h = h , ; a zero of the zeroth-order amplitude appears, which leads to the total absorption of incident light. For deeper grooves g moves away from the real axis and, according to eq. (16), the minimum value of the reflectivity increases (fig. 10). A further increase of the groove depth leads to a stronger coupling between the surface wave and the order diffracted in the air. This phenomenon has two direct consequences. First, the diffraction losses increase (fig. 7), and the imaginary part of a p rises (fig. 16). Second, above some critical value of groove depth, diffraction losses become so high that the surface wave is no longer localized at the surface; the real part of its propagation constant becomes less than unity. The pole is transferred into a zero of the zeroth-order amplitude, which is solitary, i.e., not accompanied by a pole (Popov, Mashev and Loewen [ 19891).
162
I.IGHT DIFFRACTION B Y KI:l I I I. G K A I I N G S
,'h.0.1
, 0.05
I
?
0.08
I
I
*
I
0.06
0 0
0
0.15
0.04,'
0.20
0.06.
0.08.
- 0.05 0.1
,
Fig. 15. Trajectories of the pole aP (dashed line) and the zero of the zeroth reflected order a: (dotted line) in the complex plane as a function of the modulation depth h/d,indicated with circles. The parameters of the sinusoidal silver grating correspond to those in fig. 10. (After Maystre and Neviere [1977].)
0 09
,-tl 0
h
02
5
Dl 0.40
0.44
-0.13 0 75
0.90 a)
1.05
Re(
Fig. 16. (a) Trajectory of the pole U P (heavy solid line) and zero a; (thin solid line) in the complex plane when the groove depth is increased. The dotted line represents the real axis, and the dashed line represents the cut corresponding to the change of sign of xo. (b) Magnification ( x 15) in the vicinity of point a. = (1,O). (After Popov, Mashev and Loewen [1989].)
11, I 31
PHENOMENOLOGICAL APPROACH OF GRATING PROPERTIES
163
Following the trajectory of this newly born zero, the formation of loops is observed (fig. 16). Its cross-points with the real axis correspond with other anomalies that are not within the topic of this subsection, i.e., non-Littrow perfect blazing (aA) and total absorption of light at grazing incidence (aG)(see next subsection). When the trajectory again crosses the cut defined by Re(a,) = 1, the zero becomes a pole, and a new branch of the plasmon surface wave propagation constant is obtained corresponding to the right-hand side of fig. 7. It is characterized by lower absorption losses than a surface wave on a flat surface (fig. 8b). An increase of the groove depth causes a new growth of the absorption and diffraction losses. Thus, a new loop is formed in the trajectory. In fact, the position of the cut that starts from the point a, = ( 1 , O ) is somewhat arbitrary (Maystre [ 19821, Mashev and Popov [ 19891). The only restriction is that it lies in the upper half-plane. Nevertheless, this requirement is sufficient to distinguish between poles and zeros on the real a-axis, where the experimentally realizable angles of incidence are located. The meaning of the cut is to separate physical from nonphysical solutions, the first being defined by the proper radiation conditions at infinity (e.g., Petit [ 19801). From that point of view all the zeros of the zeroth reflected order are indistinguishable and correspond to one and the same nonphysical solution of the homogeneous problem determined by the improper choice of the sign of xo (Andrewartha, Fox and Wilson [ 1979a,b] ). This conclusion has a practical application when dealing with the interaction between the different zeros, which behave like poles, demonstrating the splitting and repelling of trajectories (Mashev and Popov [ 19891). The pole tracing in fig. 16 corresponds to a surface wave carried through the zeroth order. As already mentioned, corrugation multiplies the poles : the fragments of the pole trajectory in fig. 17 correspond to a surface wave propagating to the left and excited through the first diffraction order. They are symmetrical with those in fig. 16 with respect to the position of the Littrow mount, and are accompanied by a zero (with trajectory starting from the point A/d - as when h = 0). The trajectories of both pole and zero are piecewise continuous; they are divided by the cut that starts from the point = (A/d - 1,O). As discussed in detail for fig. 10, the first cross-point a; with the real a,-axis of the zero trajectory causes a total absorption of light by shallow gratings. The formation of loops in fig. 16 also results in the formation of loops in fig. 17. As a result, several cross-points of the trajectory of a; with the real a,-axis exist. The real zero ak that is situated in the region where two orders propagate is
164
LIGHT DIFFRACTION BY RELIEF GRATINGS
0.09
5
(4
[II, I 3
h/d
0.34
0.383
-0.02
E -
-0.13
I
I
0.2
0.5
0.35 R 4 U )
Fig. 17. Similar to fig. 16, but in the vicinity of the point a, = I / d - I , corresponding to the beginning of the cut of x - . (Atler Mashev, Popov and Loewen [ 19891.)
,
not accompanied by a pole, and it causes broad anomalies that are characterized by a minimum of the specular order efficiency and an increase of the first order. Thus, a non-Littrow perfect blazing (Maystre, Cadilhac and Chandezon [ 19811) can be explained phenomenologically. The other two real zeros of the zero-order amplitude a; and a; lie below the first-order cut-off, and they describe the total absorption of incident light (the second and third minima in fig. 11).
3.2. NONRESONANCE ANOMALIES
In the preceding subsection we observed the existence of the zeroth-order zeros that are not accompanied by a pole. These are the perfect blazing in the non-Littrow mount and the “black hole” (Mashev, Popov and Loewen [ 19881) at grazing incidence, expressed phenomenologically by the corresponding crossing points in figs. 16 and 17: a k , ah, and ac. The presence of a pole can be registered by a substantial increase of the electromagnetic energy density in the near vicinity of the grating surface, as shown in 5 4. Such poles and field enhancement do not occur for the nonresonance anomalies, although their behavior in the far-field region could be rather different; some (e.g., nonLittrow perfect blazing) are broad, whereas others are narrow (fig. 18). Moreover, the diffracted energy can vary from total absorption to perfect blazing. A simple explanation can be found in the relative independence of the specific behavior of the zeros of the different orders. That is, light absorption at grazing incidence occurs when simultaneous (but independent) zeros of the zeroth order ( a c ) and first order (due to the antiblazing effect) exist for a specific value of groove depth (fig. 18), whereas blazing is characterized by a single zero of
165
PHENOMENOLOGICAL APPROACH OF GRATING PROPERTIES
\ \ (
I
l
l
1
0.5
0.6
0.7
0.8
03
/
/
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
h/d
Fig. IS. Efficiency curves as a function of the modulation depth for an aluminum grating with d = 0.50 pn, I, = 0.6328 pm, So = 87.85", TM polarization. The solid curve represents the zeroth-order diffracted energy, the dashed line represents the first-order diffracted energy, and the dotted curve represents the total diffracted energy. (a) Sinusoidal groove profile, (b) symmetrical triangular groove profile. (After Mashev, Popov and Loewen [ 19881.)
the specular order, leading to a maximum of the first order. From that phenomenological point of view non-Littrow perfect blazing is equivalent to the perfect blazing in the Littrow mount, and both phenomena can be expressed by a zero of the specular order amplitude (without a nearby pole). They can be distinguished, however, by the origin of their trajectories when groove depth is vaned. The non-Littrow perfect blazing and the light absorption at grazing incidence are connected (in a peculiar way) with the surface wave excitation along a flat interface in the groove depth regions, where such a wave is forbidden (fig. 16), whereas Littrow mount phenomena lie on a separate trajectory, as shown here. Tseng, Hessel and Oliner [ 19691 were the first to notice that a Bragg-type anomaly is manifested by a zeroth-order real zero a:. They distinguished between resonant and nonresonant anomalies, noticing that a t has a specific type of behavior when varying the groove depth around its value A,, which is responsible for perfect blazing in the Littrow mount; this zero moves along a
166
LIGHT DIFFRACTION BY RELIEF GRATINGS
[II, § 3
line perpendicular to the real a,,-axis. Their results were confirmed recently (Mashev and Popov [ 19891) by following a t over a large groove depth interval and, in particular, when h -+ 0; the zero tends toward negative imaginary infinity along the line Re(a,,) = a; defined by eq. (1). In contrast to the anomalies discussed previously, perfect blazing in the Littrow mount cannot be localized for flat surfaces. Whereas the other resonant or nonresonant phenomena can be specifically connected to a surface wave propagation along a flat interface, the zero a; lies along a definitely separated trajectory (fig. 19), starting from - ioo for a flat surface. When the groove depth is increased, a t moves towards the real axis, and the crossing point that appears for h = h,, leads to a maximum in the first-order efficiency, as shown in fig. 5. Increasing h, a t goes away from the real axis, and the zeroth-order efficiency is proportional to (Im at)’. Thus, the first-order efficiency decreases. At a groove depth value corresponding to the antiblazing effect, the zeroth-order zero a t that is lying far in the upper complex a,-half-plane is transformed into a second zero with a negative imaginary part (fig. 20). It results in a second perfect blazing in the Littrow mount for a further increase of groove depth. If we return to the issues stated at the beginning of this section, the quasiperiodic character of the grating properties as a function of groove depth remains unclear. In fact, the phenomenological approach, which consists of expressing the grating peculiarities in terms of the poles and zeros and their tracing when the groove depth is varied, enables us to transfer these peculiarities into a specific behavior of the phenomenological parameters and to draw some interesting connections between the various grating properties. It remains unknown (and unclear from the intuitive point of view), however, why the
h
0.57
0.61 Re(a)
0.65
Fig. 19. Trajectory of a; in a Littrow mount when the groove depth is varied: d = 0.5 pm, I = 0.6328 pm. The solid line represents a perfectly conducting grating and the dashed line represents an aluminum grating. (After Mashev and Popov [1989].)
11, s 31
PHENOMENOLOGICAL APPROACH OF G R A r I N G PROPERTIES
167
2.6
-8 v
-2.7
E -
- 8.0
0
0.91 h /d
1.82
Fig. 20. Groove depth dependence of the imaginary part of the Littrow mount zero a t , corresponding to the trajectory shown in fig. 19. The solid lines represents the TM polarization and the dashed line represents the T E polarization. (After Mashev and Popov [1989].)
Littrow mount zero of the zeroth-order amplitude exhibits some type of quasiperiodic behavior (fig. 20). An interesting attempt to introduce another set of phenomenological parameters was proposed by Maystre, Cadilhac and Chandezon [ 19811. A detailed analysis of the properties of the scattering matrix makes it possible to show that in the Littrow mount the logarithmic eigenvalues pof the S-matrix vary almost linearly with groove depth h (fig. 21) when only two diffraction orders propagate. A straight connection between p and q - I leads to a periodicity in the groove depth dependence of the efficiencies:
q-
I =
(17)
sin2p.
Although we believe that this approach has not received the acknowledgement it deserves, it has already proved useful. The possibility of perfect blazing in
1
0
0
2I n
4In
Hid
6/n
Fig. 21. Variation of the first-order efficiency (E) and of the phenomenological parameter p with the groove depth-to-period ratio for a sinusoidal grating with a wavelength-to-period ratio of l / d = 0.8 for the T E case. (After Maystre, Cadilhac and Chandezon [1981].)
168
LIGHT DIFFRACTION BY RELIEF GRATINGS
111, § 4
non-Littrow mounts was discovered as a result of a detailed study of the behavior of p.
6 4. Microscopic Properties of Light Diffracted by Relief Gratings The phenomenological parameters introduced in the previous section are shown to be precise tools for obtaining a more compact expression for the peculiar features of the gratings. Moreover, some of the physical reasons for these peculiarities could be found by simply tracing the behavior of the phenomenological parameters. A typical example is the total absorption of light in shallow gratings. The mechanism of its appearance becomes clear when one realizes why the existence of a zero in the propagation order amplitude is necessary to accompany the pole corresponding to a surface wave excitation. It remains unknown, however, why the quasiperiodicity of the grating characteristics is such a general property. In fact, the phenomenological approach transfers the quasiperiodicity into loops in the trajectories of the poles and zeros (figs. 16, 17) or into the linearly varying parameter p (fig. 21), but it does not reveal why such loops are formed. These loops are not physical phenomena, but are a mathematical expression of the quasiperiodicity of the far-field properties. To find a physical explanation we have to go deeper into the physical process of diffraction of light by relief gratings. It should be pointed out that “deeper” does not mean more rigorous; as already discussed, even the phenomenological results may be sufficiently accurate; deeper is used in the sense of being more physical from the heuristic point of view. Such a physical explanation of quasiperiodicity can be found in the pictures of the distributions of energy flow lines. These lines are locally tangential to the Poynting vector P, defined as
P = Re(E x H * ) ,
(18)
where the asterisk means complex conjugation. In some examples the local density of the electromagnetic field energy is also analyzed, where it has a peculiar behavior leading to specific phenomena of major importance, such as surface-enhanced Raman scattering (Reinisch and Neviere [ 19811, Metcalfe and Hester [ 19831, Yamashita and Tsuji [ 1983]), second harmonic generation (Neviere and Reinisch [ 19831, Reinisch, Chartier, Neviere, Hutley, Clauss, Galaup and Eloy [ 19831, Coutaz [ 19871, Neviere, Akhouayri, Vincent and Reinisch [ 19871, Maystre, Neviere, Reinisch and Coutaz [ 1988]), and surface plasmon luminescence (Coutaz and Reinisch [ 19851).
11, I 41
LIGHT DIFFRACTED BY RELIEF GRATINGS
169
Cases previously discussed are analyzed successively without classification into separate groups, since the properties they have in common are obvious.
4.1. PERFECTLY CONDUCTING GRATING IN LITTROW MOUNT
In the far-field zone the z-component of the electromagnetic field parallel to the grooves can be represented as a sum of incident, reflected, and diffracted orders. For TM polarization, when there are only two propagating diffracted orders in the Littrow mount, according to eqs. (3) and (7), this is the z-component of the magnetic field H,
a,exp[ik(a,x - xdy)]
=
+ b, exp [ - ik(a,x
-
+ 6, exp[ik(a,x + xdy)]
xdy)] .
(19)
When the grating is reduced to a perfectly conducting plane mirror located on the x-axis, b, = 0 and b, = a,. Next, we assume that a, = 1. Then it can easily be shown that for h # 0 [ lbrI2 + Re(br)cos(2kxdy) - Im(b,) s i n ( 2 k ~ d y ) ] , (20a)
[Re(b,b,*)cos(2ka0x) - Im(b,b;) sin(2ka0x)],
(21a)
where o = c/k, c is the velocity of light in vacuum, and t o is the vacuum permittivity. Equations (20a) and (21a) can be simplified more if we assume that the amplitudes b, and b d are real, which is almost true for shallow gratings. When the origin of the coordinate system lies on the corrugated surface, e.g., f(x) = i h sin(2nx/d), then
P,
=
~
kxd brb, cos(2ka0x).
oxon;
4.1.1. Flat sutjiuces
In the familiar case of a flat mirror, the vertical (P,,)component of the Poynting vector becomes zero, as b, = 0. This case is characterized by flow
170
LIGHT DIFFRACIION B Y RELIEF GRATINGS
PI, § 4
lines parallel to the surface (no losses occur in either medium). For TM polarization, P., has a maximum on the surface ( y = 0), and has a zero at heights given by yOTM = (2m
R + 1) __
, m = 0 , 1 , 2, . . . .
2kX: For TE polarization, P , is zero at the surface and at heights given by
P,’“ has a maximum at the level where P.TM exhibits zeros.
4.1.2. Shullow grrrtings When the groove depth is slightly greater than zero, the diffracted field amplitude also becomes nonzero. Since there is no absorption in either medium, in the very near vicinity of the grating the flow lines must follow the groove profile without osculating the surface (fig. 22a), and a curvature of the flow lines appears. In fact, eq. (21b) states that P, $?! 0, except for the vertical lines above the tops and bottoms of the grooves. The profile is defined by the equation JJ = (h/2) sin(2nxld). Thus, in the Littrow mount (2a, = l i d ) , PI, becomes zero for x = (2171 + l)d/2, according to eq. (21b). Since now 16, j < 1, each horizontal line at which P-, = 0 (eq. (20b)) is split into two lines (e.g., fig. 22b), the splitting increasing with the decrease of I b,. , i.e. with the increase of groove depth. Between each two pairs of lines, P, has an opposite sign compared with the case of the flat surface. The two dashed lines in fig. 22b correspond to the splitting of the dashed line in fig. 22a and represent the positions where P., becomes zero. As a result of this changing of the directions of P, so-called “curls” are formed around the points where 1 PI = 0. It could easily be deduced that in lossless media div P = 0; thus, energy flow lines are either closed lines or end on the boundary of the medium. The regions with closed vector lines are called “curls”, and they exist even for shallow gratings and are not typical only of light diffraction from a grating. When three waves are interfering, independently of their origin, regions where the curls exist occur throughout space, as can easily be shown from eqs. (20b) and (21b). These waves can result from the diffraction of light from a grating, and can also be radiated by three different coherent sources. Hajnal [ 19871 showed that the
11,
I 41
I.IGHT DIFFRACTED BY RELIEF GRATINGS
h/d
0.02
h/d
0 .'24
@
h/d 0 72
171
0.08
0.38
@
@
h/d 0 72
@
Fig. 22. Energy flow lines above a perfectly conducting sinusoidal grating. Littrow mount, d = 0.5 pm,I, = 0.6328 pm,TM polarization: (a) h = 0.01 pm,(b) h = 0.04 pm,(c) h = 0.12 pm, (d) h = 0.19 pm, (e) h = 0.26 pm, (f) h = 0.36 pm, (9) h = 0.36 pm and a, = 0.85, (h) h = 0.72 pm. (After Popov, Tsonev and Maystre [1990a].)
172
LIGHT DIFFRACTION BY RELIEF GRATINGS
PI, § 4
interaction of three monochromatic circularly polarized electromagnetic waves can lead to highly interesting physical phenomena and field distributions. Considering the grating again, we must point out that two different sets of curls can be found (fig. 22b), above the tops (labelled as “top curls”) and above the bottoms (labelled as “bottom curls”) of the grooves, the latter lying a little lower than the former. By increasing the depth of the grooves, the centers of the two sets of curls are shifted in the vertical direction, and the curls are enlarged, enveloping more and more flow lines. It is important to note that for TE polarization the formation of the lowest curls begins at a distance from the grating surface twice that for TM polarization, which is a direct consequence of eq. (22).
4. I .3. Perfect blazing in Littrow mounts The increase in the size of curls means that increasingly few flow lines pass from left to right, i.e., less energy is carried towards the positive direction of the x-axis (fig. 22c). As a result, the amplitude of the reflected order b, decreases, and an increase of the diffraction efficiency can be observed. It is interesting to note that the centers of the lowest top curls and lowest bottom curls lie at an almost identical height from the grating surface. For this reason, when h grows, the vertical distance between the centers of the different sets of curls also increases almost proportionally to h. When the centers of the lowest bottom curls lie on the line connecting the tops of the grooves (fig. 22d), all the curls become uniformly distributed. The centers of the top curls are localized in the vertical direction exactly between the centers of the bottom curls. The curls then occupy the entire upper medium, and no energy flow towards + x is observed, i.e., b, = 0 and perfect blazing occurs. This happens, as can easily be predicted, when h
=
Y,TM, TE
When the evanescent waves are included in the near-field zone, eq. (23) becomes only an approximation, but a very good one. It must be pointed out that although the grating becomes increasingly deeper, the far- and near-field electromagnetic field distributions are identical, except for the groove area where the picture should depend strongly on the form of the profile. Equations (22) and (23) explain why perfect TE blazing can usually be obtained at gratings almost twice as deep as those for the T M polarization: the lowest curls are formed at surfaces twice as high as those for the TM polariza-
LIGHT DIFFRACTED BY RELIEF GRATINGS
173
tion, and groove depth values twice as high are required to bring the centers of the lowest bottom curls to the line connecting the tops. When b, = 0 it can easily be shown that not only all the flow lines are closed curves (fig. 22d) but also (PIvanishes everywhere in the upper medium. As h increases, the curls appear again, but they now rotate in the opposite direction. The bottom curls are lowered with h (fig. 22e). The increasing asymmetry in the position of the two sets of curls leads to a decrease of the area they occupy. Thus, more energy lines go from left to right, and b, increases. 4.1.4. Antiblazing of gratings The unfolding of the curls continues as the lowest bottom curls settle increasingly deeply in the grooves. The vertical distance between the centers of the second bottom curls and the first top curls decreases, and the actual energy flow distribution increasingly resembles the distribution of shallow gratings, except for the groove region. When the centers of both sets of curls lie on one horizontal line, the flow distribution above the grooves is the same as that above a plane surface. Then the first (i.e. the lowest) bottom curls completely separate the flow above the grooves from the groove surface (fig. 22f). Their deformation is such that they follow the profile. Since the flow is the same as that above a flat surface, no diffraction is observed (b, = 0). The curls inside the grooves are stable and feel little influence from the angle of incidence (fig. 22g with a,, = 0.85). In this way, antiblazing of gratings can be given a physical interpretation. 4.1.5. Very deep gratings A further increase ofh leads to a repetition of the flow distribution for shallow gratings, except for the existence of a hidden curl inside each groove. These curls become gradually lower inside the grooves as h increases. For very deep grooves a second perfect blazing appears (hid = 1.08 in fig. 5), followed by a second antiblazing. The latter is characterized by two hidden curls inside each groove that separate the flow above the grooves from the bottom of the groove (fig. 22h).
174
LIGHT DIFFRACTION B Y RELIEF GRATINGS
[II, § 4
4.2. PERFECTLY CONDUCTING GRATING SUPPORTING A SINGLE DIFFRACTION ORDER
The energy flow distribution above a perfectly conducting grating, which supports only a single diffraction order, has a similar behavior. As shown in 5 2.3.1, a quasiperiodicity of the phase of the reflected wave can now be observed (fig. 9) in the far-field zone. In the near-field zone the formation of curls can again be traced when the groove depth is varied. They are localized in a layer a few wavelengths thick near the corrugated surface, as long as evanescent orders are responsible for their existence. The curls appear again on the line corresponding to IPI = 0 for a plane surface. The primary differences when compared with the case of d = 0.5 pm (fig. 22) are, first, the curls are formed only in the near-field zone; going away from the grating they become increasingly smaller, and where the evanescent diffraction order vanish, the flow becomes the same as that above a flat surface. Second, the bottom curls are formed higher than the top curls, a direct consequence of which is that the direction of the energy flow for shallow gratings in the very near vicinity of the surface is opposite to that for the far-field flow. When h increases, the top curls unfold. The increase of h moves the lowest bottom curls deeper into the grooves, and at a certain value of h they become totally hidden. This corresponds to zero phase difference between the waves reflected by the grating and the flat mirror. For this case the flow distribution above the grooves is identical to the flow above the flat surface, which can be expected considering the results for the grating supporting two diffraction orders. An additional increase of the groove depth has the same effect on the energy flow distribution as it does for shallow gratings, except that inside each groove a curl goes increasingly deeper.
4.3. PLASMON SURFACE WAVE ALONG A METALLIC GRATING
Figures 23a and b represent the energy flow lines for two different values of groove depth: 0.058 and 0.395 pm, respectively. In these cases the surface wave propagating constant UP have almost equal imaginary parts (fig. 16). The energy flow above the grooves is almost identical for the two gratings: near the surface the lines are almost parallel to it, corresponding to the physical fact that the surface wave propagates from left to right. Few lines end on the surface, leading to absorption losses. Some upper lines turn away from the surface, correspond-
175
LIGHT DIFFRACTED BY RELIEF GRATINGS
8d
8d
4d
4d
0 -
0
0
4d
8d
0
0
4d
d/2
8d
d
Fig. 23. Energy flow lines of a plasmon surface wave propagating along a shallow (a) h = 0.058 pm,and deep (b) h = 0.395 pm aluminum grating: d = 0.5 pm,I = 0.6328 pm,TM polarization. (After Popov, Tsonev and Maystre [ 1990bl.)
ing to radiation losses. As a result of both types of losses, the guided-wave amplitude decreases as it propagates along the grating, which is represented, in terms of the energy flow, as a decrease of the density of lines. For a flat surface the energy flow direction is almost parallel to the surface (small losses). As h increases, the flow lines are pushed upwards by the groove tops, and above them the electromagnetic field power density increases. Inside the grooves the lines are thinned out, and the field density decreases. Nevertheless, the normalized average field density value over one period grows with h. The increase in the absorption losses is proportional to the increase of the average field density, and is almost an order greater than the growth of the absorbing surface area (fig. 8a). This is confirmed in the case of deep gratings: they can have lower absorption losses (e.g., h/d = 0.72), half those for a flat surface. The primary difference between figs. 23a and b occurs inside the grooves: curls exist in each groove of a deep grating. Although some of the flow lines
176
“I, I 4
LIGHT DIFFRACTION BY RELIEF GRATINGS
reach the surface, closed curves in the deep grooves separate their bottoms from the energy flow above the tops, which explains why the absorption losses for deep gratings can take lower values than for shallow ones and even for a flat surface (fig. 8).
4.4. RESONANT TOTAL ABSORPTION OF LIGHT BY METALLIC GRATINGS
Total absorption of light by a metallic grating that accompanies surface wave excitation occurs for shallow, deep, and very deep grooves (fig. 11). Figure 24 shows the lines along which energy flows in the three different cases that
8d
0
4d
8d
4d
0 0
4d
8d
8d
4d
2d (f)
d
d’4
0 ‘
I
0
d/2
d
d
d/2
0 0
d/2
d
0
I
0
I
d2l
I
d
Fig. 24. Energy flow lines correspond to the total absorption of light by a sinusoidal aluminum grating: d = 0.5 pm, L = 0.6328 pm, TM polarization. (a, b) h = 0.05 pm, (c, d) h = 0.395 pm, (e, f) h = 0.6 pm. (After Popov and Tsonev [ 19901.)
I I , § 41
LIGHT DIFFRACTED
BY RELIEF GRATINGS
177
correspond to total absorption (i.e. matching the groove depth and angle of incidence for a given wavelength). Away from the grating surface (approximately above lOA), the picture is equivalent to the far-zone energy flow distribution, i.e., the evanescent orders have no influence above a distance of 10A. When total absorption of light occurs, the far zone is characterized by a uniform energy flow towards the grating that corresponds to the incident wave. As the surface is approached, the flow lines turn to the left and the line density increases. The near-zone energy flow distribution above the grooves represents a surface wave propagating to the left (the surface plasmon is excited through the first diffraction order). As can be expected, a close similarity is observed with fig. 23, except for the direction of the energy flow. The increase of the density of the flow lines leads to an enhancement of the electromagnetic field. This is usually expressed in terms of the ratio between the local optical power density
w = EonZIE12 + polHIZ
(24)
and the power density of the incident wave (assumed to be equal to unity). As is usual, po stands for the vacuum permeability. For a flat surface, this ratio (later called “enhancement”) does not exceed four. In the region of the resonant total absorption of light the enhancement can exceed several orders of magnitude. This phenomenon has been known in shallow gratings for a decade, and is believed to play a major role in the surface-enhanced Raman spectroscopy (SERS) and nonlinear excitation. Figure 25 shows the groove depth dependence of the field enhancement at the top and bottom of the grooves corres-
0
0.65 h /d
1.3
Fig. 25. Groove depth dependence of the normalized optical power density at the top (solid curve) and at the bottom (dotted curve) of the grooves of a sinusoidal aluminum grating: d = 0.5 pm, 1 = 0.6328 pm, 0, = 14.93”, TM polarization. (After Popov and Tsonev [1989].)
178
LIGHT DIFFRACIION B Y RELIEF GRATINGS
(x=d. p 3 d )
Fig. 26. 2D distribution ofthe electromagnetic power density in the vicinity ofthe grating surface, corresponding to the cases in fig. 24a, c, and e. (After Popov and Tsonev [1989].)
ponding to fig. 11. A 2D distribution of the optical power density near the grating surface is given in fig. 26 for the three cases of total light absorption. The primary difference in the values of the energy distribution is found inside the grooves. Whereas for shallow gratings the field enhancement at the top and bottom of the grooves is of an identical order (the difference does not exceed a factor of two; see, e.g., Garsia [ 1983a,b], Popov and Tsonev [ 1989]), in deep gratings the energy is localized at the tops of the groove. Phenomenologically, such an enhancement results from the existence of a pole of the scattering matrix. For the diffraction order that corresponds directly to the solution of the homogeneous problem (the first, in our case), the existence of a zero is unnecessary to compensate for the pole, since the plasmon surface wave with a propagation constant given by eq. (10) also exists for a flat surface. Thus, a substantial (resonant) increase of the amplitude of this particular order, described by eq. (1 5), could be expected. This phenomenon is illustrated in fig. 27 by the angular dependence of the first-order amplitude in the region of light absorption by shallow gratings. Moreover, since this order is evanescent far
11, I 41
(x=O,
I79
LIGHT DIFFRACTED BY RELIEF GRATINGS
y=3
.. .. .. .. .. .. ... ... (x-0.
.....
.........
POI
::
v:
......... . . . .. . . .
......
Icl I.
180
LIGHT DIFFRACTION BY RELIEF GRATINGS
0.20
0.25
PI, § 4
0.30
ahOo
Fig. 27. The amplitude 1 b ,I of the first evanescent diffracted order as a function of a. in the vicinity of the total absorption of light by a shallow aluminum sinusoidal grating: d = 0.5 pm, I = 0.6328 pm, h = 0.05 pm, TM polarization.
from the corrugated surface, the field enhancement is localized along the interface. However, the principal difference between fig. 26a on the one side and figs. 26b and c on the other can again be understood from the picture of the energy flow distribution (fig. 24), as follows. Some lines very close to the surface turn downwards and finish at the metal surface. The energy flow through the grating boundary is exactly equal to the energy flow of the incident wave. By comparing figs. 24a, c and e, it can be observed that the energy flow distribution is almost identical above the top of the grooves for the total absorption of light in shallow, deep, and very deep gratings. Figures 24b, d and f show the peculiarities of each case. First, in shallow gratings (h/d = 0.1) the flow lines follow the grating surface. They are almost parallel to the metal-air boundary. Of course, some lines terminate at the surface, since the metal is not perfectly conducting. The line density above the top of the grooves is slightly greater than that above the bottom, since less space is present above the top than above the bottom. This results in a slight variation of the electromagnetic field enhancement values along the groove (fig. 25). Second, in deep gratings ( h / d = 0.79 and h/d = 1.2) inside the grooves one or two curls are formed that separate the main energy flow from the bottom of the grooves. This separation is the reason why the electromagnetic field enhancement that accompanies the total absorption of light is exhibited at the tops but not at the bottoms of the deep grooves, in contrast with shallow gratings. Moreover, it should be remembered that in the middle of the curls in figs. 24d and f, 1 PI = W = 0, whereas no such regions are observed in fig. 24b.
I t 8 41
181
LIGHT DIFFRACTED BY RELIEF GRATINGS
4.5. NONRESONANT TOTAL ABSORPTION OF LIGHT
Figures 28a and b correspond, respectively, to the energy flow distributions above and inside the grooves in the case of the nonresonant total absorption of light at grazing incidence (fig. 18 with h/d = 0.69, 0, = 87.86'). It should be noted that this phenomenon is not strictly a total absorption of light: whereas the zeroth-order efficiency is exactly zero, the first-order efficiency is approximately 10- 3, but from a practical standpoint we can speak of the total absorption of light. The energy flow distribution up to the close vicinity of the grating surface is almost the same as that in the far-field zone; no bending of the flow lines in the direction parallel to the surface is observed, in contrast with figs. 24a, c, and d, since no surface wave is excited in this case. As a result, no line density enlargement nor electromagnetic field enhancement appears. The total change of direction in fig. 28a in comparison with figs. 24a, c, and d is due to the difference in the angles of incidence. Inside the grooves (fig. 28b) the energy flow distribution is similar to that in fig. 24d; the groove depth values show little difference (in fig. 24d h/d = 0.79, and in fig. 28 h/d = 0.69). A curl is totally hidden inside each groove. In the previous subsection a similarity exists between the pictures of the energy flow distribution above the grooves, but no similarity is observed inside them. When figs. 24 and 28 are compared, however, just the opposite occurs, i.e. the pictures inside the grooves are similar, but differ greatly above them.
d
d/2
0
0
4d
8d
0
d/2
d
Fig. 28. Energy flow lines for the nonresonanttotal absorption of light, corresponding to the case in fig. 18a with h = 0.345 pm. (After Popov and Tsonev [1990].)
182
LIGHT DIFFRACTION BY RELIEF GRATINGS
PI, 8 4
4.6. TOTAL INTERNAL REFLECTION BY DIELECTRIC GRATINGS
Perfect conductivity does not limit the generality of the results presented in and 4.2. Absorption losses cause some of the lowest flow lines to terminate on the surface, without modifying the flow distribution significantly. Moreover, similar flow distributions are obtained when a lossless dielectric grating is considered in the same mounting, supporting only two propagating reflected orders. Total internal reflection on a flat substrate-air boundary is characterized by an energy flow parallel to the interface. In air it has the same direction but with an exponentially decreasing amplitude. In the dielectric its horizontal component is equal to
$ 0 4.1
for TE polarization, and to
for TM polarization. Similarly to the perfectly conducting case, the corrugation causes the flow lines to become curved and curls to be formed around the points where 1 PI = 0. When the centers of the lowest curls are aligned with the tops of the grooves, perfect blazing occurs in the first order. There are, however, two important differences when compared with the metallic gratings. (1) Both media are characterized by positive real values of the electric permittivity, and the energy flow can extend into the air. This fact has two direct consequences: (a) the lowest curls also spread in the second medium, and are larger in diameter than the corresponding curls for the metallic substrate; (b) at the bottom of the grooves, flow lines undulate along the corrugation and separate the bottoms from the lowest curls (fig. 29). Thus, deeper grooves are required to move the centers of the curls to the depth necessary to ensure any special phenomena, such as perfect blazing and antiblazing (compare figs. 5 and 13). (2) Because of a partial penetration of the field inside the second medium even for a flat surface, the lines where P, = 0 are given by more complex expressions than in eqs. (22). Their distance from the surface is determined by
11,s41
LIGHT DIFFRACTED BY RELIEF GRATINGS
183
Fig. 29. Energy flow lines of the light diffracted by a dielectric grating in the total internal reflection regime: n , = 1.5, n 2 = I , I, = 0.55 pm,d = 0.26 pm, h = 0.24 pm, TE polarization, and e, = 450.
where the phase difference rp of the reflected light can be determined by Fresnel's formula, remembering that x i is imaginary:
Taking into account the values of the system parameters in fig. 13, it directly follows that
v,'"
> y,TE
3
(28)
in contrast with the metallic substrate. Thus, perfect blazing for TM polarization is obtained for deeper grooves (fig. 13).
184
LIGHT DIFFRACTION BY R E L I E F GRATINGS
4.7. LIGHT REFRACTION BY DEEP TRANSMISSION GRATINGS
As mentioned in 9 4.1, the process of curl formation can be explained by simply taking into account the interference between only three coherent waves. A reflection grating supporting two diffraction orders is the simplest model. Moreover, as discussed earlier, even a grating supporting a single order forms curls in the near-field zone, where evanescent orders are substantial. However, the mechanism of the diffraction of light by transmission gratings is entirely different. When a plane wave is incident on a flat interface between two lossless media, the flow lines are straight. In the upper medium their inclination depends on the angle of incidence and the surface reflectivity. In the lower medium, where only a single transmitted wave propagates, the energy flow follows the direction of the surface. Corrugation does not lead to a substantial change in the distribution of
Fig. 30. Energy flow lines of the light diffracted by a transmission sinusoidal grating (n,= 2.1, n 2 = 1,d = 0.4476 pm,h = 0.6 pm,A = 0.6328 pm,TEpolarization,andO,, = 45"),havinga99% efficiency in the first transmitted order.
111
REFERENCES
185
energy flow lines; inside the corrugated region the lines are slightly curved, depending on the amount of light diffracted in the non-zeroth transmitted orders. When almost the entire incident energy is diffracted into a single dispersive order, the energy flow in the lower medium follows its direction of propagation. This is the case with the grating, the efficiency behavior of which is shown in fig. 14. The flow distribution for the value of groove depth that ensures maximum efficiency in the first transmitted order is presented in fig. 30. It is surprising to discover that evanescent orders do not significantly modify the picture, providing only the smoothness of the curvature of lines inside the corrugated region.
Acknowledgements The author is grateful to Dr. Erwin Loewen (Milton Roy Company, Rochester) for his major role in initiating and preparing this article.
References Andrewartha, J. R., J. R. Fox and I. J. Wilson, 1979a, Opt. Acta 26(1), 69. Andrewartha, J. R., J. R. Fox and I. J. Wilson, 1979b, Opt. Acta 26(2), 197. Breidne, M., and D. Maystre, 1980, Periodic Structures, Gratings, Moire Patterns and Diffraction Phenomena, Proc. SPIE 240, 165. Chandezon, J., M. Dupuis, G. Cornet and D. Maystre, 1981, J. Opt. SOC.Am. 72, 839. Chang, K. C., and T. Tamir, 1980, Appl. Opt. 19, 282. Chang, K. C., V. Shan and T. Tamir, 1980, J. Opt. SOC.Am. 70, 804. Coutaz, J. L., 1987, J. Opt. SOC.Am. B 4, 105. Coutaz, J. L., and R. Reinisch, 1985, Solid State Commun. 56, 545. Dumery, G., and P. Filippi, 1970, C. R. Hebd. Seances Acad. Sci. 270, 137. Enger, R. C., and S. K. Case, 1983, J. Opt. SOC.Am. 73, 1113. Fano, U., 1941, J. Opt. SOC.Am. 31, 213. Garsia, N., 1983a, Opt. Commun. 45, 307. Garsia, N., 1983b, J. Electron Spectrosc. Relat. Phenom. 29, 421. Hajnal, I. V., 1987, Proc. R. SOC.London A 414, 447. Hessel, A., and A. A. Oliner, 1965, Appl. Opt. 4, 1275. Hessel, A., J. Schmoys and D. Y. Tseng, 1975, J. Opt. SOC.Am. 65, 380. Hutley, M. C., 1982, Diffraction Gratings (Academic Press, New York). Hutley, M. C., and D. Maystre, 1976, Opt. Commun. 19, 431. Loewen, E., 1983, Diffraction gratings, ruled and holographic, in: Appl. Opt. Opt. Eng., Vol. IX (Academic Press, New York) ch. 2. Madden, R. P., and J. Strong, 1958, Diffraction gratings, in: Concepts of Classical Optics, ed. J. Strong (Freeman, San Francisco, CA). Mashev, L., and E. Loewen, 1988, Appl. Opt. 27, 31. Mashev, L., and E. Popov, 1984, Opt. Commun. 51, 131.
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Mashev, L., and E. Popov, 1989, J. Opt. SOC.Am. 6, 1561. Mashev, L., E. Popov and E. Loewen, 1988, Appl. Opt. 27, 152. Mashev, L., E. Popov and E. Loewen, 1989, Appl. Opt. 28, 2538. Mashev, L., E. Popov and D. Maystre, 1988, Opt. Commun. 67, 321. Maystre, D., 1972, Opt. Commun. 6, 50. Maystre, D., 1973, Opt. Commun. 8, 216. Maystre, D., 1978a, J. Opt. Soc. Am. 68, 490. Maystre, D., 1978b, Opt. Commun. 26, 127. Maystre, D., 1980, Integral methods, in: Electromagnetic Theory of Gratings, ed. R. Petit (Springer, Berlin) ch. 3. Maystre, D., 1982, General study of grating anomalies from electromagnetic surface modes, in: Electromagnetic Surface Modes, ed. A. D. Boardman (Wiley, New York) ch. 17. Maystre, D., 1984a, Rigorous vector theories of diffraction gratings, in: Progress in Optics, Vol. XXI, ed. E. Wolf (North-Holland, Amsterdam) ch. 1. Maystre, D., 1984b, J. Opt. (France) 15, 43. Maystre, D., and M. Cadilhac, 1981, Radio Sci. 16, 1003. Maystre, D., and M. Neviere, 1977, J. Opt. (France) 8, 165. Maystre, D., and R. Petit, 1976, Opt. Commun. 17, 196. Maystre, D., M. Cadilhac and J. Chandezon, 1981, Opt. Acta 28, 457. Maystre, D., J. P. Laude, P. Gacoin, D. Lepere and J. P. Priou, 1980, Appl. Opt. 19, 3099. Maystre, D., M. Neviere and P. Vincent, 1978, Opt. Acta 25, 905. Maystre, D., M. Neviere, R. Reinisch and J. L. Coutaz, 1988, J. Opt. SOC.Am. B 5, 338. McClellan, R. P., and G. W. Stroke, 1966, J. Math. Phys. 45, 383. McPhedran, R. C., and D. Maystre, 1974, Nouv. Rev. Opt. 5, 241. Mermit, N. D., 1990, Phys. Today 11, 9. Metcalfe, K., and R. Hester, 1983, Chem. Phys. Lett. 94, 411. Millar, R. F., 1961a, Can. J. Phys. 39, 81. Millar, R. F., 1961b, Can. J. Phys. 39, 104. Millar, R. F., 1971a, Proc. Cambridge Philos. SOC.69, 175. Millar, R. F., 1971b, Proc. Cambridge Philos. SOC.69, 217. Moaveni, M. K., H. A. Kalhor and A. Afrashteh, 1975, Comput. Electron. Eng. 2, 265. Moharam, M. G., and T. K. Gaylord, 1982, J. Opt. SOC.Am. 72, 1385. Moharam, M. G., T. K. Gaylord, G. T. Sincerbox, H. Werlich and B. Yung, 1984, Appl. Opt. 23, 3214. Neureuther, A., and K. Zaki, 1969, URSl Symp. Electron. Waves, Aka Freq. 38, 282. Neviere, M., 1980, The homogeneous problem, in: Electromagnetic Theory of Gratings, ed. R. Petit (Springer, Berlin) ch. 5. Neviere, M., and M. Cadilhac, 1970, Opt. Commun. 2, 235. Neviere, M., and R. Reinisch, 1983, J. Phys. Colloq. (Paris) 44, Suppl. 12, ClO-359. Neviere, M., H. Akhouayri, P. Vincent and R. Reinisch, 1987, Proc. SOC.Photo-Opt. Instrum. Eng. 815, 146. Neviere, M., M. Cadilhac and R. Petit, 1972, Opt. Commun. 6, 34. Neviere, M., M. Cadilhac and R. Petit, 1973, IEEE Trans. Antennas Propag. AP-21, 37. Neviere, M., P. Vincent and R. Petit, 1974, Nouv. Rev. Opt. 5, 65. Pavageau, J., R. Eido and H. Kobeisse, 1967, C. R. Hebd. Seances Acad. Sci. B 264, 424. Petit, R., 1966a, Rev. Opt. 45, 249. Petit, R., 1966b, Rev. Opt. 45, 353. Petit, R., ed., 1980, Electromagnetic Theory of Gratings (Springer, Berlin). Petit, R., and M. Cadilhac, 1964, C. R. Hebd. Seances Acad. Sci. B 259, 2077. Petit, R., and M. Cadilhac, 1966, C. R. Hebd. Seances Acad. Sci. B 262, 468. Petit, R., and M. Cadilhac, 1987, Radio Sci. 22, 1247.
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Popov, E., 1989, J. Mod. Opt. 36,669. Popov, E., and L. Tsonev, 1989, Opt. Commun. 69, 193. Popov, E.,and L. Tsonev, 1990, Surf. Sci. 230, 290. Popov, E.,L. Mashev and E. Loewen, 1989, Appl. Opt. 28, 970. Popov, E., L. Mashev and D. Maystre, 1986, Opt. Acta 33, 607. Popov, E.,L. Mashev and D. Maystre, 1988, Opt. Commun. 65,97. Popov, E., L. Tsonev and D. Maystre, 1990a, J. Mod. Opt. 37, 367. Popov, E..L. Tsonev and D. Maystre, 1990b, J. Mod. Opt. 37, 379. Rayleigh, Lord, 1907, Proc. R. SOC.London A 79, 399. Reinisch, R., and M. Neviere, 1981, Opt. Eng. 20,629. Reinisch, R., G. Chartier, M. Neviere, M. C. Hutley, G. Clauss, J. P. Galaup and J. F. Eloy, 1983, J. Phys. (Paris) Lett. 44,L1007. Roumiguieres, J. L., D. Maystre and R. Petit, 1976, J. Opt. SOC.Am. 66, 772. Tseng, D. Y., A. Hessel and A. A. Oliner, 1969, URSI Symp. Electron. Waves, Alta Freq., special issue 38,82. Twersky, V., 1956, IRE Trans. Antennas Propag. AP-4, 330. Twersky, V., 1960, J. Res. Nat. Bur. Stand. D 64,715. Uretsky, J. L., 1965, Ann. Phys. (New York) 33,400. Van den Berg, P. M., 1971, Appl. Sci. Res. 24, 261. Van den Berg, P. M., 1981, J. Opt. SOC.Am. 71, 1224. Van den Berg, P. M., and J. T. Fokkema, 1979, J. Opt. SOC.Am. 69,27. Vincent, P., 1980, Differential methods, in: Electromagnetic Theory of Gratings, ed. R. Petit (Springer, Berlin) ch. 4. Wirgin, A., 1964, Rev. Opt. 43,499. Wirgin, A., 1968, Rev. Cethedec 5, 131. Wirgin, A,, 1979a. C. R. Hebd. Seances Acad. Sci. A 289, 259. Wirgin, A., 1979b, C. R. Hebd. Stances Acad. Sci. B 288, 179. Wirgin, A., 1979c, C. R. Hebd. Seances Acad. Sci. B 289,273. Wirgin, A,, 1980, Opt. Acta 27, 1671. Wirgin, A., 1981, Opt. Acta 28, 1377. Wood, R. W., 1902, Philos. Mag. 4, 396. Yamashita, M., and M. Tsuji, 1983, J. Phys. SOC.Jpn. 52, 2462. Yokomori, K.,1984, Appl. Opt. 23,2303.
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E. WOLF, PROGRESS IN OPTICS XXXI 0 1993 ELSEVIER SCIENCE PUBLISHERS B.V.
OPTICAL AMPLIFIERS BY
N. K.DUTTA and J. R. SIMPSON AT& T Bell Laboratories Murray Hill.New Jersey 07974, USA
189
CONTENTS PAGE
9 1.
INTRODUCTION
. . . . . . . . . . . . . . . . . . . 191
0 2. SEMICONDUCTOR OPTICAL AMPLIFIERS . . . . . . . 191 0 3 . FIBER AMPLIFIERS . . . . . . . . . . . . . . . . . . 207 0 4. LIGHTWAVE TRANSMISSION SYSTEM STUDIES . . . 216 REFERENCES
. . . . . . . . . . . . . . . . . . . . . . .
190
222
4 1. Introduction An optical amplifer, as the name implies, is a device that amplifies the input light signal. The amplification, or gain, can be higher than 1000 in some devices. The two principal types of optical amplifiers are the semiconductor optical amplifier and the fiber optical amplifier. For a semiconductor amplifier the amplification of the light takes place when the light propagates through a semiconductor medium fabricated in a waveguide form. For a fiber amplifier the amplification of the light occurs when it travels through a fiber doped with rare earth ions (such as Nd , Er , etc.). Semiconductor amplifiers are typically less than 1 mm long, whereas fiber amplifiers are typically 1 to 100 m long. The operating principles design, fabrication, and performance characteristics of semiconductor and fiber amplifiers are described in this chapter. For a lightwave propagating through a transmission system, as the light signal travels through the fiber, the signal weakens and becomes distorted. Regenerators are used to restore the optical pulses. Figure l a shows the block diagram of a typical lightwave regenerator. Its main components are the optical receiver, optical transmitter, and the electronic timing and decision circuits. It has been shown that optical amplifiers can nearly completely restore the original optical pulses and thus the transmission distance can be increased without using conventional regenerators. Figure l b shows an example of a semiconductor amplifier that functions as a repeater. Semiconductor amplifiers need external current to produce gain, and fiber amplifiers need pump lasers for the same purpose. Because of its simplicity, an optical amplifier is an attractive alternative for new lightwave system installations. +
8 2.
+
Semiconductor Optical Amplifiers
A semiconductor optical amplifier is a device very similar to a semiconductor laser, with similar operating principle, fabrication and design. Since the invention of the semiconductor laser in the early 1960s, extensive research and development has been carried out at various industrial laboratories and universities. The fabrication and performance characteristics of lasers fabricated 19 1
192
-
OPTICAL AMPLIFIERS
b RECEIVER
LIGHT IN
-('
DEC'S'oN ---+TRANSMITTER CIRCUIT
"11, I 2
LIGHT OUT
\ACTIVE REGION (A-1.3prn InGoAsP)
(b)
Fig. I . (a) Block diagram of a lightwave regenerator; (b) schematic of a semiconductor optical amplifier.
using both the AlGaAs/GaAs and InGaAsP/InP material systems were extensively reviewed (Kressel and Butler [1977], Casey and Panish [1978], Thompson [ 19801, Agrawal and Dutta [ 19861). The basic structure can be used to study light amplification. When the injection current is below threshold, the laser acts as a linear amplifier for incident light waves, and above threshold it undergoes oscillation. Initial optical amplifier studies were carried out on GaAs homostructure devices in the mid-1960s (Coupland, Mambleton and Hilsum [ 19631, Crowe and Graig [1964], Kosnocky and Cornely [1968]). Extensive work on AlGaAs laser amplifiers was subsequently carried on by Mukai, Yamamoto and Kimura [1985] and Nakamura and Tsuji [1981]. These amplifiers used an index guiding structure and are therefore closer to a practical device. Semiconductor optical amplifiers can be classified into two types, the Fabry-Perot (FP) amplifier and the traveling wave (TW) amplifier. A F P amplifier has considerable reflectivity at the input and output ends, which results in resonant amplification between the end mirrors. Thus it exhibits very large gain at wavelengths corresponding to the longitudinal modes of the cavity. The TW amplifier, by contrast, has negligible reflectivity at each end, which results in signal amplification by a single pass. The optical gain spectrum of a
11198 21
SEMICONDUCTOR OPTICAL AMPLIFIERS
193
TW amplifier is thus quite broad and corresponds to that of the semiconductor. Most practical TW amplifiers exhibit some small ripple in the gain spectrum that arises from the residual facet reflectivities. TW amplifiers are more suitable for system applications. Therefore, over the last few years, much effort has been devoted to fabricate amplifiers with very low facet reflectivities. These amplifier structures use special low-reflectivity dielectric coatings or have tilted or buried facets. The fabrication and performance of these devices are described later. Much recent work on semiconductor optical amplifiers was carried out using the InGaAsP material system with optical gain centered around 1.3 or 1.55 pm. The interest in these wavelength ranges is primarily due to the low loss and low fiber dispersion of silica fiber, which is extensively used as the transmission medium for the fiber optical transmission systems that are being installed throughout the world. This chapter will focus primarily on amplifiers fabricated using the InGaAsP material system. The amplifiers used in fiber transmission system applications, such as preamplifiers in front of a receiver or in-line amplifiers as a replacement of regenerators, must also exhibit equal optical gain for all polarizations of the input light. In general, the optical gain in a waveguide is polarization dependent, although the material gain is independent of polarization. This arises from the unequality between the mode confinement factor of the light polarized parallel to the junction plane (TE mode) and that of the light polarized perpendicular to the junction plane (TM mode). For thick active regions the confinement factors of the TE and TM mode are nearly equal. Hence, the gain difference between the TE and TM modes is smaller for amplifiers with a thick active region. Saitoh and Mukai [1988] have calculated this gain difference as a function of cavity length for different active layer thicknesses (fig. 2). For thicker active regions the gain difference is smaller. In addition, this difference increases with increasing cavity length, since the overall gain increases. In some applications, however, the polarization independence of the gain is not important. An example is a power amplifier following a laser. Since the light from the laser is TE polarized, the parameters of interest in this application are high TE gain and high saturation power. Optical amplifiers with multiquantum well active regions (which have strongly polarization-dependent gain) satisfy both of these requirements and are therefore ideally suited for this application. The nonisotropic nature of the optical transition in a quantum well makes the TE-mode gain of a multiquantum well amplifier much higher (> 10 dB) than the TM-mode gain.
I94
OPTICAL AMPLIFIERS
10
20
50
100
200
500
1000
LENGTH L (pm)
Fig. 2. The optical gain difference between the TE and TM mode of a semiconductor amplifier as a function of device length for different active layer thicknesses (Saitoh and Mukai [ 19881).
2.1. IMPACT OF FACET REFLECTIVITY
Two basic amplifier types are the Fabry-Perot (FP) and the traveling wave (TW) amplifier. In practice, antireflection-coated facets generally exhibit some residual reflectivity and form an optical cavity. An example of the transmission characteristics of a TW amplifier with antireflection-coated cleaved facets is shown in fig. 3. The output exhibits modulations at longitudinal modes of the cavity, and at these modes optical gain is a few decibels higher than in between modes. The phase and amplitude transfer functions of an optical amplifier can be characterized by a change in phase and in amplitude G . For an amplifier with facet reflectivities R , and R , , the gain G is given by
195
SEMICONDUCTOR OPTICAL AMPLIFIERS
30 TE
1.44
1.46
1.48
1.5
1.52
1 54
WAVELENGTH (prn)
Fig. 3. The optical gain as a function of frequency of a Fabry-PCrot optical amplifier (Saitoh and Mukai [1987]).
where G, is the single-pass gain, and the phase shift $ is given by
where @o = 2 nLn/L is the nominal phase shift, L is the length of the amplifier, is the refractive index, b is the linewidth enhancement factor, go is the unsaturated gain, I is the total internal intensity and I , is the saturation intensity. The sin2$ term in eq. (1) is responsible for the modulation of the output signal at cavity modes. The case of practical interest is one of low reflectivities, i.e. R , , R , 6 Under such situations the modulation appears as a small ripple (fig. 3) superimposed on an envelope function. The envelope function is essentially the gain spectrum of the semiconductor material. The peak-to-valley ratio of the output intensity ripple is given by n
For the ideal case R , , R , + 0, V equals 1, i.e. no ripple at cavity mode frequencies is observed in the output spectrum. The quantity I/ is plotted as a function of (reflectivity) in fig. 4 for different values of gain. A practical
196
OFTlCAL AMPLIFIERS
REFLECTIVITY (R)
Fig. 4. The ripple in the gain spectrum as a function of reflectivity for different values of gain (OMahony [1988]).
should have V < 1 dB, thus reflectivities < are needed. Three principal schemes exist for achieving these low reflectivities. They are: (i) ultralow-reflectivity dielectric coated amplifiers, (ii) buried-facet amplifiers and (iii) tiltedfacet amplifiers.
2.2. AMPLIFIER DESIGNS
2.2.1. Low-rejectivity coatings As discussed earlier, a key factor for good performance characteristics, namely, low gain ripple and low polarization selectivity, for traveling wave optical amplifiers is a very low facet reflectivity. The reflectivity of cleaved facets can be reduced by dielectric coating. For plane waves incident on an air interface from a medium of refractive index n, the reflectivity can be reduced to zero by coating the interface with a dielectric in which the refractive index equals nl/*and the thickness equals 1/4.The fundamental mode propagating in a waveguide is not a plane wave, however, and therefore the above law only provides a guideline for achieving very low (facet reflectivity by dielectric coatings. In practice, very low facet reflectivities are obtained by monitoring the amplifier performance during the coating process. The effective reflectivity then can be estimated from the ripple at the Fabry-Perot mode spacings, caused by residual reflectivity, in the spontaneous emission spectrum.
111, § 21
197
SEMICONDUCTOR OPTICAL AMPLIFIERS
WAVELENGTH (pm)
Fig. 5. The measured reflectivity as a function of wavelength (Olsson [1989]).
The result of such an experiment is shown in fig. 5. The reflectivity is very low (< only in a small range of wavelengths. Although many laboratory experiments have been carried out using amplifiers that rely only on low-reflectivity coatings for good performance, the critical nature of the thickness requirement and a limited wavelength range of good antireflection coating led to the investigation of alternate schemes as discussed below.
2.2.2. Buried-facet crtnplifier.~ The principal feature of the buried-facet (also known as the window device) optical amplifiers as compared with antireflection-coated cleaved-facet devices is a polarization-independent reduction in mode reflectivity due to a buried facet, which results in better control in achieving polarization-independent gain and gain ripple. The schematic cross section of a buried-facet optical amplifier is shown in fig. 6. The current confinement in this structure is provided by semi-insulating Fe-doped inP layers grown by the metal-organic chemical-vapor-deposition (MOCVD) growth technique on either side of the active region. The fabrication of the device involves the following steps. The first four layers are grown on a (100)-oriented n-type InP substrate by MOCVD. These layers are: (i) an n-type InP buffer layer; (ii) an undoped InGaAsP (A 1.55 pm) active layer; (iii) a p-type InP cladding layer; and (iv) a p-type InGaAsP (A 1.3 pm) layer. Mesas are then etched on the wafer along the [ 1101 direction with 15 pm wide channels normal to the mesa direction using a SiO, mask. The latter is needed
-
N
198
[III, § 2
OPTICAL AMPLIFIERS
r
ELECTRODE
- lnGoAsP
p+- InGoAsP1-
(a)
BURIED
FACET
(b)
Fig. 6. Schematic of a buried-facet optical amplifier (Dutta, Lin, Piccirilli, Brown and Chakrabarti [ 19901).
for buried-facet formation. Semi-insulating Fe-doped InP layers are then grown around the mesas by MOCVD with the oxide mask in place. The oxide mask and p-type InGaAsP layer are removed, and a p-type InP and p-type InGaAsP ( I , 1.3 pm) contact layer is then grown over the entire wafer by the vaporphase epitaxy growth technique. The wafer is processed using standard methods and cleaved to produce 500 pm long buried-facet chips with 7 pm long buried facets at each end. The facets of the chips are then antireflectioncoated using single-layer films of ZrO,. The fabrication of the cleaved-facet devices follows the same procedure as just described, except that the mesas are continuous with no channels separating them. The latter is needed for defining the buried-facet regions. The semi-insulating layer, in both types of devices, provides: (i) current confinement; and (ii) lateral index guiding. For buriedfacet devices it also provides the buried-facet region. The effective reflectivity of a buried facet decreases with increasing separation between the facet and the end of the active region. The effective reflectivity of such a facet can be calculated using a Gaussian beam approximation for the propagating optical mode. It is given by N
-
Re, = R / [ 1 + 2 S / k 0 ~ ) ,~ ]
(4)
where R is the reflectivity of the cleaved facet, S is the length of the buried-facet region, k = 27~11,where A is the optical wavelength in the medium, and w is the spot size at the facet. The calculated reflectivity using eq. (4) is plotted in fig. 7 using w = 0.7 pm and R = 0.3 for an amplifier operating near 1.55 pm. Figure 7 shows that a reflectivity of lo-, can be achieved for a buried-facet length of 15 pm. Although increasing the length of the buried-facet region decreases the reflectivity, if the length is too long, the beam emerging from the active region will
-
199
SEMICONDUCTOR OPTICAL AMPLIFIERS
w = 0.7p-n
-51-
!l
R = 0.3
CLEAVED FACET
R
'
1+ (2s/k w 2 f
,
-15,
-20 0
\ w=
SPOT FACET, SIZE
i 25
5 10 15 20 LENGTH OF BURIED FACET, s ( p n )
Fig. 7. Calculated effective reflectivity of a buried-facet region (Dutta, Lin, Piccirilli, Brown and Chakrabarti [ 19901).
strike the top metallized surface, producing multiple peaks in the far-field pattern, a feature not desirable for coupling into a single-mode fiber. The beam waist o of a Gaussian beam after traveling a distance z is given by 0 2 ( z ) = o;[1
+($>'I.
where wo is the spot size at the beam waist and A is the wavelength in the medium. Since the action region is -4 p m from the top surface of the chip, it follows from eq. (5) that the length ofthe buried-facet region must be less than 12 pm for single-lobed far-field operation. The optical gain is determined by injecting light into the amplifier and measuring the output. The internal gain of an amplifier chip as a function of current at two different temperatures is shown in fig. 8. The open circles and squares represent the gain for a linearly polarized incident light with the electric field parallel to the p-n junction in the amplifier chip (TE mode). The coupling losses between this chip and a lensed single-mode fiber were 6 d B per end. The solid circles are the measured gain for the T M mode at 40 " C .The measurements were done for sufficiently low input power ( - 40 dBm), so that the observed saturation is not due to saturated output power of the amplifier, but rather, to carrier loss caused by Auger recombination. Note that the optical gain for the T E and T M input polarizations are nearly equal. Figure 9 shows the measured gain as a function of input wavelength for TE-polarized incident light.
200
OPTICAL AMPLIFIERS
30
25 -
1
1
1
1
I
1
TE, 2O'C 0 T E , 40.C a TM.40°C
I
I
I
2ooc
I
I
o o o
..
o o
-
-
0
5
40'C
-
0
0
-
-
0
a
0
5-
I
l
l
-
w I
I
I
-+
p 15300
-
0
10-
-
-
m .
0 a
0
(3
U
I
o o o
0
2 15-
-
l
0
- 20 -m , -
2
(
0
I
I
toox
I
I
I
I
1
1
+
6
45600 WAVELENGTH
15 DO
(1)
0
Y
92
Fig. 9. Measured gain as a function of wavelength for the TE mode (Dutta, Lin, Piccirilli, Brown and Chakrabarti [1990]).
111, § 21
20 1
SEMICONDUCTOR OPTICAL AMPLIFIERS
The modulation in the gain (gain ripple) with a periodicity of 7 A is due to the residual facet reflectivity. The measured gain ripple for this device is less than 1 dB. The estimated facet reflectivity using the measured gain ripple of 0.6 dB at 26 dB internal gain is 9 x The 3 dB bandwidth of the optical gain is 450 A for this device. The light-versus-current (L-I) characteristics and the amplified spontaneous emission spectrum of a buried-facet amplifier is shown in fig. 10. The L-I curve exhibits a soft turn-on ( I , in fig. 10). At currents larger than la,most of the light output from the facets is amplified spontaneous emission, and at currents less than I , the device behaves like an edge-emitting LED (light-emitting diode). For high gain, the amplifier should be operated at currents larger than I,. The amplified spontaneous emission spectrum is shown in figs. 10b and c. Figure 10b shows that the peak of the spectrum shifts to shorter wavelengths with increasing current. The small modulation in fig. 1Oc correlates well with N
CURRENT (mA)
-65 1.52
v)
4.57 WAVELENGTH (pm)
4.62
-42 -46
I-
-20
4.5568
1.5648 1.5728 WAVELENGTH ( p m )
Fig. 10. (a) Spontaneous emission from the facet plotted as a function of current; (b) spectrum of the emission from the facet; and (c) spectrum under high resolution.
202
IIII, § 2
OPTICAL AMPLIFIERS
the modulation observed in the gain, and is caused by the residual reflectivities of the facets. Dutta, Lin, Piccirilli, Brown and Chakrabarti [I9901 showed that the gain ripple and polarization dependence of gain correlate well with the ripple and polarization dependence of the amplified spontaneous emission spectrum. Thus, the amplified spontaneous measurements (fig. lo), which are much simpler to make than gain measurements, provide a good estimate of the packaged amplifier performance. 2.2.3. Tiltedyfucet umplfirs Another way to suppress the Fabry-Perot resonant modes of the cavity is to slant the waveguide (gain region) from the cleaved facet so that the light incident on it internally does not couple back very well into the waveguide. The process essentially decreases the effective reflectivity of a tilted facet relative to a normally cleaved facet. The reduction in reflectivity as a function of tilt angle is shown in fig. 11 for the fundamental mode of the waveguide. The schematic of a tilted-facet optical amplifier is shown in fig. 12. The waveguiding along the junction plane is weaker in this device than that for the strongly index-guided buried heterostructure device (fig. 12). The weak index guiding for the structure of fig. 12 is provided by the dielectric defined ridge. The fabrication of the device follows a procedure similar to that described previously in 0 2.2.2. I
I
I
I
=
w
u
10-3cJ
h
A
4b 1; 44 STRIPE ANGLE (DEGREES)
Fig. 1 I . Calculated change in reflectivity as a function of tilt angle of the facet.
111, § 21
203
SEMICONDUCTOR OPTICAL AMPLIFIERS
(a)
(b)
Fig. 12. Schematic of a tilted-facet amplifier (Zah, Osinski, Caneau, Menocal, Reith, Salzman, Shokoohi and Lee [1987]).
Fig. 13. Measured gain as a function of injection current (Zah, Osinski, Caneau, Menocal, Reith, Salzman, Shokoohi and Lee [1987]).
The measured gain as a function of injection current for TM and TE polarized light for a tilted-facet amplifier is shown in fig. 13. Optical gains as high as 20 dB or higher have been obtained using a tilted-facet amplifier. Although the effective reflectivity of the fundamental mode decreases with increasing tilt of the waveguide, the effective reflectivity of the higher-order modes increases. This may cause the appearance of higher-order modes at the output (which may reduce fiber-coupled power significantly), especially for large ridge widths.
204
OPTICAL AMPLIFIERS
2.3. MULTIQUANTUM WELL AMPLIFIERS
As mentioned previously, multiquantum well
(MQW)amplifiers are ideally
suited for applications that d o not require polarization-independent gain, and they are capable of much higher output power than regular double heterostructure ( D H ) amplifiers. The output power of an amplifier is limited by the gain saturation power of the amplifier. If P, is the saturation power in the gain medium, the output saturation power can be approximated by Po = PJT,where r i s the confinement factor of the optical mode. For MQW amplifiers whose active region consists of a few (generally 3 or 4) quantum wells of 50-100 A thickness, the confinement factor is considerably smaller than that for a regular D H amplifier. This effect results in a higher saturation power. However, since the signal gain is given by G = e x p [ ( r g - a ) L ] , where g is the material gain, the MQW amplifiers have lower gain than D H amplifiers for the same cavity length. The gain saturation characteristics of an optical amplifier are obtained by plotting the measured gain as a function of output power, which is shown in fig. 14 for both a DH and MQW amplifier. Both devices amplify signals near
MQW AMPLIFIER
z4
(51
*0°
10
-5
-10
14
,-.
m
D
za
.
15
PSAT= 14 dBm (28 mW) 0 5 10 15 OUTPUT POWER (dBm)
0
I
Y
BURIED FACET
"
12
u
6
m U I
U w
FACET
10
8
m U -8
I
I
-6
-4
I
-2
-
1
0
1 -
2
FIBER OUTPUT POWER (dBm)
Fig. 14. Measured optical gain is plotted as a function ofoutput power for regular DH and MQW amplifiers.
SEMICONDUCTOR OPTICAL AMPLIFIERS
0-
1.47
205
1.49 1.51 1.53 WAVELENGTH (pm)
Fig. 15. Measured gain spectrum of a MQW amplifier at two injection currents (Eisenstein, Koren, Raybon, Wiesenfeld and Wegner [ 19901).
1.55 pm. The DH amplifier had a 0.4 pm thick active region, was 500 p m long, and exhibited < 1 d B gain difference between T E and T M polarizations. The MQW amplifier result is shown for the T E mode. It had 4 active layer wells 70 A thick and barrier layers also 70 A thick. Saturation intensities as high as 100 mW have been reported for M Q W amplifiers. The density of states function for electrons and holes in a quantum well is independent of energy. This results in a broad spontaneous emission spectrum and, hence, a broad gain spectrum of a MQW amplifier. The measured gain spectrum of a device at two different currents is shown in fig. 15.
2.4. INTEGRATED LASER AMPLIFIER
Since the M Q W amplifier is ideally suited for amplifying the output power of a semiconductor laser, it is useful to combine them on a single chip and
206
[Ill, § 2
OPTICAL AMPLIFIERS
thereby eliminate coupling losses. An integrated distributed Bragg reflector (DBR) laser and MQW amplifier chip fabricated by Koren, Miller, Raybon, Oron, Young, Koch, DeMiguel, Chien, Tell, Brown-Goebeler and Burrus [1990] is shown in fig. 16. The grating provides the frequency-selective feedback, which results in single-frequency operation of the laser. The MQW layers grown over the InP substrate serve as the active region for both the laser and the amplifiers. It had six active layer wells. The effectiveness of the amplifier can be seen from fig. 17, which shows the light versus current characteristics of the laser with the amplifier biased at 170 mA. The slope of the L-I curve (2 mW/mA) is about a factor of 10 higher than that for a typical DBR laser without an amplifier. AR COATING 7, LASER
GRATING
AMPLIFIER
Fig. 16. Integrated distributed Bragg reflector laser and MQW amplifier structure (Koren, Miller, Raybon, Oron, Young, Koch, DeMiguel, Chien, Tell, Brown-Goebeler and Burrus [ 19901).
AMPLIFIER CURRENT 170 ma C.W. T = 2 3 " C
S L O P E = 2.0 mWIma
10
Fig. 17. Light output from the amplifier facet is plotted as a function of laser current (Koren, Miller, Raybon, Oron, Young, Koch, DeMiguel, Chien, Tell, Brown-Goebeler and Burrus [ 19901).
s
207
FIBER AMPLIFIERS
111, 31
6 3.
Fiber Amplifiers
Amplification of light in a fiber by the interaction of a pump with a signal can be accomplished in several ways. Nonlinear optical phenomena such as Raman, Brillouin and four-wave mixing methods (Stolen [ 19791, Olsson and van der Ziel [ 19861, Aoki [ 19881) or by the stimulated emission from a n excited state of a rare earth ion within the fiber. Although the nonlinear methods have been shown to be useful, they are generally less efficient in transferring pump to signal energy. The placement of rare earth ions in the core of an optical fiber as an amplifying media was first demonstrated in 1964 by Koester and Snitzer observing a gain of 40 dB at 1.06 p m in a flashlamp, side-pumped neodymiumdoped fiber 1 m in length. The concept was revisited in 1973 by Stone and Burrus who demonstrated an end-pumped neodymium-doped fiber laser with a threshold of a few milliwatts. Furthermore, a semiconductor diode pump was suggested, bringing the idea of active fiber devices closer to practical use. Amplification of light in the wavelength region of minimum loss for a silicabased optical fiber (2 1.5 pm) using transitions of the erbium ion was first demonstrated in 1986 (Mears, Reekie, Poole and Payne [ 19861, Desurvire, Simpson and Becker [ 19871). More recently, amplification at the minimum dispersion wavelength for a standard telecommunication fiber (A 1.3 pm) using praesodymium in a fluoride host glass has also been shown (Ohishi, Kanamori, Kitagawa, Takahashi, Snitzer and Sigel [ 199 11, Durteste, Monerie, Allain and Poignant [ 19911, Carter, Szebesta, Davey, Wyatt, Brierley and France [ 19911, Miyajima, Sugawa and Fukasaku [ 19911). Given that the most mature technology in fiber amplifiers is that based on erbium, the discussions below will focus on the performance of this device.
-
-
3.1. ENERGY LEVELS
Fundamental to all rare earth-doped amplifier systems is the ability to invert the population of ions from the ground state to an excited state that acts as a storage of pump power from which incoming signals may stimulate emission. The energy level states are broadly classified as either three- or four-level systems, as shown in fig. 18. Erbium is considered a three-level system, whereas Nd3 and Pr3 are considered four-level systems. Notable in the three-level system is an absorption at the signal wavelength when the system is not inverted or underpumped. From this property the need arises for an optimum length or number of erbium ions for the pump power available (Desurvire, Simpson and +
+
208
OPTICAL AMPLIFIERS
El' OPTICAL 4-LEVEL SYSTEM (q.Nd3'@ 1060nm, 1350nm)
OPTICAL 3-LEVEL SYSTEM (eg. E r 3 + @ 1550nm)
Fig. 18. Energy diagrams for three- and four-level systems.
Becker [ 19871); i.e. if the fiber is too long, its end will not be inverted and therefore will not contribute to gain but cause loss, which will also contribute to noise. The four-level system remains transparent. Suitable pump energies for these systems include any higher-energy absorption state that would rapidly decay to the metastable state. For erbium these absorption energies correspond to wavelengths of 0.51, 0.64, 0.82, 0.98, and 1.48 pm, as can be seen in the absorption spectrum for an unpumped erbium-doped germanium-aluminasilicate glass (fig. 19). Unlike transition metal ions in glass where the host determines the resulting energy levels, rare earth ion energies are based on inner shell electrons, which are substantially host-independent. The relative merits of these pump bands are determined by the efficiency of energy transfer to the
PUMP WAVELENGTHS 1
SIGNAL
nnrw
1000
g
100
m
'D m
s 0
10
't
011
I
I
I
I
1
I
Fig. 19. Absorption spectrum for an erbium-doped silica fiber.
FIBER AMPLIFIERS
111, § 31
209
metastable state. Inefficiencies due to low absorption cross sections, excited state absorption, and other nonradiative decay mechanisms can significantly affect the device performance. The energy levels of Er3 ion in a glass fiber are shown in fig. 20. The local electric field acts as a small perturbation on the erbium ion, which results in a splitting of each level of the ion into a number of closely spaced levels (shown for 4 1 1 3 j 2 ) . The splitting between these levels is usually much smaller than the energy separation between the discrete levels of the ion. Each closely spaced level is further broadened by its characteristic lifetimes and inhomogeneities in the glass host. This results in the observation of broad absorption and fluorescence spectra. The measured spectra for erbium in a glass host for transitions between the first excited state and ground state are shown in fig. 21. The fluorescence transition from the first excited state (4113,2) to the ground state (41,5,2) is fortuitously near the low-loss transmission window for telecommunication applications. Although the transition energies are primarily determined by the erbium ion, the properties of the host material play a role in determining radiative and nonradiative lifetimes and absorption and emission cross sections. The lifetime of 4113,2 state is 10 ms. Erbium ions excited to higher levels cascade down to the 4 1 1 3 , 2 level by losing energy nonradiatively. +
-
Fig. 20. Energy levels of erbium in a glass fiber (Becker [ 19901).
210
OPTICAL AMPLIFIERS
Wavelength (nrn)
Fig. 21. Absorption and emission spectra of an erbium-doped fiber.
The 1480 nm pumping corresponds to directly pumping the first excited state, the absorption and gain spectra of which are shown in fig. 21. The most commonly used pumps at present are at 1480 and 980 nm. Pumping at 980 nm is the more efficient of the two and also results in less noise due to more complete inversion possible for this pump band.
3.2. FIBER DESIGN AND FABRICATION
3.2.1. Fiber fiibrication
The traditional methods of fabricating low-attenuation doped silica fiber are based on the reaction of halides such as SiCI,, GeCI,, POCI, and SiF, with oxygen or with an oxyhydrogen torch to form the desired mix of oxides (Li [1985], Miller and Kaminow [1988]). Processes based on the oxidation method, where reaction and deposition take place inside a silica substrate tube, are referred to as modified chemical vapor deposition (MCVD), plasma chemical vapor deposition (PCVD), and intrinsic microwave chemical vapor deposition (IMCVD). Processes based on the flame hydrolysis method, where the resulting oxide particles are collected on a rotating target and subsequently sintered, are referred to as outside vapor deposition (OVD) and vapor axial deposition (VAD). Control of the vapor compositions during the formation of
III,O 31
FIBER AMPLIFIERS
21 1
the fiber preforms allows control of the composition and refractive indices over the drawn fiber radii. Several variations on these processes to introduce rare earth dopants into the fiber core have been reviewed (Urquhart [1988], Simpson [1989], DiGiovanni [ 19901, Ainslie [ 19911). The principal difficulty in adapting these techniques is the delivery of the relatively low vapor pressure rare earth reactants, typically chorides or organic chelates. Methods which overcome these difficulties include the use of heated delivery lines, rare earth chelate sources, solution doping ofunsintered silica and sol gel dip coating. In addition, host compositions containing aluminum, not used in transmission fiber, were shown to be advantageous in allowing the rare earth to be uniformly dissolved in the glass structure, thereby reducing inefficient pump to signal conversion processes and flattening the wavelength-dependent gain near 1.55 pm. The introduction of aluminum requires methods similar to those for the rare earths, again because of low vapor pressure reactants. Fluoride glass fiber preform processing is achieved not by vapor phase processing but by batch melting of mixed particle constituents, followed by casting into rod and tube shapes and subsequent drawing into fiber. (France, Carter, Moore and Day [1987], Takahashi and Iwasaki [1991]). The low melting temperatures ( < 1000 C) and low viscosities of these glasses make this method particularly suitable. Compositions containing many fluorides such as ZrF,-BaF,-LaF,-AlF,-NaF (ZBLAN) have been chosen to diminish the tendency of this glass system to crystallize. O
3.2.2. A mplifir design The fiber amplifier configuration depends on the performance required, breaking into two broad categories of lumped, referring to short length (0.5-100 m), relatively high gain devices, or distributed, referring to amplification along the fiber span between repeaters with low signal excursion and low net gain. The lumped amplifiers are, in turn, divided between high output power and low-noise high-gain devices. In all cases the pump power and signal are combined, using a wavelength division multiplexer (WDM) based on a fourport fused fiber coupler or a miniature bulk optic (fig. 22). Configurations include combinations of co- and counter-propagating pump, with the addition of filters within the amplifier cavity and isolators or angle polished fiber ends to diminish signal reflections that can initiate lasing. High-power semiconductor lasers emitting at 0.98 and 1.48 pm are used as pump sources for fiber amplifiers. Lasers at these wavelengths have
212
OPl'lC41. AMPLIFIERS
Signal In Erbium-Doped Fiber
Isolator Filter
WDM
x splices
Signal Out
Fig. 22. Schematic of an erbium fiber amplifier.
been fabricated using InGaAs/GaAs and InGaAsP/InP material systems with peak facet powers of 200 mW (Tanbuk-Ek, Logan, Olsson, Temkin, Sergent and Wecht [ 19901, Dutta, Lopata, Sivco and Cho [ 199 1 I). Commercial erbium fiber amplifiers typically use pump powers of 20 to 100 mV.
3.3. FIBER AMPLIFIER PERFORMANCE
3.3.1. Chnrncterislics A principal characteristic for a device such as an optical fiber amplifier is
optical gain. The measured small-signal gain at two different signal wavelengths as a function of pump power is shown in fig. 23. The pump laser wavelength is 1.476 pm,and the fiber length is 19.5 m. The gain increases rapidly at pump powers near threshold, and increases slowly at high pump powers where almost all the erbium ions along the length of the fiber are inverted.
-10
0
10 20 Launched Pump Power (mW)
30
Fig. 23. Small-signal gain as a function of pump power. (Courtesy I. Zyskind.)
111, § 31
213
FIBER AMPLIFIERS
A figure of merit commonly used to describe the amplifier is the slope of the tangent to the gain versus pump power curve (dB/mW). Large values indicate a low threshold and steep rise in gain with pump. Pump efficiencies vary with pump wavelength, with record values of 5.9 dB/mW for 1.48 pm pumping, 11.0 dB/mW for 0.98 pm pumping, and 1.3 dB/mW for 0.82 pm pumping. The maximum output power from an amplifier depends on the gain saturation power. As mentioned previously, the erbium-doped amplifier behaves as a three-level system, where the lower lasing level (41,5,2) is the ground state. For a three-level system the output saturation power (i.e. the output power at which gain decreases by 3 dB) increases approximately linearly with the pump power once the pump power exceeds the power necessary to invert substantially the population of erbium ions in the amplifier. Figure 24 shows typical plots of amplifier gain as a function of the output signal power for different values of input pump powers and fiber lengths. The figure shows typical roll-off of the gain as the signal is increased. A saturation power of + 11.3 dBm or 13 mW is obtained for a 53 mW input pump. Recent experiments have shown CW output powers in excess of 100 mW from an erbium-doped fiber amplifier using a combination of both forward and backward pumping by four laser diodes providing a pump power of 345 mW (Takenaka, Okano, Fujita, Odagiri, Sunohara and Mito [ 19911). Similarly a + 21 dBm output power amplifier pumped by a diode-pumped N d : Y A G laser has also been demonstrated
-15
-10
-5 0 +5 +I0 Output Signal Power (dBm)
+I5
Fig. 24. Amplifier gain as a function of output signal power for different values of pump power (P,,,,) and fiber length (Lop,).The values of PA:: represent output power at which the gain is decreased by 3 dB (Desurvire, Giles, Simpson and Zyskind [1989]).
214
OPTICAL AMPLIFIERS
':
2.6 ' 2.4.
0 2.2.
n
I-
$
2.0.
k
1.8.
8
1.6.
0 w
z
1.4.
-c.40rn
1.2 '
-.+-
-+45m 55rn
(Grubb, Humer, Cannon, Windhorn, Vendetta, Sweeney, Leilabady, Barnes, Jedrzejewski and Townsend [ 19921). For practical system applications a wide gain spectrum and wide pump band are desirable. The measured small-signal gain at two Wavelengths as a function of the wavelength of the pump laser (near 1.48 pm) is shown in fig. 25. Note that the pump wavelength band for high gain is only 10 nm wide. A similar pump wavelength dependence of gain was reported for pump wavelengths close to 980 nm (Becker, Lidgard, Simpson and Olsson [ 19901). An important characteristic of an amplifier used as an in-line repeater or receiver pre-amplifier is the noise figure, which is defined as the ratio of the signal-to-noise ratios at the input and output of the amplifiers. The noise figure ( F ) is given by
2n,,(G - 1) 1 _Vin
+
1 -
1 ~
qin qdet
G
7
where qin, are the input and output coupling efficiencies, qdet is the detector quantum efficiency, G is the gain of the fiber amplifier, and n s pis the spontaneous emission factor. Under ideal conditions qin = 1, G D 1 and nsp = 1,
215
FIBER AMPLIFIERS A
10 8 -
m n
6-
g- 4 2-
0-2
-4
I
I
I
I
+
- -
which results in F 2nSp 3 dB. The measured values of F for 1.48, 0.98 and 0.82 pm pumping are 4.1 dB (Giles, Desurvire, Zyskind and Simpson [1990]), 3.2dB (Way, von Lehman, Andrejco, Saifi and Lin [1990], and 4.0 dB (Kimura, Suzuki and Nakagawa [ 1991]), respectively. Erbium-doped fiber amplifiers are relatively well developed. Amplifiers operating near 1.3 pm are currently being investigated (Carter, Szebesta, Davey, Wyatt, Brierley and France [ 19911, Durteste, Monerie, Allain and Poignant [ 19911). Optical gain has been reported in fluorozirconate fiber doped with 560 ppm W Pr3 and pumped at 1.007 pm. The measured gain at 1.3 pm as a function of launched pump power is shown in fig. 26. Note that although the pump power is considerably higher than that for Er -doped fiber amplifiers, these initial results are very promising. Availability of commercial grade fiber amplifiers at 1.3 pm will most likely impact the upgrade of existing optical fiber systems which have a dispersion minimum near 1.3 pm. +
+
3.3.2. Commercial erbium jiber umplijiers Many companies (e.g. Amoco, AT&T, BT&D, Corning JDS-Fitel and Pirelli) offer erbium-doped fiber amplifiers with gains from 10 to 35 dB and output powers from 0 to + 15 dBm. The devices pumped with 980 nm diodes offer an advantage of the lowest noise figures (near the 3 dB quantum limit), whereas the 1480 nm pumped devices offer an advantage from pump diode reliability with a slightly higher noise figure of 4-5 dB. High-power devices are typically bidirectionally pumped, and are correspondingly more expensive because of the added pump laser. Isolation of the amplifier from both internal and external reflections is provided by one or more opto-isolators, with external connections provided by low-reflectivitv physical-contact type.
216
OPTICAL AMPLIFIERS
8 4.
[III, § 4
Lightwave Transmission System Studies
4.1. DIRECT-DETECTION TRANSMISSION
A variety of direct-detection system experiments have been performed to test the ability of erbium-doped fiber amplifiers (EDFA) to operate at high bit rates, to amplify closely spaced optical channels simultaneously, and to test their use in extending the distance between transmitter and receiver in a long haul system. Bit rates near 20 Gb/s have been amplified, showing the very high bandwidth of the EDFA (Hagimoto, Miyamoto, Kataoka, Kawano and Ohhata [ 19901). Closely spaced wavelength division multiplexing (WDM) has been demonstrated using frequency shift keying (FSK), where up to 100 optical channels, each modulated at 622 Mb/s, have been accommodated, showing the combination of large optical bandwidth (16 nm) and 10 Ghz spaced channels in the EDFA, a property not achievable by semiconductor amplifiers (Inoue, Toba, Sekine, Sugiyama and Nosu [ 19901). The bandwidth of these amplifiers was also challenged by bits rates as high as 100 Gb/s (Izadpanah, Chen, Lin, Saifi, Way, Yi-Yan and Gimlett [ 19901). The ability to use a long chain (10000 to 21 000 km) of optical amplifiers, 300 to 500 in number, spaced by 40 km of dispersion-shifted fiber was also recently demonstrated (Bergano, Aspell, Davidson, Trischitta, Nyman and Kerfoot [ 1991al). This opens the door for the development of undersea systems that will no longer use the traditional electro-optic regenerators but, instead, use a chain of EDFAs. A list of direct-detection experiments using amplifiers is given in table 1.
4.2. COHERENT TRANSMISSION
The original promise of coherent systems was to provide a near 6 dB improvement in receiver sensitivity and closely spaced signal wavelength division multiplexing (WDM). This advantage over direct detection requires a complex receiver design. The basic attraction of the coherent system has been compromised by the application of power and pre-amplifiers based on EDFAs to direct-detection systems. The combination of both coherent detection and EDFAs has, however, been used to demonstrate the longest (nonloop) system with 25 amplifiers over 2200 km at 2.5 Gb/s (Saito, Imai, Sugie, Ohkawa, Ichihashi and Ito [ 19901). Coherent communications may regain interest for future channel selectivity using a tunable heterodyne receiver. A list of coherent transmission experiments using amplifiers is given in table 2.
TABLE1 Direct-detection transmission exDeriments. P
Y
Bit rate (Gb/s)
17.0 11.0 10.0 5.0 2.4
Length (km)
150 200 20 200 459
2.4 I .8 2.488 (FSK) 1.7 (FSK)
710 308 132 177
1.0 (FSK) 0.622 (FSK)
0 0
2.5 (Loop) 5.0 (Loop) 2.5 (Loop)
Number of optical channels
10000 9 000 21 000
2 100
Signal I (Pm)
1.54 1.536 1.536 1.54 1.549 to 1.555 1.536 1.55 1.53 1.500 to 1.513 I .54 1.548 to 1.556 1.56 1.55 1.55
Fiber
10 (pm)
Number of amplifiers
1.55 1.55 1.53 1.3 1.3
2 2 1 3 6
1.3 1.3 1.3 1.3
10 2
-
1 pwr
2
Launched signal power (dBm)
+ 9.8
+ 8.0
0.0
+ 7.7
+ 16.0 + 12.6 + 3.5
1 1
Receiver signal power (dBm) 10-’BER
Author [Ref.*]
- 24.8 - 18.0 - 27.3 - 30.0 - 30 to - 32.7 - 31.7 - 38.9 - 20.2 - 34.0
K. Hagimoto [I] A. Righetti [2] N. Henmi [3] N. Henmi [4] H. Taga [5]
- 40.0
A. Willner [lo] K. Inoue [ 1 I]
- 37.0
N. Edagawa [6] K. Aida [7] E. G. Bryant [8] D. Fishman [9]
v1
--I
1.55 1.55 1.55
324 75 175
+3
No BER
-
- 22.0
0.0
-
K. Malyon [I21 N. Bergano [ 131 N. Bergano [I41
[I] Hagimoto, Miyamoto, Kataoka, Kawano and Ohhata [1990]; [2] Righetti, Fontana, Delrosso, Grasso, Iqbal, Gimlett, Standley, Young and Cheung [ 19901; [3] Henmi, Aoki, Fujita, Suzaki, Sunohara, Mito and Shikada [ 1990al; [4] Henmi, Aoki, Fujita, Suzaki, Sunohara, Mito and Shikada [1990b]; [5] Taga, Yoshida, Edagawa, Yamamoto and Wakabayashi [1990]; [6] Edagawa, Yoshida, Taga, Yamamoto and Wakabayashi [1990]; [7] Aida, Masuda and Takada [1990]; [8] Bryant, Carter, Lewis, Spirit, Widdowson and Wright [1990]; [9] Fishman, Nagel, Cline, Tench, Pleiss, Miller, Coult, Milbrodt, Yeates, Chraplpy, Tkach, Piccirilli, Simpson and Miller [1990]; [lo] Willner, Desurvire, Presby and Edwards [1990]; [ I I] Inoue, Toba, Sekine, Sugiyama and Nosu [1990]; [I21 Malyon, Widdowson, Bryant, Carter, Wright and Stallard [1991]; [13] Bergano, Aspell, Davidson, Trischitta, Nyman and Kerfoot [1991a]; [I41 Bergano, Aspell, Davidson, Trischitta, Nyman and Kerfoot [1991b].
C
P
c1
4
N m
TABLE 2 Coherent transmission experiments. Number of optical channels
Fiber I0
(Pm)
Number of amplifiers
Launched signal power (dBm)
Receiver signal power (dBm)
Author [Ref.*]
1 pwr + Raman 25 2 1 pwr 10
+ 19.1
- 45.8
T. Sugie [I]
r
+ 8.8
- 43
+ 8.2 + 12.2
c
- 39
S. Saito [2] Y. K. Park [3] J. Augie [4] s. RP [51
$
=! 0
2.5 2.488 1.7 (FSK) 0.565 0.560 (FSK)
364 2200 419 219 1028
1
1.55
1.3
1.554 1.54 1.532 1.536
1.554 1.3 1.3 1.3
- 2.0
50.5 - 33.0 -
[I] Sugie, Ohkawa, Imai and Ito [1990]; [2] Saito, Imai, Sugie, Ohkawa, Ichihashi and Ito [1990]; [3] Park, Delavaux, Tench and Cline [1990]; [4] Auge, Clesca, Biotteau, Bousselet, Dursin, Clergeaud, Kretzmeyer, Lemaire, Gautheron, Grandpierre, Leclerc and Gabla [ 19901; [5] Ryu, Edagawa, Yoshida and Wakabayashi [ 19901.
>
5
$ W v)
111.5 41
LIGHTWAVE TRANSMISSION SYSTEM STUDIES
219
4.3. SOLITON TRANSMISSION
Soliton transmission offers the promise of very low pulse distortion over extremely long fiber spans. Soliton transmission experiments began to flourish with the introduction of EDFAs, which are well suited to solve two important requirements for soliton pulse transmission, namely, high peak power generation from the transmitter and maintenance of the high peak power over the transmission fiber span. Early in 1990 the ability to transmit 4 Gb/s solitons over a 136 km length of standard fiber (zero dispersion near 1.3 pm) was demonstrated (Olsson, Andrekson, Becker, Simpson, Tanbun-Ek, Logan, Presby and Wecht [ 19901). Especially high peak power pulses were required here for the large effective area and large fiber dispersion. Several experiments followed, using dispersion-shifted fiber tailored for soliton transmission along with a chain of EDFA’s (Mollenauer, Neubelt, Evangelides, Gordon, Simpson and Cohen [ 1990a1, Nakazawa, Kimura and Suzuki [ 19901, Andrekson, Olsson, Haner, Simpson, Tanbun-Ek, Logan, Coblentz, Presby and Wecht [ 19911). It was originally thought that low-distortion soliton transmission would require a distributed gain, as was demonstrated by the rather inefficient Raman gain mechanism. Recent theory and demonstration showed that a chain of low-gain EDFAs appropriately spaced for the soliton bit rate can perform as well as a distributed gain (Mollenauer, Evangelides and Haus [ 19911). This application eventually breaks down for bit rates on the order of 10 Gb/s and higher where repeater spacings must be impractically close. Here the distributed amplifier will likely be revisited, but this time in the form of a very low concentration distributed erbium amplifier pumped at 1.48 pm (Nakazawa, Kimura and Suzuki [ 19901, Simpson, Shang, Mollenauer, Olsson, Becker, Kranz, Lemaire and Neubelt [ 19911, Takenaka, Okano, Fujita, Odagiri, Sunohara and Mito [ 199 1I). Loop experiments have been used to demonstrate extremely long haul applications for solitons, as also shown with direct detection. Of particular note is the two-optical-channel loop experiment at 2.0 Gb/s, demonstrating the promise of multichannel WDM for undersea system applications and simultaneously showing !he ability of the system to accommodate soliton collisions within the fiber span (Andrekson, Olsson, Simpson, Tanbun-Ek, Logan, Becker and Wecht [ 19901). A list of soliton transmission experiments is given in table 3.
t4
h 0 )
TABLE 3 Soliton transmission experiments. Bit rate (Gb/s)
Number of optical channels
Number of amplifiers
BER
90 70
1 1
4
Yes Yes
19.2
100
1
Raman 2+ Raman
5.0 5.0
250 23
1 1
11 1
32 20
4.0 2.5
2.4 2.0
Length (km)
I36 9.4
12000 Loop 9 000 Loop
1
1
Ralllan 4 1 Distributed Er 480
2
225
1 1
Signal 1 (Pm)
Pulse width (PSI
Author [Ref.*]
(Pm)
Total dispersion (ps/nm)
1.55
1.51 1.53
90 252
15 5.7
P. Andrekson [I] K. Iwatsuki [2]
No
1.56
1.55
35
10
I. W. Marshall [3]
Yes Yes
1.55 1.55
1.50 1.5
550 92
27 16
M. Nakazawa [4] K. Iwatsuki [5]
r
Yes No
1.56 1.53
1.3 1.49
2,298 28.2
75 20
N. A. Olsson [6] M. Nakazawa [4]
arn
No
1.53
1.51
16.560
60
L. F. Mollenauer [7]
Yes
1.56
1.54
2 1,600
60
N. A. Olsson [8]
1.53
Fiber 10
8
i n
>
5r 72 v1
~~~
[I] Andrekson, Olsson, Haner, Simpson,Tanbun-Ek, Logan, Coblentz, Presby and Wecht [ 19911; [2] Iwatsuki, Suzuki, Nishi and Saruwatari [ 19901; [3] Marshall, Spirit, Brown and Blank [1990]; [4] Nakazawa, Suzuki, Yamada and Kimura [1990]; [5] Iwatsuki, Nishi, Saruwatari and Nakagawa [ 19901; [6] Olsson, Andrekson, Simpson, Tanbun-Ek, Logan and Wecht [ 19911; [7] Mollenauer, Neubelt, Evangelides, Gordon, Simpson and Cohen [1990a]; [8] Olsson, Andrekson, Becker, Simpson, Tanbun-Ek, Logan, Presby and Wecht [1990].
-
I I
M
P
TABLE4 Video transmission experiments. Number of optica I channels
System length (km)
Number of optical channels
Modulation
Signal
Fiber
I
20
Number of subscribers
Author [Ref.*]
(pm)
Number of amplifiers
(pm)
K. Kikoshima [l]
1 1
254 480
19 11
AM FDM FM FDM
1.552 1.552
1.3 1.55
6 6
12500 7.76 x 10’
16
9
100
FM 622 Mb/s
1.525 to 1.561
1.3
1
4096
W. Way [2]
622 Mb/s SCM FM NTSC SCM FM HDTV
1.536
1.3
1
8 192
H. E. Tohme [3]
1
8
6 1
15
1 10
3
4
P
f
z
: z v)
2 2
v)
1
6.7
34
AM VSB
1.55
1.3
1
0
42
VSB-AM NTSC
1.53
-
10
29.6
320
PAL (2.26 bs)
1.531 to
1.3
1
7 203
A. M. Hill [6]
12
28
384
PAL (2.2 Gbs)
AI = 1 nm
1.3
2
39530064
A. M. Hill [6]
16
M. Shigematsu [4]
S. Y. Huang [5]
3C 4! Fz
~~~~~~
[ l ] Kikoshima, Yoneda, Suto and Yoshinaga [1990]; [2] Way, Wagner, Choy, Lin, Menendez, Tohme, Yi-Yan, von Lehman, Spicer, Andrejco, Saifi and Lemberg [1990]; [3] Tohme, Lo and Saifi [1990]; [4] Shigematsu, Nakazato, Okita, Tagami and Nawata [1990]; [5] Huang, Cline, Upadhyayula, Tench, Lipson and Simpson [1990]; [6] Hill, Payne, Blyth, Forrester, Atkwright, Wyatt, Massicott, Lobbett, Smith and Hodgkinson [1990]. N
c!
222
OPTICAL AMPLIFIERS
4.4.VlDEO TRANSMISSION
For digital video transmission experiments, high-power EDFAs are particularly useful in boosting the output power of the expensive laser transmitters located at the central office or head end of the loop plani. The amplified signal, which may contain tens of media channels, then can be split among a number of users spaced several kilometers away. By a step-and-repeat process of amplification and splitting, one source can be distributed among literally millions of subscribers (Kikoshima, Yoneda, Suto and Yoshinaga [ 19901). A list of video transmission experiments is given in table 4. Now that erbium-doped fiber amplifiers are readily available, they will soon be introduced into commercial communications traffic. The first applications will probably be for point-to-point long haul (> 100 km) systems. Further refinements undoubtedly will be made to improve both noise figure and output powers of EDFAs as the competition grows. A need remains for an equivalent to the EDFA for 1.3 pm operation. Further development of both Nd3 +-doped silicates and Pr3 -doped fluoride materials for this application will probably be the next front. A substantial increase in the soliton-based communication method has occurred as a result of EDFAs, and this method will be pursued vigorously, most likely for future very high capacity long haul systems. +
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OPTICAL AMPLIFIERS
[Ill
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Amplifiers and Their Applications, Monterey, CA, Technical Digest Series, Vol. 13 (Optical Society of America, Washington, DC) paper TuB3. Way, W. I., S. S. Wagner, M. M. Choy, C. Lin, R.C. Menendez, H. E. Tohme, A. Yi-Yan, A. C. von Lehman, R. E. Spicer, M.J. Andrejco, M. A. Saifi and H. L. Lemberg, 1990, Proc. of OFC90, San Francisco, CA, Technical Digest Series, Vol. 1 (Optical Society of America, Washington, DC) Post-deadline paper PD21. Willner, A. E., E. Desurvire, H. M. Presby and C. A. Edwards, 1990, Proc. Optical Amplifiers and Their Applications, Monterey, CA, Technical Digest Series, Vol. 13 (Optical Society of America, Washington, DC) paper WB5. Zah, C. E., J. S. Osinski, C. Caneau, S. G . Menocal, L. A. Reith, J. Salzman, F. K. Shokoohi and T. P. Lee, 1987, Electron. Lett. 23, 990. Zyskind, J. L., C. R.Giles, E. Desurvire and J. R. Simpson, 1989, J. Lightwave Technol. 1,428.
E. WOLF, PROGRESS IN OPTICS XXXI 0 1993 ELSEVIER SCIENCE PUBLISHERS B.V.
IV
ADAPTIVE MULTILAYER OPTICAL NETWORKS BY
DEMETRI PSALTIS and YONG QIAO California Institute of Technology Department of Electrical Engineering Pasadena. CA 91 125. USA
CONTENTS PAGE
Q 1 . INTRODUCTION
. . . . . . . . . . . . . . . . . . . 229
Q 2. OPTICAL MULTILAYER NETWORK . . . . . . . . . . 231 Q 3 . IMPLEMENTATION OF FULLY ADAPTIVE LEARNING ALGORITHMS . . . . . . . . . . . . . . . . . . . . 243
0 4.
DISCUSSION AND CONCLUSIONS
. . . . . . . . . . 259
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 260 REFERENCES
. . . . . . . . . . . . . . . . . . . . . . .
228
260
5 1.
Introduction
Neural networks are massive parallel computers in which a large number of simple processing elements are densely interconnected (see, e.g., Anderson and Rosenfeld [ 19881). Typically, several hundred connections are present for each neuron. Therefore, the most difficult part in the practical implementation of neural networks is the realization of the interconnections. If the implementation is a simulation on a digital computer, the implementation of a large network becomes very time consuming because each connection needs to be realized by a digital multiplication. Moreover, for a large network, a serious problem arises with the storage and retrieval of the weights of interconnections from a mass memory. Hardware implementations can solve these problems through parallelism and the use of a distributed memory, in which the weights are stored adjacent to the neurons that they connect. In this way the logical function of the hardware directly replicates the neural network, and the network can be simulated extremely fast. Electronics and optics are the two main approaches for neural network implementation. The advantages of the electronic implementation derive from the fact that it is based on a very mature technology; hence chips that reliably simulate complex neural network functions can be easily fabricated. Problems with the electronic implementation arise when the size of the network exceeds what can be accommodated on a single chip. For large networks requiring a large number of chips, the problems associated with interconnecting the chips and sequencing the operations properly make the electronic implementation difficult. The optical implementation, on the other hand, is not based on a well-established technology, making the fabrication of optical systems relatively difficult and expensive, and controlling their characteristics more difficult. Its great advantage derives from the ability to implement optical interconnection in three dimensions. This allows us to have an architecture consisting of planes of “neurons” separated by optical systems that implement the connections between the neurons in the planes. The neural planes consist of nonlinear optoelectronic processing elements, whereas the interconnecting system typically consists of holograms and/or spatial light modulators. The ability to store and process information in three dimensions makes it relatively easy to build large optical networks within a relatively small 229
230
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[IV,§ 1
volume and with small power requirements. Typically, a large electronic chip can accommodate 104-105 weighted connections. This chapter will show that optical networks with 108-10’ connections are readily realizable. Therefore, these optical systems can be functionally equivalent to several thousand chips without the difficulties involved in connecting and synchronizing them. Adaptation is another characteristic feature of neural networks. The functionality of most neural network models is determined by the values of the weights of the interconnections. The weights are set during a learning phase, in which examples of the function that we wish to implement are presented to the network and its performance is monitored. The weights are then modified to improve the performance. A wide variety of weight modification algorithms exists, most of which can be thought of as a variation of the Hebbian law (Hebb [ 19491)
Awi, a oioj ,
(1)
where wi, is the weight connecting the ith and jth neurons, and oiand oj are the activation functions of the two neurons. This simple rule not only forms the basis of most neural network learning algorithms, but also has a direct analogy to holography. Two optical beams, interfering to form a hologram, reinforce the recorded hologram in proportion to the product of their amplitudes. If the hologram implements the interconnection between the ith and jth neurons, the Hebbian law can be realized simply by allowing light emanating from the two neurons to interfere and modify the interconnecting hologram. The recent activities in the area of optical neural networks, began with the optical implementation of a Hopfield network (Psaltis and Farhat [ 19851, Farhat, Psaltis, Prata and Paek [ 1985]), in which every neuron is connected to all the rest. Since then, many research efforts in this area have been undertaken to investigate associative memories (Anderson [ 19861, Soffer, Dunning, Owechko and Marom [ 19861, Yariv and Kwong [ 19861, Athale, Szu and Friedlander [ 19861, Abu-Mostafa and Psaltis [ 19871, Paek and Psaltis [ 19871, Guest and TeKolste [ 19871, Kinser, Caulfield and Shamir [ 19883, Lee, Stoll and Tackitt [ 19891, Paek and von Lehmen [ 1989]), high-order networks (Psaltis, Park and Hong [ 19881, Jang, Shin and Lee [ 19891, Zhang, Robinson and Johnson [ 1991]), methods of learning (Fisher, Lippincott and Lee [ 19871, Farhat [ 19871, Psaltis, Brady and Wagner [ 19881, Ishikawa, Mukohzaka, Toyoda and Suzuki [ 1990]), perceptron networks (Paek, Wullert and Patel [ 19891, Hong, Campbell and Yeh [ 1990]), feedforward multilayer networks (Wagner and Psaltis [ 19871, Psaltis and Qiao [ 1990]), and self-organizing systems (Benkert, Hebler, Jang, Rehman, Saffman and Anderson [ 19911).
IV, § 21
OPTICAL MULTILAYER NETWORK
23 1
Among the various optical neural networks, feedforward multilayer networks represent the most powerful systems, since they are capable of approximating any measurable function to any desired degree of accuracy (Hornik, Stinchcombe and White [ 19891). Optics is particularly suited for the implementation of feedforward multilayer neural networks because of the high parallelism that optics provides and the similarity between a single layer of feedforward structures and classical optical correlators (van der Lugt [ 19641). Most important is the maturing of several critical technologies, such as twodimensional spatial light modulators with light amplification and nonlinear thresholding capabilities (Bleha, Lipton, Wiener-Avnear, Grinberg, Reif, Casasent, Brown and Markevitch [ 19781) and dynamic photorefractive volume holograms (Psaltis, Brady, G u and Lin [ 1990]), which are necessary for the realization of multilayer learning networks. Specifically, such 2 D spatial light modulators can be used to simulate the action of 2 D arrays of neurons, whereas photorefractive volume holograms provide massive, parallel, dynamic interconnections between these neurons. Section 2 describes an experimental two-layer optical neural network recently built at California Institute of Technology. The system was trained for handwritten character recognition, and the experimental results will be presented. Section 3 discusses the implementation of fully adaptive learning algorithms in such a network, and addresses several key issues in its subsections: $ 3.1 describes a local learning algorithm for fully adaptive two-layer networks; $ 3.2 discusses the problem of hologram decay and describes a solution using periodic copying; and $ 3.3 presents a system that provides phase-locked sustainment of photorefractive holograms. Discussion and conclusions follow in $ 4 .
4 2. Optical Multilayer Network This section describes an experiment in which commonly available optical devices are used to implement an adaptive multilayer network (Psaltis and Qiao [ 19901). The system is a two-layer network that was trained based on Kanerva’s model of sparse, distributed memory (SDM) (Kanerva [ 1986]), which was chosen primarily because it is relatively easy to implement. The system uses photorefractive holograms as synaptic interconnections and liquid crystal light valves (LCLVs) to perform nonlinear thresholding. The first layer has random interconnection weights, which map each input pattern into a very large, sparse, distributed internal representation. The second layer
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ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV,8 2
is trained by the sum-of-outer-products rule (Kohonen [ 1984]), which associates internal representations of different classes of characters to different responses of output neurons. The trained network can recognize not only all the training patterns but also a fairly large percentage of test patterns that it has never seen before.
2.1. SYSTEM ARCHITECTURE
We will review the SDM model to point out the necessary characteristics that the optical system must incorporate. A schematic representation of a two-layer network is shown in fig. 1, which consists of an input layer globally interconnected to a hidden layer, which is interconnected through a second weighted network to an output layer. The system is trained so that the desired outputs y ( I ) ,. . , ,f M ) are produced for the respective input patterns d l ) , . . . ,d M ) . Moreover, the output d 2 )of the network should be close to y(’) when the system is presented with the input d o )close to d’).y ( J )and x(’) are real vectors of length N2 and No, respectively, with components restricted to the binary set B = { - 1, + I}. The weights of the connections between the input and hidden layers form an No x N, matrix denoted by “ ( I ) , whereas the weights of the hidden-to-output layer connections form an N, x N2 matrix W2).In general, the interconnection weights of both layers are modifiable, so that the system can be trained to perform a desired pattern transformation from the input space to the output space. In SDM, however, the first layer acts as a fixed-weight preprocessor, encoding each No-bit input into a very large N,-bit internal representation, N , 9 No. The second layer is a trainable sum-of-outerproducts network, which is programmed to recognize the higher-dimensional
Random-Weight
Outer-Product
Matrix W“)
Memory W(’)
Fig. 1. Kanerva’s sparse, distributed memory (SDM) model.
IV, § 21
OPTICAL MULTILAYER NETWORK
233
internal representations. Kanerva’s primary contribution is the specification of the preprocessor, i.e., how to map each No-bit input into a very large N,-bit internal representation in such a way as to permit the network capacity to exceed by far any linear relationship with the input dimension. This is important, since in most applications the dimension of the input, which is approximately equal to the capacity of a single-layer machine, is much smaller than the number of patterns we wish to recognize. The operation performed by each hidden neuron is thresholding. Specifically, if we denote by f o ( U ) the neuron response function with U being the input to the neuron and 8 being the threshold, then f u ( U ) is 1 if U 2 8 and 0 if U < 8. The weight matrix W ( ’ )is populated at random by + 1’s and - 1’s. The input vector to the hidden neurons is given by the matrix-vector product W ( ’ ) d 0 ) , which is thresholded by the function f o to become the output vector dl)= fo(W(l)o(o))of the hidden neurons. With 0 = No - 2r, the N,-bit word d 1 contains ) a 1 in the ith coordinate if and only if d o )is within Hamming distance Y of the ith row of W ( ” . If the parameters r and N , are set correctly, the number of 1’s in the representation d’)will be very small compared with the number of 0’s. Hence, o(I) can be considered as a sparse, distributed representation of do): sparse because there are few l’s, and distributed because several 1’s share in the representation of do). The overall SDM can be regarded as a sum-of-outer-products associative memory operating on the sparse, distributed representation of do).Let g : R N 2 + R N 2be the vector signum function, which takes the sign of each coordinate independently. Then the response of the output neuron is = g(W(2)o(1)), where the synaptic weight matrix W(’) is given by
It was shown by Chou [ 19891 that by allowing N,, the dimension of
hidden layer, to grow exponentially with the input dimension No, the capacity of the SDM can grow exponentially in No, achieving the universal upper bound of any associative memory. This is in sharp contrast to the capacity of a single-layer associative memory, which grows at most linearly with the input dimension. In terms of pattern recognition, large N , implies mapping input vectors into a higher-dimensional space, so that it is much easier to find the appropriate decision boundaries. In this way a linearly nonseparable problem can be converted into a linearly separable one at the hidden layer (Duda and Hart [ 19731). The optical implementation of a two-layer neural network trained by SDM
234
ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV,§ 2
requires both fixed and modifiable interconnection matrices. Dynamic volume holograms recorded in photorefractive crystals are very promising candidates for the implementation of such interconnection matrices because of the threedimensional storage capacity possible within the volume of a crystal (van Heerden [ 1963]), the well-studied dynamic response of photorefractive crystals and the ability to fix photorefractive holograms. Nonlinear effects, such as fanning in photorefractive crystals, generally a nuisance, are helpful for the implementation of the random interconnection matrix in the first layer. Optically addressed spatial light modulators (SLMs) with nonlinear thresholding and amplification functions can be used to simulate neural response. In our experiment, liquid crystal light valves (LCLVs) manufactured by Hughes are used both for providing the input and gain, as well as for use as thresholding devices. The basic architecture for each stage of our two-layer optical network is shown in fig. 2. The neurons are arranged in planes, with the (n - 1)th and nth neural planes being the input and output layers of the nth stage, respectively, where n can be 1 or 2. Neurons in the input plane are connected to the neurons in the output plane by means of holographic gratings recorded in a photorefractive crystal. As shown in fig. 2, the light from the ith neuron at the input, with its field amplitude 0 Y - l ) representing the response of that neuron, is collimated by a Fourier lens and then diffracted by a holographic grating. The diffracted light is focused by a second Fourier lens onto the jth neuron in the output plane. An interconnection between the ith neuron in the input plane and the jth neuron in the output plane is formed by interfering & - I ) , the light emanating
Input
output Neural Plane
Neural Plane
Fourier
Photorefractive
Lens
VolumeHologram
Fourier Lens
Training Neural Plane
Fig. 2. Basic architecture for an optical multilayer neural network.
IV, I 21
OPTICAL MULTILAYER NETWORK
235
from the ith input neuron, with t y - I ) , the light emanating from the jth neuron in the training plane. The image of the jth training neuron coincides with the jth neuron in the output plane. The interference of the input signal and the training signal redistributes photogenerated charges among local trap sites in the crystal to form a modulated space charge field, which, in turn, creates a refractive index grating through the electro-optic effect. The grating vector k,, is equal to k, - k,, where k, ( k , ) is the wave vector of the light that is emitted by the ith (jth) neuron and collimated by a Fourier lens. The strength (i.e., the weight value) of the interconnection is determined by A::), the amplitude of the refractive index modulation of the hologram. For photorefractive crystals, A;:) is proportional to the modulation depth of the interference pattern :
where I,, is the total illuminating intensity. This grating diffracts an input beam with wave vector k , into an output beam with wave vector k , if these two beams satisfy the Bragg condition
k, - k,
=
kjj.
(4)
Under this condition, the amplitude diffraction efficiency w$) of the grating is given by
w!?) .I 1 = sin ( PA?)) ,
(5)
where p is a parameter that depends on crystal properties and recording geometry. The Bragg condition [eq. (4)] is obviously satisfied if k , = kiand k, = k,. In general, however, the solution to eq. (4) is not unique, which means that more than one pair of input and output neurons can be connected by the same grating. During network training, therefore, the modification of a certain grating that connects two neurons can affect the connections between other neurons. In many neural net applications this situation is undesirable. It has been shown (Psaltis, Yu, Gu and Lee 19871, Lee, Gu and Psaltis [ 1989]), however, that by placing the neurons in the input and output planes on appropriate fractal grids, it is possible to ensure that only the ith input neuron andjth output neuron may be coupled by a grating with grating vector kji.In this case the connection between these two neurons can be modified without directly affecting the connections between other neurons. If instead of a single pair of input and training neurons, patterns of neurons are active on the fractal grids of the input
236
ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV, 8 2
and training planes, the recorded hologram is then the outer-product matrix of the input and training patterns. Exposing the hologram with a series of M different pairs of patterns yields the sum of outer products described by eq. (2).
2.2. CHARACTER RECOGNITION APPLICATION
The problem that we selected to test the operation of the system is handwritten character recognition. We created 104 training patterns by drawing characters on a 10 x 10 pixel grid (4 character patterns for each letter of the alphabet A-Z). The optical system architecture is shown in fig. 3. The interconnections between the layers were implemented with Fourier transform holograms recorded in two LiNbO, photorefractive crystals (PR1 and PR2) using an argon-ion laser (A = 514 nm). The input layer consists of a video monitor (VM) and a liquid crystal light valve (LCLVl). There are 100 input units, matching the size of the 10 x 10 grid for the character patterns. Input patterns are presented on VM by a computer, imaged onto the LCLVl by an imaging lens (Ll), and read out by the laser beam on the other side of the LCLVl. The hidden layer, implemented by a second liquid crystal light valve (LCLV2), consists of an array of approximately 300 x 300 neurons. This system has 26 output neurons, represented by 26 pixels in a charge-coupleddevice (CCD) detector array, each responding to one letter of the alphabet. The method used to train the network is a modification of the SDM model.
VM LI
RM
Fig. 3 . Optical two-layer network. VM denotes a video monitor, LCLV a liquid crystal light valve, PR a photorefractive crystal, (P)BS a (polarizing) beam splitter, RM a rotating mirror, L a lens, and S a shutter.
IV,8 21
OPTICAL MULTILAYER NETWORK
23 7
According to this method, the weights of the first layer are selected at random. The weights of the second layer are trained by presenting the training patterns at the input of the network that induce a response at the hidden layer through the random connections. If, for the current input, the desired response for an output neuron is high, the response of the hidden layer is added to the second layer weights leading to that particular output neuron. This simple procedure is repeated for all the patterns in the training set. During the training of the first layer, random dot patterns were presented at the input as training patterns. Each random dot pattern was split into two parts, and both were Fourier transformed by the lenses L2 and L3. These two Fourier-transformed random patterns were used to record a hologram that consisted of gratings of random strength. This process was repeated many times so that a volume hologram with random interconnection weights was recorded. Furthermore, in the crystal used the photorefractive nonlinearity is sufficiently strong that a laser beam passing through the crystal loses much of its power to a broad fan of light resulting from amplification of radiation scattered by imperfections in the crystal (Cronin-Golomb and Yariv [ 19851) and from asymmetric refractive index change due to nonuniformity of the incident beam (Feinberg [ 19821). This phenomenon, called beam fanning, further randomized the recorded interconnections and simultaneously drastically increased the number of hidden neurons to which input neurons are connected. To obtain maximum fanning, the writing beams in the first layer were polarized in the extraordinary direction with respect to the crystal. In our experiment each of the input neurons was connected to about lo5 hidden neurons. Therefore, the resulting weight matrix performs a dimensionalityexpanding random mapping, which is exactly what is needed in the implementation of the SDM model. After the first-layer training was completed, the random interconnection hologram was thermally fixed by heating the crystal to 100 " C for 30 minutes (Amodei and Staebler [ 1971]), and training of the second layer was then started. The goal of the second-layer training is to ensure that when a character pattern is presented at the input of the network, one of the 26 output neurons, with spatial position proportional to the order of that letter in the alphabet, will be switched on. This was achieved by training the second layer using the sum-of-outer-productsrule. During this process, the 104 training patterns were sequentially presented at the network input and randomly mapped into higherdimensional hidden representations, which were amplified and thresholded by LCLV2 and Fourier transformed by lens L5. Their Fourier transform holograms were recorded in association with plane wave references with appro-
238
[IV, § 2
ADAPTIVE MULTILAYER OPTICAL N E T W O R K S
priate propagation directions. The directions of these reference beams are chosen according to the identity of the input patterns. The reference beam transmitted through the crystal was focused by lens L6 onto a different position on the CCD detector array, with the angle of the reference beam determining the position of the focused spot. Therefore, by selecting the proper angle for the reference, the response of the hidden layer was added to the weights of the interconnections leading to the output neuron that is responsible for the current input pattern. The reference beam angle was selected by rotating a mirror, which was mounted on a motorized rotary stage controlled by the computer. The photorefractive crystal was exposed 104 times to record the desired interconnection pattern. To compensate for the hologram decay associated with multiple exposures in photorefractive crystals, an exposure schedule (Psaltis, Brady and Wagner [1988]) was followed during the learning process so that weight adaptation was done linearly, i.e., all the holograms were formed with equal strength. This leads to the implementation of the sum-of-outer-products in eq. (2). The exposure schedule can be derived as follows by considering the dynamics of photorefractive hologram formation and decay. Let A , be the index modulation amplitude of the mth hologram recorded. After a total of A4 exposures, A,=A,[l-exp(-?)]exp(-
m'=m+
I
o>.
where A, is the saturation amplitude of the index modulation recorded in the photorefractive crystal, and it depends on the modulation depth of holographic exposure, the crystal parameters, and the recording geometry; t , is the exposure time for the mth hologram; and z is the characteristic time constant for recording or erasing a hologram in the crystal. If we require A , = A , I for all m, we obtain +
[I
-
exp( -?)]exp(
-+) =
[I - exp(
-+)I,
(7)
The solution to eq. (7), for maximum diffraction efficiencies of the recorded holograms, is given by t , B z and t,
=
(,Yl)?
zln m'19
IV, I21
OPTICAL M U L I I L A Y E R NETWORK
239
which yields A,,,
=
m
A,/M,
=
1 , 2, . . . , M .
(9)
Therefore, the diffraction efficiency of each hologram, which is proportional to the square of the recorded amplitude, decays as M-'. For recording of M holograms the total exposure time is given by M
t=
C m =
t,
= t,
+T
~ M .
I
The crystal we used for the second layer was an 8 mm thick LiNbO,, doped with O.Ol~oFe. Under our experimental condition the time constant T was measured to be 425 s. During the network training, internal representations of the 104 training patterns need to be recorded in the second-layer crystal with roughly equal diffraction efficiencies. The exposure time for each of these holograms except the first can be calculated from eq. (8). For example, t, = 295 s and t,, = 8.6 s. We chose t , to be 25 min, so that t , % T and the first hologram reached the saturation diffraction efficiency. Therefore, with M = 104, the total exposure time is t = 58 min. Another important issue is the finite angular bandwidth of volume holograms. If the angular separation between the reference plane waves is too small, the presentation of any character pattern at the input may reconstruct several plane waves so that several output neurons (corresponding to these reference waves) will be turned on. This leads to cross-talk and possible misclassification. The angular separation, however, cannot be too large because of the limitation of optics. To find an appropriate angular separation, we need to examine the angular bandwidth of volume holograms in the crystal, which is given by (Collier, Burckhardt and Lin [ 19711)
Adc x
1 2ncd sin 6,
where 1is the laser wavelength in vacuum, 0, is the angle between the normal of the crystal surface and the propagation direction of the reference beam inside the crystal, and d is the hologram thickness. In our experiment the angle of incidence of the writing beams in the air is 6, = 20°, and the index of refraction of the LiNbO, crystal is n, = 2.20. Therefore, 6, can be solved from n, sin 0,
=
which gives 6,
sin do, =
8.94'. With 1 = 0.514 pm, d
(12) =
8 mm and using eq. (1 l),
240
A D A P T I V E M U L T I L A Y E R OPTICAL N E T W O R K S
[IV, § 2
AU, = 0.0054'. Finally, we can find the angular bandwidth in the air by differentiating eq. (12), which yields Atlo
=
A 8 , n , c o s 8 , / ~ 0 s 8 ~= 0.0125'.
(13)
To make sure that cross-talk due to the finite angular bandwidth is completely suppressed, we chose the angular separation between reference beams to be 0.03'. Therefore, the total angular sweep of the reference beam is 26 x 0.03 ' = 0.78 O , which is reasonable for the motorized rotary stage and at the same time guarantees that the two writing beams overlap in the crystal for all reference beam angles. Once the training is complete, the presentation of any one of the training patterns causes the second hologram to reproduce the reference beam with which it was recorded. This reconstructed beam codes, in the angle of propagation, the identity of the pattern. The final lens in the system focuses the reconstructed beam to an output neuron whose position in the output array is proportional to the angle of the reconstructed beam.
2.3. EXPERIMENTAL RESULTS
A photograph of the experimental system is shown in fig. 4. After training, all the 104 training patterns were tested and recognized correctly by the system. Figure 5 shows three examples of the input patterns, their internal representations, and the responses at the output of the optical system. The input patterns shown in fig. 5 were among those used for training the network. The bright dot in each example indicates the position of the switched-on output neuron. As can be seen, cross-talk was completely suppressed in these cases, mainly due to the drastically expanded dimensionality of hidden representations and the nonlinear thresholding operation of the neurons. We can also observe the differences among hidden representations for different input patterns. To check the generalization property of this trained network, 520 handwritten character patterns (20 patterns from each class) that were not in the training set were presented to the optical network, and the identity of each pattern was determined from the position of the output neuron that had the maximum response. Figure 6 shows some of the testing patterns, and the result is summarized in fig. 7, which gives the number of correct classifications out of 20 tests for each class. It turned out that 3 1 1 out of the 520 testing patterns were correctly classified, giving an average recognition rate of about 60"/, . This
IV,§ 21
OPTICAL MULTILAYER NETWORK
24 1
Fig. 4. Experimental apparatus.
Fig. 5. Examples of the signals at the input (top), hidden (middle), and output (bottom) layers
in the experimental system.
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ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV, I 2
Fig. 6 . Examples of the test patterns.
A B C D E F G H I J K L M N O P Q R
S T U V W X Y Z
Fig. 7. Histogram of the test results.
rate is much better than random guessing (4%), but far below that required for a useful character recognition system. The reason for the relatively poor performance on the test set is the choice of training algorithm used, specifically the fixed first-layer weights and the limited number of training cycles for the second layer. This same system can be used to implement algorithms in which both layers are fully trained in response to the training patterns, which, in computer simulations, give a much better performance. The following section discusses the important issues involved in the optical implementations of such algorithms.
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IMPLEMENTATION O F FULLY ADAPTIVE LEARNING ALGORITHMS
8 3.
Implementation of Fully Adaptive Learning Algorithms
One of the most widely used learning algorithms for training fully adaptative multilayer neural networks is the backward error propagation (BEP) algorithm (Werbos [ 19741, Rumelhart, Hinton and Williams [ 19861). Optical architectures capable of implementing the BEP algorithm were proposed by Wagner and Psaltis [1987] and Psaltis, Brady and Wagner [1988], one of which (Psaltis, Brady and Wagner [ 19881) is shown in fig. 8. It has two layers, but an arbitrary number of layers can be implemented as a straightforward extension. A training pattern d') is placed at the input plane No. The pattern is then interconnected to the intermediate (hidden) layer N, by means of the dynamic volume hologram H I . Simulating the action of an array of hidden neurons, a spatial light modulator (SLM) placed at plane N, performs a soft thresholding operation on the light incident on it to produce d ' ) ,the output of the hidden layer of neurons. Hologram H2 connects N, to the output plane N2, where another SLM performs the final thresholding to produce the response of the network to the particular input pattern. This network output d 2 )is compared with the desired response t , and an error signal S = t - d 2 )is generated at N2. The undiffracted beams from No and N, are recorded on SLMs at TI and T,, respectively. The SLMs at TI, T2 and N2 are then illuminated from the right to read out the stored signals, and the modulated light propagates back toward the left. Let s,,(") be the total input to the jth neuron in plane N,,, and w;:) be the weight of the interconnection between the jth neuron at N,, and the ith neuron at N, - for n = 1, 2. Let the function f [ . ] be the thresholding function that operates on the input to each neuron in the forward path. According to the BEP algorithm, the change of the interconnection matrix stored in H2 is given by (14)
Aw$)a S , f ' [ ~ ~ ~ ) ] o : ' ) ,
NO
J-1
L2
TI
L3
L4
T2
Fig. 8. Optical architecture for backward error propagation (BEP) learning.
244
ADAPTIVE M U L T I L A Y E R OPTICAL N E T W O R K S
[IVY§ 3
where/’[*] is the derivative o f f [ . ] . Each neuron in N , is illuminated from the right by the error signal S,, and the backward transmittance of each neuron is proportional to the derivative of the forward output. As we described previously, the hologram recorded in H , is the outer product of the activity patterns on planes N2 and T,, which means the change made in H , is that described by eq. (14). The change in the interconnection matrix stored in H , is r
1
The error signal applied to N2 produces a diffracted signal at the j t h neuron at N , , which is proportional to C, Sky’[si2) J w(k:). By setting the backward transmittance of the j t h hidden neuron to be proportional to f ’ [ s , ! ’ ) ] , the interconnection matrix in HI is modified as described by eq. (15). Although it is possible to modify the system of fig. 3 to implement the BEP algorithm, several difficult problems, involving both learning methods and hardware technologies, must be solved. In this section we address each of these issues. The overall objective is to build an adaptive optical multimayer network with compact architecture, composed of simple optical devices and allowing an unlimited number of adaptations. The hardware implementation of the BEP algorithm is complicated by the need to realize error backpropagation through the network and the need for bidirectional optical devices with different forward and backward characteristics. To overcome this problem, we describe an anti-Hebbian local learning (ALL) algorithm for two-layer networks (Qiao and Psaltis [ 19911). With this rule, weight update for a certain layer depends only on the input and output of that layer and a global, scalar error signal. We show that this learning procedure still guarantees that the network is trained by error descent. The fact that error signals need not backpropagate through the network makes this local learning rule easy to implement. The implementation of fully adapative learning algorithms typically requires the recording of thousands of holograms on each of the photorefractive crystals in optical multilayer networks. If the exposure schedule described in 0 2.2 is used in the recording of photorefractive holograms to ensure that all exposures result in holograms of the same strength, then the interconnection weights specified by the diffraction efficiencies of the holograms decay with the increase in the number of adaptations. This limits the number of training cycles that can be implemented, and therefore severely narrows the range of problems that can be solved by optical multilayer neural networks. Methods must be found to
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245
provide a nondecaying hologram for arbitrarily long training sequences. In addition, such methods must maintain the phase coherence of the photorefractive holograms so that any interconnection can be controllably enhanced or reduced. Section 3.1 describes the local learning rule for fully adaptive two-layer networks. Section 3.2 discusses the problem of hologram decay and describes a solution using periodic copying. Section 3.3 presents a system that provides phase-locked sustainment of photorefractive holograms.
3.1. ANTI-HEBBIAN LOCAL LEARNING ALGORITHM
We will describe the local learning algorithm with a feedforward two-layer network (fig. 9). The numbers of neurons for the input, first and second layers, are No, N , and N 2 , respectively. The inputs to the neurons of the nth layer are
where $7) is the weight of the interconnection between the j t h neuron in the nth layer and the ith neuron in the previous layer, and o y ) is the output of the ith neuron in the nth layer (01" being that of the ith input neuron). For the ith input neuron, 0:') = x i , where xi is the input signal. The first and second layers of neurons perform a soft thresholding operation on their inputs, forming the outputs
where the f-function is chosen as f [ x ] = tanh [XI. The desired response for the input x i , presented at the input of the network in a certain machine cycle, is given by a target vector y,, which we take to be
Fig. 9. Schematic diagram of a feedforward two-layer neural network.
246
ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV, 8 3
binary { 1, - l}. The logarithmic energy function measuring the network output error is defined as (1 k= I
+ yk)In-
+Yk
1+o p
+ (1 - yk)In- 1 -
k
It reaches its minimal value of zero only when the network output is the same as the desired response. We chose this form of error measure instead of the more commonly used quadratic error function because we found that, for our learning procedure, this energy function gave a better performance. The BEP rule changes the weights by means of gradient descent, i.e.,
where bk = yA- o:*) is the output error signal. The nonlocal nature of the BEP algorithm is due to the X,";, bkkw(k:.) factor in eq. (20), which requires that the output error Sk be propagated backwards through the same weights. We can avoid the need to backpropagate the error by adopting a reward (punish) strategy, where the function implemented by the weights of the first layer is reinforced (suppressed), if the overall response at the final output is correct (incorrect). This idea leads to the following antiHebbian local learning (ALL) algorithm for the first layer:
where y = Cr:, 6,~:~). With this rule the weight update for the first layer depends only on the input and output of that layer and a global, scalar error signal y,which can be easily evaluated at the output stage. In some cases during training, the output of a certain hidden neuron may become so close to + 1 or - I that the factor (1 - 01' ) ' ) in eq. (21) is close to zero. This causes numerical instability. One way to avoid this is to find o:,flx, the hidden neuron output that has the maximum magnitude, and normalize the right-hand side of eq. (21) by the factor (1 - ogii). The learning rule for the second layer can just follow gradient descent, since it is already a local rule. Consider the case of a single output neuron, i.e.. N, = 1. If the sign of the network output is different from that of the desired response, then y < 0. Since
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241
(1 - of"') is always positive, we obtain Aw$)oc - of')oIo), which differs from the Hebbian rule (Hebb [ 19491) in its sign and therefore we call it anti-Hebbian. Intuitively, if a sign error occurs at the output, it can be corrected by flipping the sign of the internal representation. The anti-Hebbian rule implies exactly that in such cases we should train the first layer with the flipped internal representation as its target. Ifthe network output and the desired response have the same sign but different magnitude, y becomes positive and the learning rule for the hidden layer changes to the Hebbian type, which will enhance the internal representation and increase the magnitude of the network output in the right direction. For multiple output neurons, learning in the first layer will be Hebbian if the sign of most output neurons matches the target sign (so that y is positive), and will be anti-Hebbian if the reverse is true. This new local learning rule is obviously n o longer a steepest descent rule. It is, however, still an error descent rule. Using eqs. (20) and (2 l), and assuming that the weights of interconnections between any input neuron and all hidden neurons are updated simultaneously (true in most practical situations), we obtain
which proves our claim. Computer simulations of the ALL algorithm were performed for the problem of recognition of handwritten zip-code digits provided by the US Postal Service. For comparison, the BEP algorithm was also used to solve the same problem. The handwritten zip codes were first segmented into single digits, and then each digit was reduced to fit a 10 x 10 binary pixel grid. A network of 100 input neurons (to match the 10 x 10 pixel grid), 5 hidden neurons and 3 output neurons was selected and trained to perform classification on 3 classes of handwritten digits: 3,6, and 8. Each output neuron responds to only one class. 600 digit patterns, with 200 patterns from each class, were selected. These 600 patterns were partitioned into 300 training samples, 150 validation samples and 150 test samples. The validation samples were used after each learning iteration (i.e., presentation of the whole training set) to calculate the classification error of the network. The network training stops when the classification error of the network on the validation set stops decreasing with further iteration. After the network was trained, the test samples were presented to the network to find its generalization error. For the ALL algorithm, the first layer was trained only in 1 out of 40 iterations. By doing this, we rely more on the steepest descent
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[IV, § 3
training in the second layer, and it improved learning convergence for this particular classification problem and this particular network. For given training, validation and test sets the network was trained four times with different initial conditions for both the ALL and the BEP algorithms. The same step size was used for the two algorithms for the purpose of comparison. The same step size was used for the two algorithms for the purpose of comparison. The same simulations were repeated using different training, validation and test sets obtained from different partitioning of the 600 digit patterns (the numbers of the training, validation, and test samples were still 300, 150 and 150, respectively). Therefore, there was a total of eight runs for each algorithm. For the ALL algorithm the network could converge (meaning that all the training patterns were classified correctly) in seven cases. In only one case the network fell into a local minimum and gave a training error of 1 %. For the BEP algorithm the network was able to converge in all eight cases. The average generalization errors for the ALL and BEP algorithms are 9 and 8 % , respectively. As for the average convergence rate of the two algorithms, it took 581 iterations for the BEP algorithm to converge, and 3665 iterations for the ALL algorithm. Since the amount of computation involved in each learning iteration is different for the two algorithms, however, we should also compare the convergence rate in terms of number of computational steps. It turns out that for this size of network, the number of computational steps in each iteration for BEP is about 2.4 times the number for ALL. So, based on these two comparisons, the conclusion is that ALL is only about 2.6 times slower than BEP. Considering the great hardware implementation advantage offered by the ALL algorithm, this cost seems acceptable. 3.2. WEIGHT DECAY A N D HOLOGRAM COPYING
To solve a practically significant problem, neural-net learning algorithms typically require thousands of iterations (i.e., modifications of synaptic interconnections) (LeCun, Boser, Denker, Henderson, Howard, Hubbard and Jackel [ 19891). In the optical implementation, each iteration requires an additional holographic exposure to be made in the same crystal. Therefore, a very large number of holograms must be superimposed in a learning architecture. The basic problem with writing a large number of photorefractive holograms is that during the exposure of new holograms, previously recorded holograms decay as a result of the redistribution of the charge carriers. By examining the formation and erasure dynamics of photorefractive holograms, an exposure schedule can be found that enables the recording of an arbitrary
IV, § 31
IMPLEMENTATION O F FULLY ADAPTIVE LEARNING ALGORITHMS
249
number of holograms of equal diffraction efficiency in a crystal (Blertekjaer [ 19791, Psaltis, Brady and Wagner [ 19881). According to the exposure schedule (described in 0 2.2), t , >> z and
where t,,, is the exposure time for the mth hologram and z is the characteristic time constant for recording or erasing a hologram in the crystal. In this case the diffraction efficiency of each hologram decays as M - ’. The overall diffraction efficiency of the composite hologram decays as M - ’ if the individual exposures are statistically independent. This rapid decrease of diffraction efficiency (or equivalently, decrease of interconnection weights) with M limits the number of superimposed holograms to several hundred for a reasonable signalto-noise ratio (Mok, Tackitt and Stoll [1991]). Since the number of learning cycles that can be implemented in adaptive optical networks is in general equal to the number of holograms that can be superimposed in the same crystal, this weight decay severely limits the extent to which optical networks can be trained. As a partial solution to this problem, a periodic copying scheme was devised (Brady, Hsu and Psaltis [1990]), with which part of the decrease in the diffraction efficiency of a multiply exposed hologram is recovered by periodic copying between two holographic media. The basic idea is shown in fig. 10, where a series of holograms between a reference plane wave and a set of signal beams is first recorded in a photorefractive crystal. When the diffraction efficiency of the holograms becomes unacceptably low, the recorded holograms are copied into a second holographic medium (e.g., a thermoplastic plate). These holograms are then copied back to the photorefractive crystal with a single exposure using large modulation depth. The result is a rejuvenated composite hologram with overall diffraction efficiency independent of M .
Reference
Signal
Photorefractive Crystal
Fig. 10. System for periodic copying of photorefractive holograms.
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ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV, § 3
A different problem arises, however, if the periodic copying method is to be used in optical neural networks. When the holograms are copied back to the crystal, there is no guarantee that they are in phase with the original holograms. This loss of phase tracking is problematic, since we need to enhance or reduce the holographic interconnections in a controllable manner according to some learning algorithm. The following section describes a method for sustaining multiply exposed photorefractive holograms, in a phase-locked fashion, by using a pair of phaseconjugating mirrors (Qiao, Psaltis, Gu, Hong, Yeh and Neurgaonkar [ 19911). It is shown that a steady state exists where the overall diffraction efficiency is independent of the number of holographic exposures, and the final holograms are exactly in phase with the initial ones.
3.3. PHASE COHERENCE OF THE HOLOGRAPHIC GRATINGS
The system diagram is shown in fig. 11. The primary hologram is complemented by two phase-conjugating mirrors (PCMs), which are photorefractive crystals in the four-wave mixing configuration. They must share the same pair of pump beams so that the phase-conjugate beams retain the same relative phase. The basic idea is to record a primary hologram with external beams, read out this primary hologram with the reference beam oj, and finally copy the hologram that is read out back to the same crystal using the two PCMs. For photorefractive holograms produced only by diffusion, there is a phase shift of K between the interference pattern and the corresponding hologram. When the reference beam 0, is on, and if the crystal axis is oriented properly, the interference pattern formed by the reference beam oj and the diffracted beam ii will create a hologram that is exactly in phase with the original hologram
PCM 1 Amplitude Reflectivity r
PCM 2 Amplitude Reflectivity r2
Fig. 1 I . Schematic diagram for the hologram-sustaining system with a single reference beam.
IMPLEMENTATION O F FULLY ADAPTIVE LEARNING ALGORITHMS
25 1
(Staebler and Amodei [ 19721). When these two beams are phase-conjugated (to produce the beams oJ! and t,!), the hologram that the phase-conjugate beams create is exactly in phase with the original hologram, and therefore the latter becomes enhanced and sustained.
3.3.1. Temporal response derivation We assume that the hologram is recorded with a plane wave reference, the angle of which is selected from one of several possible positions. An arbitrary signal beam can also be decomposed into a set of plane wave components. The hologram can then be described as a superposition of gratings, each being the result of the interference between thejth reference beam and the ith plane wave component of the signal beam. Let E j j , , denote the amplitude of the spacecharge field recorded in the photorefractive crystal that corresponds to the (4)-th grating. The dynamic equation describing the formation and decay of photorefractive holograms is given by (Kukhtarev, Markov, Odulov, Soskin and Vinetskii [ 19791) dEij. I T~ - - E j , + m , eJviJE, , (24) dt
,
~
where z, is the characteristic time constant, and qij is the phase difference between the signal and reference beams. The modulation depth of the interference pattern (mij) is given by mij = 2t,! oj' 11, ,
where I, is the total illuminating intensity. We ignore two-wave mixing effects in the primary hologram. For holographic formation by diffusion only, the parameters in eq. (24) are given by z,
E,
= 7;11, =
9
- j lEsl,
(26) (27)
where z; and I E, I are real parameters depending on the crystal properties and the recording geometry. The fact that these parameters are real implies that the phase of the recorded grating will not change if the phases of the recording beams remain constant. Since, according to the previous discussion, the phase of the copied hologram is locked, we can ignore the complex nature of eq. (24) and work with the magnitude of E,, which is described by
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[IV,§ 3
ADAPTIVE MULTILAYER OPTICAL NETWORKS
The amplitude diffraction efficiency of the (ij)-th grating is denoted by wij, and it is related to the space-charge field by wij= ~ i n ( P l E i j , l l ) ,
(29)
where P depends on the effective electro-optic coefficient of the crystal, the hologram thickness and the recording wavelength. If we define
and c =
28 I &
(31)
9
then a set of simplified equations is obtained:
(33)
w, = sin(yij),
where we have used eqs. ( 2 5 ) and (26). We first consider the case of a single reference beam with N gratings recorded in the crystal. With the reference beam on (see fig. 1 l), the dynamics of the PCM system are described by eqs. (32) and (33), with tl! =
Ar, wij,
0; =
Ar2J1
(34) N
-
1 w:,,
(35)
k= 1
N
N
I,
= A’
+ A2r: 1 w:;
t A’(
1-
k= I
2
w:i) r:
k= I
A is the real amplitude of the reference beam, and r , and r, are the amplitude reflectivities of the two PCMs. Substituting eqs. (34)-(36) into eq. (32), we obtain dYi/ -
dt
@ z,!
{ [. + -
(p2
- 1)
:
w;/]
k= I
,Vl/
t cpw,,
JX}, (37)
I =I
where a = 1 + l/r: and p = r,/r2. In deriving eq. (37), we have assumed that c, T:, and r , are all independent of the grating index i. This assumption is valid if the spatial bandwidth of the signal beams is small.
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I M P L E M E N T A T I O N OF FULLY A D A P T I V E L E A R N I N G A L G O R I T H M S
The steady state of the system is obtained by setting dy,/dt y . . = cpwv ”
a
=
253
0 in eq. (37):
JmJ
+ ( p 2 - 1) x,N=1 w:, .
The steady-state diffraction efficiency w$)’ can be solved from eqs. (33) and (38). Assuming low diffraction efficiencies, a sufficient condition for nonzero steady state is p > max { a / c , l} .
(39)
It can be shown, using straightforward perturbation analysis, that the steady state is stable under this condition. For example, when c = 0.2, we require p > 50 > 5. For typical photorefractive crystals, c = 0.2-10. In the case of small c, the steady-state overall diffraction efficiency satisfies the condition N
C
4 1 (to be justified later),
k= I
which implies that
The latter is actually the undepleted reference approximation. With these approximations, eq. (38) can be solved explicitly, and it yields
With the approximation wii x yv, eq. (37) also shows that all the gratings rise or decay with the same time constant, which implies that w$): w(ksI) = w$‘): w$) for any i, k, with 1 < i, k < N . Here, w:;)represents the initial value of wi,.Thus eq. (40) can be rewritten as
This property of grating strength normalization is very useful in many applications, including neural network implementation, since it effectively prevents interconnection weights from either decaying or saturating. In other words, it provides a method for calculating very long averages of exposures with a nondecaying composite hologram. If the primary hologram is formed through a sequence of M exposures using the exposure schedule described previously,
254
ADAPTIVE MULTILAYER OPTICAL NETWORKS
TABLE 1 Typical values for system parameters and steady-state diffraction efficiency in the case of a single reference beam.
0.2 0.5 I .o
-
20
8 3
2.0 2.1
1.9
0.76 5 13
then w:;)’ A C 2 for all (0)pairs. Therefore, we can see from eq. (41) that the steady-state diffraction efficiency wc)’is independent of M . For large values of c, the preceding approximations do not hold, and we have to solve eqs. (33) and (38) for the exact steady states. Table 1 shows some of the typical parameters and the corresponding steady-state values calculated numerically from eq. (38). For small values of c, the assumption Z r = , WE)’G 1 is justified. The approximation wIJ x yll, used to derive the steady-state solution given by eq. (4 I), takes into account only the first term in the expansion of the sine function in eq. (33). This approximation, however, is insufficient when the overall diffraction efficiency starts approaching its steady-state value q, . When that happens, dy,,/dr x 0 and the higher-order terms of the sine expansion cannot be ignored in the dynamic equation (37). These higher-order terms, according to our model, have an equalizing effect that will lead the system to a final steady-state where all the holographic gratings reach the same diffraction efficiency. This same steady-state diffraction efficiency can be found by solving eqs. (33) and (38). For large p and low diffraction efficiencies, this equalizing process occurs much more slowly than the grating normalization process we discussed earlier, so that in practice we usually observe the latter case as a quasi-steady state. Figure 12 shows a numerical simulation ofeq. (37), in which the priniary hologram consists of two gratings with different initial amplitudes. Initially the ratio of the strengths of the two gratings remains constant until a quasi-steady state is reached. Afterwards, the strengths of the two gratings slowly converge to a common final steady-state value. This simulation was performed with the following parameters: p = 8, a = 2, c = 0.37, yl,(0) = 0.08, y 2 / ( 0 )= 0.05 and r2A2/q! = 1.
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255
0.10
g
0.08
0.06
0.04
'
0
I
I
10
20
360
700
t
Fig. 12. Numerical simulation ofeq. (37), in which the primary hologram consists of two gratings with different initial amplitudes. The simulation was performed with the following parameters: p = 8, a = 2, c = 0.37, y,,(O) = 0.08, yz,(0) = 0.05, and r i A Z / r ;= 1.
3.3.2. Experimental demonstration The experimental system consists of an SBN crystal ( 1 mni thickness) as the primary hologram and a BaTiO, crystal for the PCMs (fig. 13). The BaTiO, crystal, with the c-axis oriented 45 O from its face, provides phase-conjugation for both the reference and diffracted signal beams. This is done by directing these beams to two separate regions of the crystal illuminated by the same pair of counter-propagating pump beams, so that the crystal acts effectively as two separate phase-locked PCMs. Two experiments have been done with this system. The first examines the dynamics of a single grating recorded in this system. Both the signal and reference beams are plane waves in this case. The phase-conjugate reflectivity of the BaTiO, PCM for the reference beam was set to be 1. Therefore, r2 = 1 and a = 2. The c coefficient for the SBN was measured to be 0.37. Figure 14 shows three experimental curves measuring the changes in diffraction efficiency with time. When the phase-conjugate reflectivity of the BaTiO, PCM was 44 for the signal beam (therefore p = r , = 6.63), condition (39) was satisfied and the system reached an overall steady-state diffraction efficiency of about 0.8452, independent of the initial condition. For comparison, the theoretical value for the steady-state diffraction efficiency is q, = 1.06% from eq. (40). The discrepancy between the experimental and theoretical results may be due to the wave-mixing effect in the SBN and the dependence of phase-conjugate reflectivity on the probe intensity. When p was
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ADAPTIVE MLILTILAYER OPTICAL N E T W O R K S
,
,T(BiEz
_____
signal
I
_ __ __ __ __ __ _ __ _ _ ----Reference crystal
Fig. 13. Experimental phase-locked hologram-sustaining system.
0.03
2
5
z
.d
0.02
c
2
3
2
0.01
c L.l
2
-0
0.00 0
100
200
300
400
500
time (seconds)
Fig. 14. Experimental results for the hologram-sustaining system. In all experiments r2 = I . For = 6.63, the same steady-state diffraction efficiency is reached when we start with either low ( 0 ) or high ( A ) diffraction efficiency. For p = 2.35, the diffraction efficiency decays to zero (0).
p
reduced to 2.35, however, the system did not have a nonzero steady state and thus the grating decayed to zero, as predicted. The second experiment investigates the steady-state behavior of multiple gratings recorded in the system. This was done by recording the Fourier transform hologram of an image, which consists of multiple gratings resulting from different spatial frequency components of the Fourier transform. Figure 15a shows the reconstruction of the image from the SBN when it was first recorded, and fig. 15b shows the steady-state hologram. Although some distortion occurs in the steady-state hologram, it can be seen that the grating
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251
Fig. 15. (a) Reconstruction of the Fourier transform hologram of an image initially recorded in the SBN crystal. (b) The steady-state response of the hologram stored in the SBN with the initial condition being a hologram of the image shown in (a).
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[IV,§ 3
normalization effect is dominant, since all spatial frequency components are roughly proportional to their initial conditions. 3.3.3. Multiple reference beams
To store information in a volume hologram, multiple reference beams are required. For the multiple reference beam case, assuming N plane wave components in the signal beam and R reference beams, there are two possible ways of sustaining them. One way is to bring in the reference beams cyclically (fig. 16), and the other is to use mutually incoherent reference beams that are on simultaneously. Both schemes lead to the same steady state. Specifically, assuming small steady-state diffraction efficiency, both are described by the same dynamic equation dw,, - r:A’
dt
z,!
R
{cp-Ra-(p’-
1)
N
1 1 w:,}wij,
I=I k=l
with the steady state given by
Similarly to eq. (4l), the grating strength normalization relationship also can be found for the multiple reference beam case
From eq. (43), the number of reference beams that can be supported is
-. --
---.&
PCM 1 Amplitude Reflectivity r l
-
.... . /
Primary Hologram
0
t’i
0;
PCM 2 Amplitude Reflectivity r 2
Fig. 16. Schematic diagram for the hologram-sustaining system with multiple reference beams.
IV,§ 41
259
DISCUSSION A N D CONCLUSIONS
TABLE 2 Typical values for system parameters and number of reference beams that can be supported (rz = 1). C
rl
R
0.5 10 10
20 20 100
5 100 500
bounded by
The right-hand side of eq. (45) reaches its maximum when r, R
=
I, in which case (46)
Some of the typical system parameters and the corresponding number of reference beams that can be supported are shown in table 2.
6 4.
Discussion and Conclusions
We described the experimental demonstration of a large two-layer network and its application to handwritten character recognition. The results are encouraging in some ways, but they also indicate that we need to make progress in several key areas before such networks can become practically useful. The optical network we described has approximately lo7 synapses or weights, and the response time of the network is approximately 10 ps, which corresponds to 10’ analog multiplications per second. If the size of the input image is increased to the full resolution of the input spatial light modulator (approximately lo5 pixels), the rate increases to 10l2multiplications per second. Finally, if a ferroelectric liquid crystal SLM is used at the input and hidden layers, the response time can be improved to approximately 10 ps (Moddel, Johnson, Li, Rice, Pagano-Stauffer and Handschy [ 1989]), which yields the very impressive rate of l O I 5 multiplications per second. This processing speed is achievable with currently available optical components, and it would be extremely difficult to duplicate with electronics. The issue, therefore, is not whether optics can in practice outperform electronics in terms of computational speed,
260
ADAPTIVE MULTILAYER OPTICAL NETWORKS
[IV
but rather, whether the high speed of optics can be put to practical use. The primary issue is whether these large optical networks can be trained effectively to solve useful and practical problems. Algorithmic and device-related problems both need to be addressed. The hologram copying method we described is a promising solution for the device dynamic range problems. As for the training of multilayer networks, it is well known that large networks require very large training sets, but it is not known how the training time scales with the size of the network and/or the training set. Understanding such algorithmic issues about the training of large optical networks is the major remaining challenge before these systems can have a practical impact.
Acknowledgements This work was supported by the Defense Advanced Research Projects Agency (DARPA) and the Air Force Office of Scientific Research.
References Abu-Mostafa, Y. S., and D. Psaltis, 1987, Sci. Am. 256, 88. Amodei, J. J., and D. L. Staebler, 1971, Appl. Phys. Lett. 18, 540. Anderson, D. Z., 1986, Opt. Lett. 11, 56. Anderson, J., and E. Rosenfeld, eds, 1988, Neurocomputing (MIT Press, Cambridge, MA). Athale, R. A., H. H. Szu and C. B. Friedlander, 1986, Opt. Lett. 11, 482. Benkert, C., V. Hebler, J . 4 . Jang, S . Rehman, M. Saffman and D.Z. Anderson, 1991, in: Technical Digest on Photorefractive Materials, Effects, and Devices, Vol. 14 (Opt. SOC.Am., Washington, DC) pp. 372-375. Bleha, W. P., L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. Casasent, H. B. Brown and B. V. Markevitch, 1978, Opt. Eng. 17, 371. Blatekjaer, K., 1979, Appl. Opt. 18, 57. Brady, D., K. Hsu and D. Psaltis, 1990, Opt. Lett. 15, 817. Chou, P. A., 1989, IEEE Trans. Inf. Theory IT-35, 281. Collier, R., C. B. Burckhardt and L. H. Lin, 1971, Optical Holography (Academic Press, New York). Cronin-Golomb, M., and A. Yariv, 1985, J. Appl. Phys. 57, 4906. Duda, R., and P. Hart, 1973, Pattern Classification and Scenes Analysis (Wiley, New York). Farhat, N. H., 1987, Appl. Opt. 26, 5093. Farhat, N. H., D. Psaltis, A. Prata and E. G. Paek, 1985, Appl. Opt. 24, 1469. Feinberg, J., 1982, J. Opt. SOC.Am. 72, 46. Fisher, A. D., W. L. Lippincott and J. N. Lee, 1987, Appl. Opt. 26, 5039. Guest, C. C., and R. TeKolste, 1987, Appl. Opt. 26, 5055. Hebb, D., 1949, The Organization of Behavior (Wiley, New York). Hong, J., S. Campbell and P. Yeh, 1990, Appl. Opt. 29, 3019. Hornik, K., M.Stinchcombe and H. White, 1989, Neural Networks 2, 359.
IVI
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Ishikawa, M., N. Mukohzaka, H. Toyoda and Y. Suzuki, 1990, Appl. Opt. 29, 289. Jang, J.-S., S.-Y. Shin and S.-Y. Lee, 1989, Opt. Lett. 14, 838. Kanerva, P., 1986, in: Neural Networks for Computing, ed. J. S. Denker (American Institute of Physics, New York) pp. 247-258. Kinser, J. M., H. J. Caulfield and J . Shamir, 1988, Appl. Opt. 27, 3442. Kohonen, T., 1984, Self-organization and Associative Memory (Springer, Berlin). Kukhtarev, N. V., V. B. Markov, S. G. Odulov, M. S. Soskin and V. L. Vinetskii, 1979, Ferroelectrics 22, 949. LeCun,Y., B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard and L. D. Jackel, 1989, Neural Computation 1, 54 I. Lee, H., X.-G. Gu and D. Psaltis, 1989, J. Appl. Phys. 65, 2191. Lee, L. S., H. M. Stoll and M. C. Tackitt, 1989, Opt. Lett. 14, 162. Moddel, G., K. M. Johnson, W. Li, R. A. Rice, L. A. Pagano-Stauffer and M. A. Handschy, 1989, Appl. Phys. Lett. 55, 537. Mok, F. H., M. C . Tackitt and H. M. Stoll, 1991, Opt. Lett. 16, 605. Paek, E. G . , and D. Psaltis, 1987, Opt. Eng. 26, 428. Paek, E. G., and A. von Lehmen, 1989, Opt. Lett. 14, 205. Paek, E. G . . J. R. Wullert 11 and J. S. Patel, 1989, Opt. Lett. 14, 1303. Psaltis, D., and N. H. Farhat, 1985, Opt. Lett. 10, 98. Psaltis, D., and Y. Qiao, 1990, Opt. Photonics News 1(12), 17. Psaltis, D., D. Brady and K. Wagner, 1988, Appl. Opt. 27, 1752. Psaltis, D., D. Brady, X.-G. Gu and S. Lin, 1990, Nature 343, 325. Psaltis, D., C. H. Park and J. Hong, 1988, Neural Networks I , 149. Psaltis, D., J . Yu, X.-G. Gu and H. Lee, 1987, in: Technical Digest ofTopical Meeting on Optical Computing (Optical Society of America, Washington, DC) pp. 129-132. Qiao, Y., and D. Psaltis, 1991, Proc. Int. Joint Con[ on Neural Networks, Vol. I (IEEE, New York) p. 457. Qiao, Y., D. Psaltis, C. Gu, J. Hong, P. Yeh and R. R. Neurgaonkar, 1991, J. Appl. Phys. 70,4646. Rumelhart, D. E., G . E. Hinton and R. J . Williams, 1986, in: Parallel Distributed Processing, Vol. 1, eds D. E. Rumelhart and J. L. McClelland (MIT Press, Cambridge, MA) pp. 318-362. Soffer, B. H., G . J. Dunning, Y. Owechko and E. Marom, 1986, Opt. Lett. 11, 118. Staebler, D. L., and J. J. Arnodei, 1972, J. Appl. Phys. 43, 1042. van der Lugt, A. B., 1964, IEEE Trans. Inf. Theory IT-10, 139. van Heerden, P. J., 1963, Appl. Opt. 2, 393. Wagner, K., and D. Psaltis, 1987, Appl. Opt. 26, 5061. Werbos, P., 1974, Ph.D. Thesis, Harvard University. Yariv, A., and S. K. Kwong, 1986, Opt. Lett. 11, 186. Zhang, L., M. G. Robinson and K. M. Johnson, 1991, Opt. Lett. 16, 45.
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E. WOLF, PROGRESS IN OPTICS XXXI 0 1993 ELSEVIER SCIENCE PUBLISHERS B.V.
OPTICAL ATOMS BY
R. J . C. SPREEUW and J . P. WOERDMAN Huygens Laboratory, Universiry of Leiden, P.O. Box 9504, 2300 RA Leiden, The Netherlands
263
CONTENTS PAGE
$ 1 . INTRODUCTION
. . . . . . . . . . . . . . . . . . . 265
$ 2 . TWO-LEVEL SYSTEMS WITH CONSTANT COUPLING
. 266
$ 3. OPTICAL BAND STRUCTURE . . . . . . . . . . . . . 279 $ 4 . FOUR-LEVEL SYSTEMS . . . . . . . . . . . . . . . . 280
8 5 . DYNAMICAL BEHAVIOR O F T H E OPTICAL ATOM . . . 283 $ 6. T H E DRIVEN OPTICAL RING RESONATOR AS A MODEL FOR MICROSCOPIC SYSTEMS . . . . . . . . 307 $ 7 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . .
317
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 317 REFERENCES
. . . . . . . . . . . . . . . . . . . . . . .
264
318
8
1. Introduction
The advent of lasers as tunable sources of coherent radiation has led to much interest in resonant and near-resonant optical phenomena in atoms. In this context the atom is often treated as a two-level system, neglecting all levels not coupled by the radiation field. Actually, the concept of a fictitious two-level atom is much older than that of the laser, dating back (at least) as far as Einstein’s work on the interaction of radiation and matter (Einstein [ 19171). An extension to three- and four-level atoms is obvious if one considers resonant phenomena dealing with two radiation fields. For reviews of optical resonance of atoms we refer to the classic book by Allen and Eberly [1987] and to the work of Yo0 and Eberly [1985]. For completeness we note that the concept of a two-level system is, of course, also at the basis of nuclear- and electron-spin resonance. In fact, many typical features of two-level systems, including those to be discussed in this chapter, were observed for the first time in spin-resonance experiments (Abragam [ 19611). In that case the interaction is based on magnetic-dipole coupling, whereas in the case of quantum optics and laser physics the interaction is of the electric-dipole type. Clearly, the physics of atomic optical resonance has an intrinsic quantum nature; nevertheless, as stated by Allen and Eberly [ 19871, the equations show much similarity to those of classical physics encountered in, e.g., in Lorentz-type dielectric theory or in electrical-engineering circuit analysis. This chapter will review recent work which shows that, in this category, a particularly attractive analogous system is formed by two (or four) coupled optical modes, a so-called “optical atom”. The attraction and convenience of the system is its macroscopic nature, which allows precise control of all parameters over ranges that are sometimes not accessible for the case of real atoms. Usually, we will consider the classical limit of the coupled optical modes; quantum effects come into play, however, if the mode intensities become weak (corresponding to small photon numbers). In fact, the transition between classical physics and quantum physics permeates much of the discussion that follows, in a way that we hope will be illuminating. Conceptually, our study of “atomic” behavior in optics is complementary to recent studies of “optical” 265
266
OPTICAL ATOMS
[V,§ 2
behavior of atoms (Carnal and Mlynek [ 19911, Keith, Ekstrom, Turchette and Pritchard [ 19911). This chapter is mainly a review of our work; it presents in a unified way the essence of the results reported by Spreeuw, Woerdman and Lenstra [ 19881, Spreeuw, Eliel and Woerdman [ 19901, Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901, Woerdman and Spreeuw [ 19901, Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901, Centeno Neelen, Spreeuw, Eliel and Woerdman [ 19911, Spreeuw and Woerdman [ 1991a,b], Spreeuw, Beijersbergen and Woerdman [ 19921 and Beijersbergen, Spreeuw, Allen and Woerdman [1992]. We have broadened the scope by connecting seemingly unrelated aspects of optics, atomic physics and even solid-state physics, based on the maxim that the same equations lead, in a sense, to the same physics. In Q 2 we establish the analogy between a quantum two-level atom and the classical limit of two coupled optical modes (“optical atom”). These notions are extended in Q 3 to optical band structure and in Q 4 to four-level systems. In Q 5 we return to two-level systems and study their dynamical behavior if one of the system parameters is modulated. This forms the motivation for Q 6, where some future experimental possibilities for the optical atom as a model for microscopic systems are explored. Conclusions are given in Q 7.
6 2. Two-Level Systems With Constant Coupling The essential idea of the optical (two-level) atom is the representation of the two levels by two distinct modes in an optical cavity, distinguishable, for example, by their polarization or direction of propagation (in a ring cavity). By a two-level system we shall mean here any system, either classical or quantum mechanical, that can be described by an anticrossing diagram or Landau- Zener diagram (fig. la). This diagram is at the basis of all the discussions about two-level systems, contributing to the understanding of seemingly diverse phenomena like Landau-Zener transitions, Rabi oscillations, and multiphoton excitations. We first discuss theoptical implementation of such a Landau-Zener diagram, and then generalize this so that we can compare other well-known two-level systems, classical and quantum mechanical.
TWO-LEVEL SYSTEMS WITH CONSTANT COUPLING
261
Fig. 1. (a) Avoided crossing governed by tuning parameter S and coupling parameter W ;(b) experimental realization of avoided crossing using the implementation of fig. 3c. (From Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901.)
2.1. AVOIDED OPTICAL CROSSINGS
For our optical implementation we consider a single longitudinal mode of an optical ring cavity. This mode has a two-fold propagation degeneracy [clockwise (cw), counterclockwise (ccw)], as well as a two-fold polarization degeneracy (x, y or o + , o- ; see fig. 2). By lifting either degeneracy, we obtain two modes with separate frequencies. We restrict ourselves to the regime where the frequency splitting of the longitudinal mode is much smaller than the free spectral range of the ring cavity, so that a two-mode description is valid. The propagation degeneracy can be removed by introducing a nonreciprocity into the cavity, for example, using the Sagnac effect by rotating the ring cavity (fig. 3a). The nonreciprocity (Sagnac effect) produces a frequency difference for the cw and ccw modes, which we shall denote 2s. By increasing this tuning parameter S , we thus pull apart the cw and ccw mode frequencies and, as a function of S , the mode frequencies yield a diagram of two straight lines that cross at S = 0 (dashed lines in fig. la). An avoided crossing is now obtained by coupling the cw and cw modes, by means of backscattering, indicated
Fig. 2. Optical ring resonator showing four degrees of freedom for the light waves, i.e., two propagation degrees of freedom (cw, ccw) and two polarization degrees of freedom ( E x ,E,,).
268
OPTICAL ATOMS
EON 2
EOY 1
E,
Fig. 3. Four implementations of a two-level system in an optical ring resonator. (a) The propagation modes are tuned by the Sagnac effect and coupled by a weak reflector R. (b) As (a), but now the Sagnac effect is simulated by sandwiching a Faraday rotator FAR between two mutually compensating quarter-wave plates (QW). (c) The polarization modes Ex and E, are tuned by electro-optic modulator EOMl and coupled by modulator EOM2. (d) As (c), but now in a standing-wave cavity. (Figures 3b and c from Spreeuw, Van Druten, Beijersbergen, EIiel and Woerdman [1990].)
symbolically by the reflecting element R in fig. 3a. The minimum frequency separation 2 W is given by the backscattering rate, i.e., the coupling strength. For a reflecting element with amplitude-reflection coefficient r (4l), the backscattering rate W is simply rc/L, where L / c is the roundtrip time in the cavity. We call W the coupling parameter. The mode splitting 2 W was observed in a nonrotating fiber ring resonator (Spreeuw, Woerdman and Lenstra [ 19881). In that experiment backscattering was supplied by Fresnel reflection from the air-glass interfaces of two aligned fiber ends, separated by a variable distance d (fig. 4a). By changing this distance piezoelectrically, the interference of the reflections from the two interfaces could be changed from destructive to constructive, thus allowing control of the net reflectivity: r cc sin 2 n d / l . The resulting change in mode splitting (for S = 0) is shown in fig. 4b. Instead of rotating the ring cavity as in fig. 3a, an alternative tuning mechanism is obtained if one constructs a composite nonreciprocal element by sandwiching a Faraday rotator between two mutually compensating quarterwave plates (fig. 3b). It is easily verified that for y-polarized light, the optical path length in the cw direction differs from that in the ccw direction by an amount proportional to the Faraday rotation angle (and thus to the magnetic field strength B). Such a composite nonreciprocal device was used to simulate a rotating ring cavity in the experiments described by Spreeuw, Woerdman
v. § 21
TWO-LEVEL SYSTEMS WITH CONSTANT COUPLING
42
44
269
46
piezo voltage (V)
Fig. 4. (a) Implementation of the weak reflector R in figs. 3a,b by aligning two facets of a single-mode fiber. The Fresnel reflections from the two facets give interfering contributions to the reflectivity, depending on the facet spacing d. (b) Mode splitting induced by this reflector, measured in the ring cavity of fig. 3b, as a function of the piezo-voltage that controls the spacing d. The splitting is expressed as a fraction of the free spectral range of the cavity. (Figure 4b from Spreeuw, Woerdman and Lenstra [1988].)
and Lenstra [ 19881, Spreeuw, Eliel and Woerdman [ 19901 and Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901. We note here that this simulation of the Sagnac effect is not complete, because for x-polarized light the opposite rotation rate is simulated (i.e., the roles of cw and ccw are interchanged (Spreeuw, Beijersbergen and Woerdman [ 19921)). In experiments one should therefore select either x- or y-polarization. An alternative for tuning and coupling the propagation modes is shown in fig. 3c, where two polarization modes are tuned and coupled using two electrooptic modulators (EOM). The axes of EOM 1 are x and y, so that, using ccw propagating light, it pulls apart the modes I ccw, x ) and I ccw, y ) , with the tuning parameter S proportional to the electric field inside EOM 1. The coupling is produced by EOM2, with axes along (x t y ) and (x - y ) ; the coupling parameter W is proportional to the electric field inside EOM2. The experimentally observed mode structure, revealing an avoided crossing, is shown in fig. lb. Note that for this polarization implementation of a two-level system it is not essential to have a ring cavity; the linear cavity shown in fig. 3d should work just as well. Obviously, in the polarization implementation the role of the two EOMs is interchangeable: we could equally well have chosen EOM2 for tuning and EOMl for coupling. In the following we refer to figs. 3a,b as the propagation implementation and to figs. 3c,d as the polarization implementation of an optical atom.
270
OPTICA I. 4r0M S
2.2. COUPLED MODES AND TWO-LEVEL SYSTEMS
The problem of two coupled classical modes in an optical resonator can be described with equations of motion for the complex mode amplitudes. These can be derived directly from Maxwell’s equations, and are well known in the context of ring-laser theory (Menegozzi and Lamb [ 19731, Chyba [ 19891). We take w - S and o + S as the frequencies of the unperturbed modes, where o i s the optical carrier frequency. This carrier will not be included in the complex mode amplitudes a and b, so that they vary slowly compared to w. Taking W for the coupling rate between the modes, the equation of motion for the mode amplitudes is i
”)(:).
w -s
dt b
The reason for appending a suffix S to the “Hamiltonian” matrix H will become clear in 3 2.3. For convenience we shall always take the coupling rate Was real (essentially by choosing a suitable reference phase.) More generally, the Hermitian matrix H , has two complex conjugate off-diagonal elements. An alternative derivation of eq. (1) is obtained if we start from the Hamiltonian of two coupled quantized modes
H
= hSata
-
hSbtb
+ h W(atb + b t a ) ,
(2)
where a ( a t ) and b (bt) are now the operators destroying (creating) a photon in the respective modes. They have the well-known commutation relations with [ u, lit] = [ b, bt ] = 1 and a vanishing commutator for every other combination. The first two terms on the right-hand side of eq. (2) describe the uncoupled modes, the optical carrier frequency again having been omitted. The third term describes the coupling processes in which a photon is removed from one mode and created in the other mode. The evolution of the optical fields is described by the Heisenberg equations of motion iu
=
1 -
h
[ a , H ] = Sa
+
Wb,
* 1 i b = - [b,H]= W a - S b , h
(3)
(4)
or, in matrix notation, eq. (1). In the classical limit (i.e., if the modes are in a coherent state containing many photons) the mode operators a and b can be treated as c-numbers and are just the complex mode amplitudes.
v, I 21
TWO-LEVEL SYSTEMS W I T H CONSTANT COlJPLlNG
27 I
The analogy with a two-level system is now obvious, since we recognize eq. (1) as the Schrddinger equation for the evolution of a two-component state vector (“spinor”). We thus identify the complex mode amplitudes with the probability amplitudes of the two-level system. The mode intensities 1 a I and I bl are proportional to the populations of the levels. Note that it is the classical limit of the coupled two-mode system that corresponds to the quantum twolevel system. If we plot the eigenvalues of the matrix H , as a function of S , we find the (fig. la). For a problem of two coupled avoided crossing: co* = ,/quantum states one would normally use energy units for the parameters S and W , so that tZ enters the equation of motion. For classical systems the energy is usually not discrete, and it is more natural to use (angular) frequency units. Of course, one could also formulate the quantum problem in frequency units by dividing all energies by A. From here on we shall leave k out ofthe equations. Clearly, eq. (1) applies to any pair of coupled harmonic oscillators. In principle, one could construct, apart from optical atoms, acoustical, electrical, mechanical atoms, etc. ; of course, every variety brings its own technical problems. We hope to make clear in this chapter that the optical variety is particularly convenient from a practical point of view.
2.3. EIGENSTATES
The eigenvectors (stationary states) of H , are cos 8 sin 8 with cot 26’ = S / W and 8 is called the mixing angle (0 < 6 < II).In a classical system a stationary state corresponds to a normal mode of oscillation. The mixing angle has a direct meaning in the experiment, since it controls the character of the normal-mode wave function. As an example, we consider the propagation implementation and probe this cavity by injecting light in the cw direction (fig. 5). The expression of the eigenstates, eq. ( 5 ) , then directly shows that the eigenmodes I + ) and I - ) will be excited by the injected cw wave with relative intensities cos28 and sin2& The depth of the absorption dips in the signal IT is, therefore, proportional to cos2 8 and sin2 8 (fig. 5a,c). Also shown is the signal I,, the intensity leaving the ring in the backward (ccw) direction (fig. 5b,d). Now the excited eigenmodes are coupled out in the ccw direction
212
OPTICAL ATOMS
s-0
S#O
Il a )
I
W
W
Fig. 5. Doublet spectra obtained in a fiber-optic ring resonator as shown. Laser light with frequency o is injected in the cw direction by means of directional coupler DC. Transmitted and reflected intensities I , and I , are measured as a function of the frequency of the injected laser light. The resonance positions (0, ,w - ) represent a vertical section through fig. la. (a, b) Tuning parameters S = 0; (c, d) tuning parameter S # 0; note that the transmission doublet is now asymmetric, whereas the reflection doublet is still symmetric (see text). (After Spreeuw, Woerdman and Lenstra [1988].)
with relative efficiencies sin28 for I + ) and cos28 for I - ) . For the strength Z we must take the product of the efficiencies for of the peaks in the signal , excitation and output coupling, i.e., cos2 8 sin28 for I + ) and sin28 cos28 for I - ). This explains why the doublet in I, is always symmetric. In the limit of small coupling (I S / W l B 1) the mixing angle approaches zero or $ K (depending on the sign of S ) , and the eigenvectors approach their asymptotic form of the uncoupled states. In the implementations of fig. 3 these are the traveling wave modes (cw, ccw) or the x- and y-polarized modes. We define the eigenstates for W = 0 as the S-modes. In the limit that the coupling is strong (I S / Wl 4 1) the stationary states are completely determined by the coupling; the mixing angle is then $ K . We call the eigenstates for S = 0 and W # 0 the W-states. In the propagation implementation of the optical atom (fig. 3b) the W-states are standing-wave modes Iy, cw + ccw) and I y, cw - ccw) . They are spatially dephased by K: the nodes of one coincide with the antinodes of the other. In the experiments reported by Spreeuw, Eliel and Woerdman [ 19901 and by Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901the coupling was produced by a dielectric perturbation, i.e., a glass plate aligned to yield reflections exactly along the optic axis of the resonator. It can be shown that one standing-wave mode has a node centered inside the dielectric and the other mode has an antinode. The frequency difference between these two W-modes can now be interpreted as arising from a difference in dielectric polarization energy. In the polarization implementa-
v. 8 21
TWO-LEVEL SYSTEMS WITH CONSTANT COUPLING
213
tion (fig. 3c) the W-states are eigenstates of EOM2: Iccw, x + y ) and I ccw, x - y ) . Again, the frequency splitting can be interpreted in terms of dielectric polarization energy, the difference now being due to a difference in refractive index for the two polarizations. In the foregoing discussion of the polarization implementation (fig. 3c), we noted that the choice of EOMl for tuning and EOM2 for coupling was arbitrary. The same physics is obtained if one chooses EOM2 for tuning and EOM 1 for coupling. The choices are related by a basis transformation. Since in H , the tuning parameter S appeared on the diagonal, the implicit choice of basis was for the S-modes. If we perform a transformation to a basis of W-modes, the matrix H , goes over into a matrix H,:
w H-=(s
s -w).
and we see that the tuning and coupling mechanisms have exchanged roles. Similarly, in our first optical implementation we could have introduced the backscattering as a mechanism that tunes the standing-wave modes and the nonreciprocal phase as a mechanism that couples them. For an instructive atomic implementation of the avoided crossing of fig. 1, we consider two atomic states that can be coupled by electric-dipole interaction. We choose atomic s and p , orbitals as the W-basis. The application of a static electric field E = (E, 0,O) causes a mixing of these states, which can be described as an off-diagonal term p E in the Hamiltonian, with p the transition dipole moment
As a function of the electric field strength, the eigenvalues show an avoided crossing. This phenomenon is known as the quadratic Stark effect, because in lowest-order approximation the atomic energy levels are shifted by an amount proportional to the square of the field strength. For zero electric field the eigenstates of the atom are simply the eigenstates of the unperturbed atom, i.e., the W-states. In the limit that the electric field is very strong, p E p Am,, the eigenstates approach the S-states, i.e., equal mixtures ofthe two unperturbed states, ( l / f i ) (s f p x ) .An impression of what the wavefunctions in these cases may look like is shown in fig. 6 . We see that the unperturbed atomic states (W-states) have no electric-dipole moment, since the density of the electron cloud I $1 is symmetric with respect
'
274
OPTICAL ATOMS
(b)
(a) W-states
S-states
Fig. 6. Polarization of an atom in a DC-electric field. (a) Unperturbed atomic s and p , orbitals ( W-states); (b) polarized atomic eigenstates in a strong DC-electric field (S-states). The S-states are linear superpositions of the W-states, and vice versa.
to the nucleus (or t+h*(x) xt+h(x)d x = 0). The eigenstates in strong field ( S states), however, correspond to a strongly polarized atom: the electron cloud has been pulled away from the nucleus. The induced dipole moment of the atom can be either parallel or antiparallel to the applied electric field. If the atom is prepared in such a polarized state and the field is then suddenly switched off, the dipole will start oscillating at a frequency wo. Conversely, an oscillating electric field containing frequency components near w,, can induce transitions. These dynamical aspects are the subject of Q 5.
2.4. THE PSEUDOSPIN PICTURE
As was shown by Feynman, Vernon and Hellwarth [1957], a two-level system can always be described as a fictitious spin-; particle. The mapping is accomplished by identifying the S-modes (the unperturbed states) with the spin-up and spin-down states for a magnetic field along the z-direction. Another way of saying this is that the Hamiltonian can be expanded in the Pauli matrices
H,
=
Wa,
i00,
+ Soz = a.a,
(8)
with 0 = ( W , 0, S ) . The Hamiltonian is, therefore, identical with the spin-; Hamiltonian - p B or a *B. It is well known that the expectation value of the spin performs a precessing motion about the magnetic field vector. We define the real vector R = (x, y , z) 3 (a), the so-called Bloch vector, so that for the
-
v, § 21
215
TWO-LEVEL SYSTEMS W I T H CONSTANT COUPL.ING
spinor (a, b ) we have x = a*b
y
+ b*a
= - i(a*b -
= 2
Rep,, ,
b*a) = - 2 Imp,, ,
(9)
z = lal’ - lb12= p I I- p,,.
with the asterisk (*) denoting complex conjugation and where pi, are the components of the (pure state) density matrix. For a normalized spinor (I a I + 1 b I = 1) the Bloch vector lies on a unit sphere: the Bloch sphere. Note that a basis transformation as discussed in 0 2.3 is now just a rotation of the Oxyz coordinate frame. The Bloch sphere of propagation modes represents the cw and ccw traveling waves on the poles and standing waves on the equator, as shown in fig. 7a. For example, a cw wave corresponds to a spinor (1,O) and thus, according to eq. (9), to a Bloch vector R = (0, 0, 1). For a standing wave the phase relation of the cw and ccw components determines the position along the equator and gives the position of the nodes of the standing wave. The Bloch sphere of polarization modes (fig. 7b) is well known under the name of Poincare sphere. This is seen most clearly if we take the circular polarizations o+ and 6- as the basis vectors (1,O) and (0, l), so that they are represented by the poles of the sphere. Linear polarizations are equal superpositions of o+ and 0 - and are represented by the equator. The relative phase of the o+ and o- components determines the position along the equator and gives the orientation of the polarization. This orientation rotates by n if one makes a full turn (2n) around the equator.
@)b( I
‘standing
..-.-
wavas
ccw
.__.--. ---+...,.
v -
linear ‘linear Dolarizations
0
Fig. 7. (a) Bloch sphere of propagation modes, representing traveling waves on the poles and standing waves on the equator; (b) Bloch sphere of polarization modes, also known as Poincart sphere, representing circular polarizations on the poles and linear polarizations on the equator.
216
OPTICAL ATOMS
The equation of motion for the Bloch vector is
-dR _
-
2R x R .
dt The equation describes the precession of the vector R about the vector R with a frequency 2 I Rl = 2 In the case of a spin-; system in a magnetic field the precession frequency is known as the Larmor frequency. Obviously the stationary states are those for which R is either parallel or antiparallel with R. Knowing this, we can identify the components of the magnetic field acting on the fictitious spin-: system. For the propagation implementation the Sagnac effect, or its equivalent in fig. 3b, can be associated with a field component B,, proportional to the Sagnac phase. Reflecting optical elements correspond to a field component in the xy-plane, the direction depending on the phase relation between the cw and ccw wave and the field strength proportional to the amplitude-reflection coefficient. Similarly, for the polarization case an optical element with circular eigenpolarizations, such as a Faraday rotator, gives a fictitious field component B,, with a field strength proportional to the Faraday rotation angle. A linear retarder, such as a quarter-wave plate, gives a field component in the xy-plane, with a direction determined by the orientation of its fast and slow axes and a strength proportional to the phase difference imposed on the fast and slow components. In view of the fact that the probability amplitudes a and b correspond to the classical limit of the mode operators, it is tempting to write down an operator version of eq. (9). This leads to the so-called Schwinger representation (see Haake, Lenz and Puri [ 19901) of angular momentum:
d m .
J,
=
f(atb + b+U),
J,
=
7(Uth - bta),
1
21
J;
= ;(.'a
- btb)
The factors have been inserted so that these operators have the correct commutation relations of angular momentum components. With these definitions the Hamiltonian of eq. (2) can be written as H = 2 R . J (cf. eq. (8)), and the Heisenberg equation of motion for J is identical with the equation of motion for the Bloch vector, eq. (10). Thus J also precesses at a frequency 2 1 Rl . The magnitude of the angular momentum is related to the total photon number:
v, 8 21 J2
= J,"
TWO-LEVEL SYSTEMS WITH CONSTANT COUPLING
+ J,' + J,'
=j
this interpretation in
( j + 1) with 2j = N = ata
211
+ btb. We shall come back to
0 6.4.
2.5. CONSERVATIVE AND DISSIPATIVE COUPLING
As discussed before, the anticrossing diagram shows that a coupling can lift a degeneracy of two modes (fig. 8a). Such a tendency to produce frequency splittings is typical for so-called conservative coupling. Conservative coupling is characterized by a Hermitian matrix as in eq. (l), so that the sum of the intensities I a \ + I bl is conserved. More generally, one can also consider situations for which this is not the case, and add to the matrix H , an antiHermitian, or dissipative, contribution: H , + H , + iG,. The eigenvalues of H , + iG, will then, in general, be complex-valued, with the imaginary part indicating the loss rate of an eigenmode (Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901, Centeno Neelen, Spreeuw, Eliel and Woerdman [ 19911). We shall discuss here only one particular example that is of practical importance in ring-laser gyroscopes (Chow, Gea-Banacloche, Pedrotti, Sanders, Schleich and Scully [1985]). In a laser gyro the tuning parameter S is the rotation rate, and a coupling of the counterpropagating traveling waves is usually not desired. Nevertheless, a weak coupling of a dissipative nature is generally present, caused, for example, by scattering of the light off dust particles or by mirror imperfections. Such a coupling can be
Of
>y n VIw
.......... ..... .......... c ,
2v
0
S
0
S
Fig. 8. Comparison of conservative and dissipative coupling. (a) Conservative coupling results in repulsion of the two levels (compare with the dashed asymptotes in fig. la); (b) dissipative coupling tends to pull together the mode frequencies, denoted by Re 1.The range 2 V for which the frequency splitting vanishes, corresponds to the locking zone of a ring-laser gyroscope. The damping rates of the modes are shown as Im 1.(After Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman 119901.)
278
OPTICAL ATOMS
described by replacing the matrix H , by
where V is now the dissipative backscattering rate. The eigenvalues of HI, are given by , I= .-/, For high rotation rates, I SI > I V1, the eigenvalues are real and denote two distinct eigenfrequencies. On the other hand, for low rotation rates, I S I < I V 1, the eigenvalues are imaginary and denote two distinct loss rates. In the latter case the real parts of the two eigenvalues are equal, so that the frequency difference has vanished. In a plot of the real part of the eigenvalues (fig. 8b), we see that dissipative coupling tends to pull the mode frequencies together rather than to push them apart, as in the case of conservative coupling. This describes qualitatively one of the causes of the so-called ‘‘locking’’ problems in laser gyros for low rotation rate (Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901, Centeno Neelen, Spreeuw, Eliel and Woerdman [ 19911, Etrich, Mandel, Centeno Neelen, Spreeuw and Woerdman [ 19921). In figs. 3a,b conservative backscattering was produced by a dielectric perturbation, viz. a glass plate aligned perpendicular to the resonator axis. In a similar way dissipative backscattering can be seen as the result of localized absorption (e.g., an absorbing layer, thin compared with a wavelength), perpendicular to the mode axis. It is obvious that a standing wave with a node at the absorber will experience a lower loss rate than a standing wave with an antinode. The difference in loss rate for the standing waves represents backscattering as in eq. (12) after a basis transformation to a basis of traveling waves. For a physical interpretation of such a basis transformation, we consider the effect of a localized absorber on a traveling wave, say clockwise. Alternatively, this traveling wave may be considered as a superposition of two standing waves, one with a node and the other with an antinode at the absorber. Now the standing-wave component with an antinode will decay, whereas the standing wave with a node remains undamped. The remaining standing wave can, in its turn, be considered as a superposition of cw and ccw traveling waves. Since, in the overall process, part of the initial cw traveling wave was converted to a ccw traveling wave, localized absorption clearly implies a scattering mechanism. For a polarization variety of dissipative coupling one may consider a partial polarizer, such as a Brewster window. If a circularly polarized light wave, say o f , passes the partial polarizer, the outgoing polarization will, in general, be
v, c 31
219
OPTICAL. B A N D STRUCTURE
elliptical, since part of one linear component has been removed. This elliptical polarization can, of course, be decomposed in its o+ and 6 - components, so that the partial polarizer can be considered as a dissipative coupling between the two circular polarizations. In the discussions of optical atoms from now on we shall consider conservative coupling only.
6 3. Optical Band Structure So far we have tacitly assumed that the frequency separation in the two-level system is small compared with the separation with all other levels. However, a two-level system is created for every longitudinal mode of the cavity. Therefore, plots like fig. 1 are actually periodic in the vertical direction, with a period equal to the free spectral range 2nc/L of the ring cavity. This implies that if the level splitting o, - w _ becomes of the order of the free spectral range, coupling may occur between different longitudinal modes. The result is then a manifold of avoided crossings, as shown in fig. 9, which was previously described under the name “optical band structure” (Lenstra, Kamp and Van Haeringen [ 19861, Spreeuw, Woerdman and Lenstra [ 19881). The Brillouin zone of the band structure is determined by the value for which the splitting induced by the tuning parameter S equals 2rcclL. In the Sagnac implementation, this value of S corresponds with a rotation rate for which the phase difference of the traveling waves after one cavity roundtrip is 2 n. This means that the cavity rotates over one optical wavelength during the time that the light makes one cavity roundtrip. The bandgap Am, in fig. 9 is determined
S Fig. 9. Extension of fig. 1 a to include a manifold of longitudinal modes (“optical band structure”). The vertical periodicity is the free spectral range of the cavity, 2 nc/L.The splitting 2 W in fig. l a plays the role of a band gap Ams. A Brillouin zone corresponds to an increase of the tuning S by a free spectral range.
280
OPTICAL ATOMS
[V,§ 4
by the coupling strength (i.e., by the amplitude-reflection coefficient r), according to the expression
Am,
2c arcsin I rl L
=-
.
For small reflection coefficients (I rl 4 1) this can be approximated by Am, x 2rc/L, or twice the backscattering rate: 2 W (cf. 2.1). In the limit that Irl tends to unity, the bandgap becomes half the free spectral range and the band structure becomes a set of equidistant horizontal lines, spaced m / L apart. We could have guessed this beforehand, of course, since in the limit of unit reflectivity ( I r l % 1) the ring cavity is actually a standing-wave cavity in disguise, and hence has a free spectral range m / L . Clearly, the same extension to a band structure applies to the polarization implementation of fig. 3c,d. In that case the Brillouin zone is determined by the full-wave voltage of the tuning EOM. This is obvious, since an EOM to which the full-wave voltage is applied does not influence the polarization of the light and must therefore have the same effect as an EOM to which no voltage at all is applied. It must be noted here that the band structure analogy also has its limitations. For example, fig. 9 differs from a conventional dispersion relation in the sense that it is not possible to have waves corresponding to different tuning parameters S present at the same time, since S is a parameter for the entire system. In contrast, in a true band structure - either for photons (Yablonovitch, Gmitter and Leung [ 19911) or electrons - where the role of S is played by the wave vector k, the photons or electrons can be distributed over many different values of k at the same time. In fact, electrons have to be, as a consequence of Pauli's exclusion principle. The distinction between fermions and bosons has more consequences, of course, some of which have been discussed by Woerdman and Spreeuw [ 19901.
8 4.
Four-Level Systems
So far we have only considered situations where either the polarization or the propagation degeneracy of a cavity mode was lifted, so that a two-level system was obtained. It is also possible to do both at the same time and thus completely remove all degeneracies. In that case we obtain a four-level system or, in the light of the preceding discussion, the corresponding band structure. Instead of doublet spectra as shown in fig. 5, quartets are observed.
28 1
to11R-LEVEL. SYSTEMS
A variety of four-level band structures has been measured in a ring cavity containing an electro-optic modulator EOM and an additional optical element (Spreeuw, Beijersbergen and Woerdman [ 19921). The voltage V,, applied to the EOM was used as a tuning parameter: the spectrum of the cavity was measured as a function of V,,,. Depending on the choice of the additional optical element, various optical band structures were obtained. A few examples are shown in fig. 10. In fig. 10a the optical element was a partial reflector (glass plate). The eigenmodes are standing waves as a consequence of the reflector and have polarization x (positive slope) or y (negative slope) as a consequence of the EOM. The crossing points in fig. 10a can be changed into anticrossings by adding an additional EOM mixing the x- and y-polarizations. The result is the band structure of fig. lob. The width of the anticrossings is determined by the coupling strength between x and y, i.e., by the voltage across the “mixing” EOM. The third example, fig. lOc, shows the band structure that results if the x- and y-polarizations are mixed and if backscattering for the x + y polarized waves is also present. The reflectivity for polarization
2z EON(x.y)
R
x EON(x.y)
EOM(xiy) R
500
1000 vEOM
(v)
Fig. 10. Experimental examples of four-level band structures. In all cases the tuning parameter is the voltage across an electro-optic modulator with axes (x, y). (a) Result when a thin reflecting etalon is added; (b) result when a modulator mixing the x - and y-polarizations is added to the configuration of (a); (c) result obtained with an element that mixes the x - and y-polarizations and selectively reflects (x + y)-polarization. (Afier Spreeuw, Beijersbergen and Woerdman [ I9921.)
282
[V, I 4
OPTICAL. ATOMS
x - y is zero. These examples illustrate the wealth of possibilities for creating
four-level systems even with relatively simple optical configurations. A formalism to describe these four-level band structures was introduced by Lenstra and Geurten [ 19901. It combines the M-matrix description employed in one-dimensional scattering problems in quantum mechanics (Merzbacher [ 19701) and Jones calculus employed in polarization optics (Jones [ 19411). In this formalism the optical field at some point P along the ring cavity (fig. l l a ) is described as a pair of Jones vectors, one describing the polarization of the cw traveling wave and the other describing the ccw wave. Combining the two Jones vectors into one single vector, results in a four-component complex vector characterizing the optical field. Every optical element is now represented by a 4 x 4 complex matrix M relating the D ) on the pair of Jones vectors ( A , B ) on the left of the element to the pair (C, right (fig. 1 lb),
where the mv are 2 x 2 submatrices. For a nonrejecting optical element the off-diagonal submatrices vanish: m,, = m21 = 0 and m,, and m2, reduce to the well-known Jones matrices. When the element does not influence the polarization, the optical element is called isotropic and the submatrices mi, are just a complex number times the 2 x 2 unit matrix I,. The submatrices can then be treated as numbers and the M-matrix as a 2 x 2 matrix. The strength of a transmission-matrix formalism lies in the representation of a sequence of optical elements by the product of their transmission matrices. In particular, for a ring cavity the matrix M,, = M,, * * M 2 M ldescribing one roundtrip will determine the eigenmode spectrum of the cavity. As discussed by Spreeuw, Beijersbergen and Woerdman [ 19921, the transmission matrices are elements of the pseudo-unitary Lie group U(2,2) if
-
Fig. 11. (a) Generalized optical ring resonator as a chain of elements M,;(b) isolated optical element with Jones vectors characterizing the polarizations of the incoming and outgoing waves. The two vectors on the lefi are related to those on the right by a 4 x 4 complex matrix M . (From Spreeuw, Beijersbergen and Woerdman [ 19921.)
v, I 51
D Y N A M I C A L BEHAVIOR OF T H E OPTICAL ATOM
283
the optical elements are loss-free. The mathematical condition imposed on M is that it preserves the metric I A 1 - I BI ’, which can also be formulated as the condition MtJM = J , where the dagger (t)denotes Hermitian conjugation and J is the metric
It was also shown that the one-to-one correspondence between group elements and loss-free optical components makes the group-theoretical approach very powerful. Every statement about the group can be directly translated into optical terms. For example, all the generators of the group U(2,2) were identified with a specific type of optical element. The commutation relations of the generators were shown to be useful for “optical engineering”, allowing the construction of nonstandard optical elements out of standard building blocks. Using the group-theoretical approach, it was shown that in ring cavities with the proper choice of optical elements symmetry arguments can be applied to infer properties of the band structure, i.e., the four-level system. For example, for the band structures discussed here time-reversal symmetry is present, implying that the eigenmodes are standing waves. It was also shown that the entire group U(2,2) can be realized with standard optical components. This suggests that almost every four-level system should be realizable in a ring resonator. To mention only one example, it is not difficult to think of a configuration that mimics the ground-state hyperfine structure of the hydrogen atom in a magnetic field. The four-level systems can also be studied in dynamical experiments if a control parameter is modulated. For the special case of only two levels, such dynamical experiments are the subject of the next section. Obviously, four-level systems will offer a much richer variety of dynamical features. For example, one could study the dynamics of three- and four-level systems driven by one or more external fields.
Ji 5. Dynamical Behavior of the Optical Atom Dynamical features of the optical ring resonator can be studied if one of the control parameters, either the coupling or the tuning parameter, is time dependent. Since the coupling and tuning parameters can be exchanged by a basis transformation, we shall for convenience always take the tuning parameter as the one varying in time. Furthermore, we restrict the discussion to the case of harmonic time variation, S ( t ) = So cos cot. In the experiments that will be
284
OPTICAL ATOMS
A socos oc
x
IV,5 5
S 0 s 0 Fig. 12. Two dynamical regimes for harmonic variation ofthe tuning parameter: S(r) = Socos or. (a) Rabi regime (w z 2 W , So ;s W); population is periodically transferred between the upper and lower mode at the Rabi frequency 0,; (b) Landau-Zener regime (So5 W).In the adiabatic limit (thick arrow) the population follows the instantaneous eigenmode ( 0 -). In the diabatic limit (thin arrow) the population crosses the gap each time the tuning parameter S(r) goes through zero.
described below the driving frequency w will be typically in the MHz range (o is not to be confused with the optical carrier frequency that was mentioned in 0 2.2). We discuss a few different dynamical regimes. In the first regime, the optical atom is driven at a frequency close to the transition frequency, w s 2 W, with small amplitude (So 5 W).The main characteristic of this regime is Rabi oscillation, which is usually described in the rotating-wave approximation (RWA). In the second regime, either the amplitude or the detuning of the driving field is so large that the RWA is no longer appropriate. An important feature of this regime is the occurrence of multiphoton transitions. In the third regime we consider large amplitudes of the driving field, leading to dynamics of the Landau-Zener type. The regimes of Rabi oscillation and Landau-Zener dynamics are indicated schematically in fig. 12. We also discuss the role of the intracavity nonlinear gain medium, which is present in some experiments. Finally, a comparison is made among a spin-f system, a two-level atom, a realistic atom with electric-dipole coupling and optical atoms.
5.1. RABI OSCILLATION IN THE ROTATING-WAVE APPROXIMATION
In this regime, the frequency of the driving field is assumed to be close to the unperturbed level spacing, o x 2 W. We also assume that the amplitude is not too large (So5 W ) , so that the driving field S(t) can be considered as a
DYNAMICAL BEHAVIOR OF THE OPTICAL ATOM
285
perturbation. In this case it is convenient to write the Hamiltonian in the W-basis : HW(0
=
(so w
socos ot
cos ot
-
w
)
This Hamiltonian is that of a two-level atom [cf. eq. (7)] with transition frequency wo = 2 W, driven close to resonance by a classical electric field with frequency o and an amplitude corresponding to a resonant Rabi frequency So. For real atoms driven by a laser field it is, at this point, common practice to make the rotating-wave approximation (RWA) in eq. (16). This terminology comes from magnetic resonance, where the oscillating field B,(t) K So cos ot is decomposed into two rotating fields, one corotating with and one counterrotatingagainst theLarmorprecession: So coswt = iSo[exp( - iwt) + exp(iwt)]. The RWA neglects the counterrotating field so that the Hamiltonian becomes
In the RWA, the equation of motion, eq. (I), can be solved analytically. The common solution method is to introduce a rotating coordinate frame in which the (fictitious) magnetic field is stationary (Abragam [ 19611, Allen and Eberly [ 19871). In this rotating frame the Hamiltonian takes the form
where A = o - 2 W is the detuning of the driving field with respect to the transition frequency. The problem has now been reduced to the static problem of Q 2, and the eigenfrequencies as a function of A or So again display an avoided crossing: The quantity OR = O, - a- is often called the generalized Rabi frequency and is the frequency at which the atom is excited and de-excited (fig. 12a). As an extension, we now address a two-level atom coupled to a single mode of the quantized radiation field. The quantum Hamiltonian reads (Graham and Hdhnerbach [ 19841): H
=
of+f + Wa,
+ ~ ( +f f t ) a x ,
(20)
and consists of three parts, describing the field, the atom and their interaction.
286
[V,§ 5
OPTICAL ATOMS
The field Hamiltonian o f t f is that of a harmonic oscillator, with j ' and f as the usual boson operators. The Hamiltonian of the unperturbed atom is H A = Wa,, using spin-: notation. The interaction term is obtained by replacing S,coswt in eq. (16) by the corresponding operator ~ ( + ff t ) , where the coupling constant K is the product of the transition dipole moment and the single-photon electric field. The interaction term shows that a transition in the atom is accompanied by a change of the field. This back reaction of the atom on the field is not present in the semiclassical Hamiltonian, eq. (16), where the field has a predetermined time dependence. The problem can again be solved analytically if the RWA is made. This is done by writing ox as the sum of the atomic raising and lowering operator: qX= u+ + o- and neglecting in eq. (20) the nonresonant terms fo- and f t o + . These terms describe virtual processes in which the atom is excited by emission of a photon or de-excited by absorption of a photon. Note that the time exp ( + iot), dependence of the remaining operators, f exp ( - i o t ) and f corresponds with the exponentials in eq. (17). The resulting Hamiltonian is referred to as the Jaynes-Cummings model (Jaynes and Cummings [ 19631):
-
HJCM=
Wftf
+ wo, + K(fa+ + .pa-).
N
(21)
We note here that this nomenclature is also used for the non-RWA variety, eq. (20). The eigenvalues of HJCMare the energies of the so-called dressed states, describing the total system of atom and field (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19881). One chooses as basis functions the eigenstates of the uncoupled field and atom with, e.g., le, n) E le) €3 I n ) denoting a state with the atom in the excited state and n photons in the radiation mode. If we plot the unperturbed energies of the states le, n) and lg,n + 1) as a function of the transition frequency oo= 2 W, we find that they cross for zero detuning (w, = o)(fig. 13). The interaction terms in HJCMnow give an anticrossing with minimum separation given by 2,-K the resonant Rabi frequency. The generalized Rabi frequency in the dressed-level diagram is given ~ m . by eq. (19) with the substitution So-+ 2 In the vicinity of an anticrossing the dressed-level diagram (fig. 13) can now, in turn, be considered as a two-level system, and we can expect resonant behavior if, e.g., the detuning is modulated around zero with a frequency close to the minimum splitting (the resonant Rabi frequency) in fig. 13. One can also take a fixed detuning and modulate the amplitude of the driving field at the detuning frequency. The problem of a two-level atom in modulated laser fields has been investigated experimentally by Chakmakijan, Koch and Stroud [ 19881 and theoretically by Ruyten [ 19901.
V, § 51
DYNAMICAL BEHAVIOR OF THE OPTICAL ATOM
281
I 0
30 0 0
Fig. 13. Energy diagram of the dressed levels for a two-level atom with transition frequency o,,, driven by a field at frequency w. Notation of the unperturbed states is such that le, n ) stands for an excited atom with n photons present in the field. Avoided crossings for wo L w and wg L 3w correspond to a one- and three-photon transition, respectively. Their precise positions are affected by a Bloch-Siegert shift.
5.1.1. Rabi experiments in the time domain Rabi oscillation in the optical atom has been observed in the time domain (Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 1990]), using the propagation implementation of fig. 3b. The setup is shown in more detail in fig. 14. The static coupling W was produced by a weakly reflecting etalon E aligned perpendicular to the resonator axis. The modulated tuning S ( t ) = So sin wt was produced by discharging a capacitor through the coil of the Faraday rotator, yielding an exponentially decaying AC magnetic field at frequency w = 277 x 1.05 M H z (fig. 15a). The detuning w - 2 W could be varied by thermal tuning of the reflectivity of the etalon E. The intensities in the two W-modes were measured by combining the cw and ccw beams ( S modes) leaking out of the cavity on a SOj50 beamsplitter. In the experiment the fiber-optic version of a beamsplitter, a so-called directional coupler, was used. The relative phase of the cw and ccw input ports of the coupler was adjusted so that the outputs corresponded with the two desired W-modes. Note that the difference of these two intensities is one component of the Bloch vector R as introduced in eq. (9). In principle, any other component of R can be measured by forming the suitable superposition of the cw and ccw waves. The ring cavity also contained an optical amplifier (a He-Ne discharge tube) in order to cancel the optical losses, and in fact, the gain was set somewhat above lasing threshold. The nonlinear gain medium did not appreciably change the dynamics on the time scale of the experiments (for a discussion see 8 5.4 or Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901).
[V,t 5
OPTICAL ATOMS
Fig. 14. Experimental setup to observe Rabi oscillation. Ml-M4: mirrors; E: reflecting etalon; FR: Faraday rotator; QW1.2: quarter-wave plates; C: capacitor; S: switch (sparkgap or MOSFET); PMTl,2: photomultiplier tubes; OSC: oscilloscope; TRIG: triggering unit; 50/50 fiber-optic directional coupler.
Theory
Experiment
0 0
10 20 30 time (,w)
u 0
20 30 time (ps)
10
Fig. 15. Rabi oscillation as observed in the optical atom of fig. 14. (a) Time derivative of driving magnetic field at o = 2 2 x 1.05 MHz. (b) Normalized intensity I,, in the initially unpopulated mode; the intensity in the other mode (W2) was 1 - I,, . Different traces correspond to different values of the detuning A = o - 2W. The theoretical curves in (d) were obtained by numerical integration of the equation of motion [eq. (l)], using the driving field shown in (c). (From Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901.)
v, I 51
D Y N A M I C A L BEHAVIOR OF T H E OPTICAL A T O M
289
The experiment was started when laser action in a single W-mode (a standing wave) was observed; at that moment (t = 0) the AC magnetic field was switched on. In fig. 15b the intensity I,, measured in the initially empty W-mode is shown, i.e., the population of the initially unpopulated level of the optical atom. We see that the population oscillates with the generalized Rabi frequency; up to 75% of the population is transferred. The Rabi oscillation is chirped because the magnetic field amplitude (aSo) decays [see eq. (19)1. The different traces in fig. 15 show that an increase of the detuning A = w - 2 W leads to an increase of the generalized Rabi frequency and a decrease of the oscillation amplitude. In the right-hand column of the same figure (fig. 15c,d), we see the theoretical result obtained by numerical integration of the equations of motion. The agreement is excellent, with only a deviation at longer timescales. The deviation could be due to the influence of the nonlinear gain medium that was present in the experiment but not in the theory (see § 5.4 for further discussion). Another contribution may come from a spurious difference in loss for the two W-modes. 5.1.2. Rabi experiments in the frequency domain Rabi oscillation in the optical atom was also observed in the frequency domain (Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 1990]), using the polarization implementation (fig. 3c). In this case no intracavity amplifier was used. A constant voltage was applied to EOM2, leading to two polarization W-modes Ix f y ) , with a frequency difference 2 W = 2 n x 9 MHz. A continuous R F driving field at o x 21c x 9 MHz was applied to EOM1. We can now think about the R F driving field as dressing the unperturbed levels of the optical atom. The dressed-level structure was probed by injecting light from a fixed-frequency He-Ne laser. The injected polarization was x, so that it had equal projections on both W-modes. The length of the ring cavity was scanned and the light intensity coupled out of the ring was measured on a photodiode. If the driving field is off, we find in the photodiode signal the two cavity resonances corresponding to the two levels of the optical atom, as in fig. 5. With the driving field on, we find that both resonances are split by the generalized Rabi frequency; i.e., both resonances are transformed into a so-called Autler-Townes doublet (Allen and Eberly [ 19871) (fig. 16). The positions of the four peaks can be directly read from the dressed-level diagram of the two-level atom (fig. 13) by looking at the four dressed levels that contain an admixture of the basis states )g,n ) and le, n ) . The Autler-Townes doublets were found to be symmetric if the driving field
290
[V, § 5
OPTICAL ATOMS
Rabi splitting
Jqy 9 MHz
AC Voltage (V)
Fig. 16. Autler-Townes structure for the driven optical atom of fig. 3c, for zero and nonzero detuning A. For A = 0, the diagram illustrates the linear dependence of the Rabi frequency on the driving field amplitude, i.e., the AC voltage across EOMl in fig. 3c. (From Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901.)
was tuned exactly on resonance (detuning A = 0) and asymmetric otherwise ( A # 0). Figure 16 also shows the observed linear dependence of the resonant Rabi frequency on the amplitude of the driving field ( A = 0).
5.2. VIOLATION OF THE ROTATING-WAVE APPROXIMATION
The RWA is an excellent approximation as long as the generalized Rabi frequency is much smaller than the transition frequency of the two-level system. For electric-dipole transitions in real atoms with transition frequencies of the order of l O I 4 Hz, this condition is usually fulfilled. For the case of magnetic-dipole transitions, consequences of the violation of the RWA were observed in optically pumped magnetic-resonance experiments (Arimondo and Moruzzi [ 19731, Cohen-Tannoudji, Dupont-Roc and Fabre [ 19731). In the optical atom, the Rabi frequency can easily be made of the order of the level splitting. The driving field across the modulator responsible for the tuning can be made sufficiently large to violate the RWA, particularly in the case of an electro-optic modulator. In the regime beyond the RWA, several new features enter the dynamics, such as distortion of the Rabi oscillation and the occurrence of Bloch-Siegert shifts and multiphoton transitions. 5.2.1. Distorted Rabi oscillation
In both the experimental and theoretical results shown in fig. 15 an interesting feature becomes visible if the first Rabi oscillation period is enlarged, as shown in fig. 17. We see a ripple at about 2 MHz or twice the transition
v , I 51
DYNAMICAL BEHAVIOR OF THE OPTICAL ATOM
Theory
Experiment
5
0
29 1
0
10
time (ps)
5
10
time (p)
Fig. 17. Enlargement of the first Rabi oscillation period in fig. 15. The ripple (-2 MHz) is at twice the driving field frequency, and is caused by the counterrotating term in the driving field frequency, thus showing that the RWA was violated. In this experiment R,/w, x 0.15 at I = 0. (After Spreeuw, Van Druten, Beijersbergen, Eliel and Woerdman [ 19901.)
which is caused by the nonresonant, or “counterrotating”, term in the driving field. The ripple on the theoretical curves disappears if the rotating-wave approximation is made. This shows that the rotating-wave approximation can be violated in the optical atom. The same ripple has previously been described in theoretical work by Silverman and Pipkin [ 19721 and by Shore [ 19901. A very drastic change in the dynamics is expected if the strength ofthe driving field is increased further. This can be seen from the numerical simulations shown in fig. 18. Going from fig. 18a to 18b, we see that for increasing amplitude of the driving field the Rabi oscillation is distorted beyond recognition. Speaking of a Rabi oscillation is then, of course, no longer appropriate. Such In1 1
05
0 IN1
1
05
0 0
5
10
time (p)
Fig. 18. (a) Numerical simulation of a distorted Rabi oscillation, the distortion with respect to a sine wave being due to the violation of the RWA. In this simulation OR/w0 = 0.5. (b) Driving field is increased further, corresponding to QR/w, = 1.0; the distortion becomes very severe, indicating that a completely different type of dynamics is entered.
292
OPTICAL ATOMS
[V.8 5
.strong driving fields violating the RWA are interesting in connection with chaos. It was predicted that chaos would occur in the Jaynes-Cummings model if the RWA was violated (Milonni, Ackerhalt and Galbraith [ 19831). In $ 6 . 3 we propose an implementation of the Jaynes-Cummings model in the optical atom. 5.2.2. Bloch-Siegert shifts and multiphoton transitions
Bloch-Siegert shifts and multiphoton transitions are well-known features associated with the breakdown of the RWA (Graham and Hohnerbach [ 19841). A Bloch-Siegert shift is a change of the transition frequency that, for the one-photon transition, can be considered as the consequence of dynamic Stark shifts produced by the nonresonant terms fa- and f which were neglected in the RWA [cf. eqs. (20) and (21)]. In second-order perturbation theory one easily calculates that these terms downshift the dressed level lg, n + 1 ) and upshift the level le, n ) by the same amount Ic’(n + 1)/200. Their crossing point in fig. 13 therefore shifts to lower values of the transition frequency 0,; this is known as the Bloch-Siegert shift. To keep the atom in resonance, one must either increase the frequency of the driving field or decrease the transition frequency of the atom. The resonance condition is o-oo=
K2(n
+ 1) ,
WO
where wo is the unshifted transition frequency. In the experiments on optical atoms a Bloch-Siegert shift should, in principle, appear in fig. 16, but due to limited cavity finesse it could not be resolved. It was observed, however, in the experiments described in $ 5.2.3. The shift as given by eq. (22) is only the lowest-order term, and for a sufficiently strong driving field more terms must be included in the calculation (Cohen-Tannoudji, Dupont-Roc and Grynberg [1988]). For increasing strength of the driving field (keeping its frequency o fixed), the effective transition frequency decreases more and more, and finally vanishes altogether; i.e., the atomic transition itself vanishes. Extensive experimental studies on the Bloch-Siegert shift were carried out in magnetic-resonance experiments on optically pumped atoms (Arimondo and Moruzzi [ 19731, Cohen-Tannoudji, Dupont-Roc and Fabre [ 19731). Another phenomenon associated with the failure of the RWA is the occurrence of multiphoton transitions, which appear in higher-order perturbation theory using the Hamiltonian of eq. (20) with the interaction term HI = ~ ( + ft)ax f as a perturbation. The appearance of multiphoton transitions can
v, 5 51
D Y N A M I C A L B E H A V I O R OF T H E OPTICAL A T O M
293
be seen directly by looking at powers of H, . For example, H: contains the terms and (ft)30- (having used that 0,' is the unit matrix), describing three-photon absorption and emission, respectively. Note that in this way only processes involving an odd number of photons are found and that these processes are absent in the RWA. In a dressed-level picture a multiphoton resonance is described as an avoided crossing between dressed levels, e.g., between ( g ,n + 2 ) and le, n - 1 ) for the three-photon case (fig. 13). It has been shown that in the vicinity of such an anticrossing a generalized RWA can be used (Graham and HOhnerbach [ 19841). In fig. 19 a numerical simulation of a generalized three-photon Rabi oscillation is shown. The precise resonance condition is affected by a Bloch-Siegert shift, and the level populations are seen to oscillate in a Rabi-type fashion. So far the discussion has been restricted to odd subharmonic resonances; the even resonances were forbidden by parity conservation (Graham and Hdhnerbach [1984]). This symmetry can be broken if an offset is added to the driving field:
f30+
S(t) =
s, + socos W t .
(23)
In such a case both even and odd subharmonic resonances occur. This situation applies to experiments on multiphoton spectroscopy of Stark levels in molecules (Meerts, Ozier and Hougen [ 1989]), where a DC and a microwave electric field were applied. Multiphoton processes are also the subject of extensive experimental study in microwave-driven Rydberg atoms (Stoneman, Thomson and Gallagher [ 19881, Galvez, Sauer, Moorman, Koch and Richards [ 19881). These experiments are usually described using Floquet theory (Shirley [ 19651). We note that the Floquet states appearing in such a description are closely related to the dressed levels discussed here. For driving fields with a large amplitude, multiphoton resonances have an appealing interpretation in terms of constructive interference of successive Landau-Zener
0
20
40
60
80
time ( p s )
Fig. 19. Numerical simulation of a generalized Rabi oscillation for a three-photon transition.
294
OPTICAL ATOMS
[V,I 5
transitions, as will be discussed in 0 5.3. Such an interpretation has also been discussed in the context of multiphoton microwave transitions in Rydberg atoms (Stoneman, Thomson and Gallagher [ 19881). 5.2.3. The optical atom beyond the rotating-wave approximation Multiphoton resonances in the optical atom were observed in the frequency domain (Beijersbergen, Spreeuw, Allen and Woerdman [ 1992]), using the polarization implementation (fig. 3c). The experimental procedure was largely the same as that described in 0 5.1.2. A coil was connected to EOM1, forming an LC circuit with the capacitance of the modulator. An R F signal at the resonance frequency of the circuit (4.35 MHz) was applied to a part of the coil, thus enhancing the R F voltage by about a factor of 13. An additional advantage of the resonant LC circuit was the suppression of (spurious) higher harmonics from the R F source. The spectrum of the ring was obtained by injecting light from a fixed-frequency (633 nm) HeNe laser through one of the mirrors and periodically scanning the length of the ring over a couple of wavelengths using a piezomounted mirror. The polarization of the injected light was chosen along one of the W-modes. The light leaking through one of the other mirrors was detected behind an analyzer transmitting the polarization of the injected light. Typical examples of the spectra are shown in fig. 20. If no R F field is applied to EOM 1, the spectrum observed is as shown in fig. 20a. A resonance occurs at a particular ring length or roundtrip phase. The resonance of the other polarization mode (fig. 20b) occurs at a different length of the ring because of the phase difference between the modes induced by EOM2. If an R F field is applied to EOMl and the transition frequency is tuned to the driving field frequency, an Autler-Townes doublet appears in the spectrum (fig. ~OC), similar to the doublets of fig. 16. The splitting is interpreted as the Rabi frequency 0,= So. When the transition frequency is changed slightly, the doublet becomes asymmetric (fig. 20d), the splitting now corresponding to the generalized Rabi frequency [ cf. eq. ( 19)]. The doublet also becomes asymmetric if the strength of the driving field is increased. To keep the doublet symmetric (i.e., to keep the optical atom in resonance with the R F driving field), one has to adjust the DC voltage across the modulator that controls the transition frequency q,.This change of the resonant transition frequency is the Bloch-Siegert shift that was discussed in 0 5.2.2. When the transition frequency is approximately three times the driving field frequency, the one-photon transition is far off-resonance, so that the doublet
v, I 51
D Y N A M I C A L BEHAVIOR OF T H E OPTICAL ATOM
295
Fig. 20. Spectra of the ring resonator. (a) Spectrum ofthe x + y polarization with zero RF field; (b) spectrum of the x - y polarization with zero RF field; (c) doublet in the x + y polarization, caused by one-photon Rabi oscillation; (d) same as (c) for small detuning from one-photon resonance; (e) doublet due to three-photon Rabi oscillation; (f) same as (e) for small detuning from three-photon resonance. (From Beijersbergen, Spreeuw, Allen and Woerdman [ 19921.)
is highly asymmetric, consisting of a large and a small peak. When the driving field amplitude is sufficiently high (S,,/oz l), the larger peak is again split, resulting in a spectrum like fig. 20e. The splitting is now the three-photon Rabi frequency 51,. By detuning from the three-photon resonance, this doublet can also be made asymmetric (fig. 20f). Again, this asymmetry can also be achieved by increasing the field strength, showing that the resonance frequency is affected by a Bloch-Siegert shift. Quantitative measurements of the phenomena just described are shown in fig. 21. In fig. 21a the one-photon doublet splitting on resonance (the Rabi frequency 0, ) is plotted as a function of the dimensionless driving field strength b / o . where b = is,, [see eq. (16)]. The symmetry of the doublet was used as a criterion for resonance. The measured values were fitted to a linear relationship. The slope of the fit served as a calibration of the field strength, which was not known precisely due to the uncertainty in the amplification of the LCcircuit on EOM1. The resulting values of the field strength were within 10% of the estimated value. The errors in the measured values of the splitting are mainly due to instabilities in the system, which caused the splittings to fluctuate. The measured one-photon resonant transition frequency is plotted in fig. 21b against b / o using the calibration obtained from fig. 21a. For comparison, two theoretical results are drawn in the same figure, one of which is the result
[V.8 5
OPTICAL ATOMS
15.0
5
-p
3 -
10.0
31
Y
5
21-
0
12.5
7.5
5.0
25 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
perturbation
0
0.25
0.50
0.75
1.00
1.25
1.50
b/o Fig. 21. The Rabi frequency and the resonant transition frequency as a function of the dimensionless driving field strength b/w [where 26 = So, see eq. (16)]. The driving field oscillates at o = 271 x 4.35 MHz. The points are experimentally observed; the lines are theoretical curves. (a) One-photon Rabi frequency; (b) one-photon resonant transition frequency; (c) three-photon Rabi frequency; (d) three-photon resonant transition frequency. (From Beijersbergen, Spreeuw, Allen and Woerdman [1992].)
according to first-order perturbation theory, and the other an analytical approximation by Swain [ 19741ofacontinued fraction given by Stenholm [ 19721. The continued-fraction solution was tested, e.g., in magnetic-resonance experiments (Arimondo and Moruzzi [ 19731). The large error-bar for large driving field amplitudes is due to the severe broadening of the resonance, as already pointed out by Cohen-Tannoudji, Dupont-Roc and Fabre [ 19731. Subsequently, the three-photon Rabi frequency and resonant transition frequency were measured by looking for resonance with coo z 3w. In fig. 21c, the three-photon Rabi frequency is plotted as a function of field strength b / o , together with the theoretical curve. The three-photon resonant transition frequency is plotted in fig. 21d, together with the theoretical curves from perturbation theory and by Swain [ 19741. As discussed in 0 5.2.2, for reasons of symmetry only odd subharmonic resonances occurred in this experiment. However, experimental evidence was
DYNAMICAL BEHAVIOR OF T H E OPTICAL ATOM
291
found of two-photon resonances when the prevailing symmetry was broken by applying a DC offset to the RF modulator EOM 1.
5.3. LANDAU-ZENER DYNAMICS
For amplitudes of the driving field that are large compared to the coupling (So 5 W), the situation is similar to a sequence of Landau-Zener crossings (fig. 12b), the true Landau-Zener problem pertaining to the situation that S increases linearly in time from - cc to + cc (Landau [1932], Zener [1932], Suominen, Garraway and Stenholm [1991]). The rate of change of S is usually expressed in the adiabaticity parameter
Y=-.
W2
dS/dt
If this parameter is large, y % 1 (or the adiabatic limit), the system remains to a good approximation in the same adiabatic state (i.e., either 1 + ) or 1 - ) ;note that the adiabatic states depend on t ) . The probability for making a transition to the other adiabatic state is given by P = exp( - A?). Adiabatic behavior is indicated by the thick arrow in fig. 12b. An adiabatic state, 1 + ) or 1 - ), changes with the parameter S i n such a way that it transforms from one S-mode to the other if S changes from a large negative to a large positieve value. The opposite limit, where y < 1, is called the diabatic limit. In this limit the transition probability P is close to unity, and the evolution is as indicated by the thin arrow. The system is then said to stay in the same diabatic state. In the language of the optical atom the diabatic basis thus coincides with the S-basis. 5.3.1. Adiabatic limit Adiabatic following in the optical atom has been demonstrated using the propagation implementation of fig. 3b (Spreeuw, Eliel and Woerdman [1990]). The optical atom was prepared in the absence of a driving field ( S = 0) in one of the adiabatic modes, i.e., one of the W-modes. This state preparation was done by injecting light from a He-Ne laser and bringing the ring resonator into resonance with the laser light. At t = 0 a slowly oscillating driving field S ( t ) was switched on (o= 2 A x 440 kHz). It was observed that as the driving field swept back and forth, the intensity (population) was transferred back and forth between the cw and ccw modes (S-modes), synchro-
298
[V.§ 5
OPTICAL ATOMS Theory
Experiment
dB/dt
-5
0
5
10
Time ( p s )
15
-5
0
5
10
15
Time ( p s )
Fig. 22. Adiabatic response of the optical atom in the propagation implementation of fig. 3b. (a) Slow harmonic variation (440 kHz) ofthe tuning parameter (magnetic field; note that the time derivative is shown). (b, c) Intensities measured in the cw and ccw directions, as a response to the driving field, showing periodic transfer of intensity between the cw and ccw modes. The exponential envelope is due to optical losses. The theoretical results (e, f) have been obtained by numerical integration of eq. ( I ) with the driving field shown in (d). The value of the coupling in this experiment was 2W = 2 n x 9 MHz. (After Spreeuw, Eliel and Woerdman [1990].)
nously with the oscillating field, as shown in fig. 22. The experiment thus demonstrates a real-time monitoring of the change of the adiabatic “wavefunction”. In terms of the Bloch-vector description of § 2.4, this means that the Bloch vector is slaved to the vector R; i.e., by changing either W or S we change R and thus the Bloch vector. The reversal of the propagation direction of the light can also be considered as a Bragg reflection off a macroscopic periodic structure (“lattice”) with the circumference of the ring as the lattice constant (Spreeuw, Eliel and Woerdman [1990]). This phenomenon has a close analogy in solid-state physics, where the ionic lattice may act as a Bragg reflector for the electronic waves. Such bandgap reflections are the basic
v, I 51
DYNAMICAL BEHAVIOR OF T H E OPTICAL ATOM
299
underlying phenomenon of so-called Bloch oscillations (Lenstra, Kamp and Van Haeringen [ 19861). In this experiment it was also possible to see deviations from adiabatic behavior. If the Landau-Zener transition probability is not negligible, the system does not remain in the same adiabatic state, but a fraction of the population will be transferred to the other adiabatic level. Since the light is then in a coherent superposition of two modes, a ripple is observed on the signals at the beat frequency (fig. 23). It was also observed that this beat frequency decreased as the driving field amplitude decayed, which can be seen clearly in fig. 24, where three enlargements for different time windows are shown. The beat was still visible after the driving magnetic field had died out (fig. 240, since the intracavity light was left in a superposition of the two W-modes, with a frequency difference 2 W. The decay of this superposition can be considered the analog of a quantum beat in the spontaneous emission from a superposition of two coherently excited atomic levels. The adiabatic response has an interesting interpretation in terms of the analog of the two-level atom discussed in $ 2.3 (fig. 6). Remembering that the S-modes in that case were highly polarized states of the atom, we see that the electric field sweeps the electron cloud from one side of the nucleus to the other
dB/dt
0
0
10
20
t i m e (bs)
Fig. 23. Typical experimental result showing a fast modulation on top of the slow adiabatic response, indicating a nonadiabatic contribution (Landau-Zener transition) to the response. In this experiment the propagation implementation of fig. 3b) was used. (a) Exponentially decaying oscillation of the driving field, i.e., the magnetic field B ; (b) cw intensity in the ring; the ccw intensity is complementary. In this experiment the value of the coupling was 2 W = 2 A x 2.3 MHz and the driving frequency was w = 2 n x 1.05 MHz.
300
P.I 5
OPTICAL ATOMS
dB/dt 0
Icr
0 0
1
2
3
7 8 9 1 0
28 29 30 31
time (ps)
Fig. 24. Enlargements of fig. 23 for three different time windows. Note that the fast modulation decreases in frequency as the driving magnetic field amplitude decays, and that the modulation is still present after the magnetic field has died out.
side. Another interesting example of adiabatic following occurs in the dressedlevel diagram shown in fig. 13. A slow change in laser detuning from A $ - So to A $ + So then causes the dressed level 1 - ) to change adiabatically from / g ,n t 1 ) into le, n ) , so that the atom is excited.
5.3.2. Multiphoton resonances
In the adiabatic limit it is still possible that a large population builds up in the initially empty adiabatic mode. Such a buildup of population occurs when the contributions from the subsequent Landau-Zener crossings all interfere constructively. A heuristic criterion for such a resonance is that the integrated phase difference of the adiabatic modes between two consecutive zero-crossings of the driving field is an odd multiple of R ,
Sb’*
[ w + ( t )- o_(t)] d t = (2m
+ l)n,
where m is an integer number and T = 2 R/OJ is the period of the driving field with zero-crossings at t = i n T . Note that in the limit of small amplitude, we can approximate [ w + ( t )- w - ( t ) ] by the constant 2 W and eq. ( 2 5 ) reduces to the familiar resonance condition for a multiphoton transition, o = 2 W/(2m+ 1). This is remarkable, since in the limit of small amplitude the Landau-Zener picture is no longer appropriate.
v, I 51
DYNAMICAL. BEHAVIOR O F ‘THE OP-TICAL ATOM
30 1
dB/dt 0
0 0
5
10
15
20
time ( p s )
Fig. 25. As fig. 23, for parameter values: 2 W = 2 n x 1.7 MHz and So z 2 n x 3 MHz at t = 0. At z 4.6 ps the slow adiabatic oscillation of I,, makes a n-phase jump, indicating that more than one half of the population has made a nonadiabatic transition.
i
An experimental result indicating that interference between successive Landau-Zener crossings is important is shown in fig. 25. This result was obtained in an experiment identical to that of fig. 23, except for a smaller value of the level splitting: 2 W = 2 n: x 1.7 MHz. We see that the intensity in the cw mode oscillates as in fig. 23. At a certain time indicated by the dashed vertical line, however, a n-phase jump occurs in the intensity oscillation. This indicates that more than one half of the population has made the transition to the initially empty adiabatic mode.
5.3.3. Dia batic limit In the diabatic limit, the constant coupling W is relatively unimportant compared to the tuning S ( t ) .This is obviously the case if So 4 Wand w 2 W. Experiments in the diabatic regime so far have not been carried out with the optical atom. Qualitatively one can consider the coupling W to have a kicked nature, since it is only important during the short time intervals that the tuning goes through zero. The population in a certain diabatic state may therefore be expected to change stepwise with each “kick”, and interference is expected between the contributions from consecutive kicks. This is a topic for future experimentation. The diabatic regime seems to be interesting in connection with chaos. In experiments on microwave-driven Rydberg atoms it has been found that in this regime quantum suppression of classical chaos becomes manifest (Galvez, Sauer, Moorman, Koch and Richards [ 19881).
302
OPTICAL ATOMS
P,§ 5
5.4. PASSIVE A N D ACTIVE RING CAVITIES
This section discusses the role of the intracavity optical amplifier that was present in some of the dynamical experiments. The reason for introducing such a gain element was to cancel the optical losses caused by other optical elements and so increase the cavity decay time z, (“photon life time”). The requirement on z, is that it is long compared with the timescale of the dynamics studied, e.g., ORz,% 1 if Rabi oscillation is studied. An ideal gain medium would be one that only counteracts dissipation and does not influence the two-level dynamics. It is well known, of course, that generally the saturation of a gain medium introduces nonlinearities that may lead to very complicated dynamics. The circumstances under which such a nonlinear contribution to the dynamics can be neglected will be discussed. An obvious method to avoid nonlinearities is to avoid saturation of the gain medium. This implies keeping the gain sufficiently below the lasing threshold of the cavity. By approaching threshold from below as close as possible, the photon lifetime can be increased tremendously. In the experiments in the adiabatic regime photon lifetimes on the order of 20 ps were obtained in this way. Even so, in these experiments saturation was not completely absent. For example, in the experiments displayed in figs. 22-25 it was observed that the photon lifetime increased when, at later times, the intracavity intensity decayed, indicating that early on the photon lifetime was limited by gain saturation. The reason is that the unsaturated gain is kept below lasing threshold, whereas in the presence of intracavity light we are dealing with the saturated gain. It is an interesting challenge to investigate how well losses may be cancelled by an intracavity amplifier. Reaching this limit will require detailed insight in laser physics. The validity of the experiments above threshold, such as those in the Rabi regime, is based on different arguments, since in that case we are dealing with a ring laser. Within the context of third-order laser theory the dynamical equations are then an extension of eq. (l),
-(
i d a) = ( H s + i G ) ( l ) , dt b where the matrix G describes the gain medium. Assuming zero detuning from the gain maximum (Menegozzi and Lamb [ 19731, Chyba [ 1989]), we have
303
DYNAMICAL BEHAVIOR OF T H E OPTICAL ATOM
where go is the unsaturated net gain, /?the self-saturation coefficient and 5 the cross-saturation coefficient. The latter can take the values 0 < < 2 and determines the nature of the mode competition. A value < 1 means that a traveling wave saturates its own gain more strongly than the gain of the other traveling wave. The gain medium will then tend to equalize the two traveling wave amplitudes u and b. This situation generally occurs in inhomogeneously broadened lasers such as gas lasers (Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901). In terms of the Bloch sphere (fig. 7a) such a gain medium will pull the Bloch vector toward the equator. Conversely, if 5 > 1, a traveling wave saturates the gain of the other traveling wave more strongly than its own gain. The Bloch vector is then pulled toward the poles. Values t > 1 occur in homogeneously broadened lasers such as dye lasers (Centeno Neelen, Spreeuw, Eliel and Woerdman [ 19911). The value 5 = 1 plays a special role, since it leads to neutral mode competition; i.e., the gain saturation is determined solely by the total intensity 1 CI 1 + I bl ’. In that case the matrix G is proportional to the unit matrix, so that the traveling waves experience the same gain (or loss). In terms of the Bloch vector a gain medium with neutral competition can only change the length of the vector, but not its direction. The gain medium stabilizes the total intensity to the value ( a ( ’ + (61’= go//?,and the matrix G vanishes. This is an ideal situation for studying the two-level dynamics in a dissipation-free system. It is well known that in a He-Ne ring laser containing only one N e isotope, the value for 5 on line center is very close to unity (Chyba [1989], Spreeuw, Centeno Neelen, Van Druten, Eliel and Woerdman [ 19901). Such an amplifier was used in the experiments described earlier in the Rabi regime. The deviation of those experimental results from theory for long timescales may indicate that the value of 5 was not exactly unity, but apparently the remaining mode competition was sufficiently weak so that it did not influence the linear dynamics at shorter timescales. In fact, a comparison of figs. 15b and 15d indicates that the nonlinear interaction has an appreciable effect only on timescales that are long compared with a Rabi-oscillation period. Further experimental proof that the below- and above-threshold situations are equivalent is given by fig. 26, the above-threshold counterpart of fig. 23; both cases yield essentially the same results for diabatic plus adiabatic response.
<
<
5.5. TWO-LEVEL ATOMS AND ELECTRIC-DIPOLE COUPLING
The experimental results prove that the driven optical ring resonator behaves similarly to a two-level atom or a spin-; system. This is, ofcourse, a mathemati-
304
OPTICAL ATOMS
. . . . .. , . . 0
10
.
,
,
. . . . .
20
30
time (ps)
Fig. 26. As fig. 23, with the gain of the intracavity amplifier set above lasing threshold and therefore without the exponentially decaying envelope offig. 23b. After the magnetic field has died out, the ccw intensity oscillates at the frequency 2W/2s = 2.1 MHz. The slow decay of this oscillation gives an indication of the timescale involved in the nonlinear dynamics due to the saturated gain medium.
cal analogy. In fact, an essential difference exists between electric-dipole interaction, encountered in the case of a two-level atom (or a real atom), and magnetic-dipole interaction, encountered in the case of a spin-: system; this difference is easily overlooked in the spin-; description of the two-level atom. In this section we point out this difference and consider the optical atom from this perspective (see Spreeuw and Woerdman [ 1991al). If we compare the simplest magnetic-dipole transition with the simplest electric-dipole transition, we find an intrinsic difference in the dimensionality of state space: the minimum number of levels involved is two for the magnetic case (spin-:) and four for the electric case (J = 0 -+ J = 1 or J = $ -+ J = 4). We choose the electric-dipole transition between a singlet (J = 0) and a triplet state (J = 1) to illustrate the consequences (fig. 27a). This four-level system can be
....... 190’
...... 19-1/2>
19+1/2>
Fig. 27. The simplest electric-dipole transitions: (a) from J = 0 to J = 1 and (b) from J = 4 to = 4. Subscripts indicate the quantum number m.Dashed lines indicate the dynamic Stark shifts by the nonresonant interaction terms when the transition is strongly driven by u+ -polarized light.
J
DYNAMICAL. B E l l t \ \ 1 0 1 < 0 1 I I l l . O P I I C A L ATOM
305
reduced to an effective two-level system by choosing a-polarized light, thereby selecting the transition with Am = 0. The atom-field interaction Hamiltonian [cf. eq. (20)] is then given by HI
=
d f + f + ) ( l g c I ) (eel + le,)
(gol)9
(28)
where 18), and le,) denote the ground and excited state, both with m = 0. The interaction Hamiltonian is identical with that of a spin-; system driven by a linearly polarized magnetic field, perpendicular to the static field B,. The dynamics violate the RWA if the driving field is strong. For the spin-: system one can choose instead to use a driving magnetic field with circular polarization and thus obey the RWA exactly. For the electric-dipole transition J = 0 4J = 1, this cannot be done, since the choice of polarization (i.e., a-polarization) has been used already to reduce the system to two levels. One might think of reducing the level scheme of fig. 27a to the two-level system defined by [ g o ) and I e ) by choosing a+-polarized light. In fact, it is sometimes stated (Allen and Eberly [1987, 0 4.71, or Shore [1990, 0 3.91) that in that case the RWA is exact. However, such a statement ignores the coupling of the r ~ +-polarized field to the transition with Am = - 1. A simple classical interpretation is that in the electric-dipole approximation the atom cannot distinguish between a classical field with polarization a + at frequency w and a field with polarization a- at frequency - w, i.e. rotating backwards in time. The a+-field therefore drives the Am = - 1 transition at a frequency - w, hence with a detuning A = - w - wo z - 2w,. The full interaction Hamiltonian reads (Loudon [ 19831) +
,
where f, and f! are the boson operators of the a+-polarized mode. For completeness we note that there is also a coupling with a--polarized light, the a- -terms are unimportant, although in principle they cause virtual spontaneous emission processes, as was pointed out recently by Crisp [ 19911. The interaction Hamiltonian shows that we have effectively a three-level system. The a+-polarized light stimulates not only transitions with Am = + 1, but also virtual transitions with Am = - 1. This causes dynamic Stark shifts of the excited state I e - ) and the ground state. One easily calculates that these shifts have the same magnitude as those describing the conventional Bloch-Siegert shift, as discussed in 3 5.2.2. The only situation for which the driving a+-field connects neither the ground
,
306
OPTICAL ATOMS
[V. § 5
nor the excited state with a third level occurs for a J = 4 -,J = transition (fig. 27b). In this case we do not have a closed two-level system, however, since excited state le, , , 2 ) can decay to the ground state lg, by means of spontaneous emission. After a few spontaneous lifetimes the atom inevitably ends up in the ground state lg, ,,*) where it no longer absorbs a+-photons (optical pumping). Of course, this need not be important if one is only interested in the dynamical behavior within a spontaneous lifetime. The difference between magnetic- and electric-dipole interaction as described above raises a question about the status of the optical atom. The optical atoms discussed so far (i.e., the implementations sketched in fig. 3) are true two-level systems. They can therefore be described as a spin-; system. Moreover, they were driven by an oscillating rather than a rotating field, so that they can also be compared with the two-level atom as described by eq. (16) or (20). Since in the optical atom all parameters in the Hamiltonian are readily accessible, the possibility to drive it with a rotating field is also present here. For example, in the polarization implementation we can choose the circular polarizations a+ and a- as the ground and excited states. A rotating driving field can then be realized by using an electro-optic modulator driven by a circularly polarized RF field (e.g., by using four electrodes as in fig. 28), and driving the two pairs of opposing electrodes with signals that are in quadrature. However, the optical atom is not restricted to simulation of a spin-; system. As discussed in Q 4, the optical atom offers in principle also the possibility of realizing four-level systems and thus implementing, e.g., the electric-dipole transitions shown in fig. 27. By realizing the appropriate resonator configurations, the optical atom can simulate a spin-; system, a two-level atom or a four-level atom.
Fig. 28. Electro-optic modulator with four electrodes, allowing the generation of a rotating RF field.
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FJ 6. The Driven Optical Ring Resonator as a Model for Microscopic Systems The experimental results reviewed in the foregoing sections establish in a sense the bona fide character and practical convenience of the optical atom as a model system. In this section we enter terra incognita and investigate to what extent this model system offers interesting experimental possibilities to address current theoretical issues in semiclassical or quantum physics of microscopic systems. 6.1. CAN ONE SIMULATE SPONTANEOUS DECAY OF THE OPTICAL ATOM?
In the foregoing discussions the optical atom appeared as a suitable classical system, displaying in its behavior certain features that are usually associated with quantum mechanics. The question arises concerning how far one can push such classical simulations and where one must expect to run into phenomena of exclusive quantum nature, i.e., without classical counterpart. A phenomenon that is usually considered of intrinsic quantum nature is spontaneous emission or decay. In this section we reflect on the possibilities of simulating spontaneous decay in the optical atom. A rather trivial form of spontaneous decay was encountered in figs. 22-25, namely, the decay of the intracavity light due to optical losses by absorption and mirror transmission. This can be looked on as spontaneous decay of a Rydberg atom: the decay from both levels is equal and exclusively to “outside” levels (fig. 29a). In the geometrical picture of 5 2.4 this type of decay gives a contribution to dR/dt [eq. (lo)], which is antiparallel to R . Therefore it changes the length of the Bloch vector (causing it to shrink to the origin), but not its direction, whereas the conservative part of the evolution (ax R) is orthogonal to R and changes only the direction of the Bloch vector. This type of decay therefore does not change the internal dynamics of the two-level system. Simulation of spontaneous decay of an open two-level system is thus apparently straightforward.
53 6 IE (4
(b)
Fig. 29. Spontaneous decay in: (a) an open two-level system and (b) a closed two-level system.
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Simulation of spontaneous decay in a closed two-level system, as shown in fig. 29b, is a different issue. Note that, in principle, spontaneous decay is expected to occur as a consequence of vacuum fluctuations inside the modulators. However, an (admittedly naive) estimate given by Beijersbergen, Spreeuw, Allen and Woerdman [ 19921 arrives at a spontaneous lifetime exceeding that of the universe by three orders of magnitude. For a simulation of spontaneous decay at a higher rate than the actual spontaneous rate, one must introduce a coupling from one level to the other but not backward. To make the problem more specific, we consider a ring resonator with cw and ccw waves detuned by an amount 2 s . A one-way coupling mechanism from cw to ccw could now be realized with a one-way mirror, the construction of which was discussed by Spreeuw and Woerdman [ 1991bl. Thus one might be able to deplete one level into the other and, hence, simulate spontaneous decay. However, in such a situation the light in the ring is at any instant of time a coherent superpositon of cw and ccw waves; i.e., the optical atom is always in a pure state. This is rather different from the usual situation in real atoms. To see this, we may consider a real atom in the excited state and wait until the probabilities of finding the atom in the excited or ground state both equal i,At that moment the atom is not in a coherent superposition of the ground and excited states (that would imply a nonzero atomic polarization), but in an incoherent superposition (Messiah [ 19701). This means that we can no longer write down the state vector of the atom, but we must describe it with the density matrix of a statistical mixture. The essential reason is that the process of spontaneous decay of real atoms produces correlations between the atom and the degrees of freedom of a heat bath. The state of the entire system is an entangled state of atom plus bath, which reduces, if a description for the atom alone is desired, to a statistical mixture of the atomic states. With the optical ring it seems that, no matter how we arrange the spontaneous decay, with the optical fields in the classical limit we should always be able to express the mode amplitudes by two complex numbers (a, b). After all, the mode amplitudes can be measured at any instant of time without appreciably affecting the system. The light is therefore always a coherent superposition of the cw and ccw waves; i.e., the state vector (a, 6) exists, and the optical atom IS in a pure state. One is therefore inclined to think that spontaneous decay in a closed two-level system cannot be simulated in this classical system. This point of view, however, is somewhat mitigated by realizing that spontaneous emission is successfully incorporated in semiclassical laser theory by simply introducing a classical stochastic polarization source term (“Langevin force”), the strength of this term being determined by quantum theory
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(Henry [ 19831, Vahala and Yariv [ 19831, Marcuse [ 19841). In addition, quantum-fluctuating laser fields were successfully simulated using stochastic modulation techniques (Elliott and Smith [ 19881). We refer to Milonni [ 19761 for a critical discussion of the merits and limitations of semiclassical radiation theory in describing spontaneous emission. Another consideration might be that spontaneous emission can be simulated numerically by letting quantum jumps occur at instants of time controlled by a random number generator (Dalibard, Castin and M ~ l m e r[1992], Dum, Zoller and Ritsch [ 19921). In some sense the optical atom could be considered as an analog computer solving a set of equations of motion. Therefore it is possible that also in this case spontaneous decay could be simulated by using a noise source to introduce stochastic behavior, or perhaps even by simulating quantum jumps at instants of time derived from a random number generator. A definitive answer to the question posed in the title of this subsection will require further study. Some aspects of spontaneous decay most likely can be simulated in the optical atom, whereas others cannot. The latter category is, of course, more interesting for those who are concerned with the twilight zone between the quantum and classical worlds.
6.2. LANDAU-ZENER CROSSING PROBLEMS
As demonstrated in 0 5.3 the optical atom allows a real-time study of Landau-Zener crossings. The reason is, of course, that the optical fields are in the classical limit so that a continuous measurement on the system need not perturb it significantly.This advantage with respect to microscopic systems that are intrinsically quantum mechanical can possibly be exploited to address issues of theoretical interest in the Landau-Zener problem. One such issue is the extreme sensitivity of the level-crossing transition probability to details of the evolution (Suominen, Garraway and Stenholm [ 19911). A related issue is the universality of transition history in the near-adiabatic limit (Berry [ 1990al). It has been predicted that the use of a so-called superadiabatic basis leads to a universal evolution of the system for t = - cc -,t = + 00, irrespective of details of the evolution of the driving parameter. It has also been predicted that the famous “Berry phase factor” (Berry [ 19841)has a cousin, which might be called the “Berry amplitude factor” (Berry [ 1990bl). The amplitude factor occurs in the quantum evolution of a system described by a complex Hermitian Hamiltonian and has, like the phase factor, a geometric origin. The Berry amplitude factor might be demonstrated
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using a Landau-Zener crossing in the optical atom. Interesting possibilities may also be offered by the four-level variety of the optical atom. It was shown theoretically that Landau-Zener transitions in a four-level system, induced by short laser pulses, can lead to high-harmonic generation (Kaminski [ 19901). Finally, considerable interest has been generated in the influence of dissipation on Landau-Zener tunneling (Ao and Rammer [ 1989]), and in the light of 0 2.5 the optical atom possibly offers an experimental entrance to some aspects thereof.
6.3. JAYNES-CUMMINGS MODEL
The Jaynes-Cummings model is a popular model system in quantum optics; it was introduced in 0 5.1, and describes a two-level atom coupled to a single mode of the radiation field. This model has drawn much theoretical interest, since it has the virtue that it allows exact analytical solutions (in the RWA) (Graham and Hdhnerbach [ 19841, Cohen-Tannoudji, Dupont-Roc and Grynberg [ 19881). Another important stimulus for this interest is that it was recently realized experimentally, by coupling a Rydberg atom to a high-Q microwave cavity (Rempe, Walther and Klein [ 19871). In that experiment the dissipation rate (set by spontaneous emission and cavity loss) was smaller than the electric-dipole coupling rate, so that the free-space behavior of the atom was significantly changed (Alsing and Carmichael [ 19911). a system described by the RWA JaynesWhen resonantly driven (ow o,,), Cummings Hamiltonian (2 1) will show a Rabi-type of back-and-forth trading of energy between the atom and the field mode. If the number of photons in the field mode is large, ( f t f ) = ( n ) 9 1, the field may be treated classically, leading to the so-called semiclassical Jaynes-Cummings model. Of course, features associated with the quantum nature of the field, such as collapse and revival of the exchange dynamics due to a quantum spread of Rabi frequencies (Rempe, Walther and Klein [ 19871) do not survive in the semiclassical limit. The coupling of field dynamics and atomic dynamics survives a classical treatment of the field only if the atom-field coupling constant IC is enlarged artificially (Milonni, Ackerhalt and Galbraith [ 19831). This can be realized, in principle, if one considers, instead of a single two-level atom, an ensemble of N such atoms ( N 9 1); in that case the transition dipole of the ensemble can be described classically. Returning now to the driven optical ring resonator, we note that it is, in principle, described by the (non-RWA) Jaynes-Cummings Hamiltonian (20)
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if the driving R F field (e.g., a LC resonant circuit) is quantized (Marcuse [ 19801): S ( t ) = So sin ut+f + f t in eq. (16). In practice, however, the R F oscillator is completely in the classical limit. The reaction of the optical atom on the RF field, although present in principle, would require extremely low noise conditions. Nevertheless, the semiclassical Jaynes-Cummings model can be realized by means of the driven optical ring resonator (Spreeuw and Woerdman [ 1991bl). The basic idea is, as in the case of a real two-level atom, to increase the coupling constant. The macroscopic nature of the system allows experimental realization of this idea, contrary to the case of a real atom where an increase of the coupling constant seems to remain a theoretical artefact. The back reaction of the atomic state on the driving field can be simulated in the optical atom by introducing a hybrid feedback loop that measures the optical field (“atomic state”) and uses this information to steer the R F driving field. The exact realization of such a feedback loop can be found from the JaynesCummings Hamiltonian (20) by looking at the Heisenberg equation of motion for the driving field:
f= i[S,H] - ’
=
-iof- i q .
(30)
We see that apart from the usual harmonic part - i w h the equation for f contains the pseudospin operator a,, describing the polarization of the atom. In the definition of the Bloch vector we see that x = (a,) contains products of the field amplitudes: a*b + b*a. These can be measured by letting the two fields interfere on a photodetector, so that the photocurrent is proportional to I a + b I ’. From this we can extract a signal proportional to the polarization oy by using an AC-coupled amplifier, blocking the DC signal la12 + I bl’ and transmitting the desired signal a*b + b*a that oscillates at the transition frequency. Figure 30 shows our proposal for an optical realization of the semiclassical Jaynes-Cummings model. The optical two-level system consists of the x- and y-polarized modes of a ring resonator, which are detuned electro-optically by an amount 2 W . The system is driven by harmonic modulation (0) of the coupling element, which is an electro-optical modulator with its axes rotated by 45” with respect to the first EOM. Some of the light is coupled out of the ring and passes a polarizer transmitting the x + y polarization. A photodetector behind the polarizer therefore measures the intensity I x + y (2. The desired signal x * y + y * x is extracted with an AC-coupled amplifier and added to the current in the R F oscillator circuit driving the coupling element. (Note that the driving voltage is represented by f, and {is proportional to the current in
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Fig. 30. Proposed implementation of the Jaynes-Cummings model. Polarization modes Ix ) and Iy) constitute an optical atom. Due to the electrical feedback loop, the state of the optical atom affects the driving field in such a way that the semiclassical Jaynes-Cummings model is simulated.
the RF circuit.) The feedback loop should be much faster than the dynamical timescale in the ring resonator; i.e., the bandwidth of the loop should be much larger than the Rabi frequency. This condition is easily satisfied, since in recent experiments the Rabi frequency was typically a few hundred kHz (fig. 15). The importance of having an experimental realization of the semiclassical Jaynes-Cummings model is that, hopefully, it will allow an experimental verification of the theoretical prediction (Milonni, Ackerhalt and Galbraith [1983]) that this system shows chaotic dynamics if it is driven so strongly that the RWA is not valid. This prediction can be seen as a consequence of the fact that the unperturbed energyftf + a, is no longer a constant of the motion if the RWA is dropped, increasing the number of degrees of freedom from 2 (the atomic inversion and the relative phase of the atomic polarization and the field) to 3, which is known to be a necessary (although not sufficient) condition for chaos. Since the optical two-level system can be driven at sufficient strength to violate the RWA, as discussed in 0 5.2, the setup shown in fig. 30 is ideally suited for a first experimental verification of chaos in the non-RWA extension of the semiclassical Jaynes-Cummings model. Particularly interesting is the experimental possibility of comparing, for the same strength of the RF driving field, a situation where the RWA is violated with one where the RWA is exact (using a rotating RF field as shown in fig. 28). Several extensions of the Jaynes-Cummings model have been reported in the theoretical literature, including versions with multiphoton excitation (Rosenhouse [ 19911) and multilevel atoms (Harms and Haake [ 19901); in addition, effects of detuning (Kujawski and Munz [1987], Hillery and Schwartz [ 1991]), intensity-dependent coupling (Rosenhouse [ 19911) and ca-
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vity damping have been addressed (Munz and Kujawski [ 19901, Alsing and Carmichael [ 19911). Many of these extensions can probably be realized experimentally by adding suitably designed feedback loops to the optical ring resonator.
6.4. DRIVEN TOP
In 0 2.4 it was shown that the two-mode system can formally be described as an angular momentum J . In this subsection we discuss how this formal equivalence can be exploited to obtain an optical model system of a popular subject of theoretical study, the so-called driven top. We shall first discuss the classical top, using the quantum mechanical formulation for convenience. In 0 6.5 we then briefly address the quantum limit. The time-dependent Hamiltonian of a periodically driven top with angular momentum operator J = (Jx, J,,, J,) can be written as k H ( t ) = - J,' 2j
+ pW(t)J,,
where k and p are coupling constants. The magnitude j of the angular momentum is a constant of motion, J 2 = j ( j + 1). The function W ( t )is periodic in time; a popular case is periodic kicking, described by W(t)=
1 s(t - n T ) . n
Using the Schwinger representation of angular momentum as introduced in eq. ( 1 l), we can rewrite the Hamiltonian of the kicked top as (Haake, Lenz and Puri [ 19901)
k H ( t ) = - (ata - btb)2 + f p W ( t ) (atb + bta). 8.i
(33)
To avoid confusion with the Jaynes-Cummings Hamiltonian introduced in 0 5.1, note that in the present case, as described by eq. (33), a classical field W ( t )drives two coupled quantized modes, whereas in the Jaynes-Cummings case described by eq. (20), a quantized field drives two coupled classical modes. A comparison of eq. ( 3 3 ) with the coupled-mode Hamiltonian ( 2 ) then tells us that we can implement the nonlinear, kicked-top Hamiltonian ( 3 1 ) by making the tuning parameter S proportional to the inversion uta - btb
3 14
OPIICAL A I O M S
Fig. 31. Proposed implementation of the driven-top model. Polarization modes 1x1 and lyl constitute an optical atom. The electrical feedback loop makes the tuning parameter S proportional to the “atomic inversion” Ix I - Iy 1
’.
(Spreeuw and Woerdman [ 1991bI). Note that the level splitting 2 s corresponds to the precession frequency of the top. An experimental implementation in the classical limit using two polarization modes is shown in fig. 31. The terms utu and btb have been replaced by the ) ~ (btb)’, which classic intensities 1 u I and I b I ’. The quartic terms ( u ~ u and appear in the Hamiltonian when we make the just-mentioned substitution for S in eq. (2), reflect that the optical mode oscillators have been made anharmonic by the feedback loop. The classical driven top shows chaotic dynamic behavior if the driving and the nonlinearity satisfy certain criteria (Haake, KuS and Scharf [ 19871). In the case of the two-mode optical ring resonator, chaotic behavior would mean a chaotic distribution of the intensity between the two modes, the total intensity being constant. In the classical limit chaos does not occur for monochromatic driving if the RWA is valid. One expects that the more nonmonochromatic the driving, the easier it is for the dynamics to become chaotic, kicked driving [eq. (32)] being the extreme example (Haake, Lenz and Puri [ 19901). Kicked driving of the ring resonator can be realized by using a large bandwidth amplifier for applying high-voltage pulses to EOM I in fig. 3 1. Chaos is also expected for monochromatic driving if the RWA is violated, which, as discussed in 5 5, can easily occur for the driven optical resonator.
6.5. QUANTUM LIMIT OF T H E DRIVEN TOP
The quantum treatment of systems that show chaotic motion in the classical limit is a popular topic of theoretical study. The motivation of these studies is
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to see how the character of classical chaos is changed if the system is made more and more quantum mechanical by changing a suitable control parameter. This field is loosely referred to as “quantum chaos”. Although there seems to be a consensus that proper quantum chaos does not exist, quite some debate exists on the nature of the transition between the classical and the quantum cases (Izrailev [ 19901, Haake [ 19911). Favorite theoretical models that are used in this context concern relatively simple quantum systems that freely evolve, except for kicks or reflections at sharply defined time or space coordinates. Examples are the kicked rotor (Casati, Ford, Guarneri and Vivaldi [ 19861) and the kicked top (Haake, KuS and Scharf [ 19871); the magnitude of the rotor or top angular momentum (in units of h ) is the control parameter mentioned earlier. Only very few experimental data are available for verification of this theoretical model work, in particular, the experiments on microwave ionization of H-atoms prepared in Rydberg states (Galvez, Sauer, Moorman, Koch and Richards [ 19881). It was recently proposed by Haake, Lenz and Puri [ 19901 to realize experimentally the kicked-top Hamiltonian by using two optical modes in a “suitable” nonlinear-optical medium. As discussed in the previous section, this can be done in the classical limit by means of a suitable feedback loop to produce the required nonlinearity. A big challenge would be to investigate experimentally the quantum limit of the optical version of the kicked top, the interesting regime being where the total number of photons in the ring resonator N = 2 j = 10-104. Smaller values of N imply complete dominance of quantum fluctuations over the universal (i.e., j-independent) properties. For larger values of N, quantum effects are presumably difficult to detect. In experiments on coherence and noise properties of lasers, quantum fluctuations can dominate technical fluctuations for photon numbers up to 103-105 (Welford and Mooradian [ 19821). The dynamical behavior of a top withJ2 = j ( j + 1)that is periodically kicked at time intervals T, was predicted to be classically chaotic on time scales 2 T Inj, and to become quasiperiodic on time scales 2 T, (Haake, KuS and Scharf [ 19871). Classical chaos is thus expected to be “dead” after j kicks; this quantum suppression of classical chaos can be seen as a consequence of quantum interference (Fishman, Grempel and Prange [ 19821, Grempel, Prange and Fishman [ 19841). An optical realization ofthe kicked top should therefore be designed such that the “window” between the two types of dynamics is as large as possible; i,e. N ( = 2 j ) should be as large as possible. Values of N in the range 10-104 correspond to such low intensities that bulk-optical nonlinearities are clearly inadequate from a practical point of view.
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The hybrid nonlinearity shown in fig. 31 also works for low intensities, since the strength of the nonlinearity is determined by the amplifier in the electronic feedback loop. In the quantum limit, however, some complications arise, the main one probably being that the hybrid nonlinearity relies on a measurement. Obviously, the photodetectors in the feedback loop would have to detect at least a few photons to have any feedback. In the quantum regime where N % 10-104, the removal of a few photons cannot be neglected. Moreover, since photons are coupled out of the ring for feedback purposes, the hybrid nonlinearity introduces an optical loss rate. At the same time the output coupler opens a channel for the input of vacuum fluctuations. Compensation for the loss by an optical amplifier is not allowed, since this would introduce noise on the quantum level. It seems likely that the driven optical top has an interesting quantum limit, although it remains unclear whether the quantum limit of the driven optical top is also the implementation of a driven quantum top. Clearly, the problems encountered here belong to the realm of quantum measurement theory.
6.6. HYBRID NONLINEAR OPTICS
As the preceding discussion shows, extending the optical atom with an electric feedback loop opens a wealth of experimental possibilities. In this way we can make the tuning and coupling parameters a function of the intracavity fields; i.e., we can make the state of the “atom” influence the parameters of the driving field, or even the atom itself. Such artificial nonlinearities have been known for a long time in a rather different context - so-called “hybrid nonlinear optics” (Smith, Turner and Maloney [ 19781, Mitschke and Fliiggen [ 19841). The generic example of this is the insertion of an electro-optic phase modulator inside a Fabry-Perot, the modulator being driven by a voltage derived from the intensity of light transmitted by the Fabry-PCrot. The refractive index inside the Fabry-Perot then becomes intensity dependent, as if the Fabry-Perot was filled with a f 3 ) nonlinear-optical medium. The general motivation of hybrid nonlinear optics is to produce a bistable device for use as an optical switch; also chaotic dynamics have been studied (Mitschke and Fliiggen [ 19841). Although our motivation is different, namely, the implementation of popular model systems of quantum optics, the advantages of using an artificial, feedback-induced nonlinearity are the same including, (a) very low optical power requirements, (b) no limitation by material response times, (c) possibility of electrical control, and (d) possibility of tailoring resonator
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characteristics. Finally, we note that hybrid nonlinearities have also been considered in the context of squeezing (Haus and Yamamoto [ 1986]), possibly offering another situation where quantum effects are important in the optical atom.
8 7.
Conclusions
We believe that the driven optical ring resonator, described in this chapter, forms an interesting system for simulating resonance physics of microscopic systems. An important advantage of the macroscopic nature of this system is that all parameters are accessible to direct experimental control, including those in which the corresponding microscopic quantum system would be difficult or impossible to vary. This will allow, we think, verification of many theoretical predictions in the field of optical resonance in parameter ranges that were previously experimentally inaccessible. One broad class of problems connected with the optical atom is involved with the transition between classical and quantum physics. We believe that the optical atom is well suited to explore how far one can push classical systems to simulate quantum ones, and so identify which phenomena are exclusively quantum mechanical in nature. The optical atom allows, in principle, exact realization of popular model Hamiltonians of quantum optics, and thus allows “ideal” experiments. Equivalently, the system can be considered as a dedicated optical analogue computer, enabling it to serve a useful purpose, given the present scarcity of microscopic realizations. One may ask whether the optical character of our dedicated analogue computer is essential or simply convenient from a practical point of view. In other words, could one equally well use “acoustic” or “electrical” instead of “optical” atoms? We think that, in principle, the other varieties could be used also, although it is less easy to think of practical hardware implementations. In this context optical technology brings important advantages.
Acknowledgements
We gratefully acknowledge M. V. Berry, F. Haake, E. R. Eliel, G. Nienhuis and R. Centeno Neelen for stimulating discussions. We thank N. J. van Druten and M. W. Beijersbergen for their contribution to the experiments. Our work is part of the research program of the Foundation for Fundamental Research on Matter, and received financial support from the Netherlands Organization for Scientific Research.
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Lenstra, D., and S. H. M. Geurten. 1990. Opt. Conllnun. 75. (13. Lenstra, D., L. P. J. Kamp and W. van Hacringcii. IYXh. Opt. C'omnlun. 60. 339. Loudon, R., 1983, The Quantum Theory of Light. 2nd I d (Clarcndoii Prcss, Oxford), Marcuse, D., 1980, Principles of Quantum Electronics (:4cadcmic Prcss. New York) p. 55. Marcuse, D., 1984, IEEE J. Quantum Electron. QE-2k 1139. Meerts, W. L., 1. Ozier and J. T. Hougcn. 19x9. J . Chcm. P h y . 90. 4681. Menegozzi, L. N., and W. E. Lamb Jr. 1Y73. Ph!s. Rev. A 8. 2103. Merzbacher, E., 1970, Quantum Mcchanics. 2nd Ed. (IG'ilcy. New York) ch. 6. Messiah, A., 1970, Quantum Mechanics (North-Holl;ad. Anistcrdnni) Secs. V.16 and V111.20. Milonni, P. W., 1976, Phys. Rep. 25, I . Milonni, P. W., J. R. Ackerhalt and H. W. Galbraith. 19x3. Phys. Rcv. Lett. 50, 966. Mitschke, F., and N. Flilggen. 1984. Appl. Phys. B 35. 5Y. Munz, M., and A. Kujawski. 1990. Europhys. Lett. 13. 103. Rempe, G., H. Walther and N. Klein. 1987. Phys. Rev. Lett. 58, 353. Rosenhouse, A., 1991, J. Mod. Opt. 38. 26Y. and references thercin. Ruyten, W. M., 1990. Phys. Rev. A 42. 4226. 4246. Shirley, J. H., 1965, Phys. Rev. B 138. 979, Shore, B. W., 1990, The Theory of Coherent Atomic Excitation (Wilcy. New York). Silverman, M. P., and F. M. Pipkin, 1972, J. Phys. B 5, 1844. Smith, P. W., E. H.Turner arid P. J. Maloney. 1978. IEEE J. Quantuni Electron. QE-14, 207. Spreeuw, R. J. C., and J. P. Woerdnian. 19Yla. Phys. Rcv. A 44. 4765. Spreeuw, R. J. C.. and J. P. Woerdnian, 19Ylb. Physica B 175, 96. Spreeuw, R. J. C., M. W. Beijersbergen and J. P: Woerdman, 1992, Phys. Rev. A 45, 1213. Spreeuw, R. J. C., R. Centeno Neelen, N. J. van Druten. E. R. Eliel and J. P. Woerdman, 1990, Phys. Rev. A 42, 4315, and references therein. Spreeuw, R. J. C., E. R. Eliel and J. P. Woerdman, 1990, Opt. Commun. 75, 141. Spreeuw, R. J. C., N. J. van Druten, M. W. Beijersbergen. E. R. Eliel and J. P. Woerdman, 1990, Phys. Rev. Lett. 65, 2642. Spreeuw, R. J. C., J. P. Woerdman and D. Lenstra, 1988, Phys. Rev. Lett. 61, 318. Stenholm, S., 1972, J. Phys. B 5, 876. Stoneman, R.C., D. S. Thomson and T. F. Gallagher, 1988, Phys. Rev. A 37, 1527. Suominen, K.-A., B. M. Garraway and S. Stenholm, 1991, Opt. Commun. 82, 260. Swain, S., 1974, J. Phys. B 7, 2363. Vahala, K., and A. Yariv, 1983, IEEE J. Quantum Electron. QE-19, 1096. Welford, D., and A. Mooradian, 1982, Appl. Phys. Lett. 40,865. Woerdman, J. P., and R. J. C. Spreeuw, 1990, in: Analogies in Optics and Micro-Electronics, eds W. van Haeringen and D. Lenstra (Kluwer, Dordrecht) ch. 9. Yablonovitch, E., T. J. Gmitter and K. M. Leung, 1991, Phys. Rev. Lett. 67, 2295. Yoo, H. I., and J. H. Eberly, 1985, Phys. Rep. 118, 239. Zener, C., 1932, Proc. R. SOC.London A 137, 696.
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E. WOLF, P R O G R E S S IN OPTICS XXXl 0 1993 ELSEVIER SCIENCE PUBLISHERS B.V.
VI
THEORY OF COMPTON FREE ELECTRON LASERS BY
G . DATTOLI, L. GIANNESSI, A. RENlERl and A. TORRE ENEA - Area INN, Dip. Sviluppo Tecnologie di Punta P.O.Box 65 - 00044 Frascati. Rome, Italy
321
CONTENTS PAGE
. . . . . . . . . . . . . . . . . . . 323
§ 1 . INTRODUCTION
§ 2 . SPONTANEOUS EMISSION BY RELATIVISTIC
ELECTRONS MOVING IN AN UNDULATOR MAGNET . 333
§ 3 . T H E FEL GAIN
. . . . . . . . . . . . . . . . . . . .
§ 4 . TRANSVERSE MODE DYNAMICS
350
. . . . . . . . . . . 364
. . . . . . . . . . . . . 370
§ 5. LONGITUDINAL DYNAMICS
§ 6 . FEL OSCILLATOR REGIME A N D T H E PULSE
PROPAGATION PROBLEM § 7.
FEL SATURATION
. . . . . . . . . . . . . . 376
. . . . . . . . . . . . . . . . . . 387
§ 8. A SIMPLIFIED VIEW O F FEL STORAGE RING
DYNAMICS . . . . . . . . . . . . . . . . . . . . . .
393
. . . . . . . . . . . . . . . . . . . . .
396
§ 9. CONCLUSION
APPENDIX A . OPTICAL CAVITY FOR T H E FEL
. . . . . . 396
APPENDIX B . UNDULATOR MAGNETS FOR T H E FEL . . . 406 REFERENCES
. . . . . . . . . . . . . . . . . . . . . . .
322
411
1. Introduction
This introduction presents a simplified and heuristic approach to the free electron laser (FEL) that frames it within the context of coherent sources. Powerful coherent light can be generated either by the stimulated emission of an inverted atomic or molecular medium or by free electrons moving in a waveguide or an undulator magnet. Whereas the first mechanism is that of a conventional laser, the second is simply a travelling wave tube (TWT). The FEL is a TWT that can operate on a wider range of frequencies (in principle from VUV down to microwaves) (Dattoli and Renieri [ 19851). In the field of coherent generation with an electron beam, the FEL introduced the possibility of overcoming all the problems connected with the miniaturization of the microwave tubes needed to operate at a shorter wavelength. The basic ingredients of a FEL are (see fig. 1.1) ( 1) the accelerator, which provides a high-energy and high-quality electron beam; (2) the undulator, namely, a series of N-S magnetic poles, arranged as in fig. 1.2; and (3) the optical cavity, needed if the system operates in the oscillator configuration. The interaction takes place in the undulator magnet (UM) that transforms the longitudinal nonradiating electron motion into transverse motion, which allows the coupling to, and the amplification of, a copropagating TE wave. In fact, as noted earlier, the basic principle of a FEL is that of a TWT, and can therefore be summarized as follows. Inside the undulator, due to the Lorentz force, the electron acquires a transverse velocity, and a copropagating TE wave can therefore couple to the electron beam and produce an energy modulation. This energy modulation transforms into a density modulation, and a coherent amplification of the input wave occurs when the electron beam is longitudinally modulated on a scale corresponding to the wavelength of the TE wave. We can, in general, distinguish two processes for the FEL oscillator. The electrons are injected in the UM and emit Bremsstrahlung radiation, which is stored, e.g.. in an optical cavity (spontaneous emission). The radiation is then amplified by the stimulated emission, and if the gain is larger than the cavity losses, the 323
324
THEORY OF COMPrON F R t t L L t C r R O N LASLRS
Fig. 1.1. FEL with electron beam recovery. The electrons, after the interaction with radiation and U M fields, are decelerated in a special item (radiofrequency linac or electrostatic decelerating section) to recover part of their energy and charge (the "decelerator" can be the accelerator itself where the electrons are injected, e.g., for a radiofrequency linac, with a decelerating phase).
system may work as an oscillator. We use classical concepts to describe the FEL, since the theoretical understanding of this type of laser does not require quantum mechanics, which, however, can provide a microscopic and perhaps more physically insightful picture of the process than that based on a full
z
aaaaaa I
-I
'4"
-+I
I
Fig. 1.2. Undulator magnet.
VI,8 11
INTRODUCTION
325
classical description. The following discussion examines the frequency selection mechanism, spontaneous emission, stimulated emission, and gain using quantum mechanics, and points out the link with other processes, such as multiphoton dynamics. However, to frame the FEL better within the laser devices, some observations on the concept of lasers are in order. Regardless of the underlying mechanism of spontaneous emission and gain, the laser sustains steady-state oscillations with a well-defined frequency. This concept is rather old, and in electrical engineering it is well known as a self-sustained amplifier with a frequency-dependent feedback. The gain must be present to overcome the losses (unavoidable in any physical system), and the frequency selectivity fixes the oscillation frequency within the working band of the amplifier. Figure 1.3 shows a scheme for this idealized amplifier (see Stenholm [ 19841). We assume that the output Y depends on the input X through the nonlinear function Y
=
A(X),
(1.1)
and the feedback signal is assumed to be X,,
=
FY,
which is linear but depends on the frequency of the signal. When the feedback is connected, the whole system satisfies Y
=
A(X
+ FY).
(1.3)
In some conditions this equation may give Y # 0 even for vanishing input signal. The loop of fig. 1.3 and eqs. (1.1)-( 1.3) contain the essential features of a self-sustained oscillator determining its own amplitude self-consistently. This will be demonstrated to be true for FEL. As noted earlier, the undulator provides an intense periodic magnetic field that induces the necessary transverse momentum, allowing the coupling of the electron beam to a copropagating TE wave. The technological aspects of the
Fig. 1.3. Loop diagram of a feedback amplifier.
326
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, I 1
U M will be discussed later, and here we note that the magnet period 1, is typically of the order of a few centimeters, the on-axis field B, is a few kG, and the length of the undulator L , = NL, (where N is the number of periods) is of the order of meters. For ultrarelativistic electrons the undulator can be treated as a radiation field with a wavelength twice the undulator period and with a photon density given by
where
tl is
the fine structure constant and r, the electron classical radius, and
is the undulator parameter* or strength. For typical parameters ( K = 1, 1, = 5 cm) the photon density associated with the undulator is photons/cm’. Thus the process of radiation emission by electrons moving in a U M can be viewed as a Compton scattering. The wavelength of the radiation emitted in the forward direction is therefore given by
1 1 = 2 (1 + P ) , 2Y2 where yis the electron relativistic factor and (1 + K 2 )is the intensity-dependent shift predicted by Brown and Kibble [ 19641. The interesting feature of eq. (1.6) is that the emission wavelength can be tuned over a wide range, simply by varying the electron energy. This is the frequency selectivity of a FEL. We discussed the mechanism underlying the spontaneous emission process. We also pointed out that the amplification process takes place when an electromagnetic field with a wavelength nearly resonant with eq. (1.6) is copropagating together with the electrons. The analysis of the amplification process can be advantageously carried out in a reference frame where the electromagnetic and undulator fields are seen by the electrons as counterpropagating waves with the same frequency. The velocity u* of this frame can be chosen according to the
* In practical units we have
INTRODUCTION
321
definition
where A[,E ( A , , / 2 y 2 ) ( 1 + K 2 ) is the wavelength of the field copropagating with the electrons. We assume that both the laser and undulator magnetic fields are quantized and helically polarized. The F E L process therefore can be reduced to the nonrelativistic dynamics of an electron moving in the field generated by two counterpropagating electromagnetic waves. The possibility of treating an undulator magnetic field as a radiation field allows one to visualize the F E L process, when the laser field is copropagating with the electron through the undulator, as a stimulated Compton scattering. The analogy is further supported by the form of the interaction term in the Hamiltonian, as reported below. Since the F E L interaction is conveniently treated in a reference frame where the electron is moving nonrelativistically, the mathematical details of the analysis are greatly simplified. Furthermore, the equivalence of all the frames connected with a Galilean transformation allows one to choose ;I frame of reference where the laser and undulator fields appear ;IS oppositely travelling waves with the same frequency. I t is worth stressing that this is a choice of convenience and can easily be relaxed as, for instance, in the case of the niultimode analysis. Using a quantum-mechanical approach, the Haniiltonian relevant to the coupled electron-field system can be written down immediately as
The first term on the right-hand side of the expression above accounts for the kinetic energy of the electron and for the electron-field interaction, ( p - P / c A )being the generalized momentum of the electron in the presence of the vector potential A = A, + A,,, due to the laser and undulator fields, and identified with the labels L and u, respectively. The other two terms represent the free-field energy, b and 4 being the annihilation and creation operators with the well-known commutation rules +
Furthermore, according to the previous discussion, the laser and undulator fields are assumed to have the same frequency (o= k c ) and opposite wave vectors ( k , = k , , ) . Taking the explicit expression of the vector potentials
328
THEORY OF COMPTON FREE ELECTRON LASERS
[VI,8 1
A , and A, in the Lorentz gauge and assuniing circularly polarized fields, the
Hamiltonian (1.8) specializes into*
A
=
P2 2m
-
+ hw(ci; ri,
t ;)
+ hw(ri,'Li, + ;) (1.10)
where the coupling constant
a is given by (1.11)
and V is the interaction volume. The first three terms on the right-hand side of the Hamiltonian (1.10) represent the energy of the electron-field system in the absence of interaction. The last one describes the dynamics of the interaction, which according to the present model of the FEL process as a stimulated scattering, gives rise to a forward and backward scattering corresponding to the creation of a laser photon, the destruction of an undulator photon with the consequent loss of 2 h k of electron momentum, and to the destruction of a laser photon, the creation of an undulator photon with the consequent gain of 2 h k ofelectron momentum. The evolution of the system is then governed by the Schrddinger equation for the wave function $(t), or equivalently by the Heisenberg equation for the electron and field operators, appropriately defined. From the Hamiltonian ( 1.10) two conservation rules can be derived, namely, 17,
+ nu = const.
and
p
+ hk(n,
-
n u ) = const.
(1.12)
The first equation states the conservation of the total number of photons (laser + undulator), and the second establishes that the total linear momentum (electron t fields) is also a constant of motion. The F E L process is therefore clear: the undulutorphotons are transferred to the laser mode, and the necessary momentum is provided by the electron. The foregoing conservation rules allow a further simplification: the quantum states, on which the Hamiltonian (1.10) acts, can be characterized by a single integer specifying the number of photons exchanged during the interaction, namely, (1.13)
* Note that the term p * A in (1.8) vanishes, since A is transverse and p is assumed to be directed along the z axis.
VI.0 11
3 29
INTRODUCTION
where nf and n," are the initial umber of photons in the laser and undulator modes, respectively, the Cr(t)are the amplitude probability that I photons can be exchanged in the process, and $ is an arbitrary phase. The Schrodinger equation yields the following differential finite-difference equation for C,: i C;
= - (v
-
&I)IC,+ p
(
J
m C,,
I
+
,/mC,
- I),
( 1.14)
where the prime means derivation with respect to the dimensionless time T = ! / A t (Ar being the interaction time) and (1.15)
In deriving eq. (1.14) we have assumed % I. The equation governing the evolution of C, is strongly reminiscent of the equation defining the amplitude probability for an n-level molecular system coupled by a laser field (Stenholm [ 19841). The parameter v plays the role of the detuning, and E (in this case linked to the electron recoil) plays the role of the anharmonic term. Methods have been developed to deal with a particular class of equations of the type (1.14) (Dattoli, Gallardo and Torre [ 1986]), namely, the so-called Raman-Nath equations developed by Raman and Nath [ 19361 for light scattering by ultrasounds. In this chapter we restrict ourselves to the solution in a very limiting, interesting case, namely, the so-called small-signal regime, i.e., when pfl41.
(1.16)
Within the framework of our analogy with multiphoton transitions, the above quantity represents the Rabi frequency, so that the condition (1.16) is a type of weak-coupling limit, which allows one to treat eq. (1.14) perturbatively,
c,
- C,o +
pC,' ,
CY(0) =
a/, (,,
CJO)
=
0.
(1.17)
Inserting eq. (1.17) into eq. (1.14), we obtain
c:'(T)= 1 , C,'(Z) = p
J
m
1 - exp(i(v - E ) O )
(1.18)
V - &
C',(T)
= -p&
1 - exp(i(v
+ E )T)
V + &
It is worth stressing that the condition (1.16) allows only one photon to be
330
[VI,§ 1
THEORY OF COMPTON FREE ELECTRON LASERS
cmitted or absorbed in the process. Furthermore, it is evident from eq. ( I . 18) that the electron loses or gains an amount of energy (A&/Af) according to whether the photon is absorbed or emitted. The average number of emitted photons is therefore given by An,-
=
1 C,‘ 1’
-
I C! , I 2 sin+(v -
v
+ &)
Expanding up to the lowest order in AnL
N
p2
(T) siniv
-
E,
E)
. (1.19)
we finally obtain
p2(2n0, + 1 ) ~-
(1.20)
We can now understand eq. (1.20) in terms of the earlier conceptual scheme. The output signal A n , is a function of the input signal n:, and even when I$ = 0, we still have an output sin: v
(1.21)
The first term is the classical spontaneous emission, and the second is the quantum-mechanical spontaneous emission, or better, the gain due to the vacuum field fluctuations. The part containing n0, will be referred to as stimulated emission or gain term, as for conventional lasers. The feedback mechanism can be easily introduced by iterating our procedure. We assume that the process occurs in an optical cavity, and therefore after each roundtrip the number of emitted photons depends on those emitted at the previous passage of the electron beam in the undulator, namely (n, being the number of roundtrips),
where N , is the number of electrons in the beam. The feedback depends on thc detuning v, and we will see below that this automatically fixes the lasing frcquency. In fig. I .4 the classical spontaneous emission function and the gain function versus v are given and it is evident that the positive gain bandwidth
33 1
INTRODUCTION
b)
0.2
-10 -8
-6
-4
-2
0
2
4
6
8
10
V
Fig. 1.4. (a) Small-signal gain and (b) spontaneous spectrum.
is Av
-
272. Equation (1.22) can be transformed into a differential equation
(1.23)
332
THEORY OF C O M P l O N I RI I F I I C l RON 1 ASERS
10’0
108
106
1 on
102
-20
0
-10
20
10 b’
Fig. 1.5. Laser spectrum versus v with (Au
=
3 cm; k
=
0.3; N,/V
=
10” cm’; N = 50; and
y = 50).
the solution of which can be immediately obtained*
(1.24) where (171.)11, = 1/28 represents the threshold value of emitted photons for which the stimulated part of the emission becomes larger than the spontaneous part. The emitted spectrum versus v is shown in fig. 1.5. With increasing 11, we see that the peak shifts from v = 0 up to v = 2.6, where the maximum of the stimulated emission is located. The feedback therefore contributes to the frequency selection within the gain bandwidth and to the self-determination of the output signal amplitude. We have not mentioned intrinsinc mechanisms, which cannot be present in the linear analysis so far discussed. We notice that with increasing p f i - the number of emitted photons also increases. For large I the anharmonic tern1 increasingly affects the detuning, thus “pulling” the region of operation outside the positive gain bandwith. The condition for the saturation
*
In deriving eq. (1.24) we neglect the term & ( a / a v ) [ s i n i v ) / ( i v ) ] * , which is always very small.
VI, 8 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
333
threshold is therefore 1, x 2 4 & ,
(1.25)
which results in a very interesting conclusion. Using the explicit definition of E, we find the saturation energy (I: = Also) is given by
(1.26) where A' and LA are the laser wavelength and undulator length in the moving frame. Equation (1.26) does not contain A, and therefore the saturation process is purely classical. Returning to the laboratory frame, we find
1 Iszmyc2- ; 2N
(1.27)
i.e., a fraction 1/2N of the total electron energy is transformed into laser energy. This fraction is the FEL efficiency. This simplified analysis only gives a partially complete picture of the FEL process as photon transfer from the UM to the laser mode, and provides a simple description of the saturation mechanism. We have not, however, included the real structure of the UM, treated as a rather abstract device, or the characteristics of the electron beam. The following sections will be devoted to a detailed description of FEL.
8 2.
Spontaneous Emission by Relativistic Electrons Moving in an Undulator Magnet
We saw that the emission process by relativistic electrons moving in a U M can first be understood as a kind of Compton scattering. This treatment gives the emitted wavelengths at different angles but does not specify the harmonic content of the radiated spectrum, the influence of the undulator parameters on the spectrum shape, or the distortion induced by the electron beam characteristics. The detailed theory of the emission in a U M is more complex than that outlined in 0 1.
334
THEORY OF COMPTON FREE ELECTRON LASERS
2.1. QUALITATIVE INTRODUCTION
The spectral properties of the radiation emitted by an electrom beam passing through an undulator magnet are generally well understood. In the past 20 year several papers and review articles appeared that described various properties of this radiation (Alferov, Bashmakov and Bessonov [ 19761, Didenko, Kozhevnikov, Medvedev, Nikitin and Ya [ 19791, Kitamura [ 19801, Krinsky [ 19801, Brown, Halbach, Harris and Winich [ 19831, Tatchyn and Lindau [ 19841, Kincaid [ 19841, Colson, Dattoli and Ciocci [ 19851, Tatchyn and Qadri [ 19851). Some of these papers assume that the electron is mono-energetic and without angular and spatial distribution. Furthermore, the undulator field pattern is assumed to be a pure sinusoid. These assumptions make the problem manageable analytically and yield approximate results for the spectral intensity. In some cases, however, nongeneral approximations can lead to large errors when the analysis is applied to off-axis emission from low-period undulators (see Tatchyn and Qadri [ 19851). Only recently serious efforts were undertaken to include the effects of energy spread and emittances, which can significantly change the spectral features. Moreover, real-life undulators can have field patterns that are significantly different from pure oscillatory functions, due to random errors in field intensity and magnetization direction, for example. The electron trajectory can therefore be nonperiodic. Not all of these features are accounted for analytically, and the numerical analysis is a necessary step. A common additional approximation consists of assuming a large distance between the source and the observer if compared with the undulator length [ far-field (FF) approximation]. Although analytical results can be obtained more easily using the FF assumption, it is unnecessary for the numerical computations, even though it continues to be used in more recent numerical analysis. Tatchyn and Qadri [ 19851). The method proposed by Kim [ 19851of calculating synchrotron radiation using the Wigner distribution is valid for both F F and near-field (NF) regimes, but so far the results have been confined to the F F case. The importance of N F effects has been emphasized when radiation is viewed directly (i.e., unfocussed) offaxis, even if the distance from the source is large compared wih the undulator length, due to the variation in phase of radiation arriving from different parts of the undulator. Its importance was also stressed by Barbini, Ciocci, Dattoli and Giannessi [ 19901,who pointed out that the N F may contribute appreciably to the on-axis emission if the electron beam has an angular divergence. These observations clearly indicate that the spectral properties of undulator radiation arc cffectcd, somctinies dramatically, by different effects whose contributions cannot be easily separated.
VI, B 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
335
This section presents general considerations of the characteristics of undulator magnet radiation in view of their importance for the laser performance itself, with particular attention to the assumptions that lead to an analytically tractable problem. The dynamics of an electron moving in an undulator magnet can be described using the classical Lorentz equation. To simplify the mathematics of the analysis, we first consider a helically polarized undulator whose field is represented by
where B,, is the on-axis peak field and A,, is the undulator wavelength. The transverse velocity components p1 = (u,/c, L?,,/C, 0) can be integrated simply, over one undulator period, by noting that the force is a perfect time derivative. The result is ypl
=
-K(cos(y
z),sin(?
z ) , ~ ) ,
where K is the undulator strength, defined in eq. (1.5). Since
p: + p ; =
(
1-
the longitudinal velocity pz can be immediately inferred. Substituting eq. (2.2) into eq. (2.3), we obtain
&%[I
-2y’1
1
(1 + K 2 )
(2.4)
The spectral and angular profiles of undulator radiation are the result of an interference process of radiation emitted at different “source points” in the undulator. Therefore, radiation emitted in the forward direction (on-axis) from source points A and B separated by the magnet period A,, will interfere constructively when the path difference d between the wave fronts from A and B is an integer number of wavelengths, namely,
336
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, !2
Inserting eq. (2.4) into (2.5), we immediately obtain the familiar expression for the output wavelength (n = 1)
I
1, = 2 (1 2Y2
+ K2)
Considering that the electromagnetic field is a periodic function of time with a number of cycles equal to the number of oscillations performed by the electron through the undulator, we have
A- =1 I
1 2N
Similarly, for radiation emitted off-axis (see fig. 2.1) we find
I ~="-I,cos~=I,,
8,
which for small angles yields
Equation (2.9) contains the dependence of the output wavelength on the electron energy, undulator magnetic field and observation angle or off-axis electron transit angle. Let us now return to eq. ( 2 . 2 ) and note that 6, = K / y is the maximum deflection angle ofthe electron trajectory from the undulator axis. The qualitative aspects of the undulator radiation depend strongly on the magnitude of the deflection angle compared with the natural opening angle 6, = l / y of synchro-
Fig. 2.1. Interference of radiation emitted at an angle Owith respect to the undulator axis in the far-field case.
VI, 8 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
331
tron radiation. For K 4 1, 0, 4 0,, and the radiation produced has rather simple properties that can be calculated in the so-called dipole regime. For K 2 I , the trajectory wiggles by more than the natural opening angle, thus leading to a more complicated radiation pattern. We also saw that the particle moves with a drift velocity &c along the z axis. Following Hofmann [ 19861, we define a Lorentz factor of drift motion (2.10) It is useful for our purposes to investigate the particle motion in a frame F' that moves with the drift velocity B,c. The Lorentz transformation
(2.11) yields (2.12) and the transverse component oscillates at a frequency w'
=
y*w,,
0,=
2nc/A,.
(2.13)
Furthermore, the undulator moves in the - z ' direction with a contracted period A: = Au/y*. The transverse harmonic motion executed by the charged particle is an oscillating dipole emitting circularly polarized radiation at the frequency w ' , with a large opening angle. Transforming back to the laboratory frame, the radiation becomes contained in a cone of opening angle l/y* in the direction z. The radiated spectrum consists of a single Doppler-shifted frequency in the laboratory frame, and is given by [see also eq. (2.6)] (2.14) When K 4 1, the radiation is mostly confined in the forward direction, and the dipole approximation is sufficient to clarify the main features of the emission process. When K 2 1, a significant amount of radiation can be emitted off-axis, and the analysis becomes more difficult and less intuitive. The off-axis radiation properties cannot be derived from the simple dynamics of a charged harmonic oscillator, and higher-order harmonic contributions become important, as will
338
[VI, I 2
THEORY OF COMPTON FREE ELECTRON LASERS
be clarified. In the plane undulator case we have higher-order harmonics emission even on-axis, which can be shown qualitatively by an analysis of the electron trajectory with the field distribution given by 5 = [0,5,) cos(2xz/lU),01 (see fig. 2.2). When K < y, the equations of motion can be easily integrated. The transverse component is given by (fir1) R,
where K locity is
=(? 2
(2.15)
c o s ( F z),O,O),
=
e ( B 2 ) ' I 2 ,?,,/2nmoc2and ( B 2 ) ' I 2
+ 2K2sin2(2
-$[1
=
B,/$.
The longitudinal ve-
(2.16)
z)]},
which, averaged over one undulator period, yields the average longitudinal velocity drift p*=1-
(1
+ K2)
(2.17)
2Y2 The particle motion in the undulator is finally specified by cos
(n,z ) , 0, 211
fi*ct
,o,cp*
+ K21, ~
8ny2
+c
:(
sin
- cos
Y
Fig. 2.2. Geometry of undulator radiation.
z)), (2.18)
VI, § 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
339
- 1 -0.5 0 0.5 1 Fig. 2.3. Motion of aI particle in the undulator as seen in a moving frame.
In this case it is also convenient to view the process in the frame moving with velocity (2.17), thus obtaining (at the lowest order in y)
(2.19)
resulting in the moving frame in the following “figure-eight’’ trajectory (see fig. 2.3): 2’2
=
8(1 + K 2 )
(2.20)
which gets broader as the value of K increases. For K e 1, the electron motion in F’ is a simple harmonic motion along the x direction; thus the analysis already presented for the helical undulator still holds. For K 2 1, the eight-like motion can be decomposed into harmonic oscillations along x ‘ with odd harmonics of the frequency y*wu, and harmonic oscillations along z’ with even harmonics of frequency. The odd harmonics are therefore radiated in the z’ direction, and the even in the x’ direction. The transformation into the laboratory frame results in radiation emitted mainly in the forward direction, where the odd harmonics are emitted on-axis and the even harmonics are confined in a cone of aperture l / y * around it (see Hofmann [ 19861 and fig. 2.4).
340
THEORY O F COMPTON FREE ELECTRON LASERS
MOVING FRAME
LABORATORY FRAME
z I \
/
\
I
.-a
Fig. 2.4. Odd and even harmonic emission mechanism.
The line shape of the radiated spectrum can also be derived using a heuristic argument. In fact, both in F' and the laboratory frame the emitted radiation consists of a wave train with N periods. Thus, due to the finite duration of the interaction, its spectral shape is given by the familiar form (2.21)
2.2. SPECTRAL BRIGHTNESS CALCULATION OF UNDULATOR MAGNET RADIATION
In view of the importance of undulator radiation in FEL physics, we will describe a detailed and carefully illustrated derivation of the undulator radiation brightness both in the linear and the helical cases. The spectral properties of the undulator radiation can be derived most easily from the Lienard-Wiechert potentials (Jackson [ 19751) d2/ - e2w2 dodf2-4ffZc
-!,I
x.
n x (n x fi)exp{io(r - y ) } d t 1 2 .
(2.22)
We analyze the linear case first, adopting a technique different from that generally used which benefits from the introduction of a new class of Bessel functions, whose properties were discussed within the framework of nondipolar scattering problems (Dattoli, Giannessi, Mezi and Torre [ 19901). In evaluating . l/y, we eq. (2.22), we will keep contributions up to the order ( K / Y ) ~Since have n=(+cos$,+sin+, 1
+-
-i+2),
1
c
K2
sin(2w"t).
(2.23)
VI, 5 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
34 1
Furthermore, within the same approximation the components of the vector product in eq. (2.22) are given by ( n x ( n x j?))I
z $ cos $ - J z K cos (out) 1
~
Y
(n x ( n x j?))2z $sin$, (n x (n x
z
(2.24)
J z K $cosq5cos(Wut)-
~
$2
Y
The oscillating part on the right-hand side of eq. (2.22) can be written as
I x
f
exp(-irno,t)
m= - x
$
x J,,(' 0, =
[
exp - irno,t n7= -
x
d?
cos$, --
-
Y
( m z")]~ Y (2.25)
where Jn,(x,y ) is a generalized Bessel function of the first kind, whose generating function is provided by
and where o 1is the first harmonic resonance frequency according to eqs. (2.9) and (2.14). The integral in eq. (2.22) must be calculated in a region where an effective acceleration exists, namely, for the interaction time ranging from 0 to T = L,/c z 2nN/w,, where L , is the undulator length. For later convenience we split the integral into three contributions, coming from the components
342
THEORY O F COMPTON FREE ELECTRON LASERS
tVI, 5 2
(2.24), namely,
(2.27)
where v,,,
=
2 RN (mwl
- w)
(2.28)
mw1
The last term in eq. (2.27)has been neglected, since it comes from the third part of eq. (2.24),and once the square modulus in eq. (2.22) is taken, its contribution to the intensity radiated per unit solid angle and frequency bandwidth is of the order of ( K / Y ) ~ . The function Ifn,(v,,,) is defined as
2 n N sin;mv,,,
-~ wu
I
exp( - iimv,) .
(2.29)
Zt“‘rn
Therefore, for large N the spectrum will consist of a series of sharp peaks centered at w,,, = m w l . Introducing the parameters
r = ( 1 + K;2 + y2i,h2j K2
9
x = 1 + K2 2K Y+$y 2 $ 2
,
z= Jz x cos $, (2.30)
the functions S::
n,
can be written as
S!:)
=
i,h cos $ J , , , ( ~ z ,- m t )
Sf:)
=
I(/ sin $ J,,,(mZ, - m t ) ,
(2.31)
where we replaced w with wn, in view of the sharpness of the resonance (2.28).
VI, § 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
343
The shifted indices (m f 1) in S(a) are due to the combination of the exponential in eq. (2.22) with oscillating term ( K / y ) cos(w,t) in the first part of eq. (2.24) Combining eqs. (2.27) and (2.22), we finally obtain the following synthetic expression for the spectral brightness in the linear case
fi
(2.32) In figs. 2.5 and 2.6 we show a three-dimensional plot (w - 0) of the brightness produced in an undulator of 5 periods with K = 0.5 and K = 1, respectively. It is evident that while K increases, the contributions of higher harmonics become more and more important. For large K the contributions deriving from the nondipolur motion grows, thus providing a significant emission of higher harmonics. The subscripts o and n refer to parallel and vertical polarizations, respectively. Keeping contributions up to, e.g., ( K / Y ) a~ longitudinal , polarization mode L should also be observed. A simple case to be analyzed is that of on-axis emission. Observing that
J,,(O,
v)=
J , 8 / 2 ( y ) for n even 0
(2.33)
fornodd ’
we find
x
tJ(m+
l)/z(mt)- J ( m - l , / 2 ( m t ) 1 2 ~(2.34)
which amounts to saying that odd harmonics only are emitted on-axis (see also figs. 2.5 and 2.6). The preceding analysis can easily be extended to the helical case. The electron trajectory can be derived observing that the motion in the z direction is uniform with velocity (2.4), and integrating eq. (2.2). The electron executes a helical orbit in the field (2.1), specified by r(t) =
[t
sin(w,t),
K c --
-
Y wu
cos(out),cp,t
1
(2.35)
The differences with respect to the linear case are evident. The electron describes a circle in the transverse plane in the moving frame F ’ . Therefore,
344
THEORY OF COMPTON FREE ELECTRON LASERS
0 Fig. 2.5. Three-dimensional spectral brightness plot from linear undulator (5 periods, K versus r = w/(2y2w,) and r$.
=
0.5)
we do not expect emission of higher harmonics on-axis. Using the same approximation as before, we find (2.36)
2
3
0 Fig. 2.6. Three-dimensional spectral brightness plot from linear undulator (5 periods, K versus r = w / ( 2 y 2 w , , and ) y$,
=
1.0)
VI, § 21
345
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
and [ n x (n
+ B ) ] ] 2 $cos+
-
K -
COS(W,f),
Y [ n x (n
+ B)Iz 2 I+I sin q5 - K-
sin w,r ,
(2.37)
Y [n x (n + B ) ] 3 2 0 Furthermore,
where q denotes the quantity q=-
w K
wu Y
$.
(2.39)
In deriving eq. (2.38), we used the generating function (2.40) for the cylindrical Bessel function of the first kind J n ( x ) . Following the same procedure as before, we obtain ~
i
n m== - r
T, = m= -
r
K {I+IIcos$Jm(q)-- Y [ e l + ~ m + l ( q ) + e - l + ~ m l ( -q ) ~ ] ~ m ( v n r ) * i K {I(/ sin +Jm(q) + - - [el+Jm + 1 (4)- e 2Y
T,gO.
i+Jm -
I(
I
~ I I~H,(v,pi) 7
(2.41)
For the same reasons discussed earlier, namely, large-N limit, we end up with d21 - 8(e yN)' dwda
c
5 (mm)'
m=l
(si; rrnvm)' 1 Zmvm
(2.42) Note that the helical undulator brightness is azimuthally independent, and
346
THEORY OF COMPTON FREE ELECTRON LASERS
0 Fig. 2.7. Three-dimensional spectral brightness plot from helical undulator (5 periods, K versus r = 0 / ( 2 y 2 w , ) and y$.
=
0.5)
as expected, only the fundamental harmonic is radiated on-axis, namely, d2I
(2.43)
Figure 2.7 shows a three-dimensional view on the plane (w - $) of the brightness (2.42).
2.3. INHOMOGENEOUS BROADENING
We have endeavored to clarify the emission and higher harmonic generation mechanisms as being the result of a single particle. We have not considered the effects induced by the real structure of the electron beam exhibiting finite energy spread and emittances. According to eq. (2.9), the resonance wavelength depends on the electron energy, observation angle, and undulator field, which is a function of the electron position in the transverse plane, as will be described below. It becomes evident that an energy different from the nominal one and a divergence or a transverse displacement of the electron will induce a frequency shift of the resonance condition, resulting in an inhomogeneous broadening of the line emitted by the beam. The frequency shift induced by an electron moving with an energy slightly different from the nominal value can be directly
VI, § 21
SPONTANEOUS EMISSION BY RELATIVISTIC ELECTRONS
347
evaluated from eq. (2.9), thus obtaining
(?)&(:). =
2
(2.44)
We can define an inhomogeneous broadening of the radiated spectrum as the RMS frequency shift over the electron beam distribution. For a Gaussian energy distribution with RMS a,, we obtain (2.45) We therefore expect the peak intensity to be decreased by a certain amount and the spectral width to become larger, since the natural width combines quadratically with the inhomogeneous contribution (2.45), i.e., (2.46) Here, p, is defined by the ratio (Am/m)E/(Am/m)o,and measures the relevance of the inhomogeneous broadening contributions with respect to the natural broadening (2.7). Figure 2.8 shows the on-axis emitted spectrum for different values of pE,and 3.5,
-24 Fig. 2.8. Undulator radiation frequency spectrum versus p e
348
THEORY O F COMPTON FREE ELECTRON LASERS
3.2 I
1.8 I
0
I
I
0.10
I
I
I
0.30
0.20 4 %
Fig. 2.9. Peak value of the spontaneous emission spectrum versus pa.
the expected behavior is clearly displayed. The peak value of the spectrum versus pLEis shown in fig. 2.9, and can be reproduced by the following simple Lorentzian form: (2.47)
Two other sources of inhomogeneous broadening that should be taken into account are due to the natural beam divergence and to its transverse dimensions. With respect to the latter point, we must emphasize that the field distribution (2.1) is valid only very close to the undulator axis and contains no information on the transverse coordinates ( x , y). When this dependence is properly included, one realizes that a beam with finite transverse dimensions explores regions with different B and thus with different K . Each electron therefore emits at different wavelengths, thus resulting, according to the previous discussion, in a further inhomogeneous broadening. A similar argument holds for the beam divergence. Since the calculation of the relevant inhomogeneous broadening is computationally rather troublesome, the interested reader is referred to appendix B. Only the final result is given here:
(2.48)
v1, I 21
349
SPONTANEOUS LMISSION H \ 1<1 I A I I V I S T I C ELECTRONS
1
a.u.
-a
-4
__
4
8 V
r
0
2
1
3
Pcw =P,=P
Fig. 2.10. (a) Effect of the emittance on the spectrum shape ( p x = 1, p,, intensity versus p x = py for p, = 0.
=
0); and (b) peak
which combine quadratically to give the total inhomogeneous broadening due to spatial and angular distribution of the electron beam. The quantities a*\, , and a,.,. are the RMS of the angular and spatial distributions, respectively. The terms h,,, depend on the magnet geometry and obey the general relation h,
+ h,, = 2 .
(2.49)
In a helical undulator we have h, = h), = 1 because f the cylindrical symmetry, and the beam is focussed in both vertical and horizontal directions. In a linear undulator, assuming the field parallel to the y axis, we have h,= -6,
h,
=
1+
6 (6-g l ) ,
(2.50)
where 6 is a small sextupolar term along the x direction inducing defocussing in the y direction (see appendix B). We introduce the beam emittance as*
The effect of the inhomogeneous broadening can be minimized by setting the transverse dimensions at (2.52)
* The definition (2.5) for the emittance is not strictly correct. For a more general definition the reader is referred to Dattoli and Renieri [I9851 and references therein.
350
THEORY OF COMPTON FREE ELECTRON LASERS
[VI,5 3
thus, finally obtaining an emittance inhomogeneous broadening given by (2.53)
The relevance of emittance contributions to the emission process is fixed by p.y.,. = (Aw/w),,.,~, (Aw/o); ’). In fig. 2.10 the effect of the emittance on the line shape is shown for different pL.r,y. The broadening and consequent peak depression (in this case also reproduced by an almost Lorentzian form) become more significant with increasing p.
8 3.
The FEL gain
3.1. LOW-GAIN REGIME
Stimulated emission occurs when the electrons radiate in the presence of a copropagating T E wave that is nearly resonant with the spontaneously emitted frequency. The F E L gain is exactly the relative intensity variation of this wave. It is easily understood that the process of stimulated emission in a F E L is similar to that in a travelling wave tube. The T E wave produces an energy modulation in the electron beam that transforms into a density modulation at the same wavelengths as the input radiation. The final result is therefore a coherent emission from each packet and thus an amplification of the input wave. Let us now put the previous description of the emission mechanism in a more quantitative form. We therefore consider a wave with a transverse electric field, propagating in the longitudinal direction
EL = E , cos(wt
- kz
+ @L),
(3.1)
where @L denotes the relative phase between the field and the electron. The field (3.1) couples to the transverse electron motion induced by the undulator, determining an energy variation governed by the equation
eEL-v,
j=-,
nZoC2
which, coupled to eq. (2.18), yields
(3.2)
Taking the time derivative of eq. (2.4), after averaging over one undulator period, we obtain 2
=
+ K2)
c 7 (1
(3.4)
YCombining eqs. (3.2) and (3.4), we find the equation providing the evolution of the coupled system (i.e., electron and radiation), namely,*
Equation (3.5) is the well-known pendulum equation of FEL (Colson [ 19771, Bambini and Renieri [ 19781). The quantity 51 acts as a coupling constant, and its magnitude determines the FEL dynamics. When 51- is small with respect to the characteristic time L,,/c of the interaction, the system is said to be operating in the small-signal regime. When they are both of the same order, the device operates in the strong-signal, or saturated, regime. The preceding conditions amount to the following inequalities: ( 1) Small-signal regime
'
+),
2
(3.6a)
or, more explicitly, eE,L,<<-
1
E2
1 -
~
2 n N K mOc2
.
(3.6b)
(2) Strong-signal regime
(k) 2
Q2
z
(3.7)
From the foregoing relation we can also obtain the amplitude of the electric field (3.1) corresponding to saturation, namely, E,[MV/m]
1
E __
y E[MeV] -
2nN K
____
L,[m]
The condition for the small-signal regime corresponds to a weakly perturbed
*
Equation (3.5) is valid in the case of a system operating with a helically polarized undulator.
352
THEORY O F COMPTON FREE ELECTRON LASERS
Fig. 3.1. FEL pendulum-like phase space. In region I (dashed region inside the separatrix) the motion is periodic (saturation). In region I1 the phase-space trajectories are open (small-signal regime). (From Bambini and Renieri [1978]).
motion. The system executes open orbits in the phase space (region I1 of fig. 3. l), and a net exchange of energy can occur between the electrons and the field. In the saturated regime, closed orbits are followed in the phase space (region I of fig. 3. l), and no net energy exchange occurs between the electrons and the field (the electron gains and loses the same amount of energy during the interaction). Therefore, the gain must be calculated in the small-signal regime. Since the energy lost by the electrons amounts to an increment of the input field intensity, we define the FEL gain as (3.9)
where W t = (1/8 n)Ei V (V is mode volume) is the energy of the input electromagnetic wave. The quantity A y, i.e., the electron energy variation, can be calculated using eq. (3.5) and a naive series expansion with (3.10)
as the expansion parameter. Therefore, using simple perturbative methods and
VI, 8 31
THE
FEL GAIN
353
averaging over the phase I&, we obtain the well-known simple expression* (3.11) where go is a quantity of paramount importance, defined as follows: g - 211 - L AFL - - I n-
C,
y
K 2 (AWw)i2 lo 1 + K 2
9
(3.12)
and I and I, are the electron beam and Alfven current (1.7 x lo4 A), respectively, C Eis the electron cross section, F is the filling factor, a phenomenological term that takes into account the transverse distribution of the field (2.1) and is defined as
(3.13)
whereas CLis the laser mode cross section. Equation (3.11) states an important result: the gain profile is proportional to the derivative of the spontaneous spectrum. This represents a general property, known as Madey's theorem (see Madey [ 1979]), which relates the gain to the spontaneous radiated spectrum. From eq. (3.1 l), plotted in fig. 3.2, we also obtain the important information that the maximum gain is located at v = 2.6 and that G(2.6)
=
0.85g0.
(3.14)
We stressed that the gain is associated with an intensity variation of the input field, and a more complete analysis should also include its phase variation given by (Dattoli and Renieri [ 19811)
(3.15) dv It is worth emphasizing that thc dispcrsivc and absorptive parts satisfy the
* Equation (3.5) can be solved exactly using elliptic functions; however, the perturbative method leads to more manageable expressions. For mathematical details the reader is referred to Dattoli and Renieri [1985].
354
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, § 3
-1 0
Fig. 3.2. Small-signal gain.
relations imposed by the Kramers-Kronig condition (3.16) where C( v)
=
( - a/av) [sin
(iv)/(f v)]’
is the gain function.
3.2. HIGH-GAIN REGIME
The FEL gain was derived with the tacit assumption that the input field can be kept constant in one undulator passage. This assumption holds true depending on whether the system operates with high or low gain. The problem can be treated without any approximation ( k . , coupling the Maxwell equation to the electron dynamics), and the equation of motion and evolution of the electric field can be written in the small-signal regime as (Bernstein and Hirshfield [ 19791, Sprangle, Tang and Manheimer [ 19801, Dattoli, Marino, Renieri and Ronianelli [ 198 la], Bonifacio, Narducci and Pellegrini [ 1983])*
d E dz
-
*
‘T
= - ing,,
s:
E ( z - z’) eiv7’z‘ dz’ .
is actually a dimensionless time normalized to the interaction time.
(3.17)
VI, § 31
THE
FEL GAIN
355
The convolution-type integral on the right-hand side of eq. (3.17) contains the “memory” of the interaction at previous times. Neglecting the z’ dependence of E in the integral, the low-gain results are immediately recovered. Equation (3.17) can be solved with the following technique. Taking repeated derivatives of eq. (3.17) with respect to z, we end up with the ordinary third-order differential equation d3 ~
dz’
E - 2iv
d2 ~
dz2
E - v2
d -
dz
E
+ i?rg,E = 0
(3.18)
The solution of eq. (3.17) therefore can be written in the form
E
c El el(v+5V,)r 3
=
(3.19)
/ = I
where the 6vl are the roots of the cubic equation
Fv2(v + 6 v ) = ng,,
(3.20)
and El are defined by the following linear system: 3
1E , = l , , = I
3
3
c6vlE,= - v , / = r
C6v?E,=v2.
(3.21)
/ = r
The gain is now defined as
i.e., the relative energy variation after one passage [the electric field amplitude E is initially normalized to unity, see eq. (3.21)]. An idea of the gain dependence versus v and g, is given in fig. 3.3, and in figs. 3.4 where the gain curve was plotted versus v for values of go ranging from to 10. For small values of go, the gain curve exhibits the already discussed antisymmetric shape, whereas for increasing go, combinations of interferential and exponential effects, contributing to the gain process, yield a significantly modified profile. Figure 3.5 shows the maximum gain G(g,)~,,,ax versus go. It is remarkable that the numerical scaling of fig. 3.5 can be reproduced by a third-order polynomial in go of the type (Dattoli, Torre, Centioli and Richetta [ 19891) G(gO)I,,,,,
0.85g,, + 0.19g;
+ 0.42 x
lO-’g;.
(3.23)
Equation (3.23) yields at least two useful pieces of information. It can be used
356
PI,§ 3
THEORY OF COMPTON FREE ELECTRON LASERS
10
-10
Fig. 3.3. Three-dimensional plot of gain versus detuning parameter and gain coefficient.
to estimate the gain deviation from the linear theory. Furthermore, since the coefficients of higher powers in go become increasingly smaller, eq. (3.23) suggests that the gain coefficient can be used as a perturbation parameter even when it exceeds unity. Using a simple series expansion, we can write (3.24) thus obtaining from eq. (3.17) the following recursive relations for the E,I:
E,,(O)
=
bn,o,
E-
I =
0.
(3.25)
The calculations of the gain up to the third order in go require the explicit derivation of E l ,2, 3 . According to Dattoli, Torre, Centioli and Richetta [ 19891, the gain function containing higher-order corrections can be written as (3.26)
2n g , ( v ) = - [2(1 - cos v ) - v sin v ] , v3 n2 g2(v) = - [84(1 - cos v) - 60v sin v + 3v2 + 15v2 cos v + v 3 sin v ] , 3 v6
0.010
,
1
-. . -
AUI 0.06
1
0.02
0
-0.02
-0.06
-0.010
-10
-6
6
-2 0 2
10
-0.10' ' ' ' ' ' ' -10 -6 -2 0 2
'
' 6
'
1 10
1.2 I
AM
0.4
08
AUI
-
g0'1.0
0.2
0 -0.2
-0.4 -10
-10
-6
-2 0 2
-6
-2 0 2
6
6
10
10
-08 -10
-10
-6
-2 0 2
6
10
-6
-2 0 2
6
10
Fig. 3.4. Gain function versus the resonance parameter v for different go.
358
THEORY OF COMPTON FREE ELECTRON LASERS
0
2
4
6
8 go
1
0
Fig. 3.5. Maximum gain versus the gain coefficient go < 10.
g3(v) =
R3 ~
60 v9
[ 11520(1 - cos v ) - 9000v sin v + 3 6 0 ~ ’
+ 2880v2 cos v + 4 8 0 v 3 sin v - 2 0 v 4 ( l + 2 cos v ) - v 5 sin v ] . (3.27)
The function g , ( v ) is the antisymmetric gain function of the linear theory, and the g2,3 ( v ) are the interferential contributions of the high-gain corrections. Perturbative and exact solutions are compared in figs. 3.6 and 3.7 for go = 5 and go = 10. The corrections due to the cubic term become significant for g,, > 5, whereas, for smaller go, the second-order correction ensures a good agreement. 10
I
-8
-
4
-
2
0
2
4
~
a
Fig. 3.6. Gain versus v : exact theory (continuous line); perturbed theory (dashed line).
THE
FEL
359
GAIN
AI/I 30
20
10
0 -5'
'
I
-8
I
'
'
-2 0
2
4
I
-4
' 8
Fig. 3.7. Gain versus v : exact theory (continuous line); perturbed theory (dashed line)
3.3. VERY HIGH GAIN REGIME
This subsection explores the gain behavior of FEL operating with very large gain coefficients (go > 10). We saw that when go increases, the gain curve loses its antisymmetric shape, and its maximum shifts from v = 2.6 towards lower values. To have a first hint in the very high gain regime we can therefore set v = 0 in eq. (3.17), thus obtaining d
-
E
=
-
E(r -
ing,
(3.28)
T') d s ' ,
ds whose solution can be cast in the form 3
E j ei"I',
~ ( s= )
(3.29)
j = I
where ai are the roots of the cubic equation a3 = - i n g o ,
(3.30)
and are given by a,
a I = i(ngo)'13,
= - (ng,,) 113 e in16
,
a3 = (ngo)1'3e1n'h. (3.31)
Furthermore, using the analog of eq. (3.21) 3
C j= I
3
Ej= 1,
1 tliEj=O, j =
I
3
C j= I
tfEj=O,
(3.32)
360
[VI,5 3
THEORY OF COMPTON FREE ELECTRON LASERS
we obtain for E ( z ) the following relation: ~ ( z2 ) f[exp [ ( ~ g ~ ) l / ~-( fi);i z]
+ exp [ - (ng0)1/3(J5 + i):
+ exp[ing,z]].
TI
(3.33)
For very large go, the field grows exponentially, and eq. (3.33) therefore yields ~ ( rz)iexp [ ( n g 0 ) l / 3 d
471 exp [ - i(ng0)1/3+21 .
(3.34)
Accordingly, the gain is given by
G z $exp[(ng,)’/’fi
TI
(3.35)
.
The inclusion of v in the high-go limit yields for the gain the following relation:
.1‘
(wo1‘I3
(3.36)
The gain curve for this case is shown in fig. 3.8.
3.4. GAIN DEGRADATION INDUCED BY INHOMOGENEOUS BROADENING
So far we have calculated the gain assuming a perfect electron beam; i.e., we have not included inhomogeneous broadening contributions of the type discussed in 0 2.3. The spontaneous emission line broadening due to nonzero emittance and energy spread induces an analogous broadening and reduction of the gain curve. The inclusion of the electron beam qualities in the self-consistent FEL equation is relatively straightforward, and eq. (3.17) is modified as follows:
d E = -ingo dz
-
s,’
dz’ z’
exp [ ivr’ - f ( ~ p ~ z ‘ ) ~ E ( z - z’) . (1 + inp,xz’)(l+ inpyz‘)
(3.37)
Equation (3.37) should be solved numerically, and the results of the integration are summarized in fig. 3.9, where we report the FEL gain versus v and go for . must stress that when the p’s increase, the gain different values of p E , X , YWe curve acquires the familiar antisymmetric shape, even for large values of go. This fact is perhaps an indication that when inhomogeneous broadenings are active, the gain regime is dominated by the linear contributions for a broader range ofg,. This is further supported by the results shown in fig. 3.10, where (AI/I),,,axis plotted versus go for different p’s.
THE
FEL GAIN
36 1
3 VMAX
2
1
0 GMAX 14000
12000 10000 8000 6000
4000
2000
0 error
0.1 .............. . . . . . . . .... ................... .. ,
-_---__--------
0
!
-0.1 I
-0.2 0
I
I
I
I
20
40
60
80
100 go
Fig. 3.8. Maximum gain versus go in the high-gain limit. (a) Position of the peak. (b) Continuous curve: exact theory; ---(practically indistinguisable from the continuous curve) obtained using eq. (3.3.6); ..* using eq. (3.35). (c) Relative error: continuous curve [exact - eq. (3.36)]/exact; _.-.- [exact - eq. (3.35)]/exact.
362
IVI, § 3
THEORY O F COMPTON FREE ELECTRON LASERS
12 -pE =o 0.p, = p" = o 0
04
-
-0 8 -20
I
I
l
0
-
p" = o 0
pe -0 0;p.
-20
I
-10
-10
1
I
0
I
10
20
10
20
-2 -20
-10
0
10
Fig. 3.9. FEL gain versus v and go for different values of
(AWfWX 10
-
10
0
-..... --
pE=oo.p.=p,=oo pE=05.p,=p,=05 pE=05.p.=pq=10
-
2
4
6 9o 8
10
Fig. 3.10. ( A I / l ),,,., plotted versus g,, for different values of p,, ,,y .
20
VI, 5 31
THE
FEL GAIN
363
We will now discuss some practical relations that allow a quick evaluation of the gain reduction induced by the p parameters. The low-gain function is modified by the inhomogeneous broadening contributions as follows: G(v9 P,;,.r,J = Re
i
:"(
-2n-
.:J
1+i-
In fig. 3.11 we show the maximum gain versus pLEfor different values of p,,.,,. The numerical scaling is well reproduced by the following relations (Dattoli, Fang, Giannessi, Richetta and Torre [ 19891):
(3.39)
Thc coefficients a and y are determined by means of a fitting procedure, and in the parameter range of fig. 3.10 are given by aO= 1 , aj
a l z -0.155,
z3 x
a 2 z 2 . 8 x lo-',
a4 z 1.2 x
a5 z 1.5 x l o - * ,
(3.40)
yz 0.157. These relations hold in the low-gain case (go < 0.5); for larger go, simple expressions of the type (3.39) are not available. An idea of the interplay between 0.9 glrna" 0.5
0.3 0.1
0
0.2
0.4
0.6
0.8
1.0
PCle
Fig. 3.1 1. Maximum gain (in v) versus ps for different values of
364
THEORY OF COMPTON FREE ELECTRON LASERS
PI,8 4
the high-gain and inhomogeneous broadening contribution is given by the gain relation G(P0 go)
+ o.19g0 = go 1 +0.85 (1.7 + 0.32g0)p:
’
(3.41)
which valid for go < 2 and 11, < 1.5. The inhomogeneous broadening effects for very large gain were discussed by Colson, Gallardo and Bosco [ 19861, to whom the reader is referred for further details.
8 4.
Transverse Mode Dynamics
4.1. ANALYTICAL APPROACH
So far the effect of finite optical beam transverse distribution has been accounted for by means of the phenomenological filling factor introduced “by hand” in the definition of the gain coefficient go. A proper analysis of the effect of the transverse distribution on FEL dynamics requires the solution of the equations of motion for FEL, including the transverse mode evolution. The first step is the modification of, for example, equations of the type (3.17) to include the transverse spatial part, which is achieved in the paraxial approximation, replacing
d -++iV: dz
+ -a , az
(4.1)
where V: = (a2/aY2) + a2/aL2 is the transverse Laplacian and go(?,) contains possible transverse coordinate dependence of the gain coefficient. One primary equation therefore will be of the type
where X, refers to the (x, y ) coordinate normalized (ALu/n)’’2,with I and L , are the central wavelength and undulator length, respectively. Equation (4.2) will be used to discuss the effect of the transverse mode dynamics on the FEL gain. Before discussing eq. (4.2) in its general form, we
VI,B 41
365
TRANSVERSE MODE DYNAMICS
will discuss the particular, but enlightening, case of the low-gain regime. We can conveniently rewrite eq. (4.2) as follows:
a -
a7
E(X,, z)
= -
aigo(X,, z) joTexp(iOTr')z' dz'
E(i,,
z),
(4.3)
where
and
go(^,, z)
=
exp(:izV:)g,(x,)
exp( -tizV:)
(4.5)
is the current operator. Equation (4.3) can now be solved using standard techniques developed in quantum mechanics. Using the operational identity
1
+5 3 [ A , [A, [A, B]]] t 3!
,
(4.6)
01,
(4.7)
* *
we obtain at the lowest order in go E(E,, 1) zz exp[ -aiV:)Ei(OT,
E,)E(sT,,
where the propagation operator
is defined as follows:
B ( D T , X I ) = ( 1 +$go(S;;,)Gl(^vT) + iO:,go(X,)l
jol
dz z s,'exp(i O T z ' ) t r dz'} .
(4.8) The complex gain function G , is an operator, which once expanded up to the first order in V,: yields for E E(SZ,, 1 ) exp[ ~ -iiV:]
1 + i(go(Xl)G,(v)
+ $go(X,)
+ i n[V:,go(x,)lS(v)
a
-G,(v)V: av
G,,0 )
1
(4.9)
366
[VI,§ 4
THEORY OF COMPTON FREE ELECTRON LASERS
where the function S( v) is specified by S(v)
=
jo'
d z z j;ei''''
T'
(4.10)
dz' .
Assuming that the gain coefficient is not Y.-dependent, fies as follows: E(x,, l ) z e x p ( - z i V L )
l a
[
1 +$go G,(v)+-
-
4av
eq. (4.9) further simpli-
G,(v)V: (4.11)
Assuming a beam which initially is Gaussian, i.e., (4.12) we obtain from eq. (4.1 I), E(F,, l)zexp(-;iV:)
(4.13) Defining the gain as
:'1
dY
[+
'
dJ I E ( x , , 1) I -
--x
G(v) =
[+;
dY
1+
dJ I E(F,, 0) I
--x
I - + x x c i Y ~+*dJlE(Y,,0)I2 - %
(4.14) we find, at the lowest order in go, C(v)
=
[
go Re C,(v) +
a a v G,(v)]. ~
(4.15)
We have found, therefore, that the gain correction depends on the normalized beam waist and on the derivative of the gain function. Another important effect associated with transverse mode dynamics is the so-called radiation focussing, which can be easily understood. Due to the
VI, I 41
TRANSVERSE MODE DYNAMICS
361
assumption of low gain, we can rewrite eq. (4.11) as follows:
C2(v) = - i
a
-
av
G( v)
The evolution operator consists of two parts, the first,
t i = exP[4goG,(v)l9
(4.17)
yields just the intensity variation due to the gain, and the second,
P,
=
exp[ -$iV:(l
- igoG2(v)],
(4.18)
is a type of free propagation operator with a small counteracting term due to the gain. The action of eq. (4.18) on an initially Gaussian beam with waist 0,) yields a Gaussian beam with a modified output spot size 0 provided by (4.19)
In the region of maximum gain, G2(v) counteracts the natural diffraction, thus providing a kind of focussing effect.* It is now useful to extend the analysis of eq. (4.1) without the restriction of low-gain regime go(.wl) = goAx,)
3
(4.20)
where ,j(x,) is the transverse current, determined by the transverse shape of the electron beam. Furthermore, we expand the field in terms of Hermite-Gauss functions (4.21)
(4.22)
*
The focussing effect becomes defocussing in the negative-gain region.
368
THEORY OF COMPTON FREE ELECTRON LASERS
The amplitudes uff,,(T)
=
-ingo
C
satisfy the equation
dTt T ) e i v ~ ‘ U , , . , ~ . ( T -
.jn~.m~;fl,m
(4.23)
T I ) ,
n’.m’
where d f f , ,
f77!;
f,7
denote the matrix elements and jflf, n 7 , ;,,, J
J - x
s st: s:s’
J n ’ . 171’ ; n . 177 =
(4.24) d’ dL $n,
m(’3
Y)j(x,L) $ n , , n , , ( X , L>
*
Furthermore, the initial conditions of ufl,m(T ) are determined by the overlapping integral
“ff,
f71(’)
=
d x dL JW, L) &. m(Y,L) .
(4.25)
The integro-differential equation (4.23) can be turned into an ordinary equation, keeping repeated derivatives with respect to T, thus obtaining
(4.26)
(4.27)
-,
an, m(0) = - i v a n , m ( O ) -
i
1 1
n ’ . m’
-
I,
an,
f77(’)
=
- ivzt’l,m ( 0 ) - i
n ’ . m’
-
d,i*.m ‘ ; n , m an. m ( 0 )
9
(4.28)
-,
d n , . m ’ ; n , m an, m ( 0 ) .
The problem of understanding the transverse mode dynamics has been reduced to that of solving the system of n x m coupled equations (4.26).
369
TRANSVERSE MODE DYNAMICS
v1, § 41
4.2. NUMERICAL RESULTS FOR A TRANSVERSALLY UNIFORM ELECTRON BEAM
Many strategies can be used to solve eq. (4.26). We will now examine the numerical results for a transversally uniform electron beam and their physical meaning. In fig. 4.1 we show the output field spot size, assuming a constant transverse current [ j ( X , ) = 11 as a function of the small-signal gain coefficient go. The results are relevant to the initial field E(F, J , 0) =
8 -
exp [ - (2’+ J’)] .
(4.29)
The gain curves exhibit the same behavior as those derived from the onedimensional wave equation (see § 3). A small reduction of the peak values and a shift of the maxima toward the higher v must be noticed*. It is worth noting that the output waist versus go is reproduced, in the range 0 < go < 10, by
o z woexp[ -(0.24 x 1O-’g0 0.72 W
+ 0.18 x
10-6gi)o.42],
(4.30)
c
0.70
0.68 0.66 0.64 0.62
0.60
0.sa 0
2
4
8
6
10
go
Fig. 4.1. Output spot size versus go.
*
For the Hermite-Gaussian mode of order n, m, the maximum gain is located at + i(n + m + 1 ) where v* denotes the maximum given by the one-dimensional approach. vmax = v*
370
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, 5 5
with oobeing the waist corresponding to the free propagation [coo E 1 for the field (4.29)].
8 5.
Longitudinal Dynamics
The competition between longitudinal modes is important for lasers operating with a mode-locked structure. The FEL exhibits a natural mode-locking when it operates with an electron beam provided by radiofrequency accelerators (see fig. 5.1). In this case the bunched beam structure induces an analogous structure on the optical field, which can be accounted for by means of an appropriate longitudinal mode expansion. Before giving the mathematical details for the analysis of FEL pulse propagation, we shall discuss qualitatively its main physical aspects. The distinctive feature of a FEL operating with a pulsed electron beam is the so-called lethargic behavior, which can be easily understood. The quantity playing a central role in the theory of FEL pulse propagation is the so-called dippage distance due to the different speeds of the electron and laser bunches. The laser pulse moves at the velocity of light, whereas the electron pulse moves at an average velocity given by eq. (2.4). As a consequence, after one undulator passage, the optical pulse will be ahead of the electron bunch by the slippage distance A
=
(1
-
B,)c
At
=
NI .
(5.1)
This kinematic effect has two consequences:
I*
CM
Fig. 5.1. Time structure of a radio frequency electron beam ?M denotes macropulse duration, T the bunch to bunch distance, q, the microspulse duration, and f the repetition frequency.
VI, 8 51
37 1
LONGITUDINAL DYNAMICS
(1) The electron and photon bunches do not overlap for all the interaction time, and (2) the front part of the optical pulse experiences less gain than the backward part, and therefore the optical bunch centroid is slowed down by the interaction. This latter effect is usually referred to as the FEL lethargic behavior. This can be properly accounted for by introducing a refractive index for the FEL (Dattoli, Hermsen, Mezi, Renieri and Torre [ 19881). The important consequence of the FEL lethargy, with respect to the design of a FEL cavity, will be discussed later. According to the previous discussion, it is easy to modify the small-signal continuous beam equation to include pulse propagation effects, in the form
d E(z, z) dz -
= - ing,(z
+ AT)
~ ' E ( z+ A T ' , z -
7').
(5.2)
In eq. (5.2) the gain coefficient contains the longitudinal dependence of the current, and the convolution integral on the right-hand side accounts for both slippage dynamics and high-gain effects. This section will be limited to the low-gain case, and as in 5 3, we will perform the simplified hypothesis that the z' dependence of go can be neglected. Equation (5.2) therefore becomes d -
dr
E(z, z)
dz' z' exp(i 0,~') E(z, z) ,
= - ingo
(5.3)
At the lowest order in go, the solution of eq. (5.3) is written as E(z, 1)
=
[+ 1
(
igoC, v - id
31 -
E(z, 0)
(5.4)
A simple and important result follows. Assuming that initially the field has a Gaussian distribution
and retaining only the first order in d in eq. (5.4),we can write E(z, 1) in a form
312
THEORY O F COMPTON FREE ELECTRON LASERS
analogous to (4.16), i.e.,
where zo = fAg0G2(v).
(5.7)
The first term in eq. (5.6)is the gain part, and z, indicates that the optical packet propagates in the undulator at the average velocity C
L'L =
A 1 + +go - ReG,(v) LU
The preceding result is particularly important, and allows the introduction of a refractive index
responsible for the optical packet slowdown and thus explaining the lethargic effect. Still neglecting the z dependence ofg, (this problem will be analyzed in $ 6), eq. (5.3) can be solved numerically, thus obtaining some useful information about the behavior of the FEL gain in the short optical pulse regime. We assume a Gaussian distribution for the optical pulse with a RMS length o,, and we introduce the parameter
where A = N1 is the slippage length. The gain behavior versus v at different pE is plotted in fig. 5.2, showing significant deviation from the single-mode gain. The gain dependence on pE is shown in fig. 5.3. In the low-gain regime this dependence is reproduced by the simple relation (5.11)
v1,s 51
313
LONGITUDINAL DYNAMICS
G
0.06 0.02 -0.02
-0.06
-0.10' ' -10
' -6
'
'
-2
'
' 2
'
' 6
'
I
-0.10 -10
10
-2
-6
V
2
6
10 V
nr
-10
-6
-2
6
2
10 V
Fig. 5.2. Gain versus v for different p E with go = 0.1.
0.95 Gmax
0.85
0.75
0.65
0.55
0
1
2
3
P€ Fig. 5.3. Maximum gain versus p e . Continuous line shows the numerical calculation and the dashed line shows the fit to eq. (5.1 1).
374
[VI,8 5
THEORY O F COMPTON FREE ELECTRON LASERS
Other relevant quantities, such as the position of the peak and the RMS length of the output laser pulse, are shown in fig. 5.4. For positivegain the bunch will experience a deceleration, and vice versa for negative gain. Furthermore, for positive or negative gain there is also a slight increase or decrease of the pulse length. These effects are all summarized in fig. 5.5, where the output pulse is compared with the input pulse. These effects become more pronounced with increasingg, and pE. Figure 5.6 shows the relevant results for a Gaussian input beam undergoing a FEL interaction with go = 5 and pE = 1. In the case of high gain, the solution of eq. (5.3) is simply (5.12)
v
0.69 -10
-6
-2
2
6
10
-10
"
-6
I
-2
'
' 2
'
' 6
10 V
V
0.08
0.75
c
0
0 04
0.73
0
0.71
-0 04
0 69
-008 -10
0.67 -6
-2
2
10
6 Y
1111111111)
-10
-6
-2
2
10
6 Y
Fig. 5.4. Optical packet centroid after the interaction and RMS width as a function of v for different values of go and pE.
VI, B 51
315
LONGITUDINAL DYNAMICS
G
0.6
0.03
Q (0 0.01
0.4
-0.01
0.2
-0.03 -0.05 -10
0
-6
-2
6
2
10
-15
-5
5
15
c
V
Fig. 5.5. (a) Gain versus v; (b) pulse shape versus longitudinal position after the interaction. Input is given by the continuous line and the output by the dashed line.
-10
-6
-2
2
6
10
-15
-S
V
5
c
15
Fig. 5.6. Same as fig. 5.5 but now with go = 5.
where G( v,) is the solution of the one-dimensional FEL self-consistent equation. Equation (5.12) clearly reflects the interplay between gain and slippage, thus providing an intuitive picture of the two competing mechanisms. More general considerations of FEL pulse dynamics will be developed in 9 6 .
316
[VI, I 6
THEORY OF COMPTON FREE ELECTRON LASERS
# 6. FEL Oscillator Regime and the Pulse Propagation Problem 6.1. PRELIMINARY CONSIDERATIONS
The lethargy creates significant problems when the FEL operates as an oscillator. Let us now return to the radiofrequency bunched structure of the electron beam shown in fig. 5.1. It was stressed that this type of structure generates an analogous pattern in an optical field. The laser field displays a mode-locked structure consisting of a series of bunches. The electron and laser bunches interact in the undulator, and then the optical bunch is reflected back by the cavity mirror. To have gain, we must be sure that, after one roundtrip, at the undulator input the laser bunch overlaps a fresh incoming electron pulse (see fig. 6.1). The synchronism between electron and optical pulses therefore requires that the electron bunch distance be equal to a cavity roundtrip, or better, that the roundtrip period be an integer multiple of the electron bunch time distance T,, namely,
where L , is the cavity length. The cavity length determined by eq. (6.1) is usually called the empty-cavity length. As already noted, the interaction induces an optical pulse velocity reduction, and therefore the synchronism is assured once the cavity is shortened by a certain amount 6L compensating the lethargy effect.
OUTPUT LASER BUNCH
ELECTRON BUNCHES
Fig. 6.1. Electron and optical bunch synchronism in a FEL oscillator.
FEL OSCILLATOR
VI, § 61
REGIME A N D PULSE PROPAGATION PROBLEM
311
We also discussed the role played by the slippage distance A. Denoting with cz the electron packet RMS length, we can state that the pulse propagation effects are more or less significant according to whether
A>a, or A < c , .
(6.2)
It is therefore useful to introduce the dimensionless parameter p,=-
A 1
(6.3)
0,
usually called the coupling parameter, which defines the relative slippage of the optical pulse on the electron bunch, and fixes the strength of the coupling between longitudinal modes (see Dattoli, Marino, Renieri and Romanelli [ 1981bl). It is clear that if Pc’
1
9
(6.4)
the optical bunch will not completely overlap the electron pulse during the interaction time. If, otherwise, PL,<
1 9
(6.5)
they will overlap most of the time. We expect, therefore, that increasing p, reduces the gain. The slippage-gain reduction can be accounted for by the simple relation
According to the preceding discussion, the gain of a FEL oscillator operating with pulses is also a function ofthe cavity mismatch aL from the nominal values (see fig. 6.2).Therefore, denoting with &he so-called desynchronism parameter
we can write a simple gain formula accounting for both lethargy and slippage dependence, namely (Dattoli, Giannessi, Fang, Renieri and Torre [ 19881) G(0, p c ) = - 0 . 8 5 g 0
where 0,
s 0.456.
378
THEORY O F COMPTON FREE ELECTRON LASERS
4
\
L2
I I /
M1
LU
M2
Fig. 6.2. FEL cavity configuration and mismatch from the empty-cavity synchronous condition.
From eq. (6.8) we also learn that the maximum gain is given by eq. (6.6), and is located at ern,, =
0,
+h '
~
1
(6.9)
The preceding results can be understood as follows: (1) The maximum gain is obtained for a value of tl (and thus of 6L), which allows the exact synchronism between an optical and an electron bunch. (2) When 8 < Om,, the cavity is not shortened enough to compensate for the velocity reduction. After one roundtrip the optical bunch is positioned behind the electron pulse and therefore experiences less gain. (3) When tl> tl,,, the cavity is shortened too much, so that after one roundtrip the optical bunch is positioned just ahead of the electron bunch. Therefore, in analogy with the previous case, the laser experiences less gain per roundtrip. So far we have tried to indicate how the different parameters should be included in a theoretical approach to give a correct description of the FEL small-signal behavior. Unfortunately, we cannot give an individual, simple gain formula to account for the slippage, lethargy and inhomogeneous broadening effects. Equation (6.8) contains only the delay (0) and coupling parameters, but does not include energy spread and emittances. Furthermore, the various effects cannot be separated into mutually independent functions. In other words, the inhomogeneous broadening affects the peak gain as well as Om,, and A0 (i.e., the interval of tl for which we have positive gain). A typical example is shown in fig. 6.3. It is evident that in the case of inhomogeneous broadening, Om,, is shifted and Atlsignificantly reduced. Since A8is linked to 6L, this means that the operating conditions of the system are more critical.
VI, 8 61
FEL OSCILLATOR
0.8I
0
319
REGIME A N D PULSE PROPAGATION PROBLEM
I
0.2
0.4
Fig.6.3. Gain as a function of 0: (a)p,
0.6
0
0.8
1 .o
= 0.5, p6 = p , = p,, = 0; and p,, = p , = py = 0.5.
(b)p,
=
0.5,
6.2. QUANTITATIVE ANALYSIS
In 0 6.1 we discussed the FEL oscillatory theory using qualitative arguments. Now we will analyze the problem of FEL pulse propagation using a more quantitative approach. Dattoli, Gallardo, Hermsen, Renieri and Torre [1988] showed that the roundtrip FEL pulse evolution can be written in the form
where E(z, t) is the laser electric field amplitude, yT specifies the cavity losses, T, is the nominal cavity roundtrip period and f(z) stands for the longitudinal distribution of the electron bunch. Equation (6.10) holds when assuming a small signal and a low gain. Strictly speaking, the time derivative should be replaced by a finite difference. The use of the derivative is justified, however, by the fact that, according to the assumption of low gain, the optical pulse is not significantly modified after each roundtrip. The physical meaning of eq. (6.10) is obvious. The left-hand side accounts for the free propagation part of the optical packet, whereas the right-hand side contains the interaction with the electron bunch, slippage and, as we will see,
380
THEORY O F COMPTON FREE ELECTRON LASERS
[VI,I 6
concurrent lethargy. Equation (6. lo), as it stands, cannot be solved analytically, but it can be solved exactly in accord with the hypothesis of small coupling or in long-bunch regime ( p c 4 1, A 6 oz).The interest in this regime is due to the fact that most of the existing FELs satisfy the preceding conditions. It has been shown that, in the long-bunch hypothesis, eq. (6.10) reduces to a Schrodinger-like equation with a complex harmonic potential. Therefore, the solution can be readily found. For an initially Gaussian-shaped optical pulse we obtain (6.11)
and the time-dependent functions g(z),
~ ~ ( =7 zO(O) ) bE(z)
= gE(O)
and uE(z) are specified by
ZJZ)
+ O(p;/’),
A[G2(v) - 01 ( Z - zo)
{ 1 + :pC[G,(v)
-
GI(v)l} (z -
TO)
+ O ( F ; / ~ ) . (6.13)
In these expressions z,(O) specifies the relative initial position of the optical packet with respect to the electron bunch, o,(O) is the RMS width of the input laser packet, and the functions G I s2, ,(v) are defined by G ~ ( v ) =-271
a
-
av
a2
G3(v)= --
a v2
(
1+
I v:)
[y
eiv/2]
aa- I
G,(v);
G a = (-i)a-’
G,, a 2 1 ,
~
ava-
(6.14)
I
where G,(v) is the complex gain function; the physical meaning of G, and G, will be discussed below (see fig. 6.4).The result states that: (1) an initially Gaussian-shaped optical pulse remains Gaussian after each round trip; (2) its amplitude grows at a rate controlled essentially by the gain function; and
FEL OSCILLATOR
VI, 8 61
381
REGIME AND PULSE PROPAGATION PROBLEM
0.20
0.40 G1
G2
-
0.20
0 -0.20 -0.40
-0.60 -8
-4
0
-8
-4
0
4
v
8
4
v
8
-8
-4
0
4
V
8
0.08
G3 0.04
0 -0.04
-0.08
Fig. 6.4. Plot of the first three G functions versus v. Continuous curve gives the real part.
(3) the electron wave pulse centroid is shifted after each roundtrip by a quantity z0(7) depending on both G, and 8. The latter point is a very important result and is a direct manifestation of the lethargy. According to whether ReG,(v) - 85 0 ,
(6.15)
the optical bunch is behind or ahead of the electron pulse. On the other hand, the synchronism condition is achieved whenever
0 = ReG,(v).
(6.16)
This result allows the calculation of the average effective velocity uL of the optical bunch interacting with the electrons in the undulator. Referring to fig. 6.1 and recalling that the distance between successive electron bunches is
382
THEORY OF COMPTON FREE ELECTRON LASERS
IVI, 8 6
just 2Lc/c, we can write the FEL synchronism condition in the form
2 L C & +u+-+L L,-6L Lc-6L c c UL C C
~-
(6.17)
thus obtaining the same results (5.8) for uL. We have noted earlier that the velocity reduction allows the introduction of a refractive index for the FEL, which allows one to understand the G’s as susceptibility functions. Note that all the G’s satisfy the Kramers-Kronig dispersion relations, namely, ImG,(v)
=
--
x
dv,
(a = 1,2,3).
(6.18)
So far we have investigated the time-dependent solutions of eq. (6.10) in accord with the hypothesis of the long-bunch regime. Now the eigenvalue problem associated with this type of equation will be discussed. As emphasized earlier, in the weak-coupling regime eq. (6.10) can be reduced to a SchrOdinger-like equation, i.e.,
--
aE
-=HE,
(6.19)
fi being a kind of Hamiltonian operator explicitly given by A = a,R+ + a2L- + a,&++ a,& + a,i.
(6.20)
aT
with
Here we have defined
1= unity operator,
(6.21)
and
(6.22) The stationary solutions of eq. (6.19) can be found by solving the following eigenvalue problem: fi$n
=
ln$n.
(6.23)
The problem requires lengthy algebra that will not be given here. However, it
FEL OSCILLATOR
VI, I 61
REGIME A N D PULSE PROPAGATION PROBLEM
383
was shown that eigenvalues and eigenfunctions of eq. (6.19) exist, and are (Dattoli, Gallardo, Hermsen, Renieri and Torre [ 19881)
(6.24) With N, being a normalization constant and H,(.) the nth Hermite polynomial. Furthermore,
(6.25)
Finally, the eigenvalues are specified by
The preceding relations state that the degeneracy among the various eigenmodes is removed by the coupling parameter p c . A brief mention of the physical meaning of these eigenfunctions is necessary. The @,, identify a set of self-reproducing configurations of the laser field, the intensity and overall phase of which can change with each roundtrip. They are known in the scientific literature as F E L supermode (SM), and can also be understood as collections of longitudinal modes having the same gain and phase variation in one roundtrip. The real part of I,I represents the SM gain, and the mode with the highest gain is the fundamental (0th SM). Figure 6.5 plots R e l o as a function of 8 for different pc.* For comparison we plotted the curve derived from a numerical analysis according to eq. (6.8).
*
-
Evaluating the G functions at v = 2.6, we obtain for the maximum gain of the fundamental 0.85(1 + $ p c ) - in agreement with eq. (6.6) and the hypothesis of small pc.
SM R e l ,
’
384
[VI, 0 6
THEORY O F COMPTON FREE ELECTRON LASERS
1 .o
0.8
0.6 0.4
0.2
0
e
8
Fig. 6.5. Fundamental SM gain plotted against 0. The dotted line gives the SM theory, and the solid line the exact numerical treatment according to eq. (6.8).
The agreement between the exact numerical solution and the one based on the long-bunch hypothesis is good and tends to be less accurate (as it must be) with increasing p c . The eigenfunctions (6.24) are relevant to the optical field spatial configurations. Their spectral counterpart can be found by taking their Fourier transform. In fact, defining
(6.27) we obtain*
* Y, is the value of the resonance parameter yielding the maximum gain; homogeneous regime.
Y, =
2.6 for the
FEL OSCILLATOR
VI, 8 61
where
REGIME A N D PULSE PROPAGATION PROBLEM
385
p, 7
1
G3
Pc
(6.29) Note that the RMS widths of the fundamental spatial and spectral SM are related by the expression a,a,
=
1.
(6.30)
The SM type of structures were identified in various FEL experiments. In particular, an analysis of the first Stanford experiment (Dattoli and Renieri [ 19851) and of the SR ACO experiment (Elleaume [ 19851) was carried out using the SM concept. We have discussed a lowest-order perturbative analysis of the FEL oscillator pulse propagation problem. The highest-order corrections require complicated calculations (see, e.g., Dattoli, Giannessi, Hermsen, Renieri, Richetta and Torre [1989]), where the expression of the maximum gain of the nth SM at second order in pc is given by ReAn(pc)E 0.85[ 1 - 2n
+ l)$pc + ( 2 n 2 + 3n + 1) ( { P ~ ) ~ ] .
(6.31)
Equation (6.31) is further evidence of the pc dependence of the FEL gain contained in eq. (6.6). In addition, eq. (6.8) yields the gain of the fundamental SM. The higher-order SM gain for any pC is provided by (see also fig. 6.6) G,(B, p,) = - 0.85g0
(1
+f
‘1
~ , ) ~ ~- +1 1 ,
(6.32)
from which it follows that: (1) The maximum gain for the individual SMs is given by Gnmax(pc)=
0.85g0
(1 + f p c ) 2 n + ’ ’
(6.33)
(2) the maxima are located at (6.34) We introduced the concept of SMs, and showed that, within the framework of the small-signal analysis, SMs can be used as a convenient expansion basis for
386
THEORY OF COMPTON FREE ELECTRON LASERS
Rd"
r
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1.0
e Fig. 6.6. Gain curve for the first four SMs: (a) n
=
0 ; (b) n
=
1; (c) n = 2; and (d) n = 3.
a FEL signal. For this reason SMs determine the expected characteristics of the optical pulse. We must emphasize that SM shapes can be strongly affected by the values of 8. Referring to the fundamental SM only, we found that at Om,, the spatial and spectral width are given by (0 < pc < 2)
(6.35)
The number of longitudinal modes forming the SM is therefore given by (6.36)
The dependence on 8 of the dimensionless optical pulse length (cTE = oE/az)is shown in fig. 6.7. It is evident that at small pc, 3Eis not sensitive to the optical cavity detuning. For increasing p c , this dependence becomes more and more pronounced. With increasing 8 (and thus with decreasing cavity length), the optical bunch tends to become longer, on the other hand, the linewidth linked to 5, by
(3>, P C
=
becomes narrower.
(6.37)
FEL
VI, I 71
387
SATURATION
3.0I UE
2.0
.
1.0
'
0
0.2
0.4
0.6
e
0.8
1.0
Fig. 6.7. Pulse width versus 0 for different p c values.
6 7. FEL Saturation In 5 3 we mentioned the problems related to the strong-signal regime and argued that, within the pendulum equation picture, saturation occurs when the Rabi frequency (see eq. (3.10)) is around unity. This section discusses in more quantitative detail the physical mechanisms leading to gain saturation. Simply stated, saturation is the by-product of two effects consequent to the interaction, 1.e.: (a) the energy loss of the electrons; and (b) the energy spread induced in the electron beam. In 8 3 we stressed that the strong-signal regime is, by definition, the regime in which the electron motion is strongly perturbed by interaction. We can thus expect that the electron energy loss is significant, and therefore that the resonance condition may be changed. As a further condition, the detuning parameter v falls outside the positive-gain region and thus the FEL process stops (see also § 1). The maximum v displacement (to stay in the region of positive gain) is of the order of (see fig. 1.4)
Av 6 2nN,
(7.1)
388
THEORY OF COMPTON FREE ELECTRON LASERS
PI,I 7
which implies a maximum energy variation of the electron beam given by 1 AE - 1 A w - A v 6E 2 o 4 n N 2N
9
which is the same as the result obtained in 0 1, within the framework of a quantum-mechanical analysis. From eq. (7.2)we infer that the laser output power is related to that of the electron beam by the relationship
Defining the FEL efficiency as the ratio of the laser output power to that of the electron beam, from (7.3)we obtain 1
(7.4)
2N
An argument based on assumption (b) leads to the same result. We have, in fact, already noted that the electron beam energy spread induces a gain reduction, given by eq. (5.39).The net gain of a FEL operating with a cavity loss yr G 1 and an inhomogeneous broadening parameter p, is*
0.85go GZ
1
+ 1.7pf - YT .
(7.5)
The system will therefore stop lasing when the induced energy spread provides an inhomogeneous broadening parameter such that
which amounts to a relative energy spread (see AEr
E
(k)"' 2N 8y,
Since in low-gain devices (go/8yT)
0 2.3) (7.7)
-
1, and assuming that the energy spread
* It must be understood that the assumption in eq. (7.5) is highly qualitative, since we neglected the contribution to saturation due to mechanism (a). We should then expect an overestimation of the saturation-induced energy spread.
VI, 8 71
FEL
389
SATURATION
is induced by the interaction, for the FEL efficiency we end up with the same results (7.4). So far we have indicated that FEL and conventional lasers exhibit different gain mechanisms, and therefore we cannot expect, a priori, that, for example, the gain saturation law is the same or almost similar in both devices. It is well known that, in conventional lasers, the gain scales with the intensity 1 according to the relation g(I)=
g ~
i + i
i = ips,
,
(7.8)
where g is the small-signal gain coefficient and 1, is the saturation intensity, defined as the intensity halving the population inversion. An anlogous relation was shown to hold for the FEL too. In the FEL case, eq. (7.8) modifies slightly and reads (Dattoli, Cabrini and Giannessi [ 19921) g(f)=
1+
0.85g0 I - aI(1 -
(7.9)
I)
The parameter a is a small quantity having a value of about 0.14. The proof of the validity of eq. (7.9) is almost direct. In fact, it is achieved by integrating the pendulum equation, and then following the self-consistent evolution of the laser field and the concomitant gain saturation to check the gain scaling versus f. The results of the analysis and a comparison between the exact theory and the phenomenological eq. (7.9) are shown in fig. 7.1. We must emphasize that
0 Fig. 7.1. Gain versus
2
4
6
8
f from eq. (7.9) compared with
1 0 I 11%
numerical simulation.
390
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, § 7
the agreement is highly satisfactory. The concept of FEL saturation intensity needs clarification. An analysis of the numerical data shows that I , for a FEL operating with a linearily polarized undulator can be parameterized as follows : 1
m0c3
(A,K[JJ])’
vo
4
’ (7.10)
The saturation intensity can be found directly from the pendulum equation by defining I , as the intensity associated to the field E for which the coupling constant in eq. (3.5) satisfies the condition (7.11)
where (7.12)
The saturated intensity of a FEL is therefore associated with the value of the electric field for which the Rabi frequency is around n.A simple expression of the final intensity as a function of I , can be obtained from eq. (7.9) by imposing that the net gain is zero, namely, (7.13)
thus obtaining I ( a - 1) + J(a - 1)2 + 4a(0.85g0(1 - yT)/yT) - 1) 1s 2a
-
(7.14)
From the preceding value, referring to the intracavity intensity, one obtains the output intensity in the form I,,,
=
YT I s ~
2a
[ ( a - 1) t J(a - 1)’ t 4a(0.85g0(l - yT)/yT) - l ) ]
, (7.15)
and from the condition (7.16)
o
FEL SATURATION
VI, 71
39 1
0.10
r* 0.08
0 06 0.04 0.02
0.1 0.2
0
0.3
0.4 0.5
0.6
go
Fig. 7.2. Optimized losses versus go.
one also yields the values of the losses that maximize the outcoupled intensity, namely,* 7;
=
0.85g0
(1
-
-
4g(l
+ 0.85g0)
(7.17)
The behavior of y+ versus go is shown in fig. 7.2. Inserting eq. (7.17) into eq. (7.15), and using c( = 0.14, we obtain, at the lowest order in go, (7.18) Using eq. (7.18) to calculate the FEL efficiency, one obtains the same result described earlier [eq. (7.3)]. The analysis developed so far is relevant to a continuous electron beam, and does not account for possible lethargic and short pulse effects. From a physical standpoint we expect that, with increasing power, the lethargy effect in a FEL operating with short pulses should become increasingly less significant, and therefore the optimum operating 0, in the saturated regime, should be different from that in the small-signal regime. The analog of eq. (6.8) for saturation should be able to explain both the gain and lethargy reduction effects. The simplest, but effective, modification of eq. (6.8) including intensity contribution is G ( O , p c , I ) = -0.85g0
* See “Note added in proof‘.
(7.19)
392
THEORY O F COMPTON FREE ELECTRON LASERS
From the preceding relation we obtain that: (a) The maximum gain versus 13is G=
0.85gu
( I + f p , ) ( ~+
(7.20)
I - a1(I - 1))
(b) The maximum gain is located at
o=
0s
(1
(7.21)
+ fp,) (1 + 1- a f ( 1 - 1))
’
Both results fulfil the physical requirements. From eq. (7.19) we can evaluate the fully saturated intensity, as a function of I3 and k c , from the condition of zero net gain [C(I3,ps; f*) = yT, yT G 13 we obtain
4a 2a
(7.22)
a - 1)
where (7.23) Typical shapes of f* versus I3 for different values of pc are shown in fig. 7.3, which also shows that the value of I3 maximizing the laser intensity can be far from that maximizing the gain. From eq. (7.22) we easily obtain (7.24)
0
0.2
0.4
0.6
0.8
1.0
e Fig. 7.3. Full saturated intensity versus 0.
1.2
VI, § 81
SIMPLIFIED VIEW OF
FEL STORAGE RING
DYNAMICS
393
Although derived from a simple extension of the gain saturation formula, eq. (7.22) fairly satisfactorily reproduces the experimental power scaling versus 0 (Newnam, Warren, Sheffield, Goldstein and Brau [1985]; see also Dattoli, Giannessi, Cabrini and Loreto [ 19921). These results are related to FEL operating with single-passage devices, namely, in a configuration using lower energy accelerators, in which the electron beam is passed once only in the optical cavity. FELs operating with recirculated electron beam (storage-ring FEL) have been successful. The saturation mechanism is slightly different from that discussed so far and will be addressed in the next section.
5 8.
A Simplified View of the FEL Storage Ring Dynamics
This section examines the FEL storage ring (SR) dynamic behavior, using simple, but substantially correct, arguments. For a more accurate mathematical analysis readers are referred to Elleaume [ 19851. In his original proposal Madey [1971] suggested that the SR was an ideal candidate for FEL operation. The SR has the advantage of operating at high energy, thus allowing, in principle, short wavelengths. The beam qualities (peak current, energy spread and emittances) are in general good. Furthermore, the electron beam recirculates often in the interaction region, and it is not lost after each interaction as in low-energy, single-pass devices. This type of natural beam recovery seems to provide greater efficiency with respect to microtron or linac operating FELs, for example. The SR FEL efficiency enhancement is just illusory, since the beam quality is degraded by the multiturn interaction. A brief description of SR particle dynamics will help to clarify this point. In the SR, electrons are kept along circular orbits by the guiding magnetic field, and the energy is supplied by the radiofrequency accelerating system. The radiofrequency cavities have a twofold role: (1) to keep the electrons up to the nominal machine energy, and, (2) to restore the energy lost by the electrons by way of synchrotron radiation in the bending magnets. The interplay between energy loss and radiofrequency energy supply gives rise to the radiation damping mechanism, typical of SR dynamics. Since the energy emitted by way of synchrotron radiation is given by
394
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, 5 8
where p is the magnet bending radius, particles with energy larger than the nominal energy E, will radiate more than those with energy less than the nominal one. We therefore expect that after a time z,, the so-called synchrotron damping time,
where T is the machine revolution period, the particle energy will shift toward the nominal energy. As already noted, the niultiturn SR FEL interaction degrades the quality of the electron beam, inducing an energy spread, and the lasing process will stop when the relative induced energy spread is of the order of the homogeneous bandwidth. A new laser pulse is available after a time of the order of zs, i.e., the time necessary for the off-energy particles to be kept to the synchronous energy. This is the reason why the SR optical power P , is proportional to the power emitted by way of synchrotron radiation. More quantitatively, we have
where N , is the number of electrons in the beam. However, a more quantitative result can be obtained. We should recall that in a S R the microbunch length is proportional to the energy spread, and therefore an increase of the energy spread amounts to a bunch lengthening and thus to a reduction of the peak current. Therefore, the gain saturation is the byproduct of two concomitant effects, the reduction due to the energy spread and that due to the lowering of the peak current. Therefore, we can write a condition of the type (7.5) to evaluate the energy spread yielding SR FEL gain saturation, namely,
where p,(O) is the inhomogeneous broadening parameter corresponding to the natural energy spread. From eq. (8.4) we obtain
We can fix the conditions under which gain saturation is dominated by bunch
VI, § 81
SIMPLIFIED VIEW OF
FEL STORAGE RING
395
DYNAMICS
lengthening or by energy spread. In the first case we have 1 go
-
(8.6)
- pe(O)4 1 , YT
and therefore, solving eq. (8.5) perturbatively, we find the a,, induced by the interaction, namely, U,
G
0.85
go - a,(O),
(8.7)
YT
which amounts to a laser power go u,(O)P,. P, z 0.85 -
(8.8)
YT
In the second case we have (8.9)
thus yielding I
(8.10)
-
with the corresponding output power (8.11) Defining the quantity go 6 = 2N 0,(0),
(8.12)
YT
and defining the SR FEL efficiency as (8.13) we find the following scaling relationship: X(6)16<,
E 0.856,
X(6)Ib*,
= m,
in agreement with the numerical results of Dattoli and Renieri [ 19811.
(8.14)
396
THEORY OF COMPTON FREE ELECTRON LASERS
[VI,I 9
4 9. Conclusion The FEL, as discussed here, is a nonconventional laser device and perhaps one of the most interesting sources of coherent radiation developed during recent years. The FEL, in its modern conception, may be considered as the completion ofthe activity, aimed at the generation of radiation by free electrons, which started during the early 1950s, and was then temporarily forgotten because of the invention of the laser. After more than a decade of activity, the FEL became such a wide field of research that covering all its features in a review paper is impossible. We have therefore selected specific topics even within the limited field of the Compton FEL. We have not discussed collective regime devices, efficiency-enhancement systems such as those based on energy recovery, or tapered undulators, nor have we mentioned the possibility of obtaining coherent radiation from free electrons with the eerenkov or Smith-Purcell types of devices. The primary focus of this chapter has been to present information in a physically insightful way with minimal attention to organic generality and mathematical rigor. We tried to clarify and emphasize the various effects of the dynamic behavior of the Compton FEL interaction, starting from the spontaneous emission and then including gain effects, transverse and longitudinal mode dynamics and, finally, saturation mechanisms. Our goal was to present simple recipes to calculate, for example, the gain when inhomogeneous broadening effects are active, or the effect of optical mode focussing induced by the interaction, and to describe the interplay between saturation and lethargy. The FEL is rapidly becoming a practical source of coherent radiation with the unique feature of its wide tunability. For the first time many scientists are faced with the problem of understanding the basic mechanisms underlying FEL dynamics and evolution. We hope that this chapter will be a useful starting point to appreciate the complexities of the topic.
Appendix A. Optical Cavity for the FEL Since the role of the optical cavity in the FEL is the same as that in conventional laser systems, there is a tendency to consider the problems associated with the resonators as already solved. However, serious troubles can arise in the design, construction and operation of an optical cavity for the FEL, for example, the problems associated with the transverse mode control due to the large cavity length required by the insertion devices (quadrupoles,
VI, APP. A1
-
OPTICAL CAVITY FOR
FEL
391
= 6.1 5 m
Fig. A.1. Typical layout of FEL operating with a low-energy accelerator (microtron). Q denotes a quadrupole magnet; UM a undulator magnet; M a bending magnet; R a mirror; D a beam dump; and d the mirror spacing.
dipoles, undulator, etc., see fig. A. l), those associated with the control of the length to compensate for the lethargic effect, and those due to mirror degradation induced by synchrotron radiation damage, which imposes severe limitations on the FEL short-wavelength operation with a storage ring (Elleaume, Velghe, Billardon and Ortega [ 19851). This Appendix discusses some standard facts about the open optical resonators commonly used for the FEL operation. As noted earlier, FEL oscillators require a resonator for storing the electromagnetic energy emitted by the electrons, thus allowing the feedback process. The resonator design is determined by the operating spectral range of the laser. Open resonators are generally adopted for operation in spectral ranges from the ultraviolet up to medium infrared. Waveguide and hybrid resonators are appropriate in the far-infrared and millimeter regions of the spectrum. We wil limit ourselves to considering the open resonators, which, in the simplest configuration, are realized by two mirrors of suitable shape placed opposite each other. Unlike ordinary atomic and molecular lasers, the FEL amplifying medium has a transverse dimension narrower than that of the optical mode. Consequently, higher-order modes can be excited to the detriment of the optical quality of the beam. On the other hand, the occasional low gain available in FELs requires very low diffraction losses. Another important requirement concerns the tunability of the laser over a wide spectral range. The working wavelength of a FEL, as noted earlier, can be continuously tuned by varying the electron energy or the undulator magnetic field ; operation on high-order harmonics is also possible. A broadband resonator is required, however, because the limitations in the useful spectral range of an optical resonator are sometimes stronger than those posed by the electron energy tunability in conventional accelerators. As a consequence, a careful design of the cavity is necessary to optimize the FEL performance.
398
THEORY OF COMF'TON FREE ELECTRON LASERS
P I , APP. A
The theory of open resonators is well documented in the literature (Yariv [ 19751, Nussbaum and Phyllips [ 19761, Solimeno, Crosignani and Di Porto [ 19851). We will note those elements which are crucial for choosing and designing a resonator and review the topic in the context of geometrical and wave optics.
A.l. RAY MATRIX AND STABILITY CONDITION
In the framework of geometrical optics the transit of a paraxial ray* through any optical element is described by the so-called transfer matrix. Representing a ray at any position z along the optical axis as a column matrix
where r and r' denote the distance of the ray from the axis and its slope with the axis, respectively. We can relate the output ray (ro, r ; ) in matrix form (fig. A.2)
where w(z) and R ( z ) are the (l/electron beam) radius and the curvature of radius of the wave front at z, respectively. The matrix elements A , B, C, and D are specified by the structure of the optical system under study.
Fig. A.2. Schematic representation of an optical system.
* By paraxial ray we mean a ray whose angular deviation from the optical axis is small enough for the sine and tangent of the angle to be approximated by the angle itself.
VI, APP. A1
OPTICAL CAVITY FOR
FEL
399
TABLE A.l
Ray matrices relevant to different geometrical mirror configurations. (a) A straight section of refractive index n :
(b) A spherical dielectric interface of radius R:
(c) A spherical mirror of curvature radius R:
In table A. 1 the ray matrices corresponding to a straight section, a dielectric interface and a mirror are reported as examples of matrices relevant to the translation, refraction and reflection, respectively, inferred straightforwardly from the laws of geometrical optics. The ray matrix of combined systems involving a variety of optical elements can be obtained by multiplying the matrices relevant to the various components. The order of the multiplication is defined by the direction of propagation. As an example, the ray-transfer matrix of a thin* lens results from multiplying the refraction matrices relevant
*
By definition, a lens is thin if r,,
= r,.
400
[VI, APP. A
THEORY OF COMPTON FREE ELECTRON LASERS
to the two limiting surfaces, thus obtaining
with .f being the focal length of the lens. It is easy to realize that the propagation of a light ray through an optical
resonator is equivalent to the propagation along a sequence of identical optical systems characterized by the ray matrix of the resonator under study (fig. A.3). The solution of the height of the ray can be readily traced (Yariv [ 19751) in terms of the block index j as
r,
=
7 sin(j6 + $),
(‘4.4)
with
tan $ =
r, sinb Drb + r,(A - cosb) ’
r, and rb being the input values (i.e., for j parameter 6 is defined by
6 z arcos [ ( A
=
0) of r and r ’ , respectively. The
+ D)].
b)
+-q-F-t I-qF ----------
Fig. A.3. The propagation of a paraxial ray through a resonator (a) is equivalent to the propagation through a periodic sequence of identical systems (b).
OPTICAL CAVITY FOR
VI, APP. A1
FEL
40 1
It is evident that if 6 is real, i.e., if
IA+Dl<2.
64.7)
the transverse displacement of a generic ray trajectory as a function of the block index ,j oscillates between T; and - F. Conversely, if the confinement condition (A.7) is violated, which amounts to 6 complex, the distance of the ray from the axis will increase as a function o f j . In the former case the cavity is called stable, and in the latter, unstable. For a resonator realized by two mirrors with curvature radii R , and R2 separated by a distance d. the ray-transfer matrix is given by (fig. A.4)
where we introduce the so-called g parameters defined by g , * 2=
1-
d --
.
Rl.2
Therefore, the stability condition (A.7) can be specified in terms of the system parameters as o
1.
(A. 10)
which in the (g,, g 2 ) plane represents the region between the axis and the two branches of the hyperbola g,g, = 1 (fig. A.5). The stable resonators are therefore represented by points located in this
402
[VI,APP. A
THEORY OF COMPTON FREE ELECTRON LASERS
91
Fig. A.5. The Boyd-Kogelnik diagram; the dashed area represents the stable region.
region; their modes (in the sense of field distributions that reproduce themselves after each roundtrip) are confined along the optical axis and are thus indifferent to the mirror sizes. Conversely, the unstable resonators are represented by points lying outside this region; their modes sweep the whole cavity and are strongly dependent on the mirror shape and size. Most FEL experiments employ two-mirror stable resonators lying in the third quadrant, where the g parameters are negative. The particular configuration within this region is determined by the details of the environment in which the resonator is to be inserted. Particularly limiting is the presence of the undulator, usually several meters long, and of the injection and extraction devices of the electron beam into and from the cavity. The large dimensions of the FEL resonators allow them to be recognized as near-concentric configurations. In a concentric resonator the curvature centers of the mirrors coincide; the symmetric version of this configuration corresponds t o g , = g, = - 1.
A.2. MODES OF A STABLE RESONATOR FREE OF DIFFRACTION LOSSES
Although the aforementioned matrix formalism provides a powerful tool to follow the evolution of a paraxial ray propagating through an optical system, no information can be inferred about the evolution of the transverse distribution
VI, APP. A1
OPTICAL CAVITY FOR
403
FEL
of the optical beam. An appropriate analysis requires solving the Helmholtz scalar equation
V2u t k2u = 0 ,
(A. 11)
the electromagnetic field being exhaustively described by the electric or magnetic transverse component in the hypothesis of paraxial propagation and small curvature of the mirrors. By assuming u(x,y,z) to be a quasi-plane wave propagating along the axis z and so change slowly with z apart from the exponential factor exp( - ikz), as solutions to eq. (A. 11) we obtain the so-called Hermite-Gauss or Laguerre-Gauss modes, according to whether we are interested in a rectangular or a cylindrical geometry (Solimeno, Crosignani and Di Porto [ 19851). The Laguerre-Gauss modes read*
x exp[ -ikz - ik
-f-t i(2p t I
t 1)tan-'
2dz) with Li, being the generalized Laguerre polynomial. The complex curvature radius is defined by 1 - 1 d z ) R(z)
R
i-, nw2(z)
(A. 12)
(A. 13)
where w(z) and R(z) are the (l/electron beam) radius of the wave front at z, respectively. They evolve with z according to
(A. 14)
where the Rayleigh distance zo is given by XWO'
z,=-.
R
(A. 15)
* The integers landp are usually referred to as the azimuthal and the radial index, respectively.
404
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, APP. A
The minimum value of w(z) (i.e., wo) is assumed to be located at z = 0, and is usually called beam waist. Gaussian beams are similar to the plane waves, the only difference being in the curvature of the wave front and in the nonuniformity of the transverse distribution of the intensity. Figure A.6 shows the intensity profiles of some axially symmetric (i.e., 1 = 0) modes. Expression (A. 12) for the modes of an optical cavity can provide a qualitative approach to the problem of minimizing the diffraction losses in designing a FEL resonator. By supposing that the mirror apertures are much larger than the undulator ones, we can minimize the diffractive effect of the undulator apertures, by locating the waist of the TEM,, mode at the middel of the undulator and supposing that the spot size at the end of the undulator is $ times the waist. Practically, we require the undulator to realize a confocal resonator, which exhibits the lowest diffraction losses. We thus obtain (A. 16)
which readily provides the values of the curvature radii of the mirrors in terms of L , and their position with respect to the waist. For a waist-mirror spacing much greater than iL,, as usually occurs in FELs, the resonator approaches the concentric configuration. Unfortunately, this configuration is located at the boundary of the stability region, and is particularly sensitive to the mirror misalignment. Finally, we stress the possibility of defining the parameters of the Gaussian modes that fit an optical resonator of specified geometry. The requirement that the field distribution self-reproduces at each roundtrip, apart from a constant complex factor, is equivalent to requiring that the complex curvature radius q
TEMOl
TEMo2
Fig. A.6. Profiles of axially symmetric Laguerre-Gaussian modes.
OPTICAL CAVITY FOR
VI, APP. A1
FEL
405
will not be affected by the propagation through the resonator, thus obtaining* Aq 9’-
Cq
+B +D
9
(A. 17)
which immediately provides R and w at the reference surface, arbitrarily chosen within the resonator. The explicit expression of R and w (Yariv [ 19751) leads again to the stability condition [eq. (A.8)], which ensures the reality of R and w .
A.3. DIFFRACTION INTEGRAL AND RAY MATRIX
As already noted, the Hermite-Gauss or Laguerre-Gauss modes can be understood as the modes of an optical resonator with mirrors having infinite transverse size. A more thorough analysis of the field distribution in an open resonator should require consideration of the finite size of the mirrors. In this context the modes stand out as the eigenfunctions of Fredholm integral equation of the first kind, deduced from Huygens’ principle (Solimeno, Crosignani and Di Porto [ 19851). For an axially symmetric resonator, in the hypothesis of small curvature of the mirrors and large mirror-mirror spacing, the eigenvalue equation becomes
(A. 18) the solutions of which provide the radial distributions of the modes with azimuthal index 1, the angular dependence of which was assumed to be exp( il$). The upper limit of integration a represents the aperture of the mirror, the surface of which was selected as reference surface. Equation (A. 18) translates in mathematical terms the self-reproduction of the field distribution over the chosen mirror at each roundtrip apart from the constant complex factor r,. It is easy, therefore, to realize that the eigenvalues r,give the diffraction losses a, and the phase shift fl, according to the relations (A. 19) The kernel K ( p , p ’ ) , whose explicit expression is not needed for the present * This relation is a consequence of the A B C D law, for which the reader is referred to the literature (Yariv [ 19751) and the self-reproduction condition.
406
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, App. B
purposes (see Solimeno and Torre [1985]), depends on the geometry of the resonator, specified by the apertures and curvature radii of the mirrors and by the mirror-mirror spacing (see fig. A.6). It is worth emphasizing that for a symmetric resonator (i.e., with identical mirrors)* or in the case where the aperture of one mirror is much larger than that of the other, the kernel K ( p , p ’ ) can be written as (Collins [1970], Siegman [ 19761, Fang [ 19791) K(p, p’)
=
i‘+’
k
{
exp - i
k (Ap” 2B
-
+ Dp2)+ ik[S(p’) + S(p)] (A.20)
where S(p) is the reference surface function [the mirror surface in the case of eq. (A.lS)]. Furthermore, A, B and D are the elements of the ray matrix relevant to the straight propagation followed by mirror reflection for a symmetric resonator and to the roundtrip propagation for a resonator with a mirror of infinite size. The kernel relevant to an asymmetric resonator is a generalization of eq. (A.20), where the up and down transits are separately treated. Expression (A.20)holds in general, and it is particularly useful for analyzing the propagation through a multielement optical system (e.g., a resonator with internal lenses, or more complicated structures). In that case A, B and D are relevant to the ray matrix of the whole system.
Appendix B. Undulator Magnets for the FEL
The undulator was introduced as the magnetic element transforming the longitudinal nonradiating electron motion into the transverse one, which allows the coupling to the transverse electric field of a copropagating electromagnetic wave. According to the preceding definition, the undulator is a rather abstract device, and far from the physical reality. The first problem arises in connection with the field distribution, which on
* Actually, the definition of a symmetrical resonator is more general, involving the generalized g parameters [25]. For the sake of simplicity, we adopt the above one.
VI, APP. Bl
UNDULATOR MAGNETS FOR THE
FEL
407
the undulator axis is required to be
where E:. , depends on the polarization of the undulator. We discuss: (i) helical case E!! = E: = B,, # 0 , (B.2) (ii) lineal case B: = B,, # 0. B: = 0, If By, , are assumed to be constant, it is readily understood that the field (B. 1 ) does not satisfy the Maxwell equation (in fact, we obtain V x B # 0). A real field would otherwise exhibit the following behavior in the transverse plane (see Blewett and Chasman [ 19761): (i) helical case
(ii) linear case:
where $ = (2n/Au)z + &. Expressions (B.3) and (B.4) are the result of an expansion up to the second order in x and y . Furthermore, the quantity 6 is a sextupolar term due to the finite width of the undulator magnet poles and is typically very small ( 1 61 4 l), but it cannot be neglected because it strongly affects the focussing properties of the undulator. We define the reference orbit ( x R )relative to perfect injection as that with the same periodicity of the field, namely,
408
THEORY O F COMFTON FREE ELECTRON LASERS
[VI,APP. B
The evaluation of the reference trajectory up to the lowest order in K / y is straightforward; in fact, we find (i) helical case
(ii) linear case
It is now easy to derive the linearized motion around the reference orbit; in fact, defining XI =
x -
XR
9
Yl
= y - YR
(B.8)
9
we obtain for x , and y I the following equations of motion: (i) helical case (x;’ = 2 ( - )n K A U Y
I
I,;
=
-2(3(,,
-
‘lu 211
.;)
-2.;)
’
(ii) linear case
(B. 10) y; = -2
(ZY(2 - 6)y, 1, Y
Equation (B.9) indicates that in the helical case the electron motion is focussed in both directions. Although eq. (B. 10) states that the characteristics of the motion depend on the sextupolar term 6, in general the field is maximum
V1, APP. BI
UNDULATOR MAGNETS FOR THE
FEL
409
on-axis (x = 0) and 6 is negative. Therefore, in the direction perpendicular to the magnetic field (in this case along x), the motion is defocussed, whereas it is focussed along the other direction. Equation (B. 10) can be solved immediately. On the other hand, eq. (B.9) can be integrated easily by introducing the following auxiliary variable: u , = xi
+ iy, .
(B. 11)
which transforms eq. (B.9) into (B. 12) or, what is the same, (B. 13) Multiplying eq. (B.12) by u:‘ and eq. (B.13) by u ; and then summing, we obtain the motion invariant (B.14) An analogous invariant can be obtained from eq. (B.10), so that, in general, we have
where, for the helical case,
h,v = h,
=
1
(B. 16)
and, for the linear case,
h,
=
6,
h,
=
2 - 6.
(B. 17)
These last results clarify the role of the parameter h x * yintroduced in 8 2. The variation &BIBof the RMS transverse magnetic field with the average displacement coordinate (x,, y , ) can also be derived, and from eqs. (B.3), (B.4), (B.6), (B.7), and (B.15) we obtain (B. 18)
410
THEORY OF COMPTON FREE ELECTRON LASERS
[VI, APP. B
We have not discussed the technological aspects of the undulators, which can be realized in many different ways, including electromagnetic and permanent-magnet-type undulators, and we will not analyze the problems underlying their practical realization (for general review see, e.g., Poole and Walker [ 19811, Halbach [ 19811, Walker [ 19851)). For the sake ofcompleteness, however, we will give a few elements relevant to the permanent-magnet-type undulators, which are the most commonly adopted solution in F E L experiments. The development of rare-earth-cobalt (REC) technology in recent years has allowed the use of these materials as the main components of high-magnetic field devices. The distinctive feature of a R E C magnet is the fact that it provides high-magnet field strength in spite of its small dimensions. Linear undulators are realized using the configurations of fig. B.l. Four R E C magnets per period, for each plane array, provide a homogeneous field pattern.* If the magnets are assumed to be infinitely wide, the on-axis field intensity can be evaluated analytically as (Poole and Walker [ 19811): B,
=
2 B r [ sin ( n / M )][l
-
exp( - y ) ] e x p [
niM
-71,
(B.19)
where B, is the permanent field, M the number of blocks per period, h the block height, and g the gap width. Helical undulators can also be realized using R E C materials (see, e.g., Halbach [ 19811).
I
9
l - ~ 4 Fig. B.1. Halbach configuration of a permanent magnet undulator.
* The half-blocks at the beginning and at the end ofthe undulator are inserted to obtain a zero integral on-axis.
VII
REFERENCES
41 1
Particular attention must be devoted to the homogeneity of the field. Therefore, random magnetization errors should be carefully eliminated by an accurate magnet selection and compensation. These errors can induce a further inhomogeneous broadening type of distortion on the radiated spectrum, which in turn, can significantly affect the laser process (for further discussion, see Ciocci, Fiorentino, Renieri and Sabia [ 19861).
Note added in proof In the limit of large I/Is,relation (7.9) looses its validity. Consequently, also eq. (7.17) derived from (7.9) may give unreliable results. A large ratio I/I, is reached when the cavity losses are small compared with the small-signal gain. For this reason, eq. (7.17) may be considered as the upper limit of the overall cavity losses maximizing the efficiency. For further insight in this topic see Dattoli, Cabrini and Giannessi [ 19911.
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[VI
Dattoli, G., J. C. Gallardo and A. Torre, 1986, J. Math. Phys. 27, 772. Dattoli, G., J. C. Gallardo, T. Hermsen, A. Renieri and A. Torre, 1988, Lethargy of laser oscillations and super modes in free electron lasers (Part I), Phys. Rev. A 37, 4326. Dattoli, G., L. Giannessi, S. Cabrini and V. Loreto, 1992, Phys. Rev. A 45, 8842. Dattoli, G., L. Giannessi, H. Fang, A. Renieri and A. Torre, 1988, Nuovo Cimento B 101, 703. Dattoli, G., L. Giannessi, T. Hermsen, A. Renieri, M. Richetta and A. Torre, 1989, Nuovo Cimento D 12, 21. Dattoli, G., L. Giannessi, L. Mezi and A. Torre, 1990, Nuovo Cimento B 105, 327. Dattoli, G., T. Hermsen, L. Mezi, A. Renieri and A. Torre, 1988, Phys. Rev. A 37, 4335. Dattoli, G., A. Marino, A. Renieri and F. Romanelli, 1981a, IEEE J. Quantum Electron. QE-17, 21 1. Dattoli, G., A. Marino, A. Renieri and F. Romanelli, 1981b, IEEE J. Quantum Electron. QE-17, 1371. Dattoli, G., A. Torre, C. Centioli and M. Richetta, 1989, IEEE J. Quantum Electron. QE-25,2327. Didenko, A. N., A. V. Kozhevnikov, A. F. Medvedev, N. M. Nikitin and V. Ya, 1979, Sov. Phys.-JETP 49, 973. Elleaume, P., 1985, Nucl. Instrum. Methods A 237, 28. Elleaume, P., M. Velghe, M. Billardon and J. M. Ortega, 1985, Nucl. Instrum. Methods A 237, 263. Fang, H., 1979, Acta Phys. Sin. 28, 450. Halbach, K., 1981, Nucl. Instrum. Methods 187, 109. Hofmann, A., 1986, SSRL ACD-NOTE 38, Stanford Synchrotron Radiation Lab. Jackson, J. D., 1975, Classical Electrodynamics (Wiley, New York) ch. 14. Kim, K. J., 1985, Proc. Int. Conf. on Insertion Devices for Synchrotron Sources, SPIE 582, 2. Kincaid, B. M., 1984, J. Appl. Phys. 48, 2684. Kitamura, H., 1980, Jpn. J. Appl. Phys. L 19, 185. Krinsky, S., 1980, Nucl. Instrum. Methods 73, 172. Madey, J. M. J., 1971, J. Appl. Phys. 42, 1906. Madey, J. M. J., 1979, Nuovo Cimento B 50, 64. Newnam, B. E., R. W. Warren, R. L. Sheffield, J. C. Goldstein and C. H. Brau, 1985, Nucl. Instrum. Methods A 237, 187. Nussbaum, A., and R. S. Phyllips, 1976, Contemporary Optics for Scientists and Engineers, ed. N. Holonyak (Prentice-Hall, Englewood Cliffs, NJ). Poole, M. W., and R. P. Walker, 1981, IEEE Trans. Magn. MAG-17, 1978. Raman, C. W., and N. S. Nath, 1936, Proc. Indiana Acad. Sci. 2,406. Siegman, A. E., 1976, IEEE J. Quantum Electron. QE-12, 35. Solimeno, S., and A. Torre, 1985, Nucl. Instrum. Methods A 237, 404. Solimeno, S., B. Crosignani and P. Di Porto, 1985, Guiding, Diffraction and Confinement of Optical Radiation (Academic Press, Orlando). Sprangle, P., Cha Mei Tang and W. M. Manheimer, 1980, Phys. Rev. A 21, 302. Stenholm, S., 1984, Foundations of Laser Spectroscopy (Wiley, New York). Tatchyn, R., and I. Lindau, 1984, NIM 14, 222. Tatchyn, R., and S. Qadri, 1985, Proc. Int. Conf. on Insertion Devices for Synchrotron Sources, SPIE 582,47. Walker, R. P., 1985, Nucl. Instrum. Methods A 237, 366. Yariv, A., 1975, Quantum Electrons (Wiley, New York).
AUTHOR INDEX
A Aberg, T., 71, 135 Abragam, A., 265, 285,318 Abu-Mostafa, Y. S., 230,260 Ackerhalt, J. R.,15, 28, 38, 39, 53-55, 72, 78, 81, 93, 104, 105, 108, 133, 136, 292, 310, 312,319 Adachi, T., 68, 136 Afrashteh, A., 145, 186 Agostini, P., 40,41,43,46,47, 133, 137 Agrawal, G. P., 192, 222 Aida, K., 217,222 Ainslie, B. J., 211, 222 Akhouayri, H., 168, 186 Alferov, D. F., 334,411 Allah, J. Y., 207, 215, 223 Allen, L., 19, 57, 64, 133, 265, 266, 285, 289, 294-296, 305,308, 318 Alsing, P., 310, 313, 318 Amodei, J. J., 237, 251,260,261 Anderson, D. Z., 230,260 Anderson, J., 229,260 Andrejco, M. J., 215, 221, 225,226 Andrekson, P. A., 219,220,222,225 Andrewartha, J. R.,163, 185 Antunes Neto, H. S., 38, 39, 58, 133 Ao, P., 310, 318 Aoki, Y.,207, 217,222,223 Arimondo, E., 290,292, 296, 318 Arkwright, J. W., 221,223 Armstrong Jr, L., 52, 53, 61, 69, 134, 136, 137 Arnold, V. I., 84, 133 Aspell, J., 216, 217, 223 Athale, R. A., 230,260 Auge, J., 218, 222
B Bambini, A,, 351, 352, 411 Bandarage, G., 66, 133
Barbini, R.,334, 411 Bardsley, J. N., 124, 131, 133 Barnes, W. L., 214,223 Bashkansky, M., 43-46.48, 50, 54, 69, 70, 134,136 Bashmakov, Yu. A., 334,411 Bayfield, J. E., 109, 110, 124, 130, 131, 133 Bebb, H.B., 13, 14, 17, 133 Becker, P. C., 207-209, 214, 219, 220,222, 223,225 Beijersbergen, M. W., 266-269,272, 281, 282,287-291,294-296,308,318,319 Benkert, C., 230,260 Bergano, N. S., 216, 217,223 Bernstein, I., 354, 411 Berry, M. V., 102, 103, 133, 309,318 Bessonov, E. G., 334,411 Bethe, H. A., 11, 133 Billardon, M., 397,412 Biotteau, B., 218,222 Bjorkholm, J. E., 20.21, 135 Blank, L. C., 220,224 Bleha, W. P., 231, 260 Blewett, J. P., 407, 41 I Bletekjaer, K., 249,260 BlUmel, R., 124, 130, 131, 134 Blyth, K. J., 221,223 Bocchieri, P., 97, I34 Bohigas, O., 102, 134 Bonch-Bruevich, A. M., 20, 134 Bonifacio, R.,354, 411 Bosco, P. M., 364,411 Boser, B., 248,261 Bousselet, P., 218,222 Bowlin, J. B., 119, 120, 137 Boyer, K., 61, 136 Brady, D., 230, 231, 238, 243, 249,260,261 Brandi, H. S.,36, I34 Brau, C. H., 393,412 Breidne, M., 148, 185 413
414
AUTHOR INDEX
Brierley, M. C., 207, 215,223 Brown, G. N., 220,224, 334,411 Brown, H. B., 231,260 Brown, L. S., 326,41 I Brown, R. L., 198-200, 202,223 Brown-Goebeler, K., 206,224 Bryant, E. G., 217,223,224 Bucksbaum, P. H., 43-46,48,50, 54, 69, 70, 134, 136 Burckhardt, C. B., 239,260 Burnett, K., 61, 73, 134, 136 Burrus, C. A., 206, 207,224, 225 Butler, J. K., 192, 224 C Cabrini, S., 389, 393, 41 I , 411, 412 Cadilhac, M., 143, 145, 146, 148, 164, 167, 186 Campbell, S., 230,260 Caneau, C., 203,226 Cannon, R. S., 214,223 Carmichael, H. J., 310, 313, 318 Carnal, O., 266, 318 Carter, S. F., 207, 211, 215, 217,223,224 Casasent, D., 231,260 Casati, G., 94,96, 98, 100, 125, 128-130, 133,134, 315,318 Case, S. K., 158, 185 Casey Jr, H. C., 192,223 Castin, Y., 309,318 Caudle, G., 24, 25, 137 Caulfield, H. J., 230,261 Centeno Neelen, R.,266, 277,278, 287. 303, 318,319 Centioli, C., 355, 356, 412 Cerjan, C., 58, 134 Chakmakijan, S., 286,318 Chakrabarti, U. K., 198-200, 202,223 Chandezon, J., 148, 151, 164, 167, 185, 186 Chang, K. C., 145, 152, 185 Chartier, G., 168, 187 Chasman, R., 407.41 1 Chen, D. N., 216,224 Cheung, N. K., 217,225 Chien, M., 206,224 Chin, S., 17, 134 Chirikov, B. V., 93, 94, 96, 100, 116, 125, 128-130, 134 Cho, A. Y., 212,223 Chou, P. A., 233,260
Chow, W. W., 277,318 Choy, M. M., 221,226 Chraplyvy, A., 217,223 Chu, S.-I., 71, 72, 134 Chyba, T. H., 270,302, 303,318 Ciocci, F., 334, 411, 411 Clarke, E.M., 24, 25, 137 Clauss, G., 168, 187 Clergeaud, C., 218, 222 Clesca, B., 218,222 Cline, T. W., 217,218,221,223,225 Coblentz, D., 219,220,222 Cohen, L. G., 219,220,224 Cohen-Tannoudji, C., 286, 290,292,296, 310,318 Collier, R.,239, 260 Collins, L. A., 58, 60, 134 Collins, S. A., 406, 41 I Colson, W. B., 334, 351, 364,411 Comella, M. J., 131, I33 Cooper, J., 66, 71, 133, 134 Corduneanu, C., 82, 134 Cornely. R.H., 192,224 Cornet, G., 151, 185 Coult, D. G., 217,223 Coupland, M. J., 192,223 Coutaz, J. L., 168, 185, 186 Cowan, R. D., 61,134 Crance, M., 43, 134 Crasemann, B., 71, I35 Crisp, M. D., 305,318 Cronin-Golomb, M., 237,260 Crosignani, B., 398,403,405,412 Crowe, J. W., 192,223 Cummings, F. W., 286,318 Cvitanovic, P., 77, 134
D Dalibard, J., 309, 318 Dattoli, G., 323, 329, 334, 340, 349, 353-356, 363, 371, 377, 379, 383, 385,389, 393, 395,411,411,412 Davey, S. T., 207,215,223 Davidovich, L., 36, 38, 39, 58, 133, 134 Davidson, C. R., 216,217,223 Day, C. R., 21 1,223 Delavaux, J.-M. P., 218, 225 Delone, N. B.. 123, 134 Delrosso, G., 217, 225 DeMiguel, J. L., 206,224
AUTHOR INDEX
Denker, J. S., 248,261 Desurvire, E., 207, 208, 213-215,217,223, 226 Di Porto, P., 398, 403, 405, 412 Didenko, A. N., 334,412 DiGiovanni, D., 211,223 DiMauro, L. F., 46,48, 50, 134 Duda, R., 233,260 Dum, R., 309,318 Dumery, G., 145, 146, 185 Dunning, G. J., 230,261 Dupont-Roc, J., 286,290, 292, 296, 310, 318 Dupuis, M., 151, 185 Dursin, A., 218,222 Durteste, Y., 207, 215, 223 Dutta, N. K., 192, 198-200, 202, 212, 222-224
E Eberly, I. H., 19, 22, 23, 52, 53, 5 5 , 57-60, 62-68,72,73,133-137,265,285,289, 305,318,319 Edagawa, N., 217, 218,223,225 Edwards, C. A., 217,226 Edwards, E., 69, 134 Eido, R., 145, 186 Einstein, A., 265,318 Eisenstein, G., 205,223 Ekstrom, C. R., 266,318 Eliel, E. R., 266-269,272, 277,278, 287-291,297,298,303,318,319 Elleaume, P., 385, 393, 397, 412 Elliott, D. S., 309, 318 Eloy, J. F., 168, 187 Enger, R. C., 158, 185 Etrich, C., 278, 318 Evangelides, S. G., 219, 220, 224
F Fabre, C., 290, 292, 296, 318 Fabre, F., 40.41, I33 Faisal, F. H. M., 34, 71, 134 Fang, H., 363, 377,406,411,412 Fano, U., 145, 159, 185 Farhat, N. H., 230,260,261 Feinberg, J., 237,260 Ferray, M., 61-63, 134 Feynman, R. P., 75, 134, 274,318 Filippi, P., 145, 146, 185 Fiorentino, F., 41 1, 41 I
415
Fisher, A. D., 230,260 Fishman, D. A., 217,223 Fishman, S., 101, 135, 315,318 FlUggen, N., 316,319 Fokkema, J. T., 145, 187 Fontana, F., 217,225 Ford, J., 85, 86, 88-91,94,96, 100, 134, 137, 315, 318 Forrester, D. S., 221,223 Fox, J. R., 163, 185 France, P. W., 207, 211, 215,223 Freeman, R. R., 43-46.48, 50, 54, 134, 136 Friedlander, C. B., 230, 260 Fujita, M., 213, 219,225 Fujita, S., 217, 223 Fukasaku, Y., 207,224
c Gabla, P., 218,222 Gacoin, P., 157, 186 Galaup, J. P., 168, 187 Galbraith, H. W., 292, 310, 312, 319 Gallagher, T. F., 293, 294,319 Gallardo, J. C., 329, 364, 319, 383,411, 412 Galvez, E. J., 123, 136, 293, 301, 315, 318 Garraway, B. M., 297, 309,319 Garsia, N., 178, 185 Gautheron, O., 218,222 Gavrila, M., 72, 73, 134, 136 Gaylord, T. K., 145, 158, 186 Gea-Banacloche, J., 277,318 Geltman, S., 57, 134 Georges, A. T., 18, 134 Gersten, J. I., 72, 134 Geurten, S. H. M., 282, 319 Giannessi, L., 334, 340, 363, 377, 385, 389, 393, 411,411,412 Giannoni, M. J., 102, 134 Gibson, G., 61, 136 Giles, C. R., 213-215,223,226 Gimlett, J. L., 216, 217, 224, 225 Gmitter, T. J., 280, 319 Goggin, M. E., 57, 104-108, 123, 124, 135, 136 Gold, A., 13, 14, 17, 133 Goldberg, A,, 57, 135 Goldstein, H., 79, 135 Goldstein, J. C., 393, 412 Gontier, Y., 16, 135 Gordon, J. P., 219, 220, 224
416
AUTHOR INDEX
Graham, R.,285, 292, 293, 310,318 Graig Jr, R.M., 192,223 Grandpierre, G., 218,222 Grasso, G., 217,225 Grempel, D. R.,101,135, 315,318 Grinberg, J., 231,260 Grubb, S.G., 214,223 Grynberg, G., 286,292, 310,318 Gu, C., 250,261 Gu,X.-G., 231, 235,261 Guameri, I.. 98, 129, 130, 133, 134, 315, 318 Guest, C. C., 230,260 GUO,D.-S., 71, 135
H Haake, F., 276, 312-315,318 Hagimoto, K., 216, 217,223 Hajnal, J. V., 170, 185 Halbach, K., 334,410,411,412 Handschy, M. A., 259,261 Haner, M., 219, 220,222 Harms, K.-D., 312,318 Harris, J., 334, 411 Hart, P., 233,260 Hata, K., 68, 136 Haus, H.A., 219,224, 317, 318 Hebb, D., 230,247,260 Hebler, V., 230,260 Heller, E. J., 132, 133, 135 Hellwarth, R.W., 274, 318 Henderson, D., 248,261 Henmi, N., 217,223 Henneberger, W. C., 72, I35 Henry, C. H., 309,318 Hermsen, T., 371, 379, 383, 385, 412 Hessel, A., 148, 160, 165, 185, 187 Hester, R., 168, 186 Hill, A. M., 221,223 Hillery, M., 312, 318 Hilsum, C., 192,223 Hinton, G. E., 243,261 Hirshfield, J. L., 354,41 I Hobson, A,, 98, 135 Hodgkinson, T. G., 221,223 Hofmann, A,, 337, 339,412 H o g , T., 98, 135 Hohnerbach, M., 285, 292, 293, 310,318 Holmes, J. K., 24, 25, 137 Hong, J., 230, 250,260, 261 Hornik, K., 23 1,260
Hose, G., 130, 135 Hougen, J. T., 293,319 Howard, R.E., 248,261 Hsu, K., 249,260 Huang, S.Y.,221,223 Hubbard, W., 248,261 Huberman, B. A., 98, 135 Humer, W. F., 214,223 Hutley, M. C., 141, 142, 154, 168, 185, 187 I Ichihashi, Y.,216, 218,225 Imai, T., 216, 218, 225 Inoue, K., 216,217,224 Iqbal, M. B., 217,225 Ishikawa, M., 230,261 Ito, T., 216, 218, 225 Iwasaki, H., 21 1,225 Iwatsuki, K., 220,224 Izadpanah, H., 216,224 Izrailev, F. M., 94,96, 100, 103, 134, 135, 315,318
J Jackel, L. D., 248,261 Jackson, J. D., 340,412 Jmg, J.-S., 230,260, 261 Jara, H., 61, 136 Javanainen, J., 58-60,62-66, 68, 72, 73, 134, 135, 137 Jaynes, E. T., 286,318 Jedrzejewski, K. P., 214,223 Jensen, R.V., 61, 73, 117-121, 125, 129, 133, 135, 137 Johann, J., 61, 136 Johnson, K. M., 230,259,261 Jones, R.C., 282,318 Jung, C., 24,25, 137
K Kac, M., 97, 135 Kalhor, H. A., 145, 186 Kaminow, I. P., 210,224 Kamifiski, J. Z., 72, 73, 134, 310,318 Kamp, L. P. J., 279,299,319 Kanamori, T., 207,225 Kanerva, P., 231, 261 Karule, E., 16, 135 Kataoka, T., 216, 217, 223 Kaufman, A. N., 103, 136
AUTHOR INDEX
Kawano, K., 216, 217,223 Keith, D. W., 266,318 Keldysh, L. V., 26, 36, 37, 135 Kerfoot, F. W., 216, 217,223 Khodovoi, V. A,, 20, 134 Kibble, T. W. B., 53, 135, 326,411 Kikoshima, K., 221, 222,224 Kim, K. J., 334,412 Kimman, J., 41-44, 135 Kimura, T., 192,224 Kimura, Y., 215,219, 220,224 Kincaid, B. M., 334,412 Kinser, J. M., 230,261 Kirschbaum, C. L., 73, 135 Kitagawa, T., 207,225 Kitamura, H., 334,412 Klein, N., 310, 319 Knight, P. L., 19,62, 73, 134, 135 Kobeisse, H., 145, 186 Koch, K., 286,318 Koch, P.M., 109-113, 119-123, 130, 133, 135-137,293, 301, 315,318 Koch, T. L., 206,224 Koester, C. J., 207,224 Kohonen, T., 232,261 Koonin, S. E., 57, 73, 74, 135, 137 Koren, U., 205, 206,223,224 Kosloff, R., 58, 134 Kosnocky, W. F., 192,224 Kozhevnikov, A. V., 334,412 Krainov, V. P., 123, 134 Kranz, K. S., 219,225 Krasinski, J., 22, 134 Kressel, H., 192,224 Kretzmeyer, P., 218,222 Krinsky, S., 334,412 Kroll, N., 24, 39, 135 Kruger, H., 24, 25, 137 Kruit, P., 41-44, 135 Kujawski, A., 312, 313, 318, 319 Kukhtarev, N. V., 251,261 Kulander, K. C., 60, 62, 63, 65, 68, 135, 137 Kupersztych, J., 43, 136 KuS, M., 314, 315, 318 Kwong, S. K., 230,261 L Lamb Jr, W. E., 270, 302, 319 Lambropoulos, P., 17, 18, 22, 134, 135
Landau, L. D., 36, 135,297, 318 Laude, J. P., 157, 186 Leclerc, E., 218,222 Lecompte, C., 18, 135 LeCun, Y., 248,261 Lee, H., 235,261 Lee, J. N., 230,260 Lee, L. S., 230,261 Lee, S.-Y., 230,261 Lee, T. P., 203, 226 Leighton, R.B., 75, 134 Leilabady, P. A,, 214,223 Lemaire, P. J., 219,225 Lemaire, V., 218,222 Lemberg, H. L., 221,226 Lenstra, D., 266, 268, 269, 272, 279, 282, 299,319 Lenz, G., 276,313-315,318 Leopold, J.G., 1 1 1 , 119, 120, 135, 137 Lepere, D., 157, 186 Leung, K. M., 280, 319 Lewis, R. B. J., 217, 223 L'Huillier, A., 41, 61-63, 68, 134-136 Li, T., 210,224 Li, W., 259,261 Li, X.F., 61-63, 68, 134, 135 Liao, P. F., 20, 21, 135 Lichtenberg, A. J., 78, 92, 93, 136 Lidgard, A., 214,223 Lieberman, M. A., 78,92, 93, 136 Lifshitz, E. M., 36, 135 Lin, C., 215,221, 225 Lin, Chinlon, 216,224 Lin, L. H., 239, 260 Lin, M. S., 198-200, 202,223,224 Lin, S., 231,261 Lindau, I., 334,412 Lippincott, W. L., 230,260 Lipson, J., 221,223 Lipton, L. T., 231,260 Lo, C. N., 221,225 Lobbett, R. A., 221,223 Loewen, E., 141, 142, 155, 157, 159, 161, 162, 164, 165, 185-187 Logan, R.A., 212, 219,220,222, 225 Loinger, A,, 97, 134 Lomprt, L. A., 13, 14, 41, 43, 61-63, 68, 134-136
417
418
AUTHOR INDEX
Lopata, J., 212, 223 Lorenz, E. N., 76, 136 Loreto, V., 393,412 Loudon, R., 17,136, 305,319
M MacKay, R. S., 91, 136 Madden, R. P., 145, 185 Madey, J. M. J., 353, 393, 412 Mainfray, G., 13, 14, 18, 40, 41, 43, 61-63, 133-136 Maloney, P. J., 316, 319 Malyon, D. J., 217,224 Mambleton, K. G., 192, 223 Manakov, N. L., 14, 17, 137 Mandel, P., 278, 318 Manheimer, W. M., 354,412 Manus, C., 13, 14, 18, 41, 43, 61-63, 134-136 Maquet, A., 66, 133 Marchesin, D., 58, 133 Marcuse, D., 309, 311, 319 Marino, A,, 354, 377, 412 Markevitch, B. V., 231,260 Markov, V. B., 251,261 Marorn, E., 230,261 Marshall, I. W., 220,224 Mashev, L., 150, 155, 157, 159, 161-167, 185-187 Massicott, J. F., 221, 223 Masuda, H., 217,222 May, R. M., 74, 136 Maystre, D., 141, 142, 146-148, 150-155, 157, 159, 161-164, 167, 168, 171, 175, 185-187 Mazur, P., 97, 136 McClellan, R. P., 145, 186 McDonald, S. W., 103, 136 Mcllrath, T. J., 43-46, 48, 50, 54, 134, 136 McIntyre, I. A., 61, 136 McPhedran, R. C., 161, 186 McPherson, A,, 61, 136 Mears, R. J., 207,224 Medvedev, A. F., 334,412 Meerts, W. L., 293, 319 Meiss, J. D., 91, 136 Menegozzi, L. N., 270, 302, 319 Menendez, R. C., 221,226 Menocal, S. G., 203, 226 Meredith, D. C., 57, 135
Mermit, N. D., 143, 186 Merts, A. L., 58, 60, 134 Merzbacher, E., 282,319 Messiah, A., 308, 319 Metcalfe, K., 168, 186 Mezi, L., 340, 371, 412 Milbrodt, M. A,, 217,223 Millar, R. F., 142, 145, 186 Miller, B. I., 206, 224 Miller, C. M., 217, 223 Miller, S. E., 210, 224 Miller, T., 217, 223 Milonni, P. W., 15, 19, 26, 28, 38, 39, 53-55, 57,62, 65, 67, 68, 72, 78, 81, 93, 104-108, 123, 124, 133, 135-137, 292, 309, 310, 312,319 Mito, I., 213, 217, 219,223,225 Mitschke, F., 3 16,319 Mittleman, M. H., 52, 72, 134, 136 Miyajima, Y.,207,224 Miyamoto, Y.,216, 217,223 Mlynek, J., 266,318 Moaveni, M. K., 145, 186 Moddel, G., 259,261 Moharam, M. G., 145, 158, 186 Mok, F. H., 249,261 Mollenauer, L. F., 219, 220, 224, 225 Mslmer, K., 309,318 Monerie, M., 207, 215, 223 Montroll, E., 97, 136 Mooradian, A., 315,319 Moore, M. W., 21 1,223 Moorman, L., 110-112, 123, 130, 135, 136, 293, 301, 315,318 Mortazawi-M, A., 123, 136 Morton, V. M., 15, 136 Moruzzi, G., 290, 292, 296, 318 Mukai, T., 192-195,224,225 Mukohzaka, N., 230,261 Muller, H. G., 41-44, 52, 53, 135, 136 Munz, M., 312, 313,318,319 N Nagel, J. A., 217,223 Nakagawa, K., 220,224 Nakagawa, M., 215,224 Nakamura, M., 192,224 Nakazato, K., 221,225 Nakazawa, M., 219,220.224 Narducci. M., 354,411
AUTHOR INDEX
Nath, N. S., 329,412 Nawata, K., 221,225 Neubelt, M. J., 219, 220, 224,225 Neureuther, A., 145, 186 Neurgaonkar, R. R., 250,261 Neviere, M., 142, 145, 146, 152, 155, 161, 162, 168, 186, 187 Newnam, B. E., 393,412 Nikitin, N. M., 334, 412 Nishi, S., 220, 224 Nodomi, R., 68, 136 Nosu, K., 216,217,224 Nussbaum, A,, 398,412 Nyman, B. M., 216, 217,223 0 Odagiri, Y., 213,219,225 Odulov. S. G., 25 1, 261 Ohhata, M., 216, 217, 223 Ohishi, Y., 207, 225 Ohkawa, N., 216,218,225 Okano, H., 213,219,225 Okita, T., 221,225 Oliner, A. A., 148, 160, 165, 185, 187 Olsson, N. A., 197, 207, 212, 214, 219, 220, 222,223,225 OMahony, M., 196,225 Oppenheimer, J. R., 35, 36, 136 Oron, M., 206,224 Ortega, J. M., 397, 412 Osinski, J. S., 203, 226 Owechko, Y., 230,261 Ozier, I., 293, 319
P Paek, E. G., 230,260, 261 Pagano-Stauffer, L. A,, 259,261 Pan, L. W., 52, 53, 69, 134, 136 Panish, M. B., 192, 223 Park, C. H., 230,261 Park, Y. K., 218,225 Parker, J., 72, 136 Patel, J. S., 230, 261 Pavageau, J., 145, 186 Payne, D. B., 221,223 Payne, D. N., 207,224 Pechukas, P., 102, 136 Pedrotti, L. M., 277,318 Pellegrini, C., 354, 411
419
Percival, I. C., 102, 111, 120, 135, I36 Perelomov, A.M., 36, 37, 57, 58, 136 Peres, A., 98, 136 Petit, R., 142, 143, 145, 146, 148, 150, 154, 163, 186, 187 Petite, G., 40, 41, 43, 46, 47, 133, 137 Phyllips, R. S., 398, 412 Piccirilli, A. B., 198-200, 202, 217,223, 224 Pinnaduwage, L. A., 124, 131, 133 Pipkin, F. M., 291, 319 Pleiss, T. C., 217, 223 Poignant, H., 207,215,223 Pomphrey, N., 102, 136 Pont, M., 72, 136 Poole, M. W., 410, 412 Poole, S. B., 207,224 POPOV,E., 150-157, 159, 161-167, 171, 175-178, 181, 185-187 Popov, V. S., 36, 37, 57, 58, 136 Potvliege, R. M., 68, 136 Power, E. A., 15, 26, 30, 136 Prange, R. E., 101, 135, 315,318 Prata, A., 230,260 Presby, H. M., 217, 219, 220,222,225, 226 Preston, R. K., 106, 108, 123, 137 Priou, J. P., 157, 186 Pritchard, D. E., 266,318 Psaltis, D., 230, 231, 235, 238, 243, 244, 249, 250,260,261 Puri, F., 276, 313-315, 318
Q Qadri, S., 334,412 Qiao, Y., 230,231,244,250,261
R Rahman, N. K., 40,41, I33 Raizer, Yu. P., 5 , 137 Raman, C. W., 329,412 Rammer, J., 310, 318 Rapoport, L. P., 14, 17, 137 Rath, O., 119-121, 135-337 Raybon, G., 205,206,223,224 Rayleigh, Lord, 145, 187 Reed, V. C., 61, 73, 134, 136 Reekie, L., 207,224 Rehman, S., 230,260 Reif, P. G., 23 1, 260 Reinhardt, W. P., 71, 72, 134 Reinisch, R., 152, 168, 185-187
420
AUTHOR INDEX
Reiss, H. R., 34, 36,46-53, 136 Reith, L. A,, 203, 226 Rempe, G., 310,319 Renieri, A., 323, 349, 351-354, 371, 377, 379, 383, 385, 395,411,411, 412 Renwick, S., 119, 120, 137 Repoux, S., 13, 14, 136 Rhodes, C. K., 61, 65, 136 Rice, R. A., 259,261 Richards, D., 119-121, 135437,293, 301, 315,318 Richetta, M., 355, 356, 363, 385, 411, 412 Righetti, A., 217, 225 Ritsch, H., 309,318 Robinson, M.G., 230,261 Romanelli, F., 354, 377, 412 Rosenfeld, E., 229,260 Rosenhouse, A,, 312,319 Roumiguieres, J. L., 148, 187 Rumelhart, D. E., 243,261 Ruyten, W. M.,286,319 Ryu, S., 218,225 Rzazewski, K., 64,134 S Sabia, E., 411, 411 Saffman, M., 230,260 Saifi, M. A., 215,216,221,224-226 Saito, S., 216, 218, 225 Saitoh, T., 193-195,225 Salam, A., 53, 135 Salpeter, E., 11, 133 Salzman, J., 203, 226 Sanchez, F., 18, 135 Sanders, M. M.,129, 133, 135 Sanders, V. E., 217,318 Sands, M., 75, 134 Saraceno, M.,133, 135 Sarukura, N., 68, 136 Saruwatari, M., 220,224 Sauer, B. E., 110-112, 123, 130, 135, 136, 293, 301, 315,318 Scharf, R., 314, 315, 318 Schey, H. M., 57, 135 Schleich, W., 277, 318 Schmit, C., 102, 134 Schmoys, J., 148, 185 Schumacher, D. W., 69, 70, 134 Schuster, H. G., 78, 137 Schwartz, J. L., 57, 135
Schwartz, R. J., 312, 318 Scully, M. O., 277, 318 Sekine, S., 216, 217, 224 Sergent, A. M., 212,225 Shakeshaft, R., 68, 136 Shamir, J., 230, 261 Shan, V., 145, 185 Shang, H. T., 219,225 Sheffield, R. L., 393,412 Shepelyansky, D. L., 105, 123, 125, 128-130, 134, 137 Shigematsu, M., 221,225 Shih, M.-L., 78, 81, 93, 136 Shikada, M.,217, 223 Shin, S.-Y., 230,261 Shirley, J. H., 293, 319 Shokoohi, F. K., 203,226 Shore, B.W., 62, 63, 65, 68, 135, 137, 291, 305,319 Siegman, A. E., 406,412 Sigel, G. H., 207,225 Silverman, M. P., 291, 319 Simpson, J. R., 207, 208, 211, 213-215, 217, 219-221,222-226 Sincerbox, G. T., 158, 186 Sivco, D. L., 212,223 Smilansky, U., 124, 130, 131,134 Smith, D. C., 6, 137 Smith, P., 221,223 Smith, P. W., 316, 319 Smith, S. J., 309,318 Snitzer, E., 207,224, 225 Soffer, B. H., 230,261 Sokol, D. W., 130, 133 Solimeno, S., 398,403,405, 406,412 Soskin, M.S., 25 1, 261 Spicer, R. E., 221,226 Spirit, D. M.,217, 220, 223, 224 Sprangle, P., 354, 412 Spreeuw, R. J. C., 266-269, 272, 277-282, 287-291,294-298, 303,304, 308, 311, 314,318,319 Staebler, D. L., 237, 251,260,261 Stallard, W. A., 217,224 Standley, R. D., 217,225 Stenholm, S., 296, 297, 309,319, 325, 329, 412 Stevens, M.J., 117, 137 Stinchcombe, M.,231,260 Stolen, R., 207,225
AUTHOR INDEX
Stoll, H.M., 230, 249, 261 Stone, J., 207,225 Stoneman, R. C., 293, 294,319 Strathdee, J., 53, 135 Stroke, G. W., 145, 186 Strong, J., 145, 185 Stroud Jr, C. R., 72, 136, 286,318 SU,Q., 62-66, 68, 72, 73, 134, 137 Sugawa, T., 207,224 Sugie, T., 216, 218,225 Sugiyama, H., 216, 217, 224 Sundaram, B., 61, 62, 65, 68, 73, 117, 124, 131, 133, 133, 135, 137 Sunohara, Y., 213,217,219,223,225 Suominen, K.-A., 297,309,319 Susskind, S. M., 125, 129, 135, 137 Suto, K., 221, 222,224 Suzaki, T., 217,223 Suzuki, K., 215, 219,220,224 Suzuki, Y., 230,261 Swain, S., 296, 319 Sweeney, K. L., 214,223 Szebesta, D., 207,215,223 Szbke, A,, 52,137 Szu, H. H., 230,260
T Tabor, M., 103, 133 Tackitt, M. C., 230,249,261 Taga, H., 217,223,225 Tagami, Y., 221,225 Takada, A., 217,222 Takahashi, S., 207, 211,225 Takenaka, H., 213, 219,225 Tamir, T., 145, 152, 185 Tanbun-Ek, T., 212, 219, 220,222,225 Tang, Cha Mei, 354,412 Tatchyn, R., 334, 412 Taylor, H. S., 130, 135 Teague, M. R., 17, 135 TeKolste, R., 230,260 Tell, B., 206, 224 Temkin, H., 212,225 Tench, R. E., 217, 218, 221,223,225 Terent’ev, M. V., 36, 37, 57, 58, 136 Thebault, J., 13, 14, 136 Thompson, G. H. B., 192,225 Thomson, D. S., 293, 294,319 Tip, A,, 52, 53, 136 Tkach, R., 217,223
42 1
Toba, H., 216, 217,224 Tohme, H. E., 221,225,226 Torre, A., 329, 340, 355, 356, 363, 371, 377, 379, 383, 385,406,411,412 Townsend, J. E., 214,223 Toyoda, H., 230,261 Trahin, T., 16, 135 Trischitta, P. R., 216, 217, 223 Tseng, D. Y., 148, 165, 185, 187 Tsonev, L., 150-154, 171, 175-178, 181, 187 Tsuji, M., 168, 187 Tsuji, S., 192, 224 Turchette, Q. A., 266,318 Turner, E. H., 316, 319 Twersky, V., 142, 187 Twu, Y., 200,224 U
Upadhyayula, L. C., 221, 223 Uretsky, J. L., 145, 187 Urquhart, P., 211, 225
V Vahala, K., 309, 319 van den Berg, P. M., 145, 146, 187 van der Lugt, A. B., 231,261 van der Wiel, M. J., 41-44, 52, 53, 135, 136 van der Ziel, J. P., 207, 225 van Druten, N. J., 266-269,272, 277, 278, 287-291, 303,319 van Haeringen, W., 279, 299,319 van Heerden, P. J., 234,261 van Leeuwen, K.A. H., 119-121, 123, 135-137 Velghe, M., 397,412 Vendetta, S. W., 214,223 Vernon Jr, F. L., 274, 318 Vincent, P., 145, 161, 168, 186, 187 Vinetskii, V. L., 251, 261 Vivaldi, F., 315, 318 Volkov, D. M., 22, 137 v. Oppen, G., 119, 120, 123, 136, 137 von Lehman, A. C., 215, 221,225,226 von Lehmen, A., 230,261 W Wagner, K., 230, 238, 243, 249,261 Wagner, S. S., 221,226 Wakabayashi, H., 217,218,223,225 Walet, N. R., 72, 136
422
AUTHOR INDEX
Walker, G. H., 85, 86, 88-91, 137 Walker, R. B., 106, 108, 123, 137 Walker, R.P.,410,412 Walther, H., 310,319 Warren, R.W., 393,412 Wasson, D. A., 73, 74, 137 Watanabe, M., 68, 136 Watanabe, S., 68, 136 Watson, K.M., 24, 39, 135 Way, W. I., 215, 216, 221,2 1,225,2 5 Wecht, K.W., 212, 219, 220,222, 225 Wegner, M., 205,223 Weingartshofer, A,, 24, 25, 137 Welford, D., 315, 319 Werbos, P., 243,261 Werlich, H., 158, 186 White, H., 231,260 Widdowson, T., 217,223,224 Wiener-Avnear, E., 231,260 Wiesenfeld, J. M., 205,223 Wilets, L., 73, 135 Williams, R.J., 243, 261 Willner, A. E., 217, 226 Wilson, I. J., 163, 185 Windhorn, T. H., 214,223 Winich, H., 334, 411 Wirgin, A., 145, 187 Woerdrnan, J. P., 266-269, 272, 277-282, 287-291,294-298,303, 304,308, 311, 314,318, 319 Wood, R.W., 141, 145, 187 Wright, J. V., 217,223,224 Wullert 11, J. R., 230, 261 Wyatt, R.,207, 215, 221,223
Y Ya, V., 334,412 Yablonovitch, E., 280, 319 Yamada, E., 220,224 Yamamoto, S., 217,223,225 Yamamoto, Y., 192,224, 317,318 Yamashita, M., 168, 187 Yariv, A., 230, 237, 260,261,309,319, 398, 400,405,412 Yeates, P.D., 217,223 Yeh, P., 230,250,260,261 Yergeau, F., 43,46, 47, 137 Yi-Yan, A., 216, 221,224,226 Yokomori, K.,158, 187 Yoneda, E., 221, 222,224 Yoo, H. I., 265,319 Yoshida, Y., 217, 218,223,225 Yoshinaga, H., 221,222, 224 Young, J., 217, 225 Young, M. G., 206,224 Yu, J., 235, 261 Yung, B., 158, 186
2 Zagury, N., 36, 134 Zah, C. E., 203,226 Zaki, K.,145, 186 Zaslavskii, G. M., 103, 137 Zel’dovich, Ya. B., 5, 137 Zener, C., 297, 319 Zernik, W., 15, 137 Zhang, L., 230,261 Zienau, S., 15, 26, 30, 136 Zoller, P.,309, 318 Zon, B. A., 14, 17, 137 Zyskind, J. L., 213-215,223,226
SUBJECT INDEX
A above-threshold ionization, 3, 34, 39, 40, 45, 60, 61.66, 68 amplified spontaneous emission, 201 amplifier, buried-facet, 196, 198 -, designs, 196 -, Fabry-PCrot, 192, 193 -, fiber, 207, 212, 214-216 -, integrated laser, 205 -, multiquantum well, 204-206 -, optical, 191, 193 -, semiconductor, 191-193 -, tilted facet, 196, 202 -, travelling wave, 192, 193 associated memory, 230, 233 attractor, 76 Auger recombination, 199 Autler-Townes doublet, 289, 294 B band structure, optical, 279 beam fanning, 237 Berry phase factor, 309 Bessel function, 340, 341, 345 Bloch equations, optical, 19, 57, 64, 97 Bloch-Siegert shift, 290, 292-295, 305 -- sphere, 104, 275, 303 -- vector, 274-276, 298, 303, 307 Bragg condition, 235 - reflector, 298 Bremsstrahlung radiation, 323 Brewster window, 278 Brillouin zone, 279, 280
Compton scattering, 326, 333 correlator, optical, 231
D diffraction efficiency, 235,253, 254, 258 Dirac equation, 22 dressed-level picture, 293 Dyson expansion, 39
E Einstein-Brillouin-Keller quantization condition, 102 electro-optic effect, 235 - modulator, 269, 281, 311, 316
F Faraday rotation, 268, 276 Feigenbaum universality, 77 Fermi golden role, 9 fiber transmission system, 193 Floquet state, 293 free-electron laser, 323-328, 333, 350-352, 354, 359, 360, 364, 370-379, 382-397, 402,404,406 Fresnel's principle, 143 - theory, 145
c Galilean transformation, 327 gauge invariance, 37 Gaussian beam, 54, 367 Gavrila states, 72 geometrical optics, 399 grating, 142, 144-147, 150, 155, 158, 161, 173, 235, 254 - anomalies, 141 -, blazed, 143 -, conducting, 153, 174 -, dielectric, 156, 157, 182 -, diffraction, 141
C
chaos, 74, 76, 77, 79, 85, 91 -, deterministic, 78 - in atomic physics, 96 -, quantum, 3, 96, 97, 108, 131, 315 character recognition, 236, 242 423
424
SUBJECT INDEX
- equation, 148 -, holographic, 250 -, metallic, 150, 151, 154, 174, 176 -, reflection, 149 -, shallow, 170 -, sinusoidal, 154, 171 -, transmission, 184 H Hamiltonian system, 78, 79 Hamilton-Jacobi system, 80, 81 harmonic generation, 61, 66, 68 Hebbian law, 230 - rule, 247 Heisenberg equation, 270 - uncertainty principle, 74 Henon-Heiles potential, 102 Hermite-Gauss function, 367 - polynomial, 383 hologram, 230, 235, 238, 243, 248, 249, 253-256.260 -, Fourier transform, 237, 257 Hopfield network, 230
L Lagrangian, 26, 27 Laguerre polynomial, 403 Landau-Zener crossing, 297,300, 301, 309, 310 -- diagram, 266 -- transition, 266, 284,294, 299, 310 Langevin force, 308 Larmor precession, 285 laser gyro, 277, 278 semiconductor, 191 Lie group, 282 LiCnard-Wiechert potential, 340 liquid crystal light valve, 231, 234, 236 Littrow mount, 143, 149-151, 153, 165-167, 169, 170, 172 logistic mapping, 75, 76 Lorentz equation, 335 factor, 337 - force, 26 - gauge, 329 invariance, 37 Lyapunov exponent, 75,79,96, 103
-.
-
M 1
inhomogeneous broadening, 346, 348, 349, 360
J Jaynes-Cummings Hamiltonian, 3 10, 3 1 1, 313 -- model, 286,292, 310-312
Madey's theorem, 353 Maxwell's equations, 142, 270, 407 Monte Carlo simulation, 120 Morse oscillator, 106-108, 123 - potential, 106 multiphoton absorption, 39 - ionization, 3, 5, 7, 1 I , 13, 14, 17 - process, 24 - resonance, 293 - transition, 3
K Keldysh amplitude, 39, 60 - approximation, 3,25,26, 31, 37-39, 53, 56-58,60,69, 132 - parameter, 34-36, 58, 110. 123 - theory, 36, 37, 45, 47, 49, 58, 60, 71 Keldysh-Reiss amplitude, 71 - - approximation, 36, 59 theory, 46,47, 5 1, 69, 72 Kepler problem, 84 Kolmogorov-Arnold-Moser theorem, 84, 85,92 Kramers-Hanneberger transformation, 72 Kramers-Kronig condition, 354, 382 Kroll-Watson theory, 25, 41
--
N neural networks, 229 -, multilayer, 243 - -, optical, 230, 231
-
P Pauli exclusion principle, 74, 280
- matrices, 274
perceptron, 230 period-doubling bifurcation, 77 phase-conjugating mirror, 250 matching, 67, 142 photoelectric effect, 9 photorefractive crystal, 238
-
SUBJECT INDEX
plasmon surface wave, 174 Poincart-Bendixson theorem, 8 1 - map, 17,78 - recurrence theorem, 97, 98 Poisson bracket, 86 ponderomotive potential, 51, 53, 56 Poynting vector, 166
Q quantum jump, 309
- measurement theory, 316 - optics, 265
425
S
Sagnac effect, 267, 269
- phase, 276
Schr6dinger equation, 22, 27, 37, 57, 61, 62, 65, 77, 97, 105, 271, 329 Schwinger representation, 276 self-organizing system, 230 soliton, 219 spatial light modulator, 234 Stark shift, AC, 20 - -, quadratic, 20, 55, 56 susceptibility, third-order, 67 synchrotron radiation, 334, 336
R Rabi frequency, 4, 7, 19, 64, 285, 286, 289, 290,295,296,312,329, 390 - oscillation, 266, 284, 289-29 1, 302, 303 - splitting, 19 Raman-Nath equation, 329 - spectroscopy, surface-enhanced, 177 Rayleigh method, 145 - range, 54, 403 ray matrix, 398-400 resonance fluorescence, 19 rotating-wave approximation, 7, 64, 284, 290,294 Rydberg atom, 4, 301 - state. 315
T turbulence, 74 two-level atom, 265,266, 270, 308
U undulator magnet, 323, 324, 333, 334, 340. 406
V Volkov state, 22, 28, 31, 53
W wavelength division multiplexing, 216
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CUMULATIVE INDEX - VOLUMES I-XXXI 11, 249 ABELBS,F., Methods for Determining Optical Parameters of Thin Films VII, 139 ABELLA, 1. D., Echoes at Optical Frequencies XVI, 71 ABITBOL, C. I., see J. J. Clair Dynamical Instabilities and ABRAHAM, N. B., P. MANDEL,L. M. NARDUCCI, Pulsations in Lasers xxv, 1 AGARWAL, G. S.,Master Equation Methods in Quantum Optics XI, 1 XXVI, 163 G. P., Single-longitudinal-mode Semiconductor Lasers AGRAWAL, IX, 235 V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion AGRANOVICH, IX, 179 ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers Synthesis of Optical Birefringent Networks IX, 123 AMMANN, E. 0.. ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, 21 1 J. A., Hamiltonian Theory of Beam Mode Propagation XI, 247 ARNAUD, BALTES,H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment XIII, I V. D. OZRINand A. I. SAICHEV, BARABANENKOV, Yu. N., Yu. A. KRAVTSOV, Enhanced Backscattering in Optics XXIX, 65 BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images I, 67 BARRETT, H. H., The Radon Transform and its Applications XXI, 217 XII, 287 BASHKIN,S.,Beam-Foil Spectroscopy BASSETT,I. M., W. T. WELFORD,R. WINSTON,Nonimaging Optics for Flux Concentration XXVII, 161 VI, 53 BECKMANN, P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and XVIII, 259 their Diffraction Patterns XXVII, 227 BERTOLOTII, M., see D. Mihalache BEVERLY 111, R. E., Light Emission from High-Current Surface-Spark Discharges XVI, 357 BJORK,G., see Y. Yamamoto XXVIII, 87 IX, 1 BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M. A., W. A. VAN DE GRIND, P. ZUIDEMA, Quantum Fluctuations in Vision XXII, 77 IV, 145 BOUSQIJET,P., see P. Rouard BROWN,G. S., see J. A. DeSanto XXIII, 1 421
428
C U M U L A T I V E INDEX
- VOLUMES
I-XXXI
BRIINNtR, W., H. PAUL,Theory of Optical Parametric Amplification and O S C i h tion xv, I BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 XI, 167 BRYNcDAHL, O., Evanescent Waves in Optical Imaging O., F. WYRowSKI, Digital holography - Computer-generated BRYNGDAHL, holograms XXVIII, 1 BURCH,J. M., The Meteorological Applications of Diffraction Gratings 11, 73 XIX, 21 1 BUTrERWECK, H. J., Principles of Optical Data-Processing CAGNAC,B., see E. Giacobino XVII, 85 CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CEGLIO, N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications XXI, 287 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 C L A I RJ., J . , C. I. ABITBOL, Recent Advances in Phase Profiles Generation XVI, 71 XIV, 327 CLARRICOATS, P. J. B., Optical Fibre Waveguides - A Review COHEN-TANNOUDJI, c . , A. KASTLER, Optical Pumping v, 1 XV, 187 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see C. Froehly XX, 63 XXVIII, 361 COOK,R. J., Quantum Jumps C O U R T ~ G., S , P. C R U V E L L I EM. R , DETAILLE, M. SAYSSE, Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects XX, 1 XXVI, 349 CREATH, K., Phase-Measurement Interferometry Techniques CREWE, A. V., Production of Electron Probes Using a Field Emission Source XI, 223 CHRISTOV, I. P., Generation and Propagation of Ultrashort Optical Pulses XXIX, 199 CRUVELLIER, P., see C. G. Courtes xx, 1 VIII, 133 C U M M I NH. S , Z., H. L., SWINNEY, Light Beating Spectroscopy DAINTY, J. C., The Statistics of Speckle Patterns XIV, 1 DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 A. TORRE, Theory of Compton Free DATTOLI,G., L. GIANNESSI, A. RENIERI, Electron Lasers XXXI, 321 DECKERJr., J. A,, see M. Harwit XII, 101 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMARIA, A. J., Picosecond Laser Pulses IX, 31 DESANTO,J. A., G . S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, 1 DETAILLE, M., see G. Courtts xx, 1 X, 165 DEXTER,D. L., see D. Y. Smith DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 XIV, 161 DUGUAY, M. A,, The Ultrafast Optical Kerr Shutter XXXI, 189 DUTTA, N. K., J. R. SIMPSON,Optical Amplifiers VII, 359 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons
CUMULATIVE INDEX
- V O L U M E S I-XXXI
429
ENGLUND, J. C., R. R. SNAPP,W. C. SCHIEVE,Fluctuations, Instabilities and XXI, 355 Chaos in the Laser-Driven Nonlinear Ring Cavity XVI, 233 ENNOS,A. E., Speckle Interferometry FANTE,R. L., Wave Propagation in Random Media: A Systems Approach XXII, 341 xxx, 1 FABRE,C., see S. Reynaud FIORENTINI, A,, Dynamic Characteristics of Visual Processes I, 253 FLYTZANIS, C., F. HACHE,M.C. KLEIN,D. RICARDand PH. ROUSSIGNOL, XXIX, 321 Nonlinear Optics in Composite Materials FOCKE,J., Higher Order Aberration Theory IV, I Measurement of the Second Order Degree of CoheFRANCON, M., S. MALLICK, rence V1, 71 FREILIKHER, V. D., S. A. GREDESKUL, Localization of waves in media with onedimensional disorder XXX, 137 FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 311 FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE,Shaping and Analysis of Picosecond Light Pulses XX, 63 FRY,G. A,, The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I , 109 GAMO,H., Matrix Treatment of Partial Coherence 111, 187 GHATAK, A. K., see M. S. Sodha XIII, 169 GHATAK, A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVIII, I GIACOBINO, E., B. CAGNAC.Doppler-Free Multiphoton Spectroscopy XVII, 85 E., see S. Reynaud xxx, 1 GIACOBINO, GIANNESSI, L., see G. Dattoli XXXI, 321 GINZBURG, V. L., see V. M. Agranovich IX, 235 11, 109 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media XXIV, 389 GLASER,I., Information Processing with Spatially Incoherent Light Applications of Optical Methods in the Diffraction GNIADEK, K., J. PETYKIEWICZ, Theory of Elastic Waves IX, 281 J. W., Synthetic-Aperture Optics GOODMAN, v111, 1 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 GREDESKUL, S.A., see V. D. Freilikher XXX, 137 XXIX, 321 HACHE,F., see C. Flytzanis HALL,D. G., Optical Waveguide Diffraction Gratings: Coupling Between Guided XXIX, 1 Modes XX, 263 HARIHARAN, P., Colour Holography XXIV, 103 HARIHARAN, P., Interferometry with Lasers XII, 101 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry XXX, 205 A, see Y. Kodama HASEGAWA, xxx, 1 HEIDMANN, A., see S. Reynaud X, 289 HELSTROM,C. W., Quantum Detection Theory
430
CUMULATIVE INDEX
- VOLUMES
I-XXXI
VI, 171 HERRIOTT,D. R., Some Applications of Lasers to Interferometry HUANG,T. S., Bandwidth Compression of Optical Images x, 1 IMOTO,N., see Y. Yamamoto XXVIII, 87 JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive V, 247 Index JACQUINOT, P., B. ROIZEN-DOSSIER, Apodisation 111, 29 JAMROZ, W., B. P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation XX, 325 IX, 179 JONES,D. G. C., see L. Allen KASTLER, A,, see C. Cohen-Tannoudji v, 1 XXVI, 105 KHOO,I. C., Nonlinear Optics of Liquid Crystals XX, 155 KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy KINOSITA, K., Surface Deterioration of Optical Glasses IV, 85 KITAGAWA, M., see Y. Yamamoto XXVIII, 87 XXIX, 321 KLEIN,M. c.,see c. Flytzanis KODAMA, Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept XXX, 205 in Fibers KOPPELMANN, G., Multiple-Beam Interference and Natural Modes in Open VII, 1 Resonators 111, 1 KOTTLER,F., The Elements of Radiative Transfer IV, 281 KOTTLER, F., Diffraction at a Black Screen, Part I: Kirchhoff's Theory VI, 331 KOTTLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory XXVI, 227 Yu. A., Rays and Caustics as Physical Objects KRAVTSOV, KRAVTSOV, Yu.A., see Yu. N. Barabanenkov XXIX, 65 I, 211 KUBOTA,H., Interference Color LABEYRIE, A., High-Resolution Techniques in Optical Astronomy XIV, 47 XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XVI, 119 LEE,W.-H., Computer-Generated Holograms: Techniques and Applications VI, 1 LEITH, E. N., J. UPATNIEKS, Recent Advances in Holography V. S., Laser Selective Photophysics and Photochemistry XVI, 1 LETOKHOV, LEV], L., Vision in Communication VIII, 343 LIPSON, H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch V, 287 of Physical Optics XXI, 69 LUGIATO, L. A., Theory of Optical Bistability XXVIII, 87 MACHIDA, M., see Y.Yamamoto XXII, 1 MALACARA, D., Optical and Electronic Processing of Medical Images VI, 71 MALLICK, L., see M. Franqon 11, 181 MANDEL,L., Fluctuations of Light Beams XIII, 27 MANDEL,L., The Case for and against Semiclassical Radiation Theory xxv, 1 MANDEL,P., see N. B. Abraham XI, 305 MARCHAND, E. W., Gradient Index Lenses Optical Films Produced by Ion-Based TechMARTIN, P. J., R. P. NETTERFIELD, XXIII, 113 niques
C U M U L A T I V E INDEX
- VOLUMES
I-XXXI
43 1
MASALOV, A. V., Spectral and Temporal Fluctuations of Broad-Band Laser XXII, 145 Radiation XXI, 1 MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings xv, 77 A,, see P. Rouard MEESSEN, VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting XXX, 261 MEYSTRE, P., Cavity Quantum Optics and the Quantum Measurement Process MIHALACHE, D., M. Bertolotti, C. Sibilia, Nonlinear wave propagation in planar XXVII, 227 structures Quasi-Classical Theory of Laser RadiaMIKAELIAN, A. L., M. I. TER-MIKAELIAN, VII, 231 tion XVII, 279 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction MILONNI,P. W., B. SUNDARAM, Atoms in Strong fields: Photoionization and XXXI, I Chaos MILLS,D. L., K. R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids XIX, 43 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design I, 31 MOLLOW, B. R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence XIX, 1 V, 199 K., Instruments for the Measuring of Optical Transfer Functions MURATA, Multilayer Antireflection Coatings VIII, 201 MUSSET,A,, A. THELEN, L. M., see N. B. Abraham xxv, 1 NARDUCCI, XXIII, 113 NETTERFIELD, R. P., see P. J. Martin NISHIHARA, H., T. SUHARA, Micro Fresnel Lenses XXIV, 1 XXV, 191 OHTSU,M., T. TAKO,Coherence in Semiconductor Lasers XV, 139 OKOSHI,T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image OSTROVSKAYA, G. V., Yu. I. OSTROVSKY, Holographic Methods in Plasma XXII, 197 Diagnostics XXII, 197 OSTROVSKY, Yu. I., see G. V. Ostrovskaya Correlation Holographic and Speckle OSTROVSKY, Yu. I., V. P. SHCHEPINOV, Interferometry x x x , 87 OZRIN,V. D., see Yu. N. Barabanenkov XXIX, 321 K. E., Unstable Resonator Modes XXIV, 165 OUGHSTUN, K. P., The Self-Imaging Phenomenon and its Applications XXVII, I PATORSKI, PAUL,H., see W. Brunner xv, 1 PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, I VII, 67 PEGIS,R. J., see E. Delano PERINA, J., Photocount Statistics of Radiation Propagating through Random and XVIII, 129 Nonlinear Media PERSHAN, P. S., Non-Linear Optics V, 83 J., see K. Gniadek PETYKIEWICZ, IX, 281 PICHT,J., The Wave of a Moving Classical Electron V, 351
432
CUMULATIVE INDEX
- VOLUMES I-XXXI
POPOV,E., Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View PORTER,R. P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems PsALTis, D., Y. QIAO,Adaptive Multilayer Optical Networks PSALTIS, D., see D. Casasent Y., see D. Psaltis QIAO, The Quantum Coherence Properties of StiRAYMER, M. G., I. A. WALMSLEY, mulated Raman Scattering RENIERI, A., see G. Dattoli REYNAUD,S., A. HEIDMANN, E. GIAcoBiNo, C. FABRE, Quantum Fluctuations in Optical Systems RICARD,D., see C. Flytzanis RISEBERG, L. A., M. J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN, H., Statistical Properties of Laser Light RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSIER,B., see P. Jacquinot RONCHI, L., see Wang Shaomin ROSENBLUM,W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye RousslGNoL, PH., see c . Flytzanis ROTHBERG, L., Dephasing-Induced Coherent Phenomena ROUARD,P., P. BOUSQUET,Optical Constants of Thin Films ROUARD, P., A. MEESSEN,Optical Properties of Thin Metal Films Ru~lNowlcz,A,, The Miyamoto-Wolf Diffraction Wave RUDOLPH, D., see G. Schmahl SAICHEV, A. I., see Yu. N. Barabanenkov SAYSSE,M., see G. Courtes SAITO,S., see Y. Yamamoto SAKAI, H., see G. A. Vanasse B. E. A., see M. C. Teich SALEH, SCHIEVE, W. C., see J. C. Englund SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings The Mutual Dependence between Coherence SCHUBERT, M., B. WILHELMI, Properties of Light and Nonlinear Optical Processes SCHULZ, G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces SCHULZ, G., Aspheric Surfaces SCHWIDER, J., see G. Schulz J., Advanced Evaluation Techniques in Interferometry SCHWIDER, Tools of Theoretical Quantum Optics SCULLY, M. O., K. G. WHITNEY, SENITZKY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical Framework
XXXI, 139 XXVII, 315 XXXI, 227 XVI, 289 XXXI, 227 XXVIII, 181 XXXI, 321
xxx,
1
XXIX, 321 XIV. 89 VIII, 239 XIX, 281 111, 29 XXV, 279 XIII, 69 XXIX, 321 XXIV, 39 IV, 145 XV, 71 IV, 199 XIV, 195 XXIX, 65
xx,
1
XXVIII, 87 VI, 259 XXVI, 1 XXI, 355 XIV, 195 XVII, 163 XIII, 93 xxv, 349 XIII, 93 XXVIII, 271 X, 89 XVI, 413
CUMULATIVE INDEX
- VOLUMES
433
I-XXXI
SHCHEPINOV, V. P., see Yu. I. Ostrovsky C., see D. Mihalache SIBILIA, J. R., see N. K. Dutta SIMPSON, SIPE,J. E., see J. Van Kranendonk SITTIG,E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self-Induced Transparency SMITH,A. W., see J. A. Armstrong SMITH, D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in Insulators SMITH,R. W., The Use of Image Tubes as Shutters SNAPP,R. R., see J. C. Englund SODHA,M. S., A. K. GHATAK, V. K. TRIPATHI, Self Focusing of Laser Beams in Plasmas and Semiconductors SOROKO, L. M., Axicons and Meso-Optical Imaging Devices Optical Atoms SPREEUW, R. J. C., J. P. WOERDMAN, STEEL,W. H., Two-Beam Interferometry STOICHEFF, B. P., see W. Jamroz STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution STROKE, Spectroscopy SUBBASWAMY, K. R., see D. L. Mills SUHARA, T., see H. Nishihara SUNDARAM, B., see P. W. Milonni 0..Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser SVELTO, Beams SWEENEY, D. W., see N. M. Ceglio H. H., see H. Z. Cummins SWINNEY, TAKO,T., see M. Ohtsu K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANAKA, TANGO,W. J., R. Q. Twiss, Michelson Stellar Interferometry TATARSKII, v. I., v. u. ZAVOROTNYI, Strong Fluctuation in Light Propagation in a Randomly Inhomogeneous Medium TAYLOR, C. A., see H. Lipson TEICH,M.C., B. E. A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN, M. L., see A. L. Mikaelian THELEN, A,, see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light K., see A. Ghatak THYAGARAJAN, TONOMURA, A., Electron Holography TORRE,A., see G. Dattoli TRIPATHI, V. K., see M. S. Sodha TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering
XXX, 87 XXVII, 227 XXXI, 189 XV, 245 X, 229 XII, 53 VI, 21 1 X, 165 x, 45 XXI, 355 XIII, 169 XXVII, 109 XXXI, 263 V, 145 XX, 325 IX. 73 11,
1
XIX, 43 XXIV, 1 XXXI, 1 XII, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63 XVII, 239 XVIII, 207 V, 287 XXVI, 1 VII, 231 VIII, 201 VII, 169 XVIII, 1 XXIII, 183 XXXI, 319 XIII, 169 11, 131
434
CUMULATIVE INDEX
- VOLUMES I-XXXI
XVII, 239 Twiss, R. Q., see W. J. Tango VI, I UPATNIEKS, J., see E. N. Leith XVIII, 259 UPSTILL,C., see M. V. Berry USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in XIX, 139 Solids XX, 63 VAMPOUILLE, M., see C. Froehly VI, 259 VANASSE,G. A., H. S A K A IFourier , Spectroscopy XXII, 77 V A N D E GRIND, W. A,, see M. A. Bouman I, 289 VAN HEEL,A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE, Foundations of the Macroscopic ElectromagXV, 245 netic Theory of Dielectric Media XIV, 245 VERNIER,P., Photoemission XXVIII, 181 WALMSLEY, I. A., see M. G. Raymer WANG,SHAOMIN, L. RONCHI,Principles and Design of Optical Arrays XXV, 279 WEBER,M. J . , see L. A. Riseberg XIV, 89 WEIGELT, G., Triple-correlation Imaging in Optical Astronomy XXIX, 293 WELFORD, W. T., Aberration Theory of Gratings and Grating Mountings IV, 241 XI11, 267 WELFORD, W. T., Aplanatism and Isoplanatism XXVII, 161 WELFORD, W. T., see 1. M. Bassett XVII, 163 W I L H E L MB., I , see M. Schubert XXVII, 161 WINSTON, R., see I. M. Bassett WITNEY, K. G., see M. 0. Scully X, 89 XXXI, 263 WOERDMAN, J. P., see R.J. C. Spreeuw WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information I, 155 X, 137 WYNNE,C. G., Field Correctors for Astronomical Telescopes WYROWSKI, F., see 0. Bryngdahl XXVIII, I YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements Using Laser Light XXII, 271 YAMAMOTO, Y., S. MACHIDA, S. SAITO,N. IMOTO,T. YAMAGAWA, M. KITAGAWA, G. BJURK,Quantum Mechanical Limit in Optical Precision Measurement and Communication XXVIII, 87 VI, 105 YAMAJI, K., Design of Zoom Lenses T., Coherence Theory of Source-Size Compensation in Interference YAMAMOTO, Microscopy VIII, 295 YANAGAWA, T., see Y. Yamamoto XXVIII, 87 YOSHINAGA,H., Recent Developments in Far Infrared Spectroscopic Techniques XI, 77 Yu, F. T. S., Principles of Optical Processing with Partially Coherent Light XXIII, 227 ZAVOROTNYI, V. U., see V. 1. Tatarskii XVIII, 207 Z U I D E M AP.,, see M. A. Bournan XXII, 77
