PROGRESS IN OPTICS VOLUME XXXIX
EDITORIAL ADVISORY BOARD G. S. AGARWAL,
Ahmedabad, India
T. ASAKURA,
Sapporo, Japan
M.V BERRY,
Bristol, England
C . COHEN-TANNOUDJT, Paris, France
v: L. GINZBURG,
Moscow, Russia
F. GORI,
Rome, Italy
A. KUJAWSKI,
Warsaw, Poland
J.
Olomouc, Czech Republic
€"A,
R. M. SILLITTO,
Edinburgh, Scotland
H.
Garching, Germany
WALTHER,
PROGRESS IN OPTICS VOLUME XXXIX
EDITED BY
E. WOLF Uniuersily of Rochesfel: N.Z, US.A.
Contributors L. ALLEN, A.A. ASATRYAN, M. BABMER, G.W. FORBES, &.A. KRAVTSOV; G. LEUCHS, T. OPATFW?, M.J. PADGETT, A. SIZMANN, D.J. SOMEFSORD, S.K. SHARMA, W. VOGEL, D.-G. WELSCH
1999
ELSEVIER AMSTERDAM. LAUSANNE .NEW YO=. OXFORD. SHANNON. SINGAPORE. TOKYO
ELSEVIER SCIENCE B,V S A R A BURGERHARTSTRAAT 25
PO. BOX 21 1 1000 AE AMSTERDAM THE NETHERLANDS Library of Congress Catalog Card Number: 61-19297 ISBN Volume XXXM: 0 444 50104 5
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PREFACE In this volume five review articles are presented dealing with topics of current research interests in optics. The fist article, by Yu.A. Kravtsov, G.W. Forbes and A.A. Asatryan, is concerned with the analytic extension of the concept of geometrical optics rays into the complex domain. The extension is intimately related to inhomogeneous (evanescent) waves, which are currently of particular interest in connection with the rapidly developing area of near-field optics. The results are also relevant to investigations of wave attenuation in absorbing media, and to the understanding of light penetration into geometrical shadow regions, excitation of surface waves and propagation of Gaussian beams. The article presents the principles, with special emphasis on the physical significance of complex rays and their applications. The second article, by D.-G. Welsch, W. Vogel and T. Opatrn9, describes recent progress in the general area of quantum-state reconstruction, particularly for extracting information about the quantum state of a given object from a set of measurements. The methods are applicable to the optical field as well as to various matter waves. Some methods of processing the measured data and many of the important experiments in this area are discussed. The next article, by S.K. Sharma and D.J. Somerford, is concerned with the scattering of light in the eikonal approximation. This approximation originated in the theory of high-energy scattering processes and in the broad area of potential scattering. From the well-known analogy between a scattering potential and the distribution of the refractive index, the eikonal approximation was later adapted to the analysis of light scattering by small particles. In this article an account is given of the eikonal approximation in the context of optical scattering, and its domain of validity is discussed. The relationship of this approximation to other approximate techniques as well as some of its possible applications are considered. The fourth article, by L. Allen, M.J. Padgett and M. Babiker, concerns the orbital angular momentum of light. The orbital angular momentum is shown to be an observable quantity which can be profitably used with certain types of light beams. The phenomenological interaction of the beams with matter in bulk is reviewed and the contributions of the orbital angular momentum to the V
vi
PREFACE
dissipative and dipole forces on atoms are calculated in detail. Orbital and spin angular momentum of light are compared and contrasted. The concluding article, by A. Sizmann and G. Leuchs, presents a review of the experimental progress made in recent years in the generation of squeezed light and in quantum nondemolition measurements in optical fibers. The rich nonlinear dynamics of quantum solitons in fibers has led to the discovery of new quantum optical effects, such as intrapulse quantum correlations. The nonlinearity of optical fibers is now used to build passive fiber devices which provide all-optical functions, such as quantum noise reduction, and it is expected that active devices will allow absorption-free measurements of optical signals. The review is also concerned with these and other promising developments in this general area. Emil Wolf Department of Physics and Astronomy University of Rochester Rochestel; N a v York 14627, USA December 1998
CONTENTS I . THEORY AND APPLICATIONS OF COMPLEX RAYS by Yu.A. KRAVTSOV(Moscow. RUSSIAN FEDERATION). (SYDNEY.AUSTRALIA) G.W. FORBESAND A.A. ASATRYAN
5 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Pioneering works 1.2. Character of wavefields described by complex geometrical optics . . . . . . . 1.3. Goals of the review . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. BASICEQUATIONS OF GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . 2.1. Eikonal, transfer, and ray equations of traditional geometrical optics . . . . . . 2.2. Rays as the skeleton for the wavefield . . . . . . . . . . . . . . . . . . 2.3. Complex form of the geometrical optics method . . . . . . . . . . . . . . 2.4. Alternative approach to phenomena described by complex rays . . . . . . . . 5 3. PROPERTlES OF COMPLEX RAYS . . . . . . . . . . . . . . . . . . . . . . . 3.1. Ray paths in the complex space . . . . . . . . . . . . . . . . . . . . 3.2. Fermat’s principle for complex rays . . . . . . . . . . . . . . . . . . . 3.3. Selection rules for complex rays . . . . . . . . . . . . . . . . . . . . 3.4. Complex rays and the saddle-point method . . . . . . . . . . . . . . . . 3.5. Complex caustics . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Spacetime complex rays . . . . . . . . . . . . . . . . . . . . . . . 3.7. Electromagnetic waves and complex rays . . . . . . . . . . . . . . . . . 3.8. Complex rays and uniform asymptotics . . . . . . . . . . . . . . . . . 5 4. COMPLEX RAYSIN PHYSICAL PROBLEMS. . . . . . . . . . . . . . . . . . . . 4.1. Complex rays inside a circular caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Wave reflection in a layered medium 4.3. Point source in a layered me&um . . . . . . . . . . . . . . . . . . . . 4.4. The vicinity of a caustic cusp in free space . . . . . . . . . . . . . . . . 4.5. Swallow-tail caustic: an example with four ray contributions . . . . . . . . . 4.6. Point source in a parabolic layer . . . . . . . . . . . . . . . . . . . . 4.7. Above-barrier reflection . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Complex rays behind a sinusoidal phase screen . . . . . . . . . . . . . . 4.9. Surface waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10. Reflection of inhomogeneous waves from an interface 4.1 1. Complex rays in weakly absorbing media . . . . . . . . . . . . . . . . . 4.12. Complex geometrical optics in other wave disciplines . . . . . . . . . . . . 5 5. GAUSSIAN BEAMSAND COMPLEX RAYS . . . . . . . . . . . . . . . . . . . . 5.1. Gaussian beams and complex sources . . . . . . . . . . . . . . . . . . vii
3 3 4 5 5 5 8 10 12 15 15 16 17 19 21 23 24 26 27 27 29 31 32 33 33 36 37 37 38 39 42 43 43
viii
CONTENTS
5.2. Another description of Gaussian beams in terms bf complex rays 5.3. Transformation of Gaussian beams in optical systems . . . . . 5.4. Diffraction of Gaussian beams . . . . . . . . . . . . . . . Q 6. DISTINCTIVE ASPECTS OF COMPLEX G E O ~ T I U COPTICS AL . . . . . . 6.1. Nonlocal properties of complex rays . . . . . . . . . . . . 6.2. Boundaries of applicability of complex geometrical optics . . . Q 7. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 44 . . . . . . . 47 . . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50 50 51 52 53 53
I1. HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION by D.-G. WELSCH (JENA,GERMANY). W. VWEL (ROSTOCK. CZECHREPUBLIC) GERMANY)AND T. OPATR@(OLOMOUC.
Q 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q 2. PHASE-SENSITIVE MEASUREMENTS OF LIGHT . . . . . . . . . . . . . . . . . . 2.1. Optical homodyning . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Basic scheme . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Quadrature-component statistics . . . . . . . . . . . . . . . . . . 2.1.3. Multimode detection . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Q function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Probability operator measures . . . . . . . . . . . . . . . . . . . 2.1.6. Positive P function . . . . . . . . . . . . . . . . . . . . . . . 2.1.7. Displaced-photon-number statistics . . . . . . . . . . . . . . . . 2.1.8. Homodyne correlation measurements . . . . . . . . . . . . . . . 2.2. Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Parametric amplification . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Measurement of cavity fields . . . . . . . . . . . . . . . . . . . . . . Q 3. QUANTUM-STATE RECONSTRUCTION . . . . . . . . . . . . . . . . . . . . . . 3.1. Optical homodyne tomography . . . . . . . . . . . . . . . . . . . . . 3.2. Density matrix in quadrature-component bases . . . . . . . . . . . . . . 3.3. Density matrix in the Fock basis . . . . . . . . . . . . . . . . . . . . 3.3.1. Sampling of quadrature-components . . . . . . . . . . . . . . . . 3.3.2. Sampling of the dlsplaced Fock-states on a circle . . . . . . . . . . 3.3.3. Reconstruction from propensities . . . . . . . . . . . . . . . . . 3.4. Multimode density matrices . . . . . . . . . . . . . . . . . . . . . . 3.5. Local reconstruction of P(a; s) . . . . . . . . . . . . . . . . . . . . . 3.6. Reconstruction from test atoms in cavity QED . . . . . . . . . . . . . . 3.6.1. Quantum state endoscopy and related methods . . . . . . . . . . . . 3.6.2. Atomic beam deflection . . . . . . . . . . . . . . . . . . . . . 3.7. Alternative proposals . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Reconstruction of specific quantities . . . . . . . . . . . . . . . . . . . 3.8.1. Normally ordered photonic moments . . . . . . . . . . . . . . . . 3.8.2. Quantities admitting normal-order expansion . . . . . . . . . . . .
65
69 69 70 72 77 81 86 88 90 92 93 94 95 100 101 106 108 108 115 118 119 122 123 124 128 131 133 134 137
CONTENTS 3.8.3. Canonical phase statistics . . . . . . . . . . . . . . . . . . . . 3.8.4. Hamiltonian and Liouvillian . . . . . . . . . . . . . . . . . . . 3.9. Processing of smeared and incomplete data . . . . . . . . . . . . . . . . 3.9.1. Experimental inaccuracies . . . . . . . . . . . . . . . . . . . . 3.9.2. Least-squares method . . . . . . . . . . . . . . . . . . . . . . 3.9.3. Maximum-entropy principle . . . . . . . . . . . . . . . . . . . 3.9.4. Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . 5 4. QUANTUM STATES OF MATTER SYSTEMS. . . . . . . . . . . . . . . . . . . . 4.1. Molecular vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 .1. Harmonic regime . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Anharmonic vibrations . . . . . . . . . . . . . . . . . . . . . 4.2. Trapped-atom motion . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Quadrature measurement . . . . . . . . . . . . . . . . . . . . . 4.2.2. Measurement of the Jaynes-Cumrnings dynamics . . . . . . . . . . 4.2.3. Entangled vibronic states . . . . . . . . . . . . . . . . . . . . 4.3. Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . . . 4.4. Atomic matter waves . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Transverse motion . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Longitudmal motion . . . . . . . . . . . . . . . . . . . . . . 4.5. Electron motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.51. Electronic Rydberg wave packets . . . . . . . . . . . . . . . . . 4.5.2. Cyclotron state of a trapped electron . . . . . . . . . . . . . . . . 4.5.3. Electron beam . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Spin and angular momentum systems . . . . . . . . . . . . . . . . . . 4.7. Crystal lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS APPENDIX A . RADIATION FIELDQUANTIZATION . . . . . . . . . . . . . . . . . . . APPENDIX B. QUANTUM-STATE REPRESENTATIONS . . . . . . . . . . . . . . . . . . B.1.Fockstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Quadrature-componentstates . . . . . . . . . . . . . . . . . . . . . . B.3. Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. s-parametrized phase-space functions . . . . . . . . . . . . . . . . . . B.5. Quantum state and quadrature components . . . . . . . . . . . . . . . . APPENDIX C. PHOTODETECTION . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX D. ELEMENTS OF LEAST-SQUARES INVERSION. . . . . . . . . . . . . . . . REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix 139 143 144 145 151 153 155 157 158 159 160 163 163 167 171 173 175 175 178 179 180 182 183 183 185 187 187 189 189 190 191 192 194 195 197 200
IJI. SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION by S.K.SHARMA (CALCUTTA. INDIA)AND D.J. SOMERFORD (CARDIFF. UK)
8 1. INTRODU~ON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. THEEIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING . . . . .
215 218 218 2.1. Preliminaries of the problem . . . . . . . . . . . . . . . . . . . . . . 2.2. The eikonal approximation . . . . . . . . . . . . . . . . . . . . . . . 219 2.2.1. Approximation from the Schroedinger equation . . . . . . . . . . . 219
X
CONTENTS
2.2.2. Approximation from the integral equation . . . . . . . . . . . . . . . 2.2.3. Propagator approximation . . . . . . . . . . . . . . . . . . . . 2.2.4. Physical picture of propagation in the EA . . . . . . . . . . . . . . 2.3. Scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Eikonal amplitude . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Glauber variant of the EA . . . . . . . . . . . . . . . . . . . . 2.4. Relationship with partial wave expansion . . . . . . . . . . . . . . . . . 2.5. Comparison with the Born series . . . . . . . . . . . . . . . . . . . . 2.6. Interpretation of the EA as a long range approximation . . . . . . . . . . . 2.7. Numerical comparisons and potential dependence of the EA . . . . . . . . . 2.8. Modified eikonal approximations: corrections to the EA . . . . . . . . . . . 2.8.1. The eikonal expansion . . . . . . . . . . . . . . . . . . . . . 2.8.2. The eikonal-Born series . . . . . . . . . . . . . . . . . . . . . 2.8.3. The generalized eikonal approximation . . . . . . . . . . . . . . . 2.9. Relationship with Rytov approximation . . . . . . . . . . . . . . . . . 0 3. EIKONAL APPROXIMATION IN OFTICAL SCATTERING . . . . . . . . . . . . . . . . 3.1. Analogy with potential scattering . . . . . . . . . . . . . . . . . . . . 3.2. Validity of scalar scattering approximation . . . . . . . . . . . . . . . . 3.3. Scattering by a homogeneous sphere . . . . . . . . . . . . . . . . . . . 3.3.1. The eikonal approximation . . . . . . . . . . . . . . . . . . . . 3.3.2. Derivation of the EA scattering function from the Mie solutions . . . . 3.3.3. Relationship with the anomalous diffraction approximation . . . . . . . 3.3.4. Corrections to the EA . . . . . . . . . . . . . . . . . . . . . . 3.3.5. Numerical comparisons . . . . . . . . . . . . . . . . . . . . . 3.3.6. One-dimensional models . . . . . . . . . . . . . . . . . . . . . 3.3.7. Backscattering in the EA . . . . . . . . . . . . . . . . . . . . 3.3.8. Vector description . . . . . . . . . . . . . . . . . . . . . . . 3.4. Scattering by an infinitely long cylinder . . . . . . . . . . . . . . . . . 3.4.1. The scattering function for normal incidence . . . . . . . . . . . . 3.4.2. Scattering by a homogeneous cylinder . . . . . . . . . . . . . . . 3.4.3. The EA from exact solutions . . . . . . . . . . . . . . . . . . . 3.4.4. Corrections to the EA . . . . . . . . . . . . . . . . . . . . . . 3.4.5. Numerical comparisons . . . . . . . . . . . . . . . . . . . . . 3.4.6. The EA as (rn - 11--* 0 approximation . . . . . . . . . . . . . . . 3.4.7. Vector formalism . . . . . . . . . . . . . . . . . . . . . . . . 3.4.8. Scattering at oblique incidence . . . . . . . . . . . . . . . . . . 3.4.9. Scattering by an anisotropic cylinder . . . . . . . . . . . . . . . . 3.5. Scattering by a coated sphere . . . . . . . . . . . . . . . . . . . . . 3.6. Scattering by a spheroid . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Scattering of light by neighboring &electric spheres . . . . . . . . . . . . 0 4. APPLICATIONS OF THE EIKONAL. APPROXIMATION . . . . . . . . . . . . . . . . . 4.1. Particle sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 .1. One particle at a time . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Suspension of particles . . . . . . . . . . . . . . . . . . . . . 4.2. Scattering by rough surfaces . . . . . . . . . . . . . . . . . . . . . . 4.3. Plasma density profiling . . . . . . . . . . . . . . . . . . . . . . . .
220 221 222 222 222 223 224 225 226 226 227 227 228 229 229 230 231 232 233 233 236 237 238 241 247 249 251 253 254 255 257 258 259 260 261 263 263 264 266 267 268 268 268 272 273 276
CONTENTS 4.4. Light scattering by.cladded fibers . . . . . . . . . . . . . . . . . . . . 4.5. Diffraction by a volume hologram . . . . . . . . . . . . . . . . . . . . 4.6. Miscellaneous applications . . . . . . . . . . . . . . . . . . . . . . . 5 5 . CONCLUSIONS AND DISCUSSIONS . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi 278 279 281 282 285
n! THE ORBITAL ANGULAR MOMENTUM OF LIGHT by L. ALLEN(COLCHESTER~ST . ANDREWS, UK). M.J. PADGETT (ST . ANDREWS. a) AND M . BABIKER(COLCHESTER, UK)
5 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 . THEPARAX~AL APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . 5 3. NONPARAXIAL LIGHTBEAMS . . . . . . . . . . . . . . . . . . . . . . . . 5 4 . EIGENOPERATOR DESCNPTION OF LASER BEAMS . . . . . . . . . . . . . . . . . 0 5 . GENERATION OF LAGUERRE~AUSSIAN MODES . . . . . . . . . . . . . . . . .
294 296 302 306 309 Q 6. OTHERGAUSSIAN LIGHTBEAMSPOSSESSING ORBITAL ANGULAR MOMENTUM . . . . . 319 5 7. SECOND-HARMONIC GENERATION ANDORBITAL ANGULARMOMENTUM . . . . . . . . 322 0 8. MECHANICAL EQUIVALENCE OF SPIN AND ORBITAL ANGULAR MOMENTUM: OPTICAL SPANNERS 324 Q 9. ROTATIONAL FREQUENCY SHIFT . . . . . . . . . . . . . . . . . . . . . . . 326 $ 10. ATOMSAND THE ORBITAL ANGULAR MOMENTUM OF LIGHT . . . . . . . . . . . . 328 Q 11. ATOMSAND MULTIPLE L A G U EGAUSSIAN R~ BEAMCON~~GLJRATIONS . . . . . . . . 342 5 12. MOTIONOF MG' M MULTIPLE BEAMCONfiGURATlONS . . . . . . . . . . . . . 345 4 13. ATOMSAND CIRCULARLY POLARIZED LIGHT . . . . . . . . . . . . . . . . . . 356 5 14. SPMaRBIT COUPLING OF LIGHT . . . . . . . . . . . . . . . . . . . . . . 363 0 15. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
V. THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS by A. SEMANNAND G. LEUCHS(ERLANGEN. GERMANY)
Q 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q 2 . HISTORICAL PERSPEC- . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 . THEOPTICAL KERREFFECT . . . . . . . . . . . . . . . . . . . . . . . . . Q 4 . QUANTUM O m c s TN FIBERS - PRACTICAL CONSIDERATIONS . . . . . . . . . . . . 4.1. Kerr-nonlinearity and power confinement . . . . . . . . . . . . . . . . . 4.2. Optical solitons in fibers . . . . . . . . . . . . . . . . . . . . . . . 4.3. Guided acoustic-wave Brillouin scattering (GAWBS) . . . . . . . . . . . . Q 5 . QUADRATURE SQUEEZING. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Properties of Kerr quadrature squeezed states . . . . . . . . . . . . . . .
375 377 380 388 388 389 393 397 397 5.1.1. Single-mode interaction Hamiltonian . . . . . . . . . . . . . . . . 397 5.1.2. Single-mode linearized approach . . . . . . . . . . . . . . . . . 400
xii
CONTENTS
5.1.3. Power enhancement with idtrashort pulses . . . . . . . . . . . . . 5.2. Experiments with continuous-wave laser light . . . . . . . . . . . . . . . 5.3. Experiments with ultrashort pulses . . . . . . . . . . . . . . . . . . . 5.3.1. Ultrashort pulses for GAWBS noise suppression . . . . . . . . . . . 5.3.2. Generation and detection of pulsed quadrature squeezing using a balanced Sagnac loop . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Generation and detection of pulsed quadrature squeezing using a linear configuration . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Experiments with solitons . . . . . . . . . . . . . . . . . . . . 5.3.5. Experiments with non-solitonic ultrashort pulses ( k ” Z 0 ) . . . . . . . 3 6. Q UANT UM NONDEMOLITION MEASUREMENTS . . . . . . . . . . . . . . . . . . 6.1. Concept and realization of a QND measurement of the photon number . . . . . 6.1.1. Cross-phase modulation as a QND interaction . . . . . . . . . . . . 6.1.2. Semiclassical approach . . . . . . . . . . . . . . . . . . . . . 6.1.3. Self-phase modulation noise in the QND measurement . . . . . . . . 6.2. Experiments with continuous-wave laser light . . . . . . . . . . . . . . . 6.3. Experiments with solitons . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Pulse preparation . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Elimination of GAWBS noise in the QND detection . . . . . . . . . 6.3.3. Quantum noise of the probe . . . . . . . . . . . . . . . . . . . 6.3.4. Recent proposals . . . . . . . . . . . . . . . . . . . . . . . . 8 7. PHOTON-NUMBER SQUEEZING. . . . . . . . . . . . . . . . . . . . . . . . 7.1. Spectral filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. Amplification and deamplification of quantum noise 7.1.2. Expehental apparatus and results: an overview . . . . . . . . . . . 7.2. Spectral filtering of picosecond pulses . . . . . . . . . . . . . . . . . . 7.3. Spectral filtering of sub-picosecond pulses . . . . . . . . . . . . . . . . 7.3.1. Noise reduction and enhancement as a function of fdter type and cut-off wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Noise reduction as a function of fiber length . . . . . . . . . . . . 7.3.3. Intrapulse spectral correlations . . . . . . . . . . . . . . . . . . 7.3.4. Spectral filtering of pulses in the normal group-velocity dispersion regime 7.4. Asymmetric fiber Sagnac interferometer . . . . . . . . . . . . . . . . . 7.4.1. Single-mode analysis of a Kerr-nonlinear interferometer . . . . . . . . 7.4.2. Considerations for pulsed squeezing . . . . . . . . . . . . . . . . 7.4.3. Pulsed photon-number squeezing from an asymmetric Sagnac loop . . . 5 8. FUTUREPROSPECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
404 406 408 408
AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF PREVIOUS VOLUMES . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVEINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
471 487 491 501
409 411 412 415 418 418 418 421 423 428 431 432 433 434 434 435 435 436 440 442 443 444 446 448 449 449 451 452 453 458 460 460
E. WOLF, PROGRESS IN OPTICS xxxD( Q 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
I THEORY AND APPLICATIONS OF COMPLEX RAYS BY
Yu.A. KRAVTSOV~, G.W. FORBES~ AND A.A. ASATRYAN’ Space Research Institute, Russian Academy of Sciences, Pvofsoyuznaya Sheet 84/32, Moscow I I781 0, Russian Federation; School of Mathematics. Physics, Computing, and Electronics, Macquarie University, Sydnq, NSW 21 09, Australia
1
CONTENTS
PAGE
0 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 9 2. BASICEQUATIONS OFGEOMETRICALOPTICS . . . . .
3
9 3 . PROPERTIES OF COMPLEX RAYS . . . . . . . . . . . . 9 4. COMPLEX RAYS IN PHYSICAL PROBLEMS . . . . . . .
15
0 5 . GAUSSIAN BEAMS AND COMPLEX RAYS . . . . . . . . 5 6. DISTINCTIVE ASPECTS OF COMPLEX GEOMETRICAL OPTICS . . . . . . . . . . . . . . . . . . . . . . . 9 7 . CONCLUSION . . . . . . . . . . . . . . . . . . . .
5
27 43
50 52
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .
53
REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
53
2
0
1. Introduction
1.1. PIONEERING WORKS
Complex rays are solutions of the ray equations of traditional geometrical optics, but correspond to extremals in the six-dimensional complex space (x’”’‘,”,y’”z’,z’’), where x’ = Re{x}, x” = Im{x}, etc. These trajectories can be used to derive both the phase and amplitude of the associated wavefield. Complex rays were first considered during the 1930s and 1940s in the theory of radio wave propagation through the lossy ionosphere (Epstein [1930a,b], Booker [1939], Bremmer [1949]), but a more general formalization was not developed until the late 1950s and early 1960s. The decisive step in understanding the analytical nature of complex rays was made by Keller [1958], who introduced the notion of complex rays to treat the area of a caustic shadow. He studied the equations for rays passing through points in the interior of a circular caustic in two dimensions and showed that such rays contact the caustic surface at complex points that lie on its analytic continuation. A year later, Seckler and Keller [19591 studied complex rays in plane-layered media, and Keller and Karal [1960] applied complex rays to the problem of surface wave excitation. Grimshaw [1968] stuhed these surface waves in more detail for particular surfaces (the sphere, the cone, and the plane with inhomogeneous impedance). Babich [ 19611 considered the analytic continuation of the wave function into the caustic shadow, and performed calculations that may be interpreted in terms of complex rays. A similar analytic continuation was also applied by Keller and Rubinow [1960] who studied eigenfunctions in both open and closed optical resonators. Complex trajectories are the quantummechanical analog of complex rays, and were studied by Maslov [1963] in connection with the quasiclassical asymptotics of solutions to the Schrodinger equation. Complex trajectories appeared there as complex solutions of the classical equations of motion in regions that are inaccessible to classical particles (i.e., areas of tunneling). Maslov [ 19641 also pointed out that complex rays may form foci and caustics. The application of complex rays within the theory of radio wave propagation through the lossy ionosphere was resumed by Budden [1961], Sayasov [1962], 3
4
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 0 1
Budden and Jull [1964], and Jones [1968, 19701. Generally speaking, in lossy media all rays become complex because the index of refraction, which enters both the eikonal and ray equations, takes complex values. The growing interest in these ideas led to the k s t review paper on complex rays, written by Kravtsov [1967a], in which the analytic nature of complex trajectories was refined and the notion of a complex focus introduced for a Gaussian beam. The idea of using a complex point source to model a Gaussian beam was considered almost simultaneously by Deschamps [1%7, 1968, 19711, Arnaud [1968, 1969b], and Keller and Streifer [1971]. Of note also, in one-dimensional problems the method of phase integrals (see, e.g., Heading [1962]) can be regarded as a precursor to complex geometrical optics; that is, complex ray methods may be viewed as a generalization of the phase integral method for three-dimensional inhomogeneous media. Similarly, the old idea of a complex angle of refraction for the evanescent component in the case of total internal reflection is a clear ancestor of complex geometrical optics. Thus this field has a long hwtory.
1.2. CHARACTER OF WAVEFIELDS DESCRIBED BY COMPLEX GEOMETRICAL OPTICS
In complex geometrical optics the direction of wave propagation is given by the gradient of the real part of the complex phase, and the direction of exponential decay of the field’s magmtude is principally determined by the gradient of the imaginary part. In nonabsorbing media these directions are orthogonal, but in lossy media they are separated by an acute angle. In both cases mhomogeneous waves can enter, and their magnitude changes exponentially on a phase front. In fact, just as homogeneous (propagating) waves are the subject of traditional geometrical optics, inhomogeneous (or evanescent) waves can be regarded as the principal subject of complex geometrical optics (Kravtsov [1967a,b], Choudhary and Felsen [1973], Felsen [1976a]). Like tradtional geometrical optics, complex geometrical optics can involve a multiplicity of rays, so that the total wavefield is then a sum of the waves associated with each ray. As emphasized by Kravtsov [ 1967a,b], when multiple complex rays are present, selection rules are typically required to exclude nonphysical solutions. Another feature of complex rays is that, unlike real rays, they can describe nonlocal (diffraction-like) processes. A clear demonstration of their nonlocal properties is provided by the example of Gaussian beams, see 9 5. It is shown in that section that complex geometrical optics provides a complete description of a Gaussian beam (Kravtsov [1967a,b], Keller and Streifer [ 19711).
4 8 7-1
BASIC EQUATIONS OF GEOMETRICAL OPTICS
5
1.3. GOALS OF THE REVIEW
Significant progress has been made in recent years in new areas of applications for complex geometrical optics - not only in optics itself, but also in other wave disciplines including microwave physics, radio wave propagation in plasmas, and elastic wave propagation. Complex geometrical optics has now been developed into an effective tool for both applied studies and more fundamental research. Although complex rays have been shown to be useful in the analysis of a variety of wave problems, their apparent intangibility has meant that they have sometimes been viewed negatively. Ironically, a moment’s reflection shows that real rays have a similarly tenuous connection to the physical world; both real and complex rays are no more than convenient analytical frameworks for studying wave phenomena. The goal of this review is to present the fundamentals of the method of complex geometrical optics and include a range of applications and examples. Our intention is that this will lift some of the aura that can surround complex rays. Section 2 outlines the basic equations for ordinary and complex geometrical optics. Section 3 is devoted to the properties of complex rays and the selection rules associated with them. These results largely follow from the application of standard asymptotic methods (e.g. stationary phase and saddle-point methods) to the Kirchhoff solutions for wave propagation. Examples of complex trajectories in different optical and physical problems are given in $ 4 . A complex ray analysis of Gaussian beam propagation is presented in more detail in $ 5 . Section 6 touches on certain distinctive aspects of complex geometrical optics, including nonlocality and applicability. These considerations serve to give some measure of the physical significance of a complex ray.
8
2. Basic Equations of Geometrical Optics
2.1. EIKONAL, TRANSFER, Ah?> RAY EQUATIONS OF TRADITIONAL GEOMETRICAL
OPTICS
The equations of geometrical optics may be found both in texts on wave theory (e.g., Born and Wolf [ 19801) and in more specialized books such as Kravtsov and Orlov [1990]. These equations essentially relate to asymptotic forms for a scalar
6
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 5 2
monochromatic field (with time dependence e-"" ') that satisfies the Helmholtz equation: Au + k2&(r)u = 0.
(2.1)
Here, k = o / c and &(r) is the medium's permittivity. The refractive index is defined by the relation E(r) = n2(r) and, for a nonabsorbing medium, Im{&(r)} _= 0. An asymptotic solution can be found by writing u(r) as the product of an oscillating factor, exp[ik$~(r)], and a slowly varying amplitude factor, A(r). Expanding the latter into an asymptotic series in negative powers of i k leads to the standard form
Equations for,)I Ao, A I , etc. follow upon substituting this form into eq. (2.1) and equating the coefficients of each power of k to zero. This approach was devised by Debye in 1911 and the details are outlined in the book by Born and Wolf [1980]. Kravtsov and Orlov [1990] also examined t h s material together with an alternative, introduced by Rytov [1938], that is based on expanding A(r) as a series in the small dimensionless parameter ,u = I/(kL), where L is a characteristic scale of the variations in the medium properties or in the wave itself. Of course, these two procedures ultimately lead to equivalent equations, specifically to the eikonal equation,
and to the transport equations for amplitudes Ao, A[, etc.:
The solutions to these equations may be expressed in terms of rays, which appear as the characteristics of the eikonal equation (2.3) or, equivalently as the
BASIC EQUATIONS OF GEOMETRICAL OPTICS
Fig. 2.1. Energy conservation in an infinitesimal ray bundle is used to determine the field amplitude. This sketch also illustrates locality principle in traditional geometrical optics.
bicharacteristics of the Helmholtz equation (2.1). It is convenient to represent the ray equations in canonical Hamiltonian form: dr dP - = iVc(r), dz dz where p = V v is the "ray momentum" and z is related to arc length by d r = &/n(r). The propagation of a simple wave then begins with the initial field represented as a function of the curvilinear coordinates 5 and r] on an initial surface, written as Q, in the form -=p,
uo = U0(E, r ] ) = 4%5?rl) exp [iW0 (5, r])]
.
(2.6)
If Q is specified in the parametric form r = r 0 ( L r]),
(2.7)
then the trajectory of the ray through ro(E, r ] ) can be written as r(Z) = &E,
r],
r),
(2.8)
where R(E, q , O ) = ro(5, r]). That is, and r] single out a ray, while z specifies the displacement along that ray. These parameters are the ray coordinates. The initial value of the ray momentum follows upon differentiating v(ro) = lyo(5,r]) with respect to 5 and r]:
The third component follows from the eikonal equation (2.3), i.e., (PO)* = &(ro), together with the prescribed sense of propagation in crossing the surface to determine the branch choice. (See fig. 2.1.) It is now straightforward to use
8
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I,
02
eq. (2.5) to fully determine the ray family of the desired field. More generally, a Green’s function can be used to propagate any field and the asymptotic results from this approach give a clear framework for certain aspects of complex geometrical optics. This is discussed further in 0 3.4. 2.2. RAYS AS THE SKELETON FOR THE WAVEFIELD
Both the phase and amplitude of the wavefield in the geometrical optics approximation follow directly from the rays. If 5 and rj are now taken to select the ray that arrives at r, the eikonal, that is’V(r) of eq. (2.2), is determined by a simple integral along t h s ray: (2.10) The zeroth approximation to the amplitude is then given in terms of A:(g, r) by (2.1 1) Here, (2.12) stands for the divergence of the ray bundle which is determined by its perpendicular cross-section do1, which is shown in fig. 2.1. Alternatively, D ( i ) can be defined as the Jacobian of the transformation from ray coordinates to Cartesian coordinates: (2.13) That is, the wavefield can now be estimated as (2.14) and t h s procedure has been referred to as “sewing the wave flesh onto the ray skeleton” (Kravtsov [ 19681, Berry and Mount [ 19721).
1, 0 21
BASIC EQUATIONS OF GEOMETRICAL OPTICS
9
In traditional geometrical optics, therefore, the rays play a dual role. First, they are orthogonal to the phase surfaces v = const, so that the phase may be calculated along the rays according to eq. (2.10). Second, the rays serve as power flow trajectories, since, in the zeroth approximation, the power flux is drected along the ray. To see this, consider
I
1 21k
-(u*Vu- u V u * ) ,
(2.15)
which (in a lossless medium) exactly obeys the conservation law (2.16)
divI= 0
and serves as a measure of power flow, similar to the Pointing vector. When each A, is real-valued in eq. (2.2), the power flux is given by 1 I = A ; p + - [(A?- 240A2)p +A1VA* -AovAI] +. . . . k2
(2.17)
In the zeroth approximation, then, I = A 20 p EIO,
(2.18)
and the power conservation law becomes divIo = div(A: p) = 0.
(2.19)
This is precisely the first of the transfer equations (2.4), since p = V q and div p = A v . According to eq. (2.17), power flow is directed along the rays in loss-free media. That is, I 11 p in both the zeroth and first approximation since I - 10= O(k-*). In a lossy medium, where E is then complex in eq. (2. l), eq. (2.16) becomes d i v l = -k
E”
(uI2.
(2.20)
Here, it is apparent that E” = ImE > 0 is responsible for absorption. A simple approximation that uses the familiar rays follows for the case of weak absorption, that is, when E” << E’ = Re E. In particular, the .right-hand side of eq. (2.20) can
10
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I,
52
be accounted for simply by including a new term in the first of the transfer equations (2.4): 2VAo . V v + A0 AV
= -k E”
Ao.
[Strictly speaking, it is effectively assumed here that E” is O(k-’ ).I Integration of this equation along the real ray path (2.5) now gives A0 = A A0e x p ( - ) k / ~ ” d z ) ,
(2.21)
0
which differs from eq. (2.11) only by the exponential attenuation due to absorption. Thus, for sufficiently weak absorption, the ray trajectories are unchanged and the decaying wavefield can be approximated by exp (ik!
E’
dz - ik
d’dr)
(2.22)
When weak absorption is accounted for to higher orders as a perturbation, the perturbed rays are then complex, as will be discussed in 0 4.1 1. 2.3. COMPLEX FORM OF THE GEOMETRICAL OPTICS METHOD
Much of complex geometrical optics deals with inhomogeneous waues. Plane inhomogeneous waves may be characterized by a complex wavevector (Burrows [ 19651) k = k’ + i k”,
(2.23)
and the wavefield then contains both oscillating and decaying factors: u(r) = exp(ik . r) = exp(ik’ . r - k” . r).
(2.24)
In general, therefore, JuJ= exp(-k” . r) need not be uniform on a phase front k‘ . r = const. For example, consider a surface wave, propagating in free space along the x-axis with a phase velocity u that is less than c so that the x-component of the wavevector, say k, = oh, exceeds the wave number k = o/c. The z-component of the wavevector is then purely imaginary:
k,
=
d-
=
ids.
(2.25)
An evanescent field of this type is generated by the tunneling of a plane wave that is totally internally reflected at an interface at z = 0. Amplitude and phase contours of such a field are shown schematically in fig. 2.2.
BASIC EQUATIONS OF GEOMETRICAL OPTICS
11
const
Fig. 2.2. Amplitude profile of a plane surface wave (shaded) and corresponding contours of the real and imaginary parts of the associated complex eikonal li, = W’ + ili,”.
Complex geometrical optics deals with the propagation of inhomogeneous, or evanescent, waves of the standard geometrical optics form, namely u(r) = A(r) eikW(‘),
(2.26)
but now W(r) is complex-valued. Such waves play an important role in many optical problems, including (1) wave propagation in lossy media, (2) surface waves, (3) wave fields near caustics, and (4) Gaussian beams and related fields. This review concentrates principally on the wavefields in caustic shadows and in lossy media, and on wavefields such as Gaussian beams. Other kinds of inhomogeneous waves are treated only briefly. The mhomogeneous wavefield of eq. (2.26) is also asymptotically expanded as in eq. (2.2). It is therefore required that E , Aj, and the real and imaginary parts of VW do not change significantly on the scale of the wavelength, A = 2 ~ / k . That is,
are necessary conditions for the validity of complex geometrical optics, where F stands for either E or Ao. Sufficient conditions are discussed in 56. Notice that the inequalities (2.27) do not prohibit changes of the field modulus, IuI = lAol exp(-kW”) that are rapid on a wavelength scale. This is consistent
12
rL § 2
THEORY AND APPLICATIONS OF COMPLEX RAYS
with the proposition that locally the wavefield has the structure of a plane inhomogeneous wave. By substituting eq. (2.2) into eq. (2.1) and equating the coefficients of the powers of k, as in traditional geometrical optics, we again obtain the eikonal equation (2.3) and transfer equations (2.4) for the amplitudes of successive approximations, but now we are looking for complex-valued solutions of these equations. The formal solutions can be presented in the traditional forms (2.10) and (2.1 l), where, naturally, the characteristics obeying the traditional ray equations (2.5) are now complex-valued. The properties of complex rays are the subject of Q 3. Although the complex rays form the geometric skeleton for the wavefield, they are no longer so simply related to the phase and to the power flux ( Q 2.5). Another distinction of complex geometrical optics is that field contributions need not have localized physical origins; that is, according to eqs. (2.14) or (2.22), each field contribution at r depends on the value of the initial wavefield at a specific point on Q. The field contributions in complex geometrical optics, on the other hand, are generally associated with particular points on the analytic extension of Q, and such an extension is essentially nonlocal in the real coordinate space. This loss of localization is the basis for the diffraction-like character of certain phenomena described successfblly by the method of complex geometrical optics (discussed fiuther in 0 6). 2.4. ALTERNATIVE APPROACH TO PHENOMENA DESCRIBED BY COMPLEX RAYS
Just as Hamilton-Jacobi theory establishes that the asymptotic solutions of geometrical optics can be derived directly fiom the eikonal and transport equations in terms of complete integrals of the former, the problems of complex geometrical optics may be approached similarly. That is, without complex rays, but by dealing directly with the complex eikonal and amplitudes in real coordinate space. It is convenient here to consider the real and imaginary parts of I),Ao, and E , writing
+
I) = I)‘ iI)”,
A0 = Ah
+ iAt,
E = E’
+ id’
(2.28)
With this, the eikonal equation in a lossy medium can be written as two realvalued equations: (2.29a,b) In loss-free media E” = 0, so eq. (2.29b) states that contours of I)’ and I)” meet orthogonally. Figure 2.2 is a simple illustration of this orthogonality.
I7
8 21
13
BASIC EQUATIONS OF GEOMETRICAL OPTICS
Equations (2.29) can be expressed simply in terms of the complex momentum p = VV =p' +," The first of the transport equations (see eq. 2.4) can be rewritten similarly as 2 (VV' . V A ; - OW" . V A ; ) +Ah AV' - A ; AV" 2 (VW" . VAh + VW' . V A ; ) +Ah AV" + A ; AV'
2
0,
(2.30a,b)
= 0.
These two equations can be expressed more simply in terms of
X' + if, since eq. (2.4) now takes the form
x = In A0
2VW. V x + A W = 0,
=
(2.31)
so eq. (2.30) becomes 2 (VW' . VX' - VV" . V X " )+ AW' = 0, 2 (OW'* VX" + VW" . OX')+ AW" = 0.
(2.32a,b)
Thus, for lossy media, the eikonal and first transport equation are equivalent to the four real equations (2.29) and (2.32). General methods to solve these systems of equations have not been developed, even for homogeneous media. The only known solutions are the plane inhomogeneouswaves of eq. (2.24). A perturbative approach for weakly inhomogeneous waves due to Choudhary and Felsen [1973, 19741 and Deschamps [I9741 was developed further by Wang and Deschamps [ 19741 and Pereira [19841, but these prove to be useful only for short propagation distances. It is interesting, nevertheless, to consider the power flow associated with a solution of the type considered here. By substituting expansion (2.2) into eq. (2.15), it is found that
I = exp(-2kV") x = I0
{ IAo l2 p 1+ k [(ALVA; 1
-
+ -111 k
1
+ -I2 + . ... k2
- A; VAL)
+ 2 (AhA:
- A;
A ; )p']
+ .. .
3
(2.33) Unlike the result in eq. (2.17), the power flux now contains a term of first order in l/k and has a global factor of exp(-2kW"). Notice also that 10is parallel to p l , which is normal to the contours of W', but that I I is not purely in this
14
THEORY AND APPLICATIONS OF COMPLEX RAYS
w“= const
+
/)( Fig. 2.3. Contours of
‘
p” and Vl1,l
I)’ and I)” in a lossy medium. The energy flow vector, Zo, is parallel to p’
= VI)’,
and V lZ01 is parallel top”
= VV”.
direction. Furthermore, because of the added exp(-2kI)”) factor, it follows [by using eq. (2.29b)I that divZo = - k ~ ”IAo12 exp(-2kW”) [l + O(k-*)]
(2.34)
in keeping with eq. (2.20) and unlike eq. (2.19). Similarly, it is found that the modulus of the power flux decreases most rapidly in the direction of p“:
V Ilo(M -2k IAoI2 Ip’Jexp(-2kI)”)p1’.
(2.35)
The expressions in eqs. (2.34) and (2.35), which have an accuracy of the order of O(l/k), are exact for the plane inhomogeneous waves of eq. (2.24). Figure 2.3 schematically shows contours of I)‘ and I)“,as well asp’ andp”, which asymptotically coincide with ZO and V 1101, respectively. Recall that in a loss-free mediump’ andpl’ are orthogonal, but according to eq. (2.29b) they are seen to meet at an acute angle in a lossy medium. The normal curves to the surfaces of constant I)‘ can be found by following p’ and are known as phase trajectories, while the curves found by following Z are the powerflux trajectories. Notice that, for absorbing media, these two trajectories differ at O(k-’), while in traditional geometrical optics the power flux trajectories deviate from the rays (i.e. the phase trajectories) only at 0 ( k 2 )see , eq. (2.17). Since the phase trajectories do not obey eq. (2.5), they may prove to be curved even in homogeneous media (see § 4.1 1). Complex rays are distinguished from the phase trajectories by following p = VI)instead of pl. (Notice that following p no longer defines a single path with a real parameter: while the step from the current position is still dr = p dz, dz can now be complex-valued; this complication is treated in 5 3.)
PROPERTIES OF COMPLEX RAYS
15
Fig. 3.1. Two real rays in the lit region near a simple caustic become two complex rays that may penetrate into the shadow region that is prohibited for real rays. Only one of the complex rays is relevant and it is represented here as a wavy line.
0
3. Properties of Complex Rays
3.1. RAY PATHS IN THE COMPLEX SPACE
Complex rays are complex solutions of the ray equations (2.5). (Such a notion enters naturally upon considering the saddle-point method of asymptotics as discussed in 6 3.4.) To appreciate their form, consider the role of complex rays for the shadow area of a simple caustic in an inhomogeneousmedium. As shown in fig. 3.1, two real rays arrive at any point r of the lit area. In t h s region, the system of three scalar equations
possesses two real solutions for g, q, t which identify two rays and the associated source points on the initial surface Q. In the caustic shadow area (progressively shaded in fig. 3.1), eq. (3.1) has no real solutions. Suppose, however, that the expressions describing Q, i.e. r = ro(E,q), can be extended analytically to complex values of E and r], and E ( X , Y , Z ) is also continued analytically into the complex space. The complex solutions of eq. (3.1) for E(r), q(r), and t ( r ) now can be identified with complex rays. As t varies from zero to t ( r ) , the vector R[E(r),q(r),z] moves from the initial source point R[g(r),q(r),01 = rOIE(r),q(r)],which may be complex, to the real point of observation, namely R [ t ( r ) ,q(r),z(r)] = r . For intermediate values of Z, R[E(r),q(r),t] traces out the complex ray in the six-dimensional space (R’, R”). Only the initial and h a 1 values of z are fixed and, within the region of analyticity, all intermediate contours connecting these points lead to identical results. Thus, it
16
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I.0 3
is possible to deform the contour used in the complex t plane for the integration of eq. (2.5). Provided any singularities are avoided, all such contours lead to the same value for the eikonal as given by eq. (2.10). Some examples are given in 5 4. (An analogous manifold of equivalent complex rays is associated with each of the familiar real rays in traditional geometrical optics, but the natural representative member is the real-valued one with a monotonic parametrization.) Complex rays arise even more directly in lossy media because the permittivity is then complex-valued.As solutions to eqs. (2.5), the rays are evidently complex in this case. Again, these rays can arrive at points in the real physical space after passing through complex intermediate points. Some of the subtleties associated with numerical treatments of eqs. (2.5) are mentioned by Jones [ 19701 and Budden and Terry [19711. In particular, in place of the two real parameters used to vary an initial point on Q and the one parameter to run along the associated ray in traditional geometrical optics, six real parameters are now required to characterize the family of complex rays: two complex parameters serve to fix a point on Q, and the ray parameter t is also complex. Furthermore, even though the initial momentum is determined as in eq. (2.9), the distinction between forward and backward propagating field components is now more complicated so that the branch choice for the third momentum component is no longer straightforward. The integration to determine the rays starts at the origin and, in the absence of singularities, can proceed along any outward radial line in the complex t plane. Since the form of the contour used to arrive at any final value of t is irrelevant in this case, these radial lines are just a convenient set of contours to cover the plane. This contour choice selects a particular representative from among what could otherwise be a tangle of equivalent complex rays. The difficulties associated with singularities, ray selection rules, and searching the six-dimensional space to find all the rays that arrive at any specified endpoint mean that much of the work in complex geometrical optics is analytical rather than numerical. 3.2. FERMAT’S PRINCIPLE FOR COMPLEX RAYS
In its usual form, Fermat’s principle identifies rays as the paths between any two fixed points, say r, and rb, that give a stationary value to
J n ds
=
j? JEodr.dr.
(3.2)
r,
The point eikonal, q(r,, r b ) , is then just the stationary value taken by this integral. As shown by Alonso and Forbes [ 19951, this idea can be generalized for the angle
I , § 31
PROPERTIES OF COMPLEX RAYS
17
eikonals, by using surfaces in place of the fixed points r, and rb, and this process is taken as being understood in what follows. Fermat’s principle has also been generalized for complex paths and absorbing media. The methods discussed by Forbes [1991] can be used to show that first, the complex eikonal equation (2.3) is just the Hamilton-Jacobi equation that follows from this extremal principle, and second, the complex rays r ( z ) satisfy eq. (2.5). This reference also gives alternative forms for eq. (2.5), and establishes that other simple variational principles are equivalent to the form given here, but avoid the square root that can be problematic in eq. (3.2). Notice that the first variations of both the real and imaginary parts of the eikonal must vanish. The second of these requirements has been regarded as a “principle of least attenuation”. Conditions of this type are discussed, for example, in the review paper by Kravtsov, Tinin and Cherkashin [1979] devoted to multipath phenomena in ionospheric radio wave propagation, and in a recent review and a book by Apresyan and Kravtsov [1996a,b] in the chapter concerning multipath phenomena in the theory of radiative transfer. It is interesting that these variational principles are related to Feynmans path integral method and that, initially, the path integral technique did not account for classically prohibited trajectories. McLaughlin [19721 incorporated complex trajectories into this method for the description of tunneling, and later Wright and Garrison [1987] applied the path integral method to the derivation of Gaussian beams. 3.3. SELECTION RULES FOR COMPLEX RAYS
As mentioned in 8 3.1, two real rays pass through every point in the lit area near a simple caustic (see fig. 3.1) and therefore it is no surprise that two complex rays reach each point in the shadow area, In nonabsorbing media the eikonals v1,2 = ?)[,* + associated with these complex rays are found to satisfy I+$ = W;, q9; = -I,!(’. That is, they are complex conjugates. (These values will be seen in the next section to be associated with the saddle points of a Kirchhoff integral.) The field contributions associated with these complex rays take the form U j = A,
exp(ikWj) = Aj exp(ik$ - kq!) , j
=
1,2.
(3.3)
One of these field contributions decays away from the caustic and the other increases exponentially. The latter is evidently nonphysical, and at this level it is natural to introduce the obvious selection rule (‘justified in the next section). When more than two complex rays are present, however, the selection rules are more difficult to formulate. The general idea, of course, is that only contributions
18
THEORY AND APPLICATIONS OF COMPLEX RAYS
/ Fig. 3.2. A cusp in a caustic that is shown as the dashed curve. Three real rays reach Q (in region I), but one real and two complex rays reach points outside the cusp, such as rS (in region 11). Only one of these complex rays, represented as a wavy line, contributes to the total wave field.
that decay away from the associated caustic should be adrmtted. For example, consider the vicinity of a caustic cusp (fig. 3.2). Inside the cusp, three real rays arrive at each point, but outside the cusp there is one real ray and two complex rays. Of these two complex rays, only the one that leads to a field contribution that decays with distance from the caustic is relevant. This simple recipe is invalid in the case of tunneling across a barrier; both terms must then be included within the gap, where increasing the distance from one boundary corresponds to approaching the other. More generally, one can consider the analytical properties of the phase (eikonal) as in the accepted procedure in the phase integral method for one-dimensional problems. (Recall that complex geometrical optics can be viewed as a generalization of the phase integral method for two or three dimensions.) Interestingly, it is possible for uniform asymptotic methods to include field contributions from the complex rays that have just been described as “nonphysical”. Thus, the total wavefield can be expressed uniformly in terms of information from all the rays. These methods can also be used as a starting point in the derivation of selection rules. The interrelations between uniform asymptotic methods and the method of complex geometrical optics are dlscussed in the book by Kravtsov and Orlov [1998], and briefly in 93.8. Note that Deschamps [1974] discusses a different type of selection rule: he proposes that rays are retained only while the associated amplitude exceeds some prescribed level.
1,
o 31
PROPERTIES OF COMPLEX RAYS
19
3.4. COMPLEX RAYS AND THE SADDLE-POINT METHOD
Diffraction integrals can be regarded as the foundation of much of geometrical optics. One option is to consider the Kirchhoff solution for a wavefield in a closed region of interest and use asymptotic estimates of both the boundary values and the Green's function. Various explicit asymptotic forms for the Green's function are treated by Alonso and Forbes [ 1997al for smoothly inhomogeneous media. The resulting form for the diffraction integral now explicitly depends only on the field value at the boundary and the properties of rays in the medium. A firther simplification is also possible. The final (and optional) step in the geometrical optics method involves applying asymptotic methods to this form for the diffraction integral. Traditional results follow by using the method of stationary phase; complex rays enter naturally when the stationary points no longer fall in the real domain. Consider the simplest example of a diffraction integral for a two-dimensional homogeneous medium: u(x,z) =
2x
exp(ikR) z uo(g)R'/2E dg,
(3.4)
m . The exact form for eq. (3.4) where uo(x) = u(x,O) and R = d involves a Hankel function (see, e.g., Born and Wolf [1980]) that has been approximated asymptotically here. If the initial field uo(g) is in geometrical optics form, so u 0 ( B = A O ( Bexp [ikWO(E)]
(3.5) the value of the integral in eq. (3.4) may be approximated by using the method of stationary phase. The exponent in the integrand is stationary at 5 = &, where 9
The last expression is precisely the equation of the ray with momentum p: = dW0/d2j that passes through ( x , z ) = (&, 0). The method of stationary phase now proceeds by expanding about E = 5'. At the lowest order, this leads to
20
THEORY AND APPLICATIONS OF COMPLEX RAYS
rL 5 3
where
and
It is readily verified that this asymptotic estimate coincides with the geometrical optics solution given in $2.2. When eq. (3.6) has multiple solutions, the asymptotic estimate is then just a sum of terms of this form, that is, (3.10) Notice the localization implicit in the field contributions:the estimate given here involves only uo(x) at x = &. However, t b s geometrical optics estimate is valid only when the integrand of eq. (3.4) is well approximated by the form used in eq. (3.7) over a region that is comparable in size to the first Fresnel zone surrounding the stationary point at x = $. The width of this zone follows from the condition that the quadratic term in eq. (3.7) reaches n,so (3.11)
It follows that the geometrical optics estimate is then valid only when
When eq. (3.6) has no real roots, the asymptotic estimate is then found by determining the complex stationary points; these are usually called saddle points because a plot of the modulus of the integrand has the form of a saddle at the stationary point. In such a case, asymptotic estimation of the integral in eq. (3.4) is performed by deforming the contour of integration from the real axis out into the complex plane. When vois real for real 5, it follows for the analytic extension that vo(g*)= [vo(g)]*, and the saddle points therefore come in conjugate pairs. The contour must be deformed to pass through the saddle point where Im{vo}> 0 (Felsen and Marcuvitz [ 19731 or Bouche, Molinet and Mittra [1997]). This is the origin of the selection rules introduced earlier in an ad hoc
1,
o 31
PROPERTIES OF COMPLEX RAYS
21
manner. Of note, the estimates that result from the saddle-point method again take the form given in eqs. (3.7-3.9), but now ES is complex-valued. Furthermore, the sum in eq. (3.10) does not include contributions from all of the saddle points. An important consequence of the disappearance of the real stationary points is the loss of the localization in the real domain; that is, uo (&) can no longer be identified with a particular real point. Of course, a variety of representations for a wavefield exists, and each has its own asymptotics. For the example considered here, the wavefield can be given in the Fourier domain, so that (3.13) where (3.14) and U(q) is the Fourier transform of the initial wavefield at z = 0:
The stationary (saddle) points for the integral in eq. (3.13) are now in the complex q plane, and the complex ray arriving at (x,z) originates at z = 0 with momentum qs, and its transverse coordinate is given in terms of the derivative of the phase of U(q).Such a situation is a characteristic also for other integral representations of the wavefield, including Maslov’s method (Maslov [1972], Maslov and Fedoriuk [1981]) and the uniform integral representation of Alonso and Forbes [1997b, 19981. In both cases, complex rays arise in a mixed momentudcoordinate space. Similarly, complex saddle points are encountered in the analysis of many wave problems, and all of them may be interpreted in terms of complex rays. 3.5. COMPLEX CAUSTICS
Like real rays, complex rays can form caustics. T h s occurs when the Jacobian,
D(t),of eq. (2.12) is zero. The caustics are .now embedded in complex space.
22
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I,
83
Formally, the equation of the complex caustic surface in the six-dimensional space (rl,rl’) may be derived by solving (3.16) for t,and substituting the result into
This leads to a four-dimensional hypersurface of the form r = rc(E,r). This hypersurface is more complicated than the familiar caustic surfaces in real space. An important step in the study of the analytical nature of complex rays and complex caustics was made by Yakushkin [1969, 19701, who introduced socalled main surfaces. These surfaces serve as a generalization of the anti-Stokes lines in the method of phase integrals. The complex phase is real-valued on the main surface, so its imaginary part changes sign there. As a result, a Stokes phenomenon takes place on the main surfaces. Thus, an abrupt change of the asymptotic behavior of the wave function occurs, like the Stokes phenomenon in the method of phase integrals. Many related important results of complex geometrical optics, or, equivalently, of the complex WIU3 method, were obtained by Maslov [1977] and by Mishchenko, Shatalov and Sternin [1990]. The same problem was studied later by Sternin and Shatalov [1996a,b], who considered Stokes phenomena in multidimensional space. In the six-dimensional complex space, the geometrical optics wavefield diverges at the complex caustic r = rc(t,o).At points in the real threedimensional subspace, rr, which are near the complex caustic, one can expect a local enhancement of the field magnitude, as discussed by Maslov [1964]. Maslov’s original example is briefly described in 9 4.1. The field at points of the real space that are near the complex caustic can be described by complex geometrical optics, provided that is sufficiently large. In particular, if the characteristic scale of the caustic zone in whch geometrical optics is invalid is written as Zc (Kravtsov [1988], Kravtsov and Orlov [1998]), then the condition for validity of complex geometrical optics in the vicinity of the complex caustic is 1r;l > 1,. In lossy media the separation of real space into lit and shadowed areas about a real caustic is now different because all rays become complex. Such a separation is acceptable for weakly absorbing media, where it is possible to use the ordinary geometrical optics method andor the corresponding uniform caustic approximations with an exponential factor describing absorption along
1, P 31
PROPERTIES OF COMPLEX RAYS
23
the ray, like that in eq. (2.22). This separation is not meaningful for strongly absorbing media, however. When E” M E’, it cannot be determined whether the exponential decay of the field is due to absorption or to displacement into the caustic shadow. 3.6. SPACE-TIME COMPLEX RAYS
In dispersive media, space-time rays characterize both spatial and temporal divergencelconvergence. An outline of the method of space-time geometrical optics is given by Lewis [1965], Kravtsov [1969], Felsen [1969, 19701, Kravtsov, Ostrovskii and Stepanov [ 19741, and Qavtsov and Orlov [ 19901. Space-time rays satisfy (3.18) where H(w,k; r,t) is a Hamiltonian for a dispersive, nonstationary, and inhomogeneous medium. The dispersion relation for the medium can be written as H(w,k; r , t ) = 0. In an isotropic and stationary medium, for example, the Hamiltonian has the form w2
H ( w , k,r) = k . k - -&(w,r). C2
(3.19)
If the space-time trajectories for a particular field are written as
where 5 = ( E l , E2,53), the field estimate at (r, t) follows upon solving eq. (3.20) for g. If there are no real solutions, the complex solutions can be associated with complex space-time trajectories which start from a complex initial point and end at (r, t). Such complex space-time rays are convenient for the description of the propagation of electromagnetic pulses in dispersive and lossy media. Early applications were demonstrated by Connor and Felsen [ 19741, Sukhy [ 19741, and Berstein [ 19751. Censor [ 19771 discussed the applicability of Fermat’s principle to space-time rays in lossy media. Orlov and Anyutin [ 19781 applied complex space-time rays in an analysis that concentrates on the propagation of Gaussian pulses. The general relations between space-time wavefields and their spectra were outlined by Felsen [1986], and Gaussian space-time beams were also
24
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 0 3
studied by Connor [1980], Arnaud [1984], Heyman and Felsen [1986, 19891, Einziger and Raz [1987], and Schatzberg, Einziger and Raz [1988]. 3.7. ELECTROMAGNETIC WAVES AND COMPLEX RAYS
Until now we have discussed only scalar waves and ignored the polarization of the electromagnetic field. As in traditional geometrical optics, specific treatments are appropriate for isotropic and anisotropic media, but few publications have addressed complex rays in anisotropic media. The eikonal equation in an anisotropic medium (Born and Wolf [1980], Kravtsov and Orlov [1990]) takes the form H
= det(q@) = 0,
(3.21)
where
In fact, eq. (3.21) does not contain the sixth power o f p and may be factored as
where
and nj(r,p/p) is the refractive index for the ordinary or extraordinary normal waves for j = 1,2. The associated rays satisfy (3.25) Having determined the ray trajectory for some fixed 5 and q, one can find the eikonal and the amplitude of the normal wave along that ray. The polarization of the normal wave at r is determined by the unit vector f (j),which satisfies q$$J) = 0. With A, W , and f for each normal wave, the total field takes the form 2
(3.26) j=I
These relations can be extended to complex rays (Kravtsov and Yashm [ 19691). Three important aspects are worth mentioning. First, since p is complex, the
I,§ 31
PROPERTIES OF COMPLEX RAYS
25
polarization of a normal wave, that is,f(j), may become elliptical where it would otherwise have been linear or circular. This may happen in either a caustic shadow or in a lossy medium. Second, in nonabsorbing media the Poynting vector S is orthogonal to the imaginary part of the momentum, that is,
s .p” = 0.
(3.27)
This result corresponds with similar properties for the scalar wave (see eqs. 2.29b and 2.33) but now S need not be parallel top’. Third, for an inhomogeneous wave in a lossless medium, S is connected with the energy density w by
dU s=-w, Bk’
(3.28)
where k = ( u / c) p = k‘ + ik” is the complex wavevector. Evidently, g = d d d k ‘ may be treated as the group velocity of the inhomogeneous wave, although eq. (3.28) is valid only for sufficiently long pulses. Other aspects of complex ray propagation in anisotropic media were treated by Budden and Terry [1971], Terry [1971, 19781, and Sukhy [1972a,b, 19741. The last paper takes into account mode coupling effects in lossy anisotropic media. An alternative formulation of complex geometrical optics for lossy anisotropic media, which explores the coupled equations for tp’ and V’’ instead of the complex eikonal equation Hj = 0, was stumed by Sukhy [1972a]. Gaussian beam propagation in anisotropic media was also investigated by Shin and Felsen [1974]. In isotropic media, since the electromagnetic field is transverse to the rays, E may be expressed as a combination of the rays’ unit normal, n, and binormal, b:
E
= qjln
+ 426.
(3.30)
The scalar amplitude, A = (qj: + qj;)”’, is determined by the transport equation (2.4), and the angle between E and n, say y = arctan(qjl/qj2), obeys Rytov’s law (Rytov [1938], Born and Wolf [1980], Kravtsov and Orlov [1990]): dY - fi da T ’
(3.3 1)
where T
=
[n (V x n) + b (V x b)]-’ 9
(3.32)
is a torsion radius. Again, as discussed by Sayasov [ 19621 and Kravtsov [ 1967a1, these results are valid for complex rays. When the unit vectors n and b are
26
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 0 3
complex, E becomes complex, which means that the electromagnetic field may acquire ellipticity even if the initial polarization in real space is purely linear or circular. Detailed analyses of such effects are not known to us. 3.8. COMPLEX RAYS AND UNIFORM ASYMPTOTICS
The relation between complex geometrical optics and integral representations of a wavefield was discussed in 5 3.4. Here, we briefly outline the interrelations with the uniform asymptotic methods of wave theory. The method of etalon functions (or etalon integrals) deals with special functions that describe the wavefield around caustics as classified by catastrophe theory (Poston and Stewart [1978], Gilmore [1981], Arnold, Varchenko and Gusein-Zade [1985, 19881, Arnold [1992]). The arguments and amplitude factors of these etalon functions are determined by equations developed by Kravtsov [1964a,b] and Ludwig [1966]. This process uses both real and complex geometrical optics results. In the case of a simple caustic, for example, the uniform asymptotic approximation to the wavefield can be expressed in terms of the Airy function A@) (Kravtsov [1964a,b], Ludwig [ 19661, White and Pedersen [ 198I], Kravtsov and Orlov [1983, 19981): u = fi k”6 [Ql Ai(k2’35.)- ik-1’3Q2 Ai’(k2’3<)] exp(ikx-id4).
(3.33)
x
Here, and are given in terms of the two values of the associated eikonal, that is, I)\and @, by
while QI and
Q2
are related to the geometrical optics amplitudes:
Ql = (-c)”4(A
1
+ Itlz),
Q 2 = (-<)”4(A
-
L42).
c
(3.35)
These expressions are valid in both the lit area, where < 0, and in the shadow region, > 0. Notice that this method involves all the complex solutions of geometrical optics - including those described earlier as being “nonphysical”. T h s W h e r demonstrates that rays are not simply the classical trajectories along which energy propagates, but they also contain information about what would generally be regarded as diffraction phenomena: both real and complex rays provide the skeleton for the wavefield. Maslov’s asymptotic theory (Maslov [1972], Maslov and Fedoriuk [1981]) takes a different approach, where the wavefield is partially constructed in
<
1 , s 41
COMPLEX RAYS IN PHYSICAL PROBLEMS
27
coordinate space, and the balance in momentum space. Away from caustics, the results of Maslov's method are equivalent to those of geometrical optics, including both real and complex rays. The same is true for Orlov's method of interference integrals (see Kravtsov and Orlov [I9981 and the review by Avdeev, Demin, Kravtsov, Tinin and Yarygin [ 19881).This is related to a recently developed method (Alonso and Forbes [ 1997b, 19981) that constructs asymptotic estimates directly in phase space. In fact, the explicit dressing of the classical phase space curve that is used in this method can be interpreted in terms of complex rays. This can be seen from the fact that the method effectively collects each coordinate and its conjugate momentum together as a single complex number. From this, it follows that points in phase space that do not lie on the classical phase space curve correspond precisely to complex rays, that is, to complex values of the parameter used to generate the curve.
0
4. Complex Rays in Physical Problems
4.1. COMPLEX RAYS INSIDE A CIRCULAR CAUSTIC
One of the first examples of complex geometrical optics concerned the field inside a circular caustic (Keller [1958]). The amplitude and phase of the wavefield in the vicinity of a circular caustic of radius a can be found by representing the caustic in a parametric form:
The eikonal on the caustic is a linear function of a:
When r > a, the eikonal at x = r cos cp, y = r sin cp may be calculated by using eq. (2. lo), which in a homogeneous medium with a permittivity E = 1 gives V ( r ) = VO@)+ t(r),
where z is the distance between the point of observation r and the tangent point at the caustic. As shown in fig. 4.1, there are two tangent rays and the corresponding values of a and z are a
q2= cp farccos -, r
z1,2= F -..
(4.3)
28
[I, § 4
THEORY AND APPLICATIONS OF COMPLEX RAYS
4
X
*-.. ........,*...Fig. 4.1. Two real rays tangent to a circular caustic.
The two values of the eikonal are v I , ~=
v:,,+ 21,2= ag, f
(4.4)
The Jacobian D ( T )is equal to -z, so that the amplitudes A1,2 are given by C 2 -1/4 (4.5a) ~ ~ = - = ~ ( r ~ - u ) ,
@m
-
C
rn
- e-in/2 C ( r 2 - - a2) -114 --i41
(4.5b)
The resulting field outside the caustic therefore takes the form u = A I eik@l + A2eik@
For points inside the caustic, however, a1,2, z1,2, and a a1,2= g, f i Arccosh-, T ~ =,T ~ i m , r v1,2 = ag,
f i (a Arccosh!
r
-
V1.2
are all complex: (4.7)
m),
where Arccosh(s) = In (s + fi) , and the amplitudes A1,2 are given by A~ = ceini4(a2 - r
2 -1/4
)
,
= ce-'3~'4 (a2 - y
2 -1/4
)
.
(4.8)
Contours of phase, that is, of v', correspond to radial lines, whereas the amplitude contours are concentric circles.
29
COMPLEX RAYS IN PHYSICAL PROBLEMS
I , § 41
The complex rays in this case are given by x 1 , ~= x0(a1,2)+ T P ~ ~ u ) ,
+ ~pyO(a1,2),
~ 1 . 2= y0(a1,2)
where x0 and yo are the coordinates of the tangent points: x 0(
0 ~ 1 , ~=) a cos a1,2= a
yO(~rl,z) = a sin
=a
[;
-
sin cp
i
cos.]
9
and p," = -sin a and p j = cos a. Thus, the rays start at the complex tangent points rp,2,but finish at the real point r = (x, y ) = (r cos q, r sin cp). It is easy to check that ry,2both sit on the circular caustic given by x2 +y 2 = a', or x12
- x112
+
y12 -y1t2 = a2
,
xlxtt+ y y
= 0.
(4.9)
Since Vtt is positive only for V l , the field inside the caustic is given by just one term: u =~~e'kPl
= C ( r 2 - a2) -1'4 e-in/4 (4.10) If the arbitrary constant C is taken to be C = eix'4(2nk)-1/2,eqs. (4.6) and (4.10) are just the Debye asymptotics for the exact solution, which takes the form u = Jh(kr) exp(ikaq),
(4.1 1)
where J,(x) is a Bessel function. For single-valuedness, ka must be an integer, and such a quantization condition (related, more generally, to area in phase space) arises whenever a ray family closes on itself as in this example. Notice that the exponential decay of the wavefield inside the caustic changes to a power law near the origin. That is, for r 4 0, Arccosh(a/r) approaches ln(2a/r), so exp[-ka Arccosh(a/r)] is proportional to rka.According to Maslov [1964], this weaker attenuation replaces the exponential because of the brightening associated with the focusing of the complex rays. 4.2. WAVE REFLECTION IN A LAYERED MEDIUM
In the medium E ( X , Z ) = EO - EI z, where EO and ~1 are real and positive, a wavefield can be determined from its initial form in the plane z = f. Suppose, for
30
[I, 8 4
THEORY AND APPLICATIONS OF COMPLEX RAYS
example, that 5; < E O / E I and uo(x,5;) = Ao exp(iloc no sin 8), where no = \/Eo-E;4. The ray starting from xo = l j , zo = 5; is described by
(For the details see Kravtsov [I9687 or Kravtsov and Orlov [1990].) Two rays arrive at every point in the lit area and the corresponding values of 5 and z are found by solving eqs. (4.12) for E and z: ljt,2 = x - r1,2 no sin 6,
r1,2 =
2 {no cos 8 El
1/2
[(no cos 6)’ - El (z - <)I
}. (4.13)
rt) and (&,r2) coincide for points on the caustic which is given by z = z, : = 5; + (no cos 8)2/&1. (El,
In the shadow of the caustic, i.e., for z > z,, all these values become complex:
&,z
= x - zl,znosin 8,
2
r1,2 = - {nocos 8 El
5 iy} ,
(4.14)
where
The corresponding complex eikonals and amplitudes are
(4.15)
Again, only the solution that decays away from the caustic should be retained, and the associated field takes the form U I=
A exp (ikWI)
3~1
8+2 (nocos 8)’] - 2k y ’ } 3 €1 (4.16) This expression corresponds to the asymptotic form of the exact solution given by an Airy function. (The two-term field u = ul+ u2 in the lit area also matches thls exact solution.)
Y
31
COMPLEX RAYS IN PHYSICAL PROBLEMS
1 , s 41
(a)
(b)
(c)
, and c1,2 on the complex (t', t") plane. These generate different Fig. 4.2. (a) Six paths a l , ~ bl,Z complex trajectories (b) X(z)and (c) Z(t)that reach the real point (x,z). Paths al, bl and CI all correspond to the same complex ray that leads to an exponentially decaying wave field behind the caustic.
Some of the properties of complex rays discussed in 0 3 can be illustrated with this example. Figure 4.2a shows six different paths u1,2, b1,2, and c1,z in the complex z plane, and the corresponding projections of the complex rays in (X'"'') and (Z', 2") are given in figs. 4.2b and 4 . 2 ~As . z moves from 0 to TI, or to z2 = T ; , X ( z ) and Z(t)are generally complex-valued, and they become real-valued only at the end of each trajectory. Each of these paths with common endpoints leads to the same value for the eikonal. The fact that two values of z and are associated with each point (x,z) means that the field can be taken to be defined on a two-sheeted surface with a branch point at z = z,. A branch cut from this point is quite similar to those used in the phase integral method in ID problems (Heading [1962]). 4.3. POINT SOURCE IN A LAYERED MEDIUM
Consider a point source at xo = 0 and zo = 5' in the same medium used for the previous example. The equations for the associated rays follow from eqs. (2.5):
In place of in eqs. (4.12), 8 now plays the role of the ray parameter. As presented by Kravtsov and Orlov [1990, 19981, solving these equations for t and sin 6 leads to two solutions: (4.18)
32
THEORY A N D APPLICATIONS OF COMPLEX RAYS
[I,
04
where
These two solutions coincide whenf2 parabolic caustic: &o-Elz=
El (2no)
=
0, and this gives the equation for the
x2.
(4.20)
When ( x , z ) lies above this caustic, A becomes negative and 21,2 and 131.2 are then complex. Thus, the rays that penetrate the caustic shadow leave the source at complex angles. Only the first solution leads to an exponential decay in the shadow area:
Together with the Companion expression for the field in the lit region, t h s result matches the asymptotic form of the exact solution given by Kormilitsyn [1966]. Notice in this case that the complex rays begin and end at points in the real space, but at complex-valued angles. 4.4. THE VICINITY OF A CAUSTIC CUSP IN FREE SPACE
We mentioned in 5 3.3 that there are three real rays inside the cusp of a caustic and anticipated that there are two complex rays outside the cusp. T h s can be verified by simple calculations when the cusp is of the form xo(5) = aE3,
~ ' ( 5=)bE2.
(4.21)
Measuring the eikonal from the cusp's beak at 5 = 0, it is readily shown that 1 = 27a2 [(9a2E2+ 4b2)3'2- (2b)3] ,
vo(E)
and that when the point of observation is (x,z ) , the equation for the ray parameter 5 is just z x (4.22) E3-3-E+2- =O. b a Let d be the discriminant of eq. (4.22):
Inside the cusp (area I in fig. 3.2), d is negative and all three roots 51,2,3(x,z) of eq. (4.22) are real. At the caustic, d = 0 and two of these roots coincide. Outside
1,
P 41
33
COMPLEX RAYS IN PHYSICAL PROBLEMS
the cusp, where d > 0 (area I1 in fig. 3.2), there is a complex conjugate pair of roots for E. One of the associated values for the eikonal corresponds to a decay away from the caustic, and this should be used, along with the contribution from the real ray, for the wavefield description outside the cusp. 4.5. SWALLOW-TAIL CAUSTIC: AN EXAMPLE WITH FOUR RAY CONTRIBUTIONS
A swallow-tail caustic like that presented in fig. 4.3 arises, for example, when a point source is placed in a homogeneous half-space z < 0 with E = EO that adjoins a layered medium with E ( Z ) = EO - E ~ Z for , z > 0 (Orlov [1966], Kravtsov and Orlov [1990, 19981). There exist four rays in region I, although only one ray is shown in fig. 4.3. Two real rays arrive at each point in region 11, and the other pair of rays prove to be complex. All four rays are complex in region 111. When region I shrinks to zero and ultimately vanishes, only two rays continue to be real. When the displacement of the caustic loop into the complex space is sufficiently small, a localized brightening of the field is exhibited in the real space. This is an example of the manifestation of the effects of complex caustics. tZ
(4
(b)
Fig. 4.3. Point source placed near linear layer (a) forms a caustic of swallow tail type (b) (Orlov [1966]).
4.6. POINT SOURCE IN A PARABOLIC LAYER
The permittivity profile &(Z) = Q
+ &* z2
(4.23)
gives an example where an infinitely large number of rays can reach a given point even for simple initial fields (Kravtsov [1967a]). For a barrier-like layer,
34
THEORY AND APPLICATIONS OF COMPLEX RAYS
I
lo
X
>
Fig. 4.4. (a) A point source placed into a plane layer of parabolic profile forms a real caustic beneath the barrier and a complex caustic above the barrier, and generates an i&te number of complex rays above the real caustic. (b) A point source placed into a potential well of parabolic profile generates an infinite number of real rays, corresponding to waveguide propagation. Just a few of the rays are shown here. EO is taken to be negative and ~2 is positive. If a point source sits at (xo,zo) and zo < z-, where zk = * ( - E o / E ~ ) ' / ~ so that ~ ( q =) 0, the summit of the caustic occurs exactly at z = z- (fig. 4.4a). The ray leaving the source at an angle 8 from the z-axis is described by
x = xo + t no sin 8, z=
cos 8 sinh(EA/2t)+zo cosh(#2t),
(4.24) (4.25)
and these can be used to find t and 8 for the ray that reaches (x,z). Below the caustic, two real solutions exist, but there are none in the shadow area. However, the transcendental equations (4.24H4.25) possess an infimte number of complex solutions at all points. The infinite number of solutions may be appreciated by considering the related problem of a point source placed within a parabolic well where EO > 0 and ~2 < 0. If the point source is located in the zone where E ( Z ) > 0, the trigonometric t)replace the hyperbolic functions in eq. (4.25). functions C O S ( E ~and '~~) It is easy to see in this case that the equation for z possesses an infinite number of real roots for any points within this waveguide, and each corresponds to a real ray. A few such rays are shown in fig. 4.4b. These rays map to complex rays in the case of a potential barrier. Moreover, since the real rays form an
I , $ 41
COMPLEX RAYS IN PHYSICAL PROBLEMS
35
infinite number of real caustics in the waveguide case, it is reasonable to expect an infinite number of complex caustics for the potential barrier. To determine the wavefield at (x,z), it is first necessary to solve eq. (4.25), which is transcendental in t. An analytical solution can only be obtained for x = xo, in which case eq. (4.24) states that 8 = 0. (When x z X O , 0 is complexvalued.) With z > z+ and 8 = 0, eq. (4.25) leads to an infinite set of solutions for t:
zm=z
-i(2m+ 1)x , m= 0 ,1 ,2 ,. . ..
{In z&+
(4.26)
ZO&+&q
This result was also obtained by McLaughlin [1972] through an asymptotic calculation of the path integral. The complex-valued eikonal corresponding to tmis given by (4.27) and the associated Jacobian and amplitude satisfy
D, = -z, A,
=B
d=exp(anirn),
(4.28a,b)
[ z i e ( z ) ~ -exp(-ixm). ~/~
Here, B is the strength of the point source, that is, u x Bexp(ikR)/R, when )r- rol -+ 0. Negative values of m must be excluded so that exp(ik decreases for z > z+, and the total field behind the barrier then takes the form
R
vm)
05
A , exp(ik
u=
vm).
(4.29)
m=O
The ratio of the (m + 1)th term to the mth term is proportional to exp(-2&), where (4.30) A factor of exp(-&) accounts for the tunneling attenuation in a single pass. When & >> 1, therefore, only the leading term is needed in eq. (4.29): u(x0,z) x
B
[T:E(Z)]-'/~
exp(ikv- &),
(4.31)
36
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 8 4
where q is just the sum of the optical path between the source and the barrier and from the barrier to the point of observation:
(4.32) More detailed analysis of this problem was performed by Kaloshin and Orlov [1973] who noticed that the caustic behind the barrier proves to be complex. The degree of embedding of this caustic into the complex space depends on the position of the source. When zo + 0, this caustic becomes real. Holford [1981] obtained some new solutions of a reduced wave equation for layered media, and Holford's examples of tunneling phenomena may also be interpreted in terms of complex rays.
4.7. ABOVE-BARRIER REFLECTION
Analysis of above-barrier reflection may ,J performed by using an example like that in Q 4.6, but now % > 0 and &2 > 0 in eq. (4.23). The point source is assumed to be placed at (xo,zo), with zo > 0, and the field on the vertical line x = xo is then given by an expression similar to that in eq. (4.31): u(xo,z)= - i B [r;~(z)]-~'~exp(-61)
where exp(-61) is the attenuation due to above-barrier reflection and (4.34)
The path of integration in eq. (4.34) encloses the two zeros of ~ ( z )ki:-. When & and 61 of eqs. (4.31) and (4.33) are much smaller than unity, many complex rays must be taken into account. Summing the contributions of all these rays by using Poisson's formula was performed by Yakushkin [ 19691 for the case of a source placed inside a parabolic layer with EO < 0.
1,
5 41
COMPLEX RAYS IN PHYSICAL PROBLEMS
31
4.8. COMPLEX RAYS BEHINLI A SINUSOIDAL PHASE SCREEN
Propagation of a normally incident plane wave through a sinusoidal phase screen gives a phase modulation of VO(E)= g sin(KE), (4.35) whch gives another example of a problem with an infhte number of complex rays. Near the screen, only one ray reaches each point and all other rays are complex. Beyond the first caustic cusp, two more real rays appear. As the distance from the screen increases, more of the complex rays become real. Figure 4.5 shows the increase in the number of real rays.
Fig. 4.5. The real caustics generated by a sinusoidal phase screen mean that the number of rays that reach any point depend on how far that point is from the screen. This is mapped schematically here where the Roman numerals @ve the number of real rays in a given area.
4.9. SURFACE WAVES
Keller and Karal [1960] and Grimshaw [1968] were the first to study surface waves on the basis of complex geometrical optics. As an example, suppose that the phase at the plane z = 0 in a space where E 3 1 for z > 0 is given by V O ( B= Y 5, Y > 1. (4.36) The phase velocity, Uph = c / y , is then smaller than c. This initial field readily leads to complex values of 5 (initial value of x-coordinate) and z for the ray that reaches (x, 2): (4.37)
38
THEORY AND APPLICATIONS OF COMPLEX RAYS
The eikonal is now found from eq. (2.10) to be ~ = ~ O + r = y x + i Z ~ ~ , whch corresponds to the surface wave propagating along the x-axis: u = A exp (ik yx -
kzdm).
(4.38)
This is just one example of the result in eq. (2.24), where the complex wavevector is now
(
k = ky, k
J-1
1 - y 2 =(ksin8, kcoso),
with 8 = arcsin y = x/2 + i Arccoshy. Thus, the surface wave has a natural description in terms of complex rays, and this can be applied most simply for the case of total reflection of a plane wave. 4.10. REFLECTION OF INHOMOGENEOUS WAVES FROM AN INTERFACE
Consider the case of an inhomogeneous wave that is reflected at a perfectly conducting surface S. By requiring the sum of the incident and reflected wave to be zero at S, that is
the same relations as for a homogeneous wave result:
These expressions give initial values for Vrefl and Arefl at S, as used by Choudhary and Felsen [1973, 19741 while considering the scattering of a plane inhomogeneous wave and a Gaussian beam from a cylinder. When complex ray optics is used, these boundary conditions are appIied not only on the physical surface, but also on its analytic continuation. Such an approach was proposed by Yakushkin [19701 and developed later by Deschamps [ 19741. Yakushkin studied the 2D problem of a linear source radating near an infinitely long circular cylinder. The analytic extension of this cylinder takes the form of eq. (4.9). Complex rays from the line source can undergo multiple reflection from this extended surface and the resulting wavefield is therefore a sum of the multiply reflected waves. Just as a real ray intersects the cylinder
I , § 41
COMPLEX RAYS IN PHYSICAL PROBLEMS
39
twice and only one of the intersections is physically significant, it is necessary to rule out some of the complex ray intercepts. Deschamps [1974] proposed rules for this purpose that are based on consideration of reflected rays in the limit of a homogeneous plane incident wave. A decomposition of the wavefield into creeping waves results by using Poisson’s formula on this infinite sum and evaluating the result at the real point (x,y). This is another type of diffraction effect that can be modelled by complex geometrical optics. At an interface of two dielectric media, the phase condition (4.40a) is still valid, but a coefficient of reflection must be introduced in eq. (4.40b):
Just as for real rays, the reflection coefficient depends on the angle of incidence, and R(8) is determined by analytic continuation. The complex angle 8 = 8’ + i0“ follows from (4.42) where p = V v is the momentum of the incident ray and n is a normal to the surface S . 4.1 1. COMPLEX RAYS
IN WEAKLY ABSORBING MEDIA
We have already seen that all the rays become complex in absorbing media. Many applications involve weakly absorbing media, however, and as indicated in $2.2, approximate solutions for the rays, eikonals, and wave amplitudes can then be obtained by perturbation of the real geometrical optics results. Bennet [1974, 19781 studied complex and real ray tracing in weakly absorbing media, as did R.M. Jones [1970], D.S. Jones [1978], Censor [1977, 19811, and Sukhy [1981]. Since some studies of complex rays in anisotropic media were already listed in 0 3.7, we consider only the scalar problem here. Different results follow, depending on whether the eikonals or the rays are hndamental to the perturbation. A perturbative solution of the eikonal equation is used in traditional geometrical optics when a mehum is close to a case for which the rays can be found in closed form, for example a weakly inhomogeneous medlum. These methods have been discussed by Kravtsov and Orlov [1990] and directly apply here if we simply take E’ to be the unperturbed permittivity and regard E” as a small
40
THEORY AND APPLICATIONS OF COMPLEX RAYS
perturbation. In the zeroth approximation the unperturbed eikonal (ie., determined by integration along the associated real rays:
[I,
84
wo)
is
t
(4.43)
= JE‘dz, 0
where d t = &/&. It also follows from the results of section 9 of Kravtsov and Orlov [1990], that the first and second corrections are given by T
T
(4.44a,b) 0
0
where the integration is again performed along the unperturbed real ray r ( t ) . Notice that ‘1y1 is purely imaginary and % is purely real. Arsaev and Kinber [1968] derived these results for &, ql and @ directly from eqs. (2.29) by considering E” and V’’ to be small perturbations. The first correction corresponds to the exponential attenuation of the wavefield as given in eq. (2.21). The second-order correction modifies the rays themselves. If the real component of the momentum (i.e., p’ = VV’) is expanded as a series in powers of E”,
then (4.45) This result characterizes the deviation of the phase trajectory from the unperturbed direction p;, and Arsaev and Kinber [1968] used this to deduce that the phase trajectories bend in the direction of the gradient of the relative losses V(E”/E’). The details of the ray bending, however, are complicated by the fact that (VV;)’ in eqs. (4.44) and (4.45) depends not only on E”/E’ but also on the initial form of the wavefield. Since thls approach is based on the point eikonal, it is invalid at caustics and certainly cannot account for any caustics introduced by the perturbation itself. On the other hand, perturbation of the rays is not limited in this way and again proceeds by analogy with the results presented by Kravtsov and Orlov [1990],
1,
o 41
COMPLEX RAYS M PHYSICAL PROBLEMS
41
but now for weakly absorbing media. This application was discussed by Zhu and Chun [1994]. Again, the ray is expanded in powers of E”, so that we have
r( z)
= ro( z)
+ rl (z) + r2( z) + . . . ,
(4.46)
where ro(z) is the real ray. In keeping with the results given above, q(t)proves to be purely imaginary, whereas r2(2) is purely real. For the final point of the trajectory to be real, the complex ray generally originates from a complex point on the initial surface. The associated complex ray parameters 5 and r] of the point of origin follow from eq. (4.46), and the first-order shifts A5 and Au prove to be purely imaginary and proportional to the integral of E”. As a simple example, consider a plane wave incident at an angle 60 to the normal of the interface with a weakly absorbing homogeneous half space (z > 0) with permittivity E = E’ + id’, E’ >> E”. The initial phase on the interface can be written as vo= 5; sin 60, and the solutions of the ray equations take the form x = E+px z,
2
= p z z,
(4.47)
where (4.48) From eqs. (4.47) and (4.48) the initial complex point 5; for the ray that reaches a given real point of observation, say ( x , z ) , is found to be
5; = x - p x z = x - Px - z = E‘+i~’’. Pz
To first order in
el’,
(4.49)
it follows that (4.50)
Since characterizes the familiar real ray for the case of a loss-free medium, A5; = i5;“ gives the shift into complex space generated by the absorption. A focused incident field is another useful example to consider. Notice that this
42
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I,
P4
general method is valid also for the case of media with gain (E” < 0) and is well suited for numerical work since only real ray tracing is required. 4.12. COMPLEX GEOMETRICAL OPTICS IN OTHER WAVE DISCTPLINES
Complex ray methods are widely used outside optics. In fact, problems in microwave physics have driven the development of some of these methods with many valuable contributions made by Felsen and Deschamps, with their respective co-workers. The field contributions associated with complex rays for waves generated by structured antennas and the interaction of inhomogeneous and Gaussian micro-wavefields with various structures have been examined, for example, by Belanger and Couture [1983], Heyman and Felsen [1983], Montrosset and Orta [1983a,b], Ghlone, Montrosset and Felsen [1984], Ghione, Montrosset and Orta [1984], Gao and Felsen [1985], Brown [1987], Einziger, Haramaty and Felsen [ 19871, Ikuno [19871, El-Hewie [ 19881, Ikuno and Felsen [1988a,b], Maciel and Felsen [1989], Hovenac and Lock [1992], Ikuno, Ohmori and Nishimoto [1993], and Goto, Yukutake and Ishihara [1995]. An analysis of the advantages and disadvantages of complex ray methods for microwave antennas is included in the publications by Montrosset and Orta [ 1983a,b]. Another important application is to the propagation of radio waves in a plasma, and we mention a few relevant papers here. Terry [1971, 19781 developed and applied complex ray methods for these lossy systems to study ion cyclotron whtstlers in the ionosphere. Smith [1973] studied the angular diameter of the so-called “Ellis window” in the ionosphere by using a complex ray method. A summary of the techniques for ray tracing in a stratified plasma was published by Budden [ 19891, and similar applications were treated by Andrianov and Sekistov [1978] and Wang [1984]. BravoOrtega and Glasser [ 19911 applied complex geometrical optics to propagation through an inhomogeneous magnetized plasma and presented an innovative numerical treatment of the amplitude equation. A recent paper by Zernov and Lundborg [1996] on high-frequency radio wave propagation in a disturbed ionosphere accounts for diffraction by localized random inhomogeneities in the neighborhood of caustics. Elastic and visco-elastic waves in acoustics and seismology bring distinctive boundary conditions at interfaces. For example, D.S. Jones [ 19781 analyzed the acoustics of splitter plates and Westwood [1989] studied the reflection of acoustic waves from a fluid-fluid interface. Collins and Westwood [1991] compared various results with solutions generated by a complex ray model for long-range acoustic wave propagation in the ocean with large bottom slopes
1,
o 51
GAUSSIAN BEAMS AND COMPLEX RAYS
43
and mhomogeneities in sound speed. Smith and Tew [ 19951 studied ultrasonic surface wave excitation and determined the reach of complex surface rays. Zhu and Chun [1994] described an efficient method for complex ray tracing, and ray perturbation, in inhomogeneous media and provided an accurate description of wave attenuation in a realistic Earth structure. Hearn and Krebes [1990a,b] considered complex rays in viscoelastic media, and emphasized that the angle between the local direction of wave travel and the direction of maximal amplitude attenuation should be taken into account in the processing of seismograms. As mentioned in 6 3, complex classical trajectories (Maslov [1963]) enter quantum-mechanical problems involving molecular collisions (see Miller [19741) as well as barrier penetration and caustic shadows (e.g., see Nussenzveig [1992]). More recently, complex paths appeared in the problem of tunneling between two regular phase space regions that are separated by a chaotic layer (Doron and Frishat [ 19951). Although the wavefunction in quantum mechanics cannot attenuate due to absorption, complex potentials arise, for example, in inelastic neutron scattering (Schiff [1968]) or for resonant atomic tunneling through a laser beam (Zhang and Sanders [1994], Tribe, Zhang and Sanders [1996]). The imaginary part of the potential is introduced in such cases to account for loss to other channels by inelastic scattering.
8 5. Gaussian Beams and Complex Rays 5.1. GAUSSIAN BEAMS AND COMPLEX SOURCES
It is widely recognized that paraxial Gaussian beams correspond to spherical waves (cylindrical in 2D) with their center of symmetry shifted into the complex domain. For example, a point source in 2D at ro = (0,zo) generates a cylindncal wave of the form L
u(r) M -exp(ikR),
A?
where r = (x, z) and R = [(z- ~ the expression in eq. (5.1) to u(x,z) M
0
+)x2]~ 1’2.
ik(z - ZO) +
~
The paraxial approximation converts
2(2 - zo)
44
THEORY AND APPLlCATIONS OF COMPLEX RAYS
With zo = ikw; Gaussian beam: u(x, z ) =
=
[I,
55
i a , R is complex and eq. (5.2) then describes a paraxial
(z - ia)*I2
(5.3)
The beam’s waist falls in the plane z = 0 where the field is proportional to exp(-x2/2w;), and a is its Rayleigh range. For a spherical wave the field from a point source is just L
u(r) = - exp(ikR), R
(5.4)
and, with zo = i a, the paraxial approximation gives an axially symmetric Gaussian beam: (5.5)
This use of complex sources was introduced repeatedly in the late 1960’s. Kravtsov [1967a,b] showed that the 2D Gaussian beam of eq. (5.3) has its focus at ( x , z ) = (0,ikw;). Similarly, Deschamps [1967, 1968, 19711 and Keller and Streifer [1971] identified eqs. (5.3) and (5.5) as the paraxial approximations of fields from complex point sources. Arnaud [1968, 1969bl arrived at the same conclusion from a different approach. The relationship between Gaussian beams and complex point sources was later examined by Couture and Belanger [ 19811, and then in a singularity-free form by Sheppard and Saghafi [1998]. Izmest’ev [1971], Ito [1973], Shin and Felsen [1977aJ, Hashimoto [1987], and Lindell and Nikoskinen [1987] showed that it is also possible to embed multipoles in a complex space to describe higher-order Gaussian modes. It0 [ 19741 described the vectorial Gaussian beam by fields of current sources at a complex location. Wu [1985] further generalized this idea by considering the analytical extension of the Green’s function for an inhomogeneous medium. Finally, we observe that Arnaud [1984], Einziger and Raz [19871, Heyman and FeIsen [1986, 19891, and Schatzberg, Einziger and Raz [ 19881 analyzed pulsed Gaussian beams by using this approach. 5.2. ANOTHER DESCRIPTION OF GAUSSIAN BEAMS IN TERMS OF COMPLEX RAYS
An initial wavefield with a Gaussian profile can be propagated directly by applying complex geometrical optics. This analysis was -first performed by
1,s 51
45
GAUSSIAN BEAMS AND COMPLEX RAYS
Kravtsov [1967a,b] for the 2D case, and by Keller and Streifer [1971] for the 3D case, and clearly demonstrates the capabilities of complex ray methods in the modeling of diffraction effects. Suppose, in the 2D case, that the initial field is given at the plane z = 0 by u0 (5,o)= exp(-$)
=exp(-g),
where wo >> A, so a = k wi >> W O .Equation (2.6) now gives the initial conditions for the complex geometrical optics solution as
This form is appropriate for the case where a is fixed as k becomes large. If wo were to be held fixed instead, the Gaussian profile must then be taken as the initial amplitude and the eikonal vo(5)is therefore identically zero:
Traditional geometrical optics now gives parallel rays, all of the form x = E, and the eikonal corresponds to a plane wave, that is, t ) ~= z. Thus, geometrical optics yields
(
;i;)
uGo(x, z ) = uo(x,Oleikz= exp ikz - -
,
(5.9)
and this result is valid for z << a. The solution where a is fixed, on the other hand, gives a full description of the beam in terms of complex rays. In free space, the rays are given by x = e+p,"z,
z =pz"z,
(5.10)
where the components of the initial momentum are obtained from eq. (5.7) by
The ray parameters for the ray that reaches (x,z) follow from eqs. (5.10):
(5.1 1)
46
rk 4 5
THEORY AND APPLICATIONS OF COMPLEX RAYS
This leads to a quartic in
5, but when
1E1 << a,
(5.12)
eq. (5.11) is approximately linear in Ezx(l+;) -1 .
E, and the solution is given by (5.13)
In the near zone (i.e. for z << a) inequality (5.12) means that eq. (5.13) is valid when 1x1 << a. When z >> a, the validity condition becomes 1x1 << z. In the intermediate zone, where z is about a, these two conditions are equivalent. It follows that eq. (5.13) is adequate for most purposes. Nevertheless, the exact roots of eq. (5.11) give insight into the ray selection rules, etc., especially when the medium has complex permittivity. Once is determined, it follows that t satisfies
(5.14) which, together with
v = v0(E)+ t, yields
vz!c+z(l-g) 2a
= z + g (I+:).
(5.15)
By using eq. (5.13), this becomes
(5.16) Finally, the amplitude follows directly from eq. (2.1 l), i.e. A and the resulting wavefield takes the form
=
(1 + iz/a)-"2,
(5.17) Thus, complex geometrical optics gives a result that has precisely the same form as eq. (5.3). Moreover, this solution coincides exactly with a diffraction result based on the Fresnel approximation to a Kirchhoff integral, that is,
(5.18) with uo(E,O) given by eq. (5.6). The same is true for an axially symmetric Gaussian beam, and even for Gaussian beams with arbitrary astigmatism. All these cases proceed similarly.
1,
P 51
GAUSSIAN BEAMS AND COMPLEX RAYS
41
5.3. TRANSFORMATION OF GAUSSIAN BEAMS IN OPTICAL SYSTEMS
It follows from eqs. (5.3) and (5.5) that the width, w(z), and the radius of curvature of the phase, R(z), of a Gaussian beam in any cross-section are given by (5.19) 2
R(z)=z-z0+*
4
z - zo
=(z-zo)
(5.20)
where the waist lies in the plane z = zo. Together with the additional factor in each of eqs. (5.3) and ( 5 9 , these completely characterize the beam. Notice that this factor gives a phase shift that is simply tan-'[a/(z - ZO)]in 2D and twice this for 3D. Many of the results concerning Gaussian beams can be expressed conveniently in terms of a single complex parameter that measures axial displacement from the complex source point, namely
q =z-(zo+ia).
(5.21)
This was suggested by Deschamps and Mast [1964] and Kogelnik [1965a,b]. Note that Kogelnik, and some others, define the beam width with an additional factor of Accordmg to eqs. (5.19) and (5.20), the real parameters R and w are related to q by
a.
_1 --- 1 + -. i q
R
kw2
(5.22)
When a beam passes through a thin lens of focal length f,the q of the beam changes simply: the beam width remains unchanged, but the radius of the phase curvature before and after, say R I and R2, are related by (5.23) If we ignore global phase offsets, the transformation law for q is formally the same: we are now imaging a complex point source according to (5.24) This means that the standard algorithms of traditional geometrical optics can be applied to the solution of many optical problems dealing with the propagation
48
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, 5 5
and diffraction of Gaussian beams. Such beam transformations in optical systems were performed by Deschamps and Mast [1964], Kogelnik [1965a,b], Kogelnik and Li[1966], Arnaud [1969a,b, 1974, 1976, 19851, Deschamps [1972], Self [1983], Jakob [1984], Jakob and Unger [1984], and Kujawski [1989], to name but a few. Notice also that, since eq. (5.24) is equally valid for a mirror, many results related to optical resonators can be readily obtained by means of complex rays (Arnaud [1969a,b, 1974, 1976, 19851). A 2D Gaussian beam may be written as
u(x, z ) =
-&
exp (ikq
+
5) ,
(5.25)
and Arnaud and Kogelnik [1969] and Arnaud [1969b, 19851 showed that two 2D beams can be combined into a 3D astigmatic Gaussian beam in the form (5.26)
u(r) =
+
where qx = z - (zoX iu,) characterizes the waist size and location in the x-z plane, and qy does so for the y-z plane. When ZO, = zoy and wo, = way, the beam has the form given by eq. (5.5). Each q is transformed according to eq. (5.24). Note also that Arnaud [1968, 1969b, 19851 associated complex rays with a Gaussian beam in a somewhat different way: He characterized a 2D Gaussian beam by a single complex ray, or by a pair of real rays. Arnaud’s rays characterize the q of a Gaussian beam (so astigmatic beams are described by two complex rays of Arnaud’s type). Further analyses of astigmatic beams were presented more recently by Lu, Feng and Cai [1993, 19941. The problem of Gaussian beam transformation at an interface has also been considered. Beam reflection at a plane interface was treated by Shin and Felsen [1977b] and by Lu, Felsen, Ruan and Zhang [1987] who accounted for the lateral shift of a beam under total reflection. The reflection and transmission of a beam at a curved dielectric interface was studied by Ra, Bertoni and Felsen [1973], Felsen [1976b], Ruan and Felsen [1986], and Ruan and Zhou [1989]. Shin and Felsen [1978] studied multiple reflection of Gaussian beams, and Kudou and Yokota [199I] analyzed reflection and transmission of Hermite-Gaussian beams incident on a curved dielectric layer. Norris [19861 considered the propagation of a Gaussian beam through a spherical interface. Analytic continuation of the interface into complex space is common in this work.
1,
s 51
5.4.
GAUSSIAN BEAMS AND COMPLEX RAYS
49
DIFFRACTION OF GAUSSIAN BEAMS
The representation of a Gaussian beam as a spherical or cylindrical wave from a complex source gives an effective method for the solution of a wide class of diffraction problems. Given an analytical solution for diffraction of a real spherical or cylindrical wave on hfferent bodies, the solution for the associated diffraction problem involving a Gaussian beam can be solved simply by displacing the source into the complex domain. This idea was put forward by Gowan and Deschamps [19701 who considered the half-plane and cylinder as examples. Such extensions involve reflected and refracted complex rays where the point of incidence on the interface generally lies on the analytic extension of the surface into complex space. Furthermore, the Fresnel coefficients of reflection and refraction are also calculated for complex angles of reflection and refraction as indicated in $4.10. The problem of Gaussian beam diffraction by a half-plane was studied in detail by Deschamps, Lee, Gowan and Fontana [1979] and Takenaka and Fukumitsu [ 19821. Generally two complex rays arrive at each point of observation: the direct ray from the complex point source, and the diffracted complex ray from the edge of the half-plane. The contribution of the diffracted ray can be calculated by a generalization for complex rays of the geometric theory of diffraction. Pereira [1984] suggested referring to this theory as the “complex geometric theory of diffraction”. The impressive accuracy of the results for this problem was reported by Takenaka and Fukumitsu [1982]. Another simple illustration of this method involves diffraction by a circular aperture. In this case, Osterberg and Smith [1961] and Heurtley [1973] gave closed forms for the diffracted field on axis for a point source, placed at z = zo, and these are convenient for analyzing the diffraction of Gaussian beams. These solutions are also valid for a complex point source at z = zo + ia, where zo now characterizes the waist location of the Gaussian beam and the width there satisfies a = k wi.This means that the direct ray and all the diffracted rays in the interpretation of the solution start from the complex source. More generally, it is possible to extend the results for the boundary-diffracted wave given recently by Forbes and Asatryan [ 19981, where the diffraction of a spherical wave is reduced to a one-dimensional, singularity-free integral over the rim of aperture. In conclusion, we note that Gaussian beams have other applications in the theory of diffraction. For example, arbitrary but sufficiently smooth wavefields can be decomposed effectively into a sum of Gaussian beams and this process is convenient for wave propagation in inhomogeneous media (e.g. see Popov [1982], Cerveny, Popov and Psencik [1982], Norris [1986], and Babich and
50
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, § 6
Popov [1989]). This is closely related to the methods developed .recently by Alonso and Forbes [ 19981.
9
6. Distinctive Aspects of Complex Geometrical Optics
6.1. NONLOCAL PROPERTIES OF COMPLEX RAYS
As mentioned at the end of 52.3 and illustrated in many of the preceding applications, the field contributions associated with complex rays are not tied to the initial field value at a specific real point as in eq. (2.14). This is partly responsible for the diffraction phenomena exhibited by the wavefields connected with complex rays. For example, this is evident in the treatment of Gaussian beams given in § 5.2, and is demonstrated in the analysis of wave penetration into caustic shadows. The nonzero field in a caustic shadow is caused by a diffraction process that amounts to tunneling of the wave from the lit zone. To illustrate the nonlocality of complex rays, we expand the initial wavefield in a power series in the imaginary parts of the ray coordinates E(r) and ~ ( r ) :
It follows that uo(E,q) at a complex point depends on u o ( r ,q') and its derivatives. Therefore, a local process in the complex space is, in fact, nonlocal in real coordinate space. In a weakly absorbing medium, for example, the initial point is shifted into complex space, which increases the area of influence in the vicinity of the real point r O ( r v'). , In other words, the region on the initial surface that contributes significantly to the diffracted field at Y grows with the introduction of weak absorption. Bertoni, Felsen and Hessel [1971] measured the extent of this region of influence by using a Gaussian filter in the associated diffraction integral, which was examined in detail by Kravtsov [1988]. The filter is centered on the real ray of interest, and when the filtering window is made sufficiently wide, the field is essentially unchanged by the window. The degree to which the window width can be reduced without significant impact is used to define the extent of the region of influence. The same idea can be used effectively for complex rays as demonstrated by Asatryan and Kravtsov [1988]. The integral remains in the real
1,
P 61
DISTINCTIVE ASPECTS OF COMPLEX GEOMETRICAL OPTICS
51
coordinate space and is centered on the real point ro(g’,u’). The results serve to quantify both ray localization and distinguishability,and help to make complex geometrical optics more concrete. Equation (6.1) exposes the difficulties in dealing with cases where the initial field, or its derivatives, are either discontinuous or known only from experimental data. Similar difficulties arise with the analytic extension of ~ ‘ ( 5u), and ~ ( r ) , and this is one of the principal limitations of complex geometrical optics. Such matters have been treated by Wang and Deschamps [1974] in relation to the example of wave reflection from finite mirrors. 6.2. BOUNDARIES OF APPLICABEITY OF COMPLEX GEOMETRICAL OPTICS
Necessary conditions for the validity of complex geometrical optics were described in 9 2.3. Sufficient conditions can be expected to follow, by analogy, from those for traditional geometrical optics. (Recall that the field is estimated in geometrical optics as a sum of contributions from each of the rays through the point of interest.) These conditions are stated in terms of a ray’s Fresnel volume, which is a generalization of the familiar first Fresnel zone (see Kravtsov [1988] or Kravtsov and Orlov [1990]). The ray family associated with the initial field specified on Q defines a geometrical phase at all points reached by these rays. This is just q ( r ) of eq. (2.10). The Fresnel volume associated with an observation point r, is made up of all points for which
where V(r,r,) is just the optical distance between r and r,. Notice that q(ro) may be multi-valued, and then each ray through r, has its own branch that is to be used in eq. (6.2). The volume associated with each ray typically takes the form of a tube enclosing that ray. The sufficient conditions for validity are based upon consideration of the application of the stationary phase method to the underlying diffraction integrals. These admittedly heuristic criteria state that, first, a field contribution is valid if the properties of the medium, the geometrical wave amplitude, and V v are all essentially constant over any cross-section of the Fresnel volume for the associated ray and, second, the Fresnel volume for each of these rays must not overlap significantly with any of the others. Although this statement of the criteria effectively ignores the diffracted rays of the geometrical theory of diffraction and remains somewhat vague, these conditions give usehl guidelines for the validity of the field estimates of geometrical optics. For complex
52
THEORY AND APPLICATIONS OF COMPLEX RAYS
[I, § 7
geometrical optics the saddle-point method is used as a generalization of the stationary phase method, and the saddle scale would therefore naturally play the role of the Fresnel scale in defimng the relevant volume. The added dimensions associated with the complex space, the flexibility in the contour of integration through the saddle point, and the freedom in choosing the contour in the complex z plane that defines the ray all add to the difficulty of a useful generalization for these criteria. Asatryan and Kravtsov [1988] have made initial steps in this direction, but this area represents yet another aspect of geometrical optics, both real and complex, that merits further development.
0
7. Conclusion
The methods of complex geometrical optics have been refined and extended significantly over the last thxty years, and they represent useful tools in modem wave theory. Many of the standard asymptotic methods in current use were developed on a heuristic basis, and complex rays have joined t h s collection. They extend the reach of these methods to cover wave processes associated with tunneling, caustic shadows, media with loss or gain, and inhomogeneous (evanescent) waves. As a whole, these methods may be referred to as the “heuristic theory of diffraction” (HTD) which is a unification of a variety of asymptotic methods aimed at sewing the wavefield onto its associated ray skeleton. The development of many of the elements of HTD is based on analysis of the exact solutions of diffraction theory for canonical problems, and it involves generalizations to obtain approximate, ray-based solutions for problems that cannot be solved rigorously. HTD includes the geometrical theory of diffraction (GTD) by Keller [1958, 19621 and Keller and Levy [1963], and the closely related physical theory of diffraction (PTD) by Ufimtsev [1971, 19911. Both theories use the exact and asymptotic results of the wave theory (e.g., plane wave diffraction by straight edges, wedges, spheres or circular cylinders) and heuristically extend the results to objects of more general shapes (James [1980], Bouche, Molinet and Mittra [ 19971). The “uniform theory of diffraction” (UTD), by Kouyoumjian and Pathak [1974] and others, generalizes GTD to make it uniformly applicable at the boundary of lit and shadowed regions. HTD also embraces the uniform methods for the description of fields at caustics: the method of etalon functions (Kravtsov [1964a], Ludwig [1966], Kravtsov and Orlov [1983, 1998]), Maslov’s approach (Maslov [ 1972]), and Orlov’s method of interference integrals (Avdeev, Demin, Kravtsov, Tinin and Yarygin [1988], Kravtsov and Orlov [1998]).
I1
REFERENCES
53
As emphasized by many authors, notably by Einziger and Felsen [1982], Felsen [1976a, 1984, 1985a,b], and Zernov and Lundborg [1996], the complex method of geometrical optics has become a recognized member of this extended family. In the recent book by Bouche, Molinet and Mittra [1997], complex geometrical optics was outlined along with other asymptotic methods of electromagnetics. This review cannot cover all optical problems where complex geometrical optics has been useful. In particular, we did not touch on the nonlinear optical problems examined by, for example, Nasalski [1994, 1995, 1996a,b], nor did we mention the complex analysis of phenomena in optical fibers (Vassallo [1985], Hashimoto [1992]) and in laser diode modules (Kudou, Yokota and Fukumitsu [1990]). We have also not included the treatment of light scattering in randomly inhomogeneous media (Kopilevich and Frolov [ 1980, 19821). However, we have attempted to illustrate that complex rays represent a useful tool for the asymptotic analysis of wavefields. As Connor [1980] emphasized, “at the expense of adding at least one additional dimension to any problem, and making even more tenuous the connection between intuition and the technique of solution, the region of applicability of geometrical optics techniques can be extended to a much greater variety of problems”.
Acknowledgments
The authors are grateful to Dr. AS. Kryukovskii for valuable advice, and to Professors W? Shevshchenko and D.M. Sazonov, cochairmen of the Moscow Seminar on Wave Physics, for an opportunity to present this material at the seminar. We also thank the Australian Research Council and Macquarie University, Sydney, Australia, for their support for ow collaboration. We are grateful to 0. Zvereva and N. Komarova for their help in preparing the manuscript.
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E. WOLF, PROGRESS IN OPTICS XXXIX 0 1999 ELSEVIER SCIENCE l3.V ALL RIGHTS RESERVED
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION BY
DIRK-GUNNAR WELSCH Friedrich-Schiller-Universitiith n a , Theoretisch-PhysikalischesInstitut, Max-Ken Platz I , 07743 Jena, Germany
WERNERVOGEL Universitat Rostock, Fachbereich Physik, Arbeitsgruppe Quantenoptik, Universitatsplatz 3, 1 805 1 Rostock, Germany
AND
TOMAS OPATR& Palacklj University, Faculty of Natural Sciences, Svobody 26, 77146 Olomouc, Czech Republic
63
CONTENTS
PAGE
Q 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .
65
Q 2 . PHASE-SENSITIVE MEASUREMENTS OF LIGHT . . . . .
69
Q 3 . QUANTUM-STATE RECONSTRUCTION . . . . . . . . .
100
Q 4 . QUANTUM STATES OF MATTER SYSTEMS . . . . . . .
157
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . .
187
Appendu A . RADIATION FIELD QUANTIZATION . . . . . . .
187
Appendix B. QUANTUM-STATE REPRESENTATIONS . . . . . .
189
Appendu C. PHOTODETECTION . . . . . . . . . . . . . . .
195
. . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
197
Appendix D. ELEMENTS OF LEAST-SQUARES INVERSION
64
200
0 1.
Introduction
Since the experimental demonstration of tomographic quantum-state measurement on optical fields by Smithey, Beck, Raymer and Faridani [1993] based on the theoretical work of Vogel and Risken [1989], a boom in studying quantum-state reconstruction problems has been observed. Numerous workers have considered various schemes and methods for extracting all knowable information on the state of a quantum system from measurable data, and early work has been recovered. In the history of quantum theory, the concept of quantum state - including quantum-state measurement - has been a matter of intense discussion. The state of a quantum object is commonly described by a normalized Hilbertspace vector 1 Y ) or, more generally, by a density operator @ = C,pi( Y,)(Ti/ which is a Hermitian and non-negative valued Hilbert-space operator of unittrace. The Hilbert space of the object is usually spanned up by an orthonormalized set of basic vectors ( A ) representing the eigenvectors (eigenstates) of Hermitian operators 2 associated with a complete set of simultaneously measurable observables (physical quantities) of the object I . The eigenvalues A of these operators are the values of the observables which can be registered in a measurement. Here it must be distinguished between an individual (single) and an ensemble measurement (i.e., in principle, an infimtely large number of repeated measurements on identically prepared objects). Performing a single measurement on the object, a totally unpredictable value A is observed in general, and the state of the object has collapsed to the state ( A ) according to the von Neumann’s projection definition of a measurement (von Neumann [1932]). If the same measurement is repeated immediately afier the first measurement (on the same object), the result is now well predictable - the same value A as in the first measurement is observed. Obviously, owing to the first measurement the object has been prepared in the state ] A ) .Repeating the measurement many times on an identically prepared object, the relative rate at which the result A is
’
For notational convenience we write k,without further specifymg the quantities belonging to the set. 55
66
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
“1, § 1
observed approaches the diagonal densityLmatrix element ( Al@lA)as the number of measurements tend to infinity. Measuring {AI@IA) for all values of A , the statistics of d (and any function of d) are known. To completely describe the quantum state, i.e., to determine all the quantumstatistical properties of the object, knowledge of all density-matrix elements ( AI@IA‘) is needed. In particular, the off-diagonal elements essentially determine the statistics of such sets of observables k that are not compatible with 2 ([d,b] # 0) and cannot be measured simultaneously with d. Obviously, the statistics of 3 can also be obtained directly - similar to the statistics of 2 - from an ensemble measurement yielding the diagonal density-matrix elements (BI @IB) in the basis of the eigenvectors IB) of B . Now one can proceed to consider other sets of observables which are not compatible with 2 and h, and the rather old question arises of which and how many incompatible observables must be measured in order to obtain all the information contained in the density matrix of an arbitrary quantum state. Pauli [1933] had already addressed the question of whether or not the wave function of a particle can be reconstructed from the position and momentum probability distributions; i.e., the diagonal densitymatrix elements in the position and momentum basis. Roughly speaking, there have been two routes to collect measurable data for reconstructing the quantum state of an object under consideration. In the first, whch follows closely the line given above, a succession of (ensemble) measurements is made such that a set of noncommutative object observables is measured which carries the complete information about the quantum state. A typical example is the tomographic quantum-state reconstruction mentioned earlier. Here, the probability distributions p(x, cp) = (x, cpI@Ix,cp) of the quadrature components i ( c p ) of a radiation-field mode for all phases cp withm a JG interval are measured; i.e., the expectation values of the quadraturecomponent projectors Ix, cp)(x, cpl. In the second, the object is coupled to a reference system (whose quantum state is well known) such that measurement of observables of the composite system corresponds to “simultaneous” measurement of noncommutative observables of the object. In this case the number of observables (of the composite system) that must be measured in a succession of (ensemble) measurements can be reduced drastically, but at the expense of the image sharpness of the object. As a result of the additional noise introduced by the reference system, only fuzzy measurements on the object can be performed, which just makes a “simultaneous” measurement of incompatible object observables feasible. A typical example is the Q function of a radiation-field mode, which is given by the diagonal density-matrix elements in the coherent-state basis, Q(a)= r1 (al@la). It can already be obtained from
11,
0 11
INTRODUCTION
67
one ensemble measurement of the complex amplitude a, which corresponds to a fuzzy measurement of the "joint" probability distribution of two canonically conjugated observables i ( q ) and .?(q+ m'2) of the object. Since the sets of quantities measured via the one or the other route (or an appropriate combination of them) carry the complete information on the quantum state of the object, they can be regarded, in a sense, as representations of the quantum state, which can be more or less close to (or far from) familiar quantum-state representations, such as the density matrix in the Fock basis or the Wigner function. In any case, the question arises of how to reconstruct from the measured data specific quantum-state representations (or speclfic quantumstatistical properties of the object for which a direct measurement scheme is not available). Again, there have been two typical concepts for solving the problem. In the first, equations which relate the measured quantities to the desired quantities are derived and tried to be solved either analytically or numerically in order to obtain the desired quantities in terms of the measured quantities. In practice, the measured data are often incomplete; i.e., not all quantities needed for a precise reconstruction are measured2, and moreover, the measured data are inaccurate. Obviously, any experiment can only run for a finite time, which prevents one, in principle, from performing an infinite number of repeated measurements in order to obtain precise expectation values. These inadequacies give rise to systematic and statistical errors of the reconstructed quantities, which can be quantified in terms of confidence intervals. In the second, statistical methods are used from the very beginning in order to obtain the best a posteriori estimation of the desired quantities on the basis of the available (i.e., incomplete andor inaccurate) data measured. However, the price to pay may be high. Whereas in the fist concept linear equations are typically to be handled and estimates of the desired quantities (including statistical errors) can often be sampled directly from the measured data, application of purely statistical methods, such as the principle of maximum entropy or Bayesian inference, require nonlinear equations to be considered, and reconstruction in real time is impossible in general. The aim of this review is to familiarize physicists with the recent progress in the field of quantum-state reconstruction and draw attention to important contributions in the large body of work. Although in much of what follows we consider the reconstruction of quantum states of travelling optical fields, the basic-theoretical concepts also apply to cavity fields and matter systems.
To compensate for incomplete data, some a priori knowledge of the quantum state is needed.
68
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[K
§ 1
Section 2 reviews the experimental schemes which have been considered for quantum-state measurement on optical fields, with special emphasis on optical homodyning (92.1). It begins by formulating the basic ideas of fourport homodyne detection of the quadrature components of single-mode fields ($92.1.1 and 2.1.2) and then proceeds to extend the scheme to multimode fields and multiport homodyne detectors (0 2.1.3), which can also be used for “simultaneous” measurement of noncommuting signal observables ($9 2.1.42.1.6). Section 2.1.7 returns to the four-port scheme in 0 2.1.1 and explains its use for measuring displaced Fock states. Homodyne correlation measurements are shown to yield insight in phase-sensitive field properties even in the case of low detection efficiencies (9 2.1.8). After addressing heterodyne detection (0 2.2) and parametric amplification (9 2.3), typical schemes for quantum-state measurement in high-Q cavity QED by test atoms are outlined (9 2.4). Section 3 reviews typical methods for reconstruction of quantum-state representations and specific quantum-statistical properties of optical fields from sets of measurable quantities carrying the complete information on the quantum state. In 00 3.1-3.5 we focus on the reconstruction of quantum-state representations from quantities measurable in homodyne detection of travelling optical fields. The next subsection (9 3.6) presents methods of reconstruction of quantum-state representations of high-Q cavity fields from measurable properties of test atoms probing the cavity fields, and in 3 3.7 alternative proposals are outlined. Section 3.8 then addresses the problem of direct reconstruction of specific quantities from the measured data - an important problem with respect to quantities for which direct measurement schemes have not been found so far. Whereas in 993.1-3.8 it is assumed that all the data needed for a precise reconstruction of the desired quantities can be measured precisely, at least in tendency, in 9 3.9 statistical methods for processing smeared and inaccurately measured incomplete data are outlined with the aim to obtain optimum estimations of the quantum states. Section 4 summarizes methods for measurement and reconstruction of quantum-state representations of matter system, with special emphasis on optical methods and experimental demonstrations. In Q 4.1 the problem of reconstruction of the vibrational quantum state in an excited electronic state of a two-atom molecule is addressed. Typical schemes for determining the motional quantum state of trapped atoms are outlined in 9 4.2, and in 9 4.3 the problem of quantumstate measurement on Bose-Einstein condensates is considered. Sections 4.4 and 4.5,respectively, present schemes suitable for determining the motional quantum state of atom- and electron-wave packets, respectively. In the debate on quantum-state measurement, which is nearly as old as quantum mechanics,
11, 5 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
69
spin systems have attracted much attention, because of low-dimensional Hilbert space. Section 4.6 outlines the main ideas in this field. Finally, 6 4.7 explains the basic ideas of Compton and X-ray scattering for measuring the single-particle quantum state of crystal lattices.
8 2.
Phase-Sensitive Measurements of Light
The statistical properties of a classical radiation field are known when the amplitude statistics and the phase statistics are known. Whereas the amplitude statistics can be obtained from intensity measurements, determination of the phase statistics needs phase-sensitive measurement. Obviously, the same is true in quantum optics. In order to obtain the complete information about the quantum state a radiation field is prepared in, interferometric measurements which respond to amplitude and phase variables must be performed. 2.1. OPTICAL HOMODYNING
Homodyne detection has been a powerful method for measuring phase sensitive properties of travelling optical fields which are suitable for quantum-state reconstruction, and a number of sophisticated detection schemes have been studied. In the four-port basic scheme, a signal field is combined, through a lossless beam splitter, with a highly stable reference field that has the same midfrequency as the signal field. The reference field, also called the local oscillator, is usually prepared in a coherent state of large photon number. The superimposed fields impinge on photodetectors, the numbers of the emitted and (electronically processed) photoelectrons being the homodyne detection output (for the basic ideas, see Yuen and Shapiro [1978a,b, 19801, Shapiro and Yuen [1979], Shapiro, Yuen and Machado Mata [1979]). The observed interference fringes, which vary with the phase difference between the two fields, reflect the quantum statistics of the signal field and can be used - under certain circumstances - to obtain the quantum state of the signal field. The homodyne output can be given hlly in terms of the joint-event probability distribution of the detectors in the output channels. In balanced homodyning, difference-event distributions are measured. In particular, the difference-event statistics measurable by a perfect four-port homodyne detector directly yields the quadrature-component statistics of the signal field, which has offered novel possibilities of quantum-state measurement.
70
HOMODYNE DETECTION A N D QUANTUM-STATE RECONSTRUCTION
I
5 dt
- ADC - n2
Jdl
-mc-nn,
tK § 2
1. A signal pulse field E s interferes with a shorter local oscillator pulse EL at a 50/50 beam splitter. The local oscillator phase 9 determines which quadrature amplitude of the signal is detected. The superposed fields are detected with high-efficiency photodiodes having response times much longer than the pulse durations. The photocurrents are integrated and sampled by analog-todigital converters, to yield pulse photoelectron numbers nl and n2. (After Raymer, Cooper, Cannichael, Beck and Smithey [1995].)
Let us start the analysis with the four-port scheme and first restrict our attention to single-mode fields mode-matched to the local oscillator (fig. 1).3 The action of a lossless beam splitter (as an example of a linear four-port coupler) on quantum fields has been studied widely (see, e.g., Richter, Brunner and Paul [1964], Brunner, Paul and Richter [1965], Paul, Brunner and Richter [1966], Yurke, McCall and Klauder [1986], Prasad, Scully and Martienssen [1987], Ou, Hong and Mandel[1987a], Fearn and Loudon [1987], Campos, Saleh and Teich [1989], Leonhardt 119931, Vogel and Welsch [1994], Luis and Sanchez-Soto [1995]), and it is well known that the input-output relations can be characterized by the SU(2) Lie algebra. In the Heisenberg picture, the photon destruction operators of the outgoing modes, 2; ( k = I, 2), can be obtained from those of the incoming modes, i k , by a unitary transformation, 2
k'
=
I
where U11 and U2l are the transmittance and reflectance of the beam splitter from one side, and U22 and U12 are those from the other side. The unitarity of Here and in the following, spatial-temporal modes are considered. They are nomonochromatic in general and defined with a particular spatial, temporal and spectral form (App. A).
11, § 21
71
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
the scattering matrix placed on it:
I
ukkl = ukk’
le’pkk’ implies that the following restrictions are
The relations (2) and (3) ensure that the bosonic commutation relations for the photon creation and destruction operators are preserved. Using the angular momentum representation, the unitary transformation (1) can be given by
6:
=
Pt,
where the operator = e-iaL3e-i/&
(4)
P reads as4 e-iyi3,-ic~?
(5)
Here, fi = k~ + 2 2 is the operator of the total photon number (kk the operators
e, ;(ata, + a:a,) , i, =
=
& (a$, a:;,) -
= iE$k),
and
, e 3 = ;(e$3, - a$,) ,
(6) obey the familiar commutation relations of angular-momentum operators. In the Schrodinger picture the photonic operators are left unchanged, but the state is transformed. Let @ and @ be the input- and output-state density operator of the two modes, respectively. It is easily seen that if they are related to each other as
then the two pictures lead to identical expectation values of arbitrary field quantities. Let us now assume that the operation of the photodetectors in the two output channels of the beam splitter can be described by means of the standard photodetection theory as outlined in Appendix C; i.e., the numbers of photoelectric events counted in a given time interval (integrated photocurrents) are proportional to the numbers of photons falling on the detectors. From The parameters in eq. (5) are related to those in eqs. (2), (3) as a = ;(PI 1 - ~2 + cpzl It n), Y= - p??2 - @ I + cp12 F n),cos2(p/2) = I U I I12, sin2(p/2) = 1 ~ 212, 1 6 = l ( r p i l + (~22).
72
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI,
§2
eq. (C.8) the joint-event distribution Qf measuring ml and m2 events in the two output channels 1 and 2, respectively, is then given (in the Heisenberg picture) by
where
with ir; being given by eq. (1) [or eq. (4)]. Note that for q = 1 (i.e., perfect detection), P,, ,,, is nothing more than the joint-photon-number distribution of the two outgoing modes. In particular, when the reference mode is prepared in a coherent state la^), then in the expectation values of normally ordered operator functions, irk can be replaced with (ir = &I);
6;
=
ukl 6 + uk2 aL .
(10)
2.I .2. Quadrature-component statistics Let us now turn to the question of what information on the quantum statistics of the signal mode can be obtained from the homodyne output. From eq. (9), together with eqs. (10) and (3), we easily find that the output photon number 2L can be rewritten as
being the quadrature-component operator, and
We see that 2; consists of three terms. The first and the second terms represent the photon numbers of the local oscillator and the signal, respectively. More
K 0 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
73
interestingly, the third term that arises from the interference of the two modes is proportional to a quadrature component of the signal mode, i(q),the rapidly varying optical phase mt that uslally occurs in a field strength (quadrature component) is replaced with the phase parameter cp. In particular, when the signal and the local oscillator come from the same source, then the phase parameter cp can be controlled easily, so that the dependence on cp of i ( v )can be monitored; e.g., by shifting the phase difference between the signal and the local oscillator in the input ports of the beam splitter in fig. 1. If a balanced (i.e., 50%:50%) beam splitter is used, the relation IU,2]= lUkl I = 1/& is valid, and hence the difference photon number is a signal-mode field strength P(cp), i?;
-
i?;
=
P(cp);
i.e., it is proportional to a signal-mode quadrature component. T h s result suggests that it is advantageous to use a balanced scheme and to measure the difference events or the corresponding difference of the photocurrents of the detectors in the two output channels in order to eliminate the intensities of the two input fields from the measured output. The method is also called balanced homodyning and can be used advantageously in order to suppress perturbing effects due to classical excess noise of the local oscillator (Mandel [1982], Yuen and Chan [1983], Abbas, Chan and Yee [1983], Schumaker [1984], Shapiro [1985], Yurke [1985], Loudon [1986], Collett, Loudon and Gardiner [19871, Yurke and Stoler [1987], Yurke, Grangier, Slusher and Potasek [1987], Drummond [ 19891, Blow, Loudon, Phoenix and Shepherd [ 19901, Huttner, Baumberg, Ryan and Barnett [ 19921). In particular, when the local oscillator is much stronger than the signal field, then even small intensity fluctuations of the local oscillator may significantly disturb the signal-mode quadrature components that are desired to be observed. For example, if the quadrature-component variance is intended to be derived from the variance of events measured by a single detector, the classical noise of the local oscillator and the quantum noise of the signal would contribute to the measured data in the same manner, so that the two effects are hardly distinguishable. Since identical classical-noise effects are observed in the two output channels, in the balanced scheme they eventually cancel in the measured signal due to the subtraction procedure. Strictly speaking, from the arguments given above it is only established that for chosen phase parameters the mean value of the measured difference events or photocurrents is proportional to the expectation value of a quadrature component of the signal mode. From a more careful (quantum-mechanical)analysis it can be
74
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
0.6
0.4
0.2
0
-10
0
lo
-10
An
0
10
Fig. 2. Difference statistics PA^ for a single-photon state, quantum efficiency 9 = 0.75, and various photon numbers of the local oscillator: laLl2 = (a) 0, (b) 0.5, (c) 5, (d) 10. The solid curves represent the smeared quadrature-component distributionp(x,rp; a), eq. (19), which is a convolution ofp(x, rp) with p(x; q). (After Vogel and Grabow [1993].)
shown that in perfect balanced homodyning the quadrature-component statistics of the signal mode are indeed measured, provided that the local oscillator is ~ ~ compared with the mean number of signal sufficiently strong; i.e., ~ C L isL large photons (Carmichael [19871, Braunstein [19901, Vogel and Grabow [19931, Vogel and Welsch [1994], Raymer, Cooper, Carmichael, Beck and Smithey [1995], Munroe, Boggavarapu, Anderson, Leonhardt and Raymer [1996]). The probability distribution of the difference events,
can be derived from the joint-event distribution (8) as
Examples of the dependence of the difference-event distribution PA,,, on the local-oscillator strength and the detection efficiency are shown in fig. 2. In the limit of the local oscillator being sufficiently strong, so that / a ~ is1 large ~ compared with the average number of photons in the signal mode, the difference-
11, § 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
75
event distribution PA,,, can be given by (assuming that the two detectors have the same quantum efficiency r ] ) (18)
with the phase q being determined by the phase parameters of the apparatus and of the local oscillator according to eq. (14). In eq. (18), p(x, q;r ] ) is a convolution of the quadrature-component distribution of the signal mode, p(x, q)= ( x , q l p l x , q) [cf. Appendix B], with a Gaussian5:
p(x; r ] )
=
, - r> / exp (-&) I 1-r]
In particular, for perfect detection, r] = 1, the measured difference-event dlstribution PA,,, is identical with the dlfference-photon-number distribution. In this case p(x; r ] ) reduces to a 6 function; i.e., p(x, q ;r] = 1) = p(x, q),and hence Phm is (apart from a scaling factor) exactly a quadrature-component distribution p ( x , q ) of the signal mode. Note that when the local oscillator is sufficiently strong, then Arn/(filaLI)in eq. (18) is effectively continuous. In other words, single-photon resolution is not needed in order to measure the (continuous) quadrature-component statistics with high accuracy. In thls case, highly efficient linear response photodiodes can be used, which do not discriminate between single photons, but nearly reach 100% quantum efficiency (Polzik, Carri and Kimble [ 19921). The Gaussian p(x; r ] ) [with dispersion u2 = (1 - r])/(2r])] obviously reflects the noise associated with nonperfect detection ( r ] < 1). In particular, for 50% quantum efficiency (i.e., q = 1/2), the functionp(x; r ] ) is the (phase-independent) quadrature-component distribution of the vacuum (a2 = 1/2). Obviously, the measured quadrature-component distribution p(x, q;r ] ) does not correspond to the quadrature component 2 ( q ) of the signal mode, but it corresponds to the
This result is obtained directly (in the limit of a strong local oscillator) from the basic equations (8, 17) of photocounting theory (Vogel and Grabow [1993]), which include the effects of imperfect detection.
76
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI, § 2
quadrature component i(q;q) of a superposition of the quadrature component of the signal, i ( q ) ,and that of an additional (Gaussian) noise source, IzN(cp),
so that f ( q ;q) is effectively a combined two-mode quadrature component. In particular, eq. (21) reveals that the effect of nonperfect detection can be modelled, assuming a (virtual) beam splitter is placed in front of a perfect detector, since eq. (2 1) corresponds exactly to a beam-splitter transformation (0 2.1.1) (Yurke and Stoler [1987], Leonhardt and Paul [1993b, 1994a,b]). In this case, the fields are only partly detected, together with some fraction of vacuum noise introduced through the “unused” input ports of the beam splitters6 . If the signal field is not perfectly mode-matched to the local oscillator mode, the non-mode-matched part can give rise to additional noise in the measured quadrature-component distributions of the mode-matched signal (Raymer, Cooper, Carmichael, Beck and Smithey [1995]). It can be shown that when the local oscillator is strong enough to dominate the mode-matched signal field but not necessarily the nonmatched signal, then the detection efficiency reduces to
where R i - i s the mean number of photons in the nonmatched signal; i.e., 0 is replaced with fi in eq. (18). As expected, fi approaches 0 only in the limit laLI2 >> Z B .
In terms of the characteristic functions of the distributions involved, the content of eqs. (18H20) can be given by
Here,
is the characteristic h c t i o n of the measured probability distribution of the scaled difference events Am/(qlaLI), and Y ( z , q;q) and @( p; s), are respectively, the characteristic functions, i.e., the Fourier transforms, of the (owing to For a discussion of eqs. (19) and (20) in terms of moments of the quadrature components.?(q;q) and 2(cp>,see Banaszek and Wbdkiewicz [1997a].
11,
0 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
77
nonperfect detection) smeared quadrature-component distribution p(x, q ;v) and the s-parametrized phase-space function P(a ;s) [see eqs. (B.26)7 and (B.23)]. Equation (23) reveals that in the case of perfect detection the characteristic function of the (scaled) difference-event distribution is equal, along a line in the complex plane, to the characteristic function of the Wigner function, which can be regarded as a suitable representation of the quantum state of the signal mode. Hence, when the difference-event statistics are measured on a sufficiently dense grid of phases q within a n interval, then all knowable information on the signal-mode quantum state can be obtained. The result is obviously a consequence of the fact that knowledge of the quadrature-componentdistribution for all phases within a n interval is equivalent to knowledge of the quantum state; see Appendix B.5 (Vogel and Risken [1989]). 2.1.3. Multimode detection Relations of the type given in eq. (23) are also valid for (hgher than four-port) multiport homodyning (Walker [1987], Kiihn, Vogel and Welsch [1995], Ou and Kimble [1995]). Let us consider a linear, lossless 2(N + 1)-port coupler (fig. 3) and assume, e.g., that N spatially separated signal modes (channels 1,. . . ,N ) and a strong local-oscillator mode (channel N + 1) of the same frequency are mixed to.obtain (N + 1) output modes impinging on photodetectors8. The input and output photon operators, & and i?;, respectively, are then related to each other by a SU(N + 1) transformation, extendmg eq. (1) to
k'
=
I
(k = 1,. . . ,N + 1). Note that any discrete finite-dimensional unitary matrix can be constructed in the laboratory using devices, such as beam splitters, phase shifters and mirrors (Reck, Zeilinger, Bernstein and Bertani [1994], for further readings, see Jex, Stenholm and Zeilinger [ 19951, Mattle, Michler, Weinfurter, Zeilinger and Zukowski [1995], Stenholm [1995], Torma, Stenholm and Jex [ 19951, Torma, Jex and Stenholm [ 19961). Simultaneous detection of the output
'
The characteristic function Y(z,cp; Q) of p(x, cp; Q) is defined according to eq. (B.26) with p(x, cp; Q) in place of p(x, cp). It is seen to be related to the characteristic function Y ( z , rp) of ~ ( x rp) , as Y(Z, rp; a) = exp[-$(v-' - 1)z21~ ( z rp). , All the modes are assumed to be matched to each other.
78
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
..
...
Fig. 3. Scheme of homodyne 2(N + 1)-port detection. N signal modes i k ( k = 1,. . . ,N ) of frequency w and a strong local oscillator &L of the same frequency are combined by a lossless 2(N + 1)-port device to give N + 1 output modes (Ukk,: unitary transformation matrix of the device). Simultaneous detection of the N + 1 output modes yields the ( N + 1)-fold joint count distribution P,, ,,,,,* N . (After Vogel and Welsch [1995].)
modes yields an (N + 1)-fold joint event distribution P,,, ,,.,,m N , . With regard to difference-event measurements, let us consider the scaled difference events, +
-
-
Here, I = 1,. . . ,k - 1, k + 1,. . . , N + 1, the reference channel being denoted by I;. It can be shown that when the photon number I GILl2 of the coherent-state local oscillator is much greater than unity and dominates all the signal modes at the detectors then the characteristic function of the joint probability distribution
',
Apart from this condition, the analysis also allows the local oscillator to be in other than coherent states (Walker [1987], Leonhardt [1995], Kim and Sanders [1996], for measuring the degree of squeezing in a signal field by using squeezed local oscillators, see p m and Kumar [1994]).*This can be done by writing the local-oscillator state & in the form @L = D(ar)0; Dt(aL), where D(aL) is the coherent displacement operator and 6; is a state that remains finite as the complex amplitude IaLI becomes large.
11, § 21
79
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
of the scaled difference events, Q*({ y,)), can be related to the characteristic function of the s-parametrized multimode phase-space function of the signal field, @({&);s), n = 1,..., N , as”
where equal detection efficiencies have been assumed ( q k N+ I
N+ I
k= I
k=l
=
Qk’= r)), and
(Kuhn, Vogel and Welsch [1995]). Note that when N = 1 (four-port apparatus) and Iuk21 = lukll = 1 / a , then eq. (27) [together with eq. (28)] reduces to Eq. (23). It should be pointed out that joint measurements on combinations of multiports of the type described above can also be used in order to detect (groups of) correlated modes of different frequencies. In particular, the combination of two balanced four-port schemes can be used to measure the joint quadraturecomponent distribution p(xl,x2, qI,q2) of a (correlated) two-mode signal field for all values of the two phases ql and ql within n intervals in order to determine the two-mode quantum state (Raymer, Smithey, Beck, Anderson and McAlister [ 19931). From eqs. (27) and (28) it can be seen that for perfect detection the characteristic function of the N-fold joint scaled difference-event distribution is nothing other than the characteristic function of the Wigner function of the N-mode input radiation field at certain values of the N complex arguments Pn, and it corresponds to the characteristic function of a joint quadrature-component distribution. To obtain all knowable information on the N-mode quantum state, i.e., the joint quadrature-component distributions for all relevant phases or the complete Wigner function, the arguments /3, must be allowed to attain arbitrary complex values. This may be achieved by means of an appropriate succession of (ensemble) measurements or one (ensemble) measurement including in the apparatus appropriately chosen reference modes whose quantum states are known (or a combination of the two methods) (Vogel and Welsch [1995]). In
lo Note that @({g};s) = exp[C, sl&12](b({&})),where b({&}) is the N-mode coherent displacement operator, and the Fourier transform of @({&}; s) is the s-parametrized N-mode phasespace function.
80
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
111, 8 2
the first case, phase shifters in the apparatus may be used to appropriately vary (similar to the four-port scheme) the scattering matrix Ukk’from measurement to measurement. In the second case, the scattering matrix is left unchanged, but additional reference inputs are used, so that a joint quadrature-component distribution of signal and reference modes is effectively measured (see § 2.1.4). Instead of measuring the N-fold joint difference-event statistics in a 2(N + 1)port scheme, the information on the N-mode signal field can also be obtained from the difference-event statistics measured in a standard four-port scheme with controlled signal-mode superposition (Opatrn9, Welsch and Vogel [ 1996, 1997a,b], Raymer, McAlister and Leonhardt [19961, McAlister and Raymer [ 1997a,b]). In the scheme, which applies to both spatially separable and nonseparable modes, the signal input is formed as a superposition of all the signal modes. Controlling the expansion coefficients in the superposition, from the measured difference-eventdistributions the same information on the quantum statistics of the N-mode signal-field can be obtained as from the N-fold joint difference-event distribution measured in a 2(N + 1)-port scheme. If the signal modes are separated spatially, they can be superimposed using an optical multiport interferometer such that each of the signal modes is fed into a separate input channel of the interferometer and the superimposed mode in one of the output channels is used as the (signal) input of a four-port homodyne detector. If the signal-field is a pulse-like mode, the signal pulse can be combined, through the beam splitter in the four-port scheme, with a sequence of short local oscillator pulses in order to analyse the signal pulse in terms of the modes associated with the local-oscillator pulses. In this case, the superposition of the component modes to be measured is achieved by a proper formation of the local oscillator pulses. To illustrate the method, let us consider the simplest case when a (correlated) two-mode field is perfectly detected (fig. 4). Then the measured quantitity is a weighted sum of the quadrature components i1 (ql)and &(cpz) of the two modes, 2 = i l ( r p l ) cos a +i*(cpz) sin a,
(29)
where the phases rpl and cpz and the relative weight [superposition parameter a, a E (O,n/2)] are assumed to be controlled in the measurement. For spatially separated modes (fig. 4a), the phases can be controlled by phase shifters placed in front of a beam splitter used for mixing the modes, and the parameter a can be controlled by the transmittance (reflectance) of the beam splitter. If the quadrature statistics of a signal pulse are measured in a set of short local-oscillator pulses (fig. 4b), then the parameter a can be controlled by
11,
81
0 21
61
b
i z
Fig. 4. Two possible schemes for reconstructing two-mode density matrices via measurement of combined quadrature components. (a)*Two modes (61 and 6 2 ) are mixed by a beam splitter and one of the interfering output modes (b) is used as signal mode in balanced homodyning in order to measure the sum quadratures of the two modes (LO is the strong local oscillator). (b) A signal pulse and a sequence of two short (strong) local-oscillator pulses with envelopes F l ( t ) and Fz(t) are superimposed in balanced homodyne detection. (After Opatml, Welsch and Vogel [1997b].)
changing the mutual intensities of the two pulses. Again, the phases can be controlled interferometrically’ I . Let p ~ ( x, lx 2 , q 1 , @ ) and ps(x, a, VI, qb) be, respectively, the joint quadrature-componentdistribution and the (measured) sum quadrature-component distribution of the two modes, and YJ(ZI, z2, q ~@) , and Ys(z, a, cp~,@) the corresponding characteristic functions. It can be shown that
In other words, the joint-quadrature component distribution for all phase values (within ?t intervals) can be obtained from the sum quadrature component distribution for all phase values and all values of the weighting factor, which shows the possibility to obtain the two-mode quantum state from the measured statistics of the summed quadratures. The effect of nonperfect detection can be taken into account according to eq. (23).
2.1.4. Q function As mentioned in 52.1.3, the 2N-fold manifold of data necessary for a reconstruction of the quantum state of an N-mode signal field can be obtained by including additional reference modes in the detection scheme in fig. 3 whose quantum states are known, such as the vacuum inputs in “unused” input channels.
’
If signal and local oscillator come from different sources, only the phase difference 6rp = rpl - QZ! can be controlled.
82
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
111, 8 2
Let us suppose that the 2(N + 1)-port apparatus is extended to a 2(2N + 1)port device, N input channels being "unused". Since the vacuum inputs are not correlated to each other nor to the signal input, the characteristic function of the quantum state of the (2N)-mode input field can be factorized: @ ( { P n } ; s) =
@({Pi}; S) @({pi); J=N+ I
1,. . . , N , signal channels; j = N + 1,. . . , 2 N , vacuum channels). Equation (28) (with 2N in place of N) can then be inverted easily in order to obtain 2N real arguments yr of the characteristic function QA({ yr}) of the measured distribution for any (freely chosen) N complex arguments of the characteristic function @({/3,}; s)' of the N-mode signal-quantum state. In other words, in the extended detection scheme there is a one-to-one correspondence between the measured (2N)-fold joint difference-event distribution and the quantum state of the N-mode signal field (Vogel and Welsch [ 19951). Obviously, one local oscillator per group of (equal-frequency) signal modes and one additional reference input per signal mode are sufficient for measuring a joint difference-eventdistribution that can be regarded as a measure of the signalmodequantum state. Needless to say, both the number of local oscillators and/or the number of additional reference inputs may be increased, and the quantum state of the signal may of course be expressed in terms of an appropriately chosen joint difference-event distribution recorded in such an extended measurement scheme. In the case of a single-mode signal field, a six-port scheme is the minimum in order to determine the quantum state of the signal (Zucchetti, Vogel and Welsch [ 19961, Paris, Chizhov and Steuernagel[1997]). The situation is quite similar to classical optics, in which the six-port scheme is the minimum in order to determine the complex amplitude (Walker [1987]). Let us consider a 3 x 3 coupler and assume that the reference mode is in the vacuum state (fig. 5). In this case, it can be shown that application of eqs. (27), (28) and (31) yields: (i
=
@ ( P ; s ) = exp(+slP12) @(PI = exp{4IBI2 [ s + ~ - r l ) ~ - l ] } QAbl,Y2),
where P is relatated to y1 and y2 as j? = ii[(@*
- 1)yl + (4 - l)y2] e"cL,
(32) (33)
= exp(-i2x/3) [@(B) 3 @(P;O)]. Equation (33) reveals that 1 - 2q-I) = Q~(yl,y2); i.e., measurement of the joint differenceevent distribution PA,,],,L\,,,~ in balanced six-port homodyning is equivalent to
with @
@(@;s =
K 0 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
PD3
83
i
unused oort
Fig. 5 . Scheme of balanced six-port homodyning for the detection of the Q function. Three input fields (signal, local oscillator, vacuum) are combined by three symmetric beam splitters BSi (i = 1,2,3) and a (-n/2) phase shifter, where two (1/2 : 1/2) beam splitters (BSI and BS3) and one (2/3 : 1/3) beam splitter (BS2) are used. The joint difference statistics is recorded by the detectors > eq. (34)]. (After Zucchetti, PDi in the output channels [psdc(D13,023) corresponds to F ' A ~ , , A ~in Vogel and Welsch [1996].)
measurement of a phase-space function P(a;s) of the signal mode (in nonorthogonal coordinates in the phase space):
with
and s = 1 - 2q-I [@ = exp(-in/3)]. The function P(a;s = 1 - 2q-I) can be regarded as a smoothed Q function of the signal mode, which approaches the Q function as the quantum efficiency rj goes to unity. In particular, for perfect detection ( q = 1) the Q function of the signal mode is measured (in non-orthogonal phase-space coordinates).
84
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRLICTION
PI,
52
Identifying in eq. (B.25) s' with 1 - 2q-', this equation can be regarded as a prescription for obtaining other phase-space functions P ( a ; s ) from the measured one. When s < 1 - 2q-I is valid, then the /3 integration in eq. (B.25) can be performed separately to obtain P(a;s) as a convolution of P ( a ;s = 1 - 2 ~ ~with ' ) a Gaussian, which reveals that all the phase-space functions whch are typically broader than the measured one, can be obtained simply by convolving the measured distribution with a Gaussian. In the opposite case, when s > 1 - 2q-I, then the integration over y must be done fist in eq. (B.25). However, experimental inaccuracies may be exploding via the inverse Gaussian and prevent a stable reconstruction of P(a;s) with reasonable precision. Since the maximum value of s for an always stable reconstruction, i.e., s = 1 - 2q-', tends to minus unity as q goes to unity, the upper boundary of s corresponds to the Q function. As mentioned above, higher than six-port homodyne detectors can also be used in order to measure the signal-mode quantum state in the phase space. Among them, eight-port homodyne detection has been studied widely (Walker and Carroll [1984], Walker [1987], Lai and Haus [1989], Noh, Fougkes and Mandel [1991, 1992a,b], Hradil 119921, Freyberger and Schleich [1993], Freyberger, Vogel and Schleich [1993a,b], Leonhardt and Paul [1993a], Luis and Peiina [1996a]). Let us consider two modes which are superimposed, in accordance with eq. (l), by a 50%:50% beam splitter, and assume that one of the incoming modes, say, the second is in the vacuum. It can then be shown that the joint probability distribution of the two outgoing quadrature components .?i(q)and .?i(q+ n/2) is the (scaled) Q function of the !%st incoming mode 1 2 ,
(Lai and Haus [1989], Leonhardt and Paul [1993a]; for details, see also Leonhardt and Paul [ 19951, Leonhardt [1997c]). Hence, using the two outgoing modes of the beam splitter as incoming signal modes in two separate balanced four-port homodyne detectors and measuring (for a n/2 phase difference) the joint difference-event statistics of the two homodyne detectors, the Q function of the original signal mode is obtained, provided that perfect detection is
Equation (36) can be proved correct, applying the beam splitter transformation (1) and expressing the characteristic function of the two-mode (outgoing) quadrature-component distribution p(x{ ,x;, rp, rp + n/2) in terms of the characteristicfunction of the single-mode (incoming) Q function Qc.1).
85
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
2’
I
rL
fi
’ 0
2-
5
D5
CI
I
3
\
L
\
3 #
X/4
I
1
D3
1 4
Fig. 6. Balanced homodyne eight-port detection scheme for measuring the Q function, using 50%:50% beam splitters and a M4 phase shifter. The signal is fed into port 1 and the strong local oscillator is fed into port 2, and vacuum is fed into the ports 1’ and 2’. The joint differenceis measured in the channels 3 and 4 (Aml) and 5 evenrprobability distribution P L \ ~ , , L \eq. ~ ?(37), , and 6 (Am2). (After Vogel and Welsch [1994].)
accomplished 13. Altogether the setup forms a balanced eight-port homodyne detector, with a signal input, a vacuum input and two local oscillator inputs. Alternatively, the beam splitter and the two four-port homodyne detectors can be combined into an eight-port apparatus with two vacuum inputs and one local oscillator input (fig. 6 ) . A straightforward calculation shows (similar to
l 3 Note that ?(q) and L(rp + d 2 ) play the same role as position and momentum of a harmonic oscillator in quantum mechanics, because of [.?(rp),i(q+ n/2)] = i. Therefore, the notations 4 = i ( q ) and j E 2(rp + n/2) are also frequently used. Equation (36) is an example of (rp) “simultaneous” measurement of a pair of conjugate quantities. Actually, the quantities 4; G i’, and 3 Li(rp + n/2) are measured, which can be regarded as the conjugate quantities il = 21 (rp) and $1 = 21(rp + n/2), respectively, “smoothed” by the introduction of additional (vacuum) noise necessary for a simultaneous (but approximate)measurement of 41 andjl . Accordingly, an additional uncertainty is introduced in the measurement, and it can be shown that the uncertainty product for 4; and?; is twice that of the measurements of 41 a n d j l made individually, Aq’,Api 2 1 (Arthurs and Kelly [1965],for a review, see Stenholm [1992]).
a;
86
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
[II, 8 2
the six-port homodyne detector) that measurement of the joint difference-event distribution PA,~,A,,,~ again yields the phase-space function P (a ;s = 1 - 277-I),
(Freyberger and Schleich [19931, Freyberger, Vogel and Schleich [ 1993a,b], Leonhardt and Paul [1993b], Vogel and Welsch [1994], D'Ariano, Macchiavello and Paris [1995], Kochahski and Wodkiewicz [1997]). Compared with the six-port scheme, PA,,,,,A~~ is the (scaled) function P (a ;s = 1 - 2q-I) in an orthogonal basis [cf. eqs. (34) and (37)]. The balanced eight-port homodyne detector was first used by Walker and Carroll in order to demonstrate the feasibility of measuring the components of the complex amplitude of optical signals, extending earlier microwave techniques (Walker and Carroll [ 19841). Finally, it was proposed to measure the Q function by projection synthesis, mixing the signal mode with a reference mode that is prepared in a quantum state such that, for appropriately chosen parameters, the joint-photon-number probabilities in the two output channels of the beam splitter realize the coherentstate projector fi(a)= n-'la)(a1for truncated signal states (Baseia, Moussa and Bagnato [ 19971). The method was first introduced to synthesize (for truncated states) the _phase-state projector fi(@) = I@)(#[ (Barnett and Pegg [1996]; cf. 53.8.3). 2.1.5. Probability operator measures
As mentioned, the reference mode with which a signal mode is mixed must not necessarily be in the vacuum state in order to obtain, in principle, all knowable information on the quantum state of the signal mode. If the reference mode is allowed to be prepared in a quantum state &, then a joint measurement of the (n/2-shifted) quadrature-components of the interfering fields can be regarded as a realization of a complex amplitude measurement (Walker [1987]). Each & implies a probability operator measure (POM) over the complex amplitude which can be characterized by a positive valued Hermitian operator:
with
K § 21
87
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
The joint probability density prob(a) of obtaining a result a from a measurement described by this POM l4 is prob(a) = ( f i ( a ) ).
=
Re a + i Im a
(40)
Note that from the properties of B(a) it follows that prob(a) 2 0 and J d 2 aprob(a) = 1 . The operational probability density distribution prob(a) in eq. (40), which is also called propensity (Popper [19821), can be given by a convolution of the Wigner function W ( a ) of the signal prepared in a state $ with the Wigner function F V R ( ~ )of the reference mode prepared in a state @R, prob(a) =
J’ d2PWR(P
-
a)W(P)
(41)
(Husimi [1940], Arthurs and Kelly [1965], Kano [1965], Wootters and Zurek [1979], O’Connell and Rajagopal [1982], Rajagopal [1983], Wodkiewicz [1984, 1986, 19881, Takahashi and Sait6 [1985], Walker [1987], Lai and Haus [1989], Hradil [1992], LaloviC, DavidoviC and BijediC [1992], Stenholm [1992], DavidoviC and LaloviC [ 19931, Leonhardt [1993], Leonhardt and Paul [1993b, 19951, Chaturvedi, Agarwal and Srinivasan [1994], Raymer [1994], Buiek, Keitel and Knight [1995a,b], Paris, Chizhov and Steuernagel [ 19971, Wiinsche and Buiek [ 19971). Obviously, the reference mode acts as a filter and smoothes the Wigner function of the signal. A filter system of this type is also called quantum ruler (Aharonov, Albert and Au [ 19811). It is needed in order to resolve the noncommutative quadrature components of the signal. The particular realization of the filter strongly influences the outcome of the measurement. The class of phase-space functions that can be obtained includes the s-parametrized functions with s -1. In particular, detection of the Q function implies that & = lO)(Ol, so that f i ( a ) = n-‘Ia)(al,and hence prob(a) = n-’(al@la)= Q(a). This is the case when vacuum is fed into the reference channels in the homodyne detection schemes in figs. 5 and 6, and the signal is mixed with it. If the signal is mixed with a squeezed vacuum, then the POMs of the form given in eq. (38) with & = 5(E)lO)(Olkt(E) can be realized
<
l 4 The concept of POM was introduced in quantum theory in order to generalize the familiar quantum probability theory based on orthogonal projectors (Davies and Lewis [1970], Holevo [1973], Davies 119761, Helstrom [1976]). In particular, it can be used for describing “simultaneous” measurement of noncommuting observables of an object, the corresponding POM being related to a quantum measurement of commuting observables in some extended Hilbert space. Note that for nonperfect detection ( q < l), the photocounting formula ((2.8) is also a POM.
88
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, 0 2
(Walker [1987], Lai and Haus [1984], Leonhardt [1993], Kim and Sanders [ 19961, Kochahski and W6dkiewicz [1997]) Is. Probability operator measures over the complex amplitude and the photon number can also be defined such that the joint probability density for an outcome a = Re a + i Im a and n is given by
where h ( a ,n) must be a positive valued Hermitian operator that can be used to resolve the idendity,
1
d 2 a h ( a , n ) = 1.
(43)
A probability operator measure of this form can be realized by ten-port homodyne detection (Luis and Peiina [1996b]) 16. In the scheme, a signal mode is fed into one of the input channels of an unbalanced beam splitter whose second input channel is “unused”. The mode in one of the output channels of the beam splitter (say, the transmitted signal) is used as the input mode in an eight-port homodyne detector, and a photodetector is placed in the other output channel in order to measure the photon-number statistics of the mode in this channel 1 7 , together with the eight-port complex-amplitudemeasurement. The realized POM is related closely to photon-added coherent states, j l ( a , n ) =g,(a)~+~la)(al iin,
(44)
where g,(a) = n-’t-2 exp[la12(1 - tr2)](1 - t2)”/n!, with t being the (absolute value of the) transmittance of the beam splitter (0 < t < 1). Note that when t + 1, then f i ( a , n ) 4 n-’la)(aJSno; i.e., the familiar coherent-state POM is realized. 2. I . 6. Positive P function It is well known that for s > -1 the s-parametrized phase-space functions P ( a ; s ) (see Appendix B.4) do not necessarily exist as positive functions, and It should be noted that these POMs can also be realized in heterodyne detection (3 2.2) (Yuen and Shapiro [1980], Yuen [1982]). Further, they can be realized in an unbalanced homodyne scheme with vacuum input but not equal-part signal-beam splitting (Leonhardt [1993, 1997~1). l 6 The ten-port scheme can be regarded as a reduced twelve-port scheme, the latter being suited for measuring simultaneously the Stokes parameters of a two-mode field. l 7 For a direct measurement of the photon-number statistics, see $2.1.7.
11, § 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
89
for s > 0 they are not necessarily well-behaved. The latter also applies to the P function, which is used widely to calculate averages of normally ordered (i.e., measurable) quantities. In order to avoid using singular functions, generalized P representations may be used (Drummond and Gardiner [1980]; see also Gardiner [1983, 1991]), such as the positive P function1*,
The possibility of measuring the single-mode Q function in perfect (sixport or eight-port) homodyne detection also offers the possibility of measuring the quantum state of the mode in terms of the positive P distribution using more involved multiport homodyne detection schemes (Agarwal and Chaturvedi [1994]). Let us again consider a signal mode and a reference mode which are superimposed by a 50%:50% beam Splitter and assume that the reference mode is in the vacuum. It can then be shown that the joint Q function of the two outgoing modes, Q(al ,a2), is related to the Q function of the signal mode, Q(a),as l9
which reveals that Q(a1, a2) is nothing but the (scaled) positive P function of the signal mode,
(
e ( a l , a2) = 4~ a = &al,
p=ha;) .
(47)
Hence, if each of the two output modes of the beam splitter is used as an input mode of a multiport apparatus (such as the six- or eight-port homodyne detector outlined in 0 2.1.4) that measures the Q function, then measurement of the joint Q function of the two modes yields the positive P function of the signalmode under study. Needless to say, for imperfect detection a smeared positive P function is measured. The positive P function is an example of a measurable phase-space function2’ that is defined as a fimction of two complex amplitudes a and p (per mode).
I8
Note that
6 = s d 2 a s d2a’ la)(a’*I ((a’*la))-IP(a,a’) in this representation.
’’Equation (46)can be proved correct, applying the beam splitter transformation (1) and expressing
the characteristic function of the two-mode (outgoing) Q function Q(al,a2) in terms of the characteristic function of the signal-mode (incoming) Q function Q(a).Note that @ a , ,9 )is the two-mode s-parametrized phase-space function P( al ,a2;s = -1). For a method suggested to measure the positive P function of a quantum-mechanical particle, see Braunstein, Caves and Milburn [1991].
’’
90
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, § 2
The concept can also be extended - similar to 9 2.1.5 - to other than vacuum reference inputs (in the beam splitter and/or the multiport homodyne detectors) in order to obtain generalized phase-space functions defined in a four-dimensional phase space. A simple example is the smeared positive P function mentioned above. 2.1.7. Displaced-photon-number statistics Let us return to the four-port homodyne detection scheme (fig. 1) and answer the question of which quantity is measured when a signal mode is mixed with a local oscillator mode by an unbalanced beam splitter and only a single-channel homodyne output is measured. We rewrite eq. (10) as 2;
=
UkI(2-a) = Uklb((X)2bt(a),
(48)
where
and apply the photocounting formula (C.5).The probability of detecting m events in the kth output channel can then be given by
where ;(a) = b ( a ) i b t ( a )is the displaced photon-number operator of the signal mode, and q = I Ukl l2q0 (with rlr, being the quantum efficiency of the detector used). Equation (50) reveals that the observed probability distribution P,,, is nothing but the displaced photon-number distribution of the signal field measured with quantum efficiency r]. In particular, when a signal and a strong local oscillator, laL[+ 00, are mixed by a beam splitter with high transmittance, I UII I = I U22 I + 1, and low reflectance, IU21 I = I U12 I + 0, such that the product I U12I IaLI is h t e , then for high quantum efficiency (q + 1) the displaced photonnumber probability distribution of the signal is measured (Wallentowitz and Vogel [1996a], Banaszek and Wodkiewicz [1996], Paris [1996a]):
P,,
+
p d a ) = (m,alcjlm, a ) ,
( 5 1)
where J m , a ) = b(a)lm) are the displaced photon-number states. It should be noted that for chosen m the quantity p , ( a ) as a function of a can be
11, § 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
91
regarded (apart from the factor m-‘) as a propensity for the signal-mode complex amplitude, which can also be measured in multiport homodyning (see 0 2.1.5). For chosen a it is an ordinary probability for the displaced signal-mode photon number, which can already be obtained from the four-port detector out%ined here. In order to obtain in this scheme p , ( a ) as a function of a, a succession of (ensemble) measurements must be performed. Hence, measurement of the displaced-signal-mode photon-number statistics as a function of the complex parameter a is equivalent to measurement of the signal-mode quantum state, and it is expected that it yields more data than the minimum necessary for reconstructing it (8 3). In contrast to balanced homodyning, measurement of the displaced photonnumber statistics in unbalanced homodyning requires highly efficient photodetectors which can discriminate between n and n + 1 photons in order to resolve the discrete nature of the photon number. Presently, such detectors are not available. Photomultipliers and streak cameras can discriminate between single photons, provided that the field does not contain more than about 10 photons, but the quantum efficiency of about 10-20% is extremely low. Currently available avalanche photodiodes operating in the Geiger regime may reach about 80% quantum efficiency (Kwiat, Steinberg, Chiao, Eberhard and Petroff [ 1993]), but they do not discriminate between single photons. They can only indicate the presence of photons, because of saturation. The problem may be overcome using multichannel coincidence-event measurement techniques, also called photon chopping. In particular, it was proposed to use highly efficient avalanche photodiodes and a beam splitter array to divide the number of readout photons among the photodiodes, so that none is likely to receive more than one photon (Ho, Lane, La Porta, Slusher and Yurke [1990], Song, Caves and Yurke [1990], Paul, TO&, Kiss and Jex [1996a])21. Alternatively, it was proposed to directly defocus the field to be measured onto an array of photodiodes (Wallentowitz and Vogel [1996a]). Let us assume that the mode to be detected enters one of the input ports of a balanced linear 2N-port apparatus (the other input ports being “unused”), and multiple coincidences are measured at the output, placing avalanche photodiodes in the N output channels. If the signal mode contains less than N + 1 photons, then there is a one-to-one correspondence between the measured coincidence-
For detection of squeezing via coincidence-event measurement, see Janszky, Adam and Yushin [1992]; for Fock state detection and preparation, see Paul, Torma, Kiss and Jex [1996b].
21
92
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, 0 2
event distributionP , ( N ) and the photon-number distributionp n of the signal. To be more specific, it can be shown that 22
n=m
where
and pmln(N)= 0 if n < m (Paul, Torma, Kiss and Jex [1996a]). Here, pmln(N) is the probability of registration of m clicks under the condition that n photons are present. The probabilities pmlq(N)form an upper triangular matrix A,,, = pml,(N)which can be inverted in order to calculate p , from j , ( N ) , N
m=n
2.1.8. Homodyne correlation measurements
As already mentioned, the accuracy with which a (quantum) field can be measured by a homodyne detector is limited by the overall quantum efficiency of the device [see, e.g., eqs (19) and (20)]. To overcome this limitation, one may perform homodyne correlation measurements (Ou, Hong and Mandel [ 1987b1). In contrast to ordinary balanced homodyning, the (time-delayed) intensity correlation between the two outgoing fields is measured. In particular, the information on squeezing is obtained from the time dependence of the measured correlation function. Since the measured coincidence events are proportional to the product of the detection efficiencies of the two detectors, small quantum efficiencies may reduce the measured signal (which could be compensated by longer measurement times), but do not smooth out the desired information. However, a drawback of the method is that the classical noise of the local oscillator is not balanced out; i.e., even small relative fluctuations of the (strong) local oscillator may prevent the quantum noise effects of a weak signal from being measured.
22
Here it is assumed that the balanced 2N-port realizes a unitary transformation UN = U2 @ U N / ~ .
11, § 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
93
It was therefore proposed to use a. weak local oscillator whose intensity is comparable to that of the signal (Vogel [ 1991, 19951). In this case, the classical noise of the (highly stablized) local oscillator may be reduced below the level of the quantum fluctuations of the signal. Moreover, simultaneous measurement of different kinds of correlation hnctions of the signal field is possible. To illustrate this, let us consider the difference between the measured second-order intensity correlation function G(2)(t, t + t) for short and long delay times t, AG'2'(t)= G'2'(t,t ) - r+m lim G'2'(r, t + t),
(55)
and restrict our attention to stationary fields, so that the time argument t can be omitted. Decomposing AG(*) with respect to powers of the local-oscillator amplitude EL, one may observe the following effects. The zeroth-order term yields the normally ordered intensity ( I ) fluctuation of the signal,
The second-order term is related to the normally ordered electric-field variance of the signal,
where k(cp)corresponds to the quadrature-component operator i ( c p ) , q being the phase difference between signal and local oscillator. Eventually, the first-order term represents the correlation between the two signal-field observables,
Note that all these quantum-statistical moments can be separated from each other by using their dependences on the phase shift cp (for the measurement of the corresponding spectral properties, see Vogel [ 19951). 2.2.
HETERODYNE DETECTION
It is worth noting that multiport homodyning for measurement of the complex amplitude is equivalent to (four-port) heterodyne detection (Yuen and Shapiro [1980], Yuen [1982], Yuen and Chan [1983], Shapiro [1985], Shapiro and Wagner [1984], Caves and Drummond [1994]). In the scheme, an optical field is combined, through a beam splitter, on the surface of a photodetector with a
94
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[H, 8 2
strong local oscillator whose frequency q is offset by an amount Aw from that of the signal mode in the optical field (Aw<< q).The measured photocurrent is filtered electrically in order to select the complex valued component at frequency Aw. The classical statistics of t h s component correspond, under certain circumstances, to the quantum statistics of the two-mode operator
where the subscripts S and I are used to denote the signal mode at frequency ~0 + Aw and the imaging mode at frequency 00 - Aw,respectively. Obviously, the imaging mode can be used to probe the signal mode. To show this, we first note that the measured quantity c? can always be brought into the form
where 61 and &Z are commuting Hermitian operators having a quantum-mechanical joint probability density p ( a )= p ( q , a2), which can be found easily through its characteristic function,
In particular, if the imaging mode is in the vacuum, then measurement of p ( a ) can be seen to yield the Q hnction of the signal mode, because
(cf. Appendix B.4). 2.3. PARAMETRIC AMPLIFICATION
Finally it should be mentioned that linear amplification of a signal mode and measurement of the complex amplitude of the amplified signal may be regarded as an equivalent of the homodyne measurement of the complex amplitude of the original signal (Bandilla and Paul [1969, 19701, Paul [1974]). If the signal mode is strongly enhanced such that it behaves classically, the conjugated quadrature components can be measured simultaneously. The scheme can be realized by (nondegenerate) parametric amplification, in whch the photon destruction
11, § 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
95
operators 2; and 2; of the amplified.signa1 and idler modes, respectively, are obtained from the SU( 1,l) input-output relations:
where g is the (linear) signal-gain factor (for details, see, e.g., Mollow and Glauber [1967a,b], Caves and Schumaker [ 19851, Yurke, McCall and Klauder [ 19861). It can be shown that when the signal mode is mixed with an idler mode that is initially in the vacuum state and the strong-amplificationlimit is realized, i.e., g + co,then the Q function of the original signal mode can be inferred from the measured distribution function for the complex amplitude of the amplified signal mode by appropriately rescalhg this distribution (Leonhardt [ 19941; see also Leonhardt and Paul [1995]). It is worth noting that a degenerate parametric amplifier that operates as a squeezer [2[ -+ ei@2gin eq. (63)] may be used to (partially) compensate for the detection losses in balanced four-port homodyne detection for measuring quadrature components (Leonhardt and Paul [ 1994al; 0 3.9.1), since it makes it possible to amplify a quadrature component with no increase of noise (Caves [1982]). Further, parametric amplification can be used to determine the quantum state of the signal mode by direct photodetection of the amplified signal if the idler mode is initially prepared in, e.g., a coherent state, without restriction to the Q function and without restriction to the strong-amplification limit (Kim [ 1997al; 0 3.5). In particular, when the idler mode is prepared in a strong coherent state such that it can be treated classically, 21 + al, then - with regard to the photocounting formula ((2.5) for detecting the amplified signal - eq. (63) takes the form of eq. (48), with a = -[(g - l)/g]”’aT in place of eq. (49). In this case, eq. (50) also applies to the detection of the amplified signal, where now i = i s and 17 = g 1 7 ~ ~ ~ . 2.4. MEASUREMENT OF CAVITY FIELDS
Let us now consider the possibilities of phase-sensitive measurements of highQ cavity fields. The realization of such fields, for example in a micromaser
Note that eq. (63) implies that the signal and idler operators in the photocounting formula (C.5) are subject to different orderings, so that eq. (48) does not apply in general.
23
96
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
Fig. 7. Set-up for the detection of a high-Q cavity mode. The Rydberg atoms are prepared in the state le) in the box CB and cross a microwave zone R- (source Sz) between CB and C which mixes the (excited) state ) . 1 and the (ground) state 1s) before the atoms enter the cavity C. A second microwave pulse (source Sz) applied between C and D in the zone R+ again mixes the states le) and lg) before the atoms reach the detector D, which renders it possible to detect linear superpositions of these two states. The microwave source S1 can be used to feed into the cavity a (classical) field. (After Brune and Haroche [1994].)
(Meschede, Walther and Miiller [1985], Rempe, Walther and Klein [1987], Brune, Raimond, Goy, Davidovich and Haroche [1987]), necessitates a strong suppression of cavity losses. That is, only a very small fraction of the field escapes from the cavity and could be used, e.g., in homodyne detection. Since the quantum efficiency of such a scheme is very low, it is impossible, in general, to extract the (complete) quantum statistics of the field inside the cavity from the homodyne data measured outside the cavity. To overcome the problem, methods have been developed which use atoms that travel through the cavity and probe the field inside. The system that is directly observed in the experiment is not the cavity field itself but the atoms after their interaction with the field. Under certain circumstances, the measured atomic occupation probabilities can then be related uniquely (in a more or less direct way) to the quantum state of the cavity field (Q 3.6). Let us consider a measurement scheme typically used in Rydberg-atom superconducting cavity QED (see, e.g., Brune and Haroche [ 19941 and references therein)24.The atoms which cross the cavity are prepared, after leaving an oven and before entering the cavity, into a superposition of Rydberg states) . 1 and 18) of different energies (fig. 7). After preparation of the state le), which involves laser and radiofrequency excitation, a microwave (Ramsey) zone can be used to
z4 Here circular Rydberg atoms are used in which one electron is placed on a highly excited energy level with large value of principal quantum number n, and the angular momentum takes its maximum value (I = m = n - 1). For n M 50, the atomic-dipole decay time is of the order of magnitude of lO-’s. The cavity damping time is of the order of magnitude of to lo-’ s for superconducting cavities at subKelvin temperatures. The Q factor defined as the ratio of the cavity frequency to the cavity damping rate (inverse damping time) then reaches values of about lo8 to 10”. The Rabi frequency, which is typically of the order of magnitude of 10 to lOOkHz, is large compared with the atomic and the cavity decay rates, so that relaxation can be disregarded in first approximation (strong coupling regime).
u, 0 21
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
91
apply a resonant pulse to the atoms, mixing the state le) with another state 1 8 )of different energy. In this way an initial atomic state of the form c,le) + cglg)can be injected into the cavity. The-atoms then interact with the cavity mode while crossing it. After leaving the cavity, the atoms can again cross a microwave zone in order to realize state mixing before they enter a detector for measuring their energies by field ionization. This detection can be made energy-sensitive and one can thus count the atoms in Ie) and Ig). The dependence of the measured data on the various parameters characterizing the experimental conditions (such as state mixing and interaction time) can then be used to gain insight into the quantum state of the cavity mode (Vogel, Welsch and Leine [1987], Bardroff, Mayr and Schleich [1995], Bardroff, Mayr, Schleich, Domokos, Brune, Raimond and Haroche [ 19961). The cavity and the atomic transition can be tuned to coincide (resonant regime) or slightly detuned (dispersive regime), with a frequency difference 6w such that any exchange of energy between atom and field is made impossible. In the resonance regime, the system can be described (in its simplest version) by the Jaynes-Cummings model (Jaynes and Cummings [1963], Paul [1963]; for a review, see, e.g., Shore and Knight [1993]). The Hamiltonian consists of three terms describing the free two-level (circular Rydberg) atom, the free cavity mode and the atom-field coupling, the latter being given by
A'
= AK
(6+& + &+6-).
(65)
Here, 6+= le) (81and 6- = 1 8)(el are the atomic raising and lowering operators, respectively, and K is the coupling constant between the atom and the cavity mode (Q = 2~ is the vacuum Rabi frequency, which is the rate at which the atom and the empty cavity exchange a photon at exact resonance.) In the dispersive regime, the atom-field interaction produces a dephasing of the field and also dephases the atom's wave function by an angle depending upon the number of photons in the cavity and on the quantum state of the atom. The nonresonant interaction can be described by an effective Hamiltonian,
The phase difference induced by the interaction of the two parts of the system can be measured when the two microwave zones in fig. 7 are active. It can be shown that the probability that a probe atom which is initially prepared in the state le) is finally detected in the state Ig) (i.e., after the combined action of the two microwave zones on it) oscillates as a function of the microwave
98
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
tI1, § 2
frequency, the position of the interference (Ramsey) fringes being dependent on the additional dephasing produced by the cavity mode. It is worth noting that the Ramsey interferometer outlined here can serve as an apparatus for a quantum non-demolition measurement (QND) (Braginsky, Vorontsov and Khalili [1977]; see also Braginsky and Khalili [1992]) of the photon number in the cavity (Brune, Haroche, Lefevre, Raimond and Zagury [ 19901, Brune, Haroche, Raimond, Davidovich and Zagury [ 19921). When in the dispersive regime an additional microwave generator is used in order to resonantly couple a classical (strong) oscillator to the cavity mode, then the Wigner function of the cavity mode can, in principle, be measured directly (Lutterbach and Davidovich [1997]). Owing to the coupling of the classical oscillator to the cavity mode, a displacement in phase space of the (initial) quantum state of the cavity mode is performed such that the density operator @ is The atom$ initially prepared in state Ie) cross the replaced with b+(a)@b(a)25. first Ramsey zone where they see a x/2 pulse, so that Ie) + 2-”2 (14 + ei%d). The atoms then interact dispersively with the cavity field, and after that they undergo a n/2 pulse in the second Ramsey zone before detection. Let P, and Pg be the probabilities of detecting a probe atom in the states 1.) and ig), respectively. It can be shown that the difference between these probabilities is (67) where qj = (Q2/6)ris the additional dephasing of the atoms in the cavity, with T being the interaction time. In particular, if qj = = x/2, then
which reveals that the Wigner hnction is indeed detected, provided that during the (ensemble) measurement the phases are controlled and each probe atom meets the cavity field in exactly the same state. The method of displaced states26 can also be used for quantum-state measurement in the resonance regime, without phase-sensitive state mixing (Wilkens and Meystre [ 19911, Bodendorf, Antesberger, Kim and Walther [1998]). Here, the probe atoms are prepared in one of the two considered
Note that this coupling acts in a similar way as the mode mixing at an unbalanced beam splitter outlined in $2.1.7. 26 An analogous method can be used in order to determine the quantum state of the center-of-mass motion of trapped ions (cf. 4.2.2).
25
99
PHASE-SENSITIVE MEASUREMENTS OF LIGHT
class. field screen
I resonator
I
Fig. 8. Atomic-deflection measurement scheme. A classical field prepares the two-level atoms in a superposition Ig) + eiqle) of the ground state Ig) and the excited state 1.). The so-prepared atoms pass a narrow slit that con6nes them to a region Ax << 1 centered around the node of the standing light field at n = 0. Atter the resonant interaction with the field the deflected atoms travel to a screen put up far away from the resonator. The kpatial distribution on t h i s screen reflects the momentum distribution of the atoms right after the interaction. (After Freyberger and Herkommer [1994].)
states Ie) and Ig) or in a statistical mixture of them. The measured two-level occupation statistics of the atoms as a function of the interaction time andor the displacement parameter contain all knowable information on the quantum state of the cavity mode. Themethod of probing a cavity field using two-level atoms which travel through the cavity can also be extended to other than purely two-level probe atoms. So it was proposed to completely transfer, by adiabatic passage, the quantum state of the field onto an internal Zeeman submanifold of an atom (Walser, Cirac and Zoller [1996]). Utilizing a tomography of atomic angular momentum states by Stern-Gerlach measurements (Newton and Young [1968]), t h ~ angular s momentum state can then be determined uniquely, by a finite number of magnetic dipole measurements. The spatial variation of the cavity mode implies a dependence on the atomic position of the Rabi frequency, which offers the possibility of probing the quantum state of the mode via atomic deflection (Akulin, Fam and Schleich [1991], Herkommer, Akulin and Schleich [1992], Freyberger and Herkommer [1994]). In the scheme in fig. 8 a narrow beam of resonant atoms is proposed to pass through a node region of the mode, where the interaction Hamiltonian can be assumed to be a linear function of the atomic position. The cavity photons repulse the atoms so that the transverse momentum distribution of the atoms is changed during their passage through the cavity, When the cavity mode is (initially) in a pure state, then the measurable change provides (for appropriately prepared atoms) all the information needed for reconstructing it.
100
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, § 3
To extend the method to the reconstruction of mixed states, the effects of the classical field and the cavity field on the atoms can be combined such that the classical field travels orthogonally with respect to the cavity mode and both fields overlap in the region where the atoms cross them (Schneider, Herkommer, Leonhardt and Schleich [19971). In particular, when the classical field is sufficiently strong, then it can play a similar role as the local oscillator in balanced four-port optical homodyning. It is worth noting that under certain circumstances the transverse momentum distribution of the atoms which is observed in the dispersive regime is nothing but a smeared quadraturecomponent distribution of the cavity mode (§ 3.6.2).
0 3.
Quantum-State Reconstruction
ai
a/)
When in an experiment a sequence of quantities (21 = can be measured, such that the density operator of the system under study can be given by
8 = C"(aj), i
then all knowable information on the quantum state of the system can be obtained in principle (Fano [ 19571).27 In practice, it must be ensured that enough data can be measured and that they can be decoded with reasonable accuracy and acceptable effort - a task which is not trivial in general. In particular, if the measured data are related directly to a smeared quantum-state representation in the phase space, it may be an effort to derive from it the quantum state in another basis for studying specific quantum-statistical properties. Typical examples of complete and overcomplete basis sets of observables for which measurement schemes have been designed are the generalized projectors di + f i ( x , q ) = n - ' ( x , q ) ( x , q l [see eq. (B.31)] a n d j i -, f i ( a ) = n-'la)(al [see eq. (B.20) with s = -I], respectively, for the quadrature-component states and the coherent states. If a scheme is not designed for measuring at least a [in
27 In connection with the study of the Pauli problem (Pauli [1933]), the sufficient set of observables that has to be measured in order to obtain the complete information on the quantum state was also called the quorum of a quantum-state measurement (Park and Band [1971]). It was shown that for the one-dimensional motion of a particle a quorum can be given by all symmetrized products of powers of position and momentum (cf. 8 3.8.1) or alternatively, all time derivatives of all powers of the position (Band and Park [1979], Park, Band and Yourgrau [1980]).
I L § 31
QUANTUM-STATE RECONSTRUCTION
101
the sense of eq. (69)] complete set .of observables, then some a priori knowledge of the quantum state is required in order to compensate for the lack of data.
3.1. OPTICAL HOMODYNE TOMOGRAPHY
From 52.1.2 we know that in a succession of (phase-shifted) balanced fourport homodyne measurements the quadrature-component distribution p(x, cp) of a signal mode can be obtained for various values of the phase cp, provided that 100% quantum efficiency is realized. The quantum state is then known when p(x, cp) is known for all values of cp within a n interval (Vogel and Risken [1989]). That is to say, all quantum-statistical properties can be obtained from the quadrature-component distributions measured in a n interval. In particular, p(x, cp) can be used to reconstruct the Wigner function. From eq. (B.28) (s = 0) it can be derived that p(x, cp)
=
/
dy W(xcos cp -y sin cp,x sin cp + y cos cp),
(70)
where the definition W(q,p) = 2-' W [ a = 2-"2(q + ip)] is used. Equation (70) reveals that the quadrature-component distributions can be regarded as marginals of the Wigner function. An integral relation of the form given in eq. (70) is also called Radon transformation (Radon [1917]). Inversion of the Radon transformation (70) yields the Wigner function in terms of the quadraturecomponent distribution2*.From eq. (B.29) (s = 0) the Wigner function is derived to be29
Performing the z integral first would lead to an integral kernel that is not wellbehaved. To overcome this difficulty in the numerical calculation, regularization
28 Inverse Radon transformation techniques are well known from tomographic imaging (for mathematical details, see, e.g.,Natterer [1986]). In quantum mechanics, the problem of tomographic reconstruction of the Wigner function of a particle moving in one dimension was first addressed by Bertrand and Bertrand [1987]. 29 For phase-randomized states, see Leonhardt and Jex [1994].
102
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
[II, 8 3
techniques can be used. In the filtered back projection algorithm, the z integral is truncated such that
with
J -z(
(zc > 0). Here, rapid variations of the quadrature-component distributions which correspond to frequencies higher than the cut-off frequency z, are suppressed; i.e., the exact distributions are effectively replaced with somewhat smeared ones. In the pioneering experimental demonstration of the method also called optical homodyne tomography (Smithey, Beck, Raymer and Faridani 11 9931, Smithey, Beck, Cooper, Raymer and Faridani [1993]), a pulsed signal field is superimposed with a pulsed coherent-state field much stronger than the signal. The quadrature-component distributions are measured for a squeezed signal field and a vacuum signal field. The squeezed field is generated by using a walk-off compensated, travelling-wave optical parametric amplifier. The generated downconversion signal centered at 1064 nm consists of two orthogonally polarized fields, the signal and idler, and has a bandwidth estimated to be lo4 times that of the laser pump (532 nm, 300 ps, 420 pulses per second). The laser pump and the local oscillator field (1064 nm, 400 ps) are obtained from a common laser source, and each local-oscillatorpulse contains a mean number of photons of about 4 x lo6. The interfering fields are detected with high quantumefficiency (- 85%) photodiodes, and the resulting current pulses are temporally integrated and subtracted. The measurements and reconstructions are performed for a squeezed signal field and for a vacuum signal field, figs. 9 and 10, the field mode detected being selected by the spatial-temporal mode of the localoscillator field. The method of optical homodyne tomography was also extended subsequently to the continuous wave-regime, including squeezed vacuum with a high degree of quantum noise reduction (Breitenbach, Muller, Pereira, Poizat, Schiller and Mlynek [ 19951, Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961) and bright squeezed light (Breitenbach and Schiller [ 19971, Breitenbach, Schdler and Mlynek [ 19971). As mentioned in 8 2.1.2, the measured quadrature-component distributions do not correspond, in general, to the true signal mode, but they must be regarded as the distributions of a superposition of the signal and an additional noise source
11, § 31
103
QUANTUM-STATE RECONSTRUCTION
h
0.8
**.
X a
**=.
0.6
0.3
'
0.00
0.79
1.57
236
0.6
I 03
3.14
Phase q Fig. 9. (a) In balanced four-port homodyne detection measured quadrature-component distributions at various values of the local-oscillator phase [P&,) corresponds to p(x, rp)]. (b) Variances of quadrature components vs local-oscillator phase: circles, squeezed state; triangles, vacuum state. In the experiment 4000 repeated measurements of the photoelectron difference number at 27 values of the relative phase rp are made. (After Smithey, Beck, Raymer and Faridani 119931.)
[eq. (21)], because of non-perfect detection. Substituting in eq. (70) for p(x, ip) the measured distributions p ( x , ip; q) with q < 1 [eq. (19)] and performing the inverse Radon transform on them, the Wigner function of a noise-assisted signal field is effectively reconstructed. Equivalently, the reconstructed Wigner function can be regarded as an s-parametrized phase-space function of the true signal field, however with s < 0. The characteristic function Y ( z , ip; q) ofp(x, ip; q ) typically measured when q < 1 and the characteristic function @ ( p ; s ) of the phasespace function P(a;s)are related to each other according to eq. (3.27), with Y ( z ,ip; q) in place of Y ( z ,47) and s - 1 + q-' in place of s in the exponential [cf. eq. (23) and footnote 71. Hence, making in the exponentials in eqs. (B.28) and (B.29) for s the substitution s - 1 + 8'yields the relations between P(a;s)
104
tQ § 3
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
and p(x,cp; q). In particular, eq. (B.29) can be used to obtain any phase-space function P(q,p;s) = 2-‘P[a = 2-”2(q + ip); s] from the measured quadraturecomponent distributions in principle30:
x IzI exp[iz(q cos cp + p sin cp - x)]p(x,cp; q)
1
(74)
.
Obviously, when s = 1 - q-’, then eq. (74) takes the form of eq. (71); i.e., replacing in eq. (70) p(x, cp) with p(x,cp; q), inverse Radon transformation yields the signal-mode phase-space function P(q,p;s = 1- q-‘) in place of the Wigner function,
p(x,cp; q) =
/
dy P(x cos cp - y sin cp, x sin cp + y cos cp; s = 1 - q-I).
(75)
The proposal was made to combine squeezing and balanced homodyning such that generalized quadrature-components
for all real parameters p and Y can be measured, and to reconstruct the quantum state from the corresponding distributions p ( X , ,u,Y) (Mancini, Man’ko and Tombesi [1995], D’Ariano, Mancini, Man’ko and Tombesi [1996], Man’ko [1996], Mancini, Man’ko and Tombesi [1997])31.In fact, i ( p , Y) can be related to i ( c p ) since it represents a scaled quadrature-component,
When s > 1 - q-’, then in eq. (74) an inverse Gaussian occurs which may lead to an artifical enhancement of the inaccuracies of the measured data, so that a stable deconvolution might be impossible and the noise dominates the reconstruction of P ( q , p ; s ) . This effect is not observed when s < 1 - q-’ , and a stable reconstruction with reasonable precision of P(a;s) for s 6 1 - 8’ may therefore be expected to be feasible for any quantum state (see also $3.9). In particular, reconstruction of the Q function is always possible if q > 1/2. When s < 1 - q-’, then in eq. (74) the z integral can be performed first to obtain P ( q , p ; s )in a form suited for application of sampling techniques (0 3.3.1):P ( ~ , Ps); = :J drp J d r K ( q , p , x ,rp; s; V ) P ( X , rp; v), with K(q,p,x, rp;s; 11) being a well-behaved integral kernel. 3 1 For an application of the method (also called symplectic tomography) to trapped-ion quantum state reconstruction, see Mancini, Man’ko and Tombesi [1996], Man’ko [1997]. 30
105
QUANTUM-STATE RECONSTRUCTION
?
2.0
2.0 -
I
1 .o
1.0.
;i
& 0.0 -
-1 .o -
-1 .o
~
-%:o
-1.0
0:o
X
1.0
2.0
-2-90 -1.0
0:o
X
1.0
2.0
Fig. 10. Wigner distributions reconstructed from the measured quadrature-component distributions for (a,h) a squeezed state and (c,d) a vacuum state, viewed in 3D and as contour plots, with equal numbers of constant-height contours [ W ( X ,P ) corresponds to W(q,p)].The reconstruction is performed by using inverse Radon transformation according to eq. (72). (After Smithey, Beck, Raymer and Faridani [ 19931.)
with
(cf
9 2.1.2 and Appendix B), which implies that
Hence, measurement of a particularly scaled quadrature-component distribution by means of an ordinary homodyne detector already yields all scaled quadrature components. Performing the analysis with a variable scaling parameter, it can be
106
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, 5 3
shown that eqs. (70) and (71), respectively, can be given in terms of p ( x , p, Y ) by32 p ( x , p, Y ) =
‘J J J dk
dq
dp e-ik(x-qlc-pv) W (4 , P ) ,
and
Comparing eq. (81) with eq. (71), we see that in any case a three-fold Fourier transformation is required in order to obtain the Wigner function from the homodyne data. Note that z in eq. (81) can be chosen arbitrarily, which reflects the above mentioned fact of overcomplete data. 3.2. DENSITY MATRIX IN QUADRATURE-COMPONENT BASES
The reconstructed Wigner function can be used in order to calculate the density matrix in a quadrature-component basis (Smithey, Beck, Raymer and Faridani [ 19931, Smithey, Beck, Cooper, Raymer and Faridani [19931). The definition of the Wigner function given in Appendix B.4 can be rewritten, on expanding the density operator in a quadrature-component basis, as
with x = q cos cp + p sin cp,
y
= -q
sin Q, + p cos q,
(83)
which for q = 0 is nothing but the well-known Wigner formula (Wigner [ 19321). Hence the density matrix in a quadrature-component basis can be obtained by Fourier transforming the Wigner function: (x - X I , cpIP[x+ X I , cp)
=
J
dy e-2‘y’J’W(xcos cp - y sin cp,x sin cp +y cos cp).
Note that eq. (84) reduces to eq. (70) for x’
32 For
(84) =
0.
a detailed discussion of the transformation properties, see also Wiinsche [1997].
K 9 31
QUANTUM-STATE RECONSTRUCTION
107
It is worth noting that the reconstryction of the density matrix from the homodyne data can be accomplished with two Fourier integrals, avoiding the detour via the Wigner function (Kiihn, Welsch and Vogel [1994], Vogel and Welsch [ 19941). Writing the density-matrix elements as
‘S
(x-x’, qI@Ix+x’,q) =
dze-irzY(z,x’,q),
(85)
the characteristic function Y(z,x’, cp) can be shown to be the characteristic function of a quadrature-component distribution, Y(z,x’, 9)= Y(Z, @I,
(86)
with 1/2
Z=Z(z,x’)= [ z ~ + ( ~ x ’ ) ~ ] and
@=q-$n+arg(2x’+iz).
(87)
Hence, the density-matrix elements can be obtained from the quadraturecomponent distributionsby means of a ’simple two-fold Fourier tran~formation~~ :
It is worth noting that eq. (88) can be used to obtain the density matrix in different representations, by varying the phase of the quadrature component defining the basis. Further, eq. (88) can also be extended, in principle, to imperfect detection, expressing Y(z, q) in eq. (85) in terms of Y ( z ,q;9) (cf. footnote 7; -for details, see Kiihn, Welsch and Vogel [1994], Vogel and Welsch [1 9941). For the numerical implementation of the reconstruction based on eq. (88) spline-expansion techniques can be used (Zucchetti, Vogel, Tasche and Welsch [19961)34, (x--x’, V I @ I X + X ’ , q) N CKrnn(x,x’, q)p(Zrn+l,@n+l),
(89)
m,n
where Krnn(x,x’,q) = (2n)-1 ~ & & , m [ i ( & + l , x ’ )BA,,,(Sk+l)~b,k(-X), l
(90)
k
with z, = -2x’cot(@, - q) [q...Jk), Fourier transform of B(...)(x)]. As an illustration of this method, in fig. 11 the reconstructed density matrices in the For the two-fold Fourier transformation that relates the density-matrix elements to p ( x , p , v), eq. (SO), see D’Ariano, Mancini, Man’ko and Tombesi [1996]. 34 Choosing a finite set of nodes { x , } , an approximate spline function f , ( x ) of f ( x ) is given by = C , f ( x , + ~ f B ~ ~ , , , ~where ( x ) , gAxn,n(X) = ( x - x,)/Ax, if x , < x < x,+l, B A ~ ~ , , L =~) (x,+2 -x)/Ax,+l if x,+l < x Q x , , ~ , and BA,,,(x) = 0 elsewhere, and Ax, = x,+l - x , ; for mathematical details of spline expansion, see de Boor [1987]. 33
108
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II,
03
Fig. 1 1 . (Real) density matrix (n,rp1&’,rp) of a squeezed state reconstructed from measured quadrature-component distributions in (a) the ‘‘position” basis ( q = 0) and (b) the “momentum” basis (rp = d 2 ) . The homodyne data were obtained by T. Coudreau, A.Z. Khoury and E. Giacobino, using the experimental setup reported by Lambrecht, Coudreau, Steinberg and Giacobino [1996]. In the experiment, the quadrature-compoventswere measured at 48 phases, and at each phase 7812 measurements were performed.
“position” basis, cp = 0, and the “momentum” basis, cp = n/2, of a squeezed vacuum state are shown. The homodyne data were recorded by T. Coudreau, A.Z. Khoury and E. Giacobino. In the underlying experimental scheme the squeezing effect is obtained in a probe beam that interacts with cold atoms in a nearly single-ended cavity (Lambrecht, Coudreau, Steinberg and Giacobino [ 199611. The phases cp = 0 and Q, = n/2 which define the quadrature-component bases in fig. 11 coincide with the phases of minimal and maximal field noise, respectively. 3.3. DENSITY MATRIX IN THE FOCK BASIS
Stimulated by the tomographic reconstruction of the Wip e r function (9 3.1) much effort has been made to obtain the density matrix in the Fock basis from measurable data as direct as possible. Let us again start with analysing balanced four-port homodyning. 3.3.1. Sampling of quadrature-components
The problem of reconstruction of the density matrix in the Fock basis from the quadrature-component distributions can be solved, in principle, by relating the density-matrix elements to derivatives of the Q function (cf. 0 3.3.3) and expressing the Q function in terms of the quadrature-component distributions, using eq. (B.29), with s = -1 (D’Ariano, Macchiavello and Paris [1994a,b]). An equivalent formalism, which is suited for practice and which can also be applied
11,
8 31
QUANTUM-STATE RECONSTRUCTION
109
to the reconstruction of the density matrix in other than the photon-number basis, is based on the expansion of the density operator as given in eq. (B.31). In the photon-number basis, this equation reads as 35
The integral kernel (also called pattern function)
with
is studied in detail in a number of papers (D’Ariano [1995], D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], Leonhardt, Munroe, Kiss, Richter and Raymer [1996], Richter [1996a], Wunsche [1997]). The function fmn(x)(fig. 12) is well behaved, and it is worth noting that it can be given by (Richter [1996a], Leonhardt, Munroe, Kiss, Richter and Raymer [1996])
(x) for En, > En,where Wrn(x) and and qm(x) are the for Em < En, andfmn(x) =Em regular and irregular solutions of the harmonic-oscillator Schrodinger equation for a chosen energy value En,3 6 . Equation (91) reveals that the density matrix in the Fock basis can be sampled directly from the measured quadrature-component statistics, since emncan be regarded as a statistical average of the (bounded) sampling function Kmn(x,q) (D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], Leonhardt, Munroe, Kiss, Richter and Raymer [ 19961, Leonhardt [ 1997~1, D’Ariano [ 1997a1). In an experiment each outcome x of .?(q),with q E [0,n), contributes individually to so that @,n,nn gradually builds up during the data collection. That is to say, emflcan be sampled from a sufficiently large set
35 For the relation between the density-matrix elements and p(x, p, v), eq. (80),see D’Ariano,
Mancini, Man’ko and Tombesi [1996]). 36 Strictly speaking, the irregular ( i t . , not normalizable) function rp,(x) must be chosen such that Qnc& QAcp, = Z/n (for details, see Leonhardt [ 1997~1).Note thatfmn(x)is not determined uniquely.
110
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
P, § 3
06 01 u2
0 42 -0.4
.O 6 0
2
J
ro
.a.75
2
4
Fig. 12. Pfots of some typical pattern functions fmn(x) (lines) along with the products of regular x) lines). In (a) the diagonal pattern function with n = m = 4 is wave functions v n ( x ) ~ , , , , , ( (dashed depicted. Oscillations between -2/n and +2/nare clearly visible in the classically allowed region. Then the function swings over and decays like x-*. In (b) the off-diagonal pattern function with n = 1, m = 4 is depicted. It is less oscillating than the diagonal pattern h c t i o n and it decays faster in the forbidden zone. In (c) and (d) highly oscillating pattern functions are shown for (c) n = rn = 25 and (d) n = 20, m = 50. (After Leonhardt, Munroe, Kiss, Richter and Raymer [1996].)
of homodyne data in real time, and the mean value obtained from different experiments can be expected to be normal-Gaussian distributed around the true value, because of the central-limit theorem. Moreover, the sampling method can also be used to estimate the statistical error (see also § 3.9.1). Experimentally, the method was applied successfully to the determination of the density matrix of squeezed light generated by a continuous-wave optical parametric amplifier (Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961, Breitenbach and Schiller [ 19971, Breitenbach, Schiller and Mlynek [ 19971). In the experiment the spectral component of the photocurrent in a small band around a radiofrequency SZ is measured (overall quantum efficiency 82%). In this case the measurement is on a two-mode quadrature-component i ( q ) = eiQR[2(w + Q)e-'P+ 2(w - Q)e'"], 4 R being the phase of the
-
K 8 31
QUANTUM-STATE RECONSTRUCTION
111
Photon number n Fig. 13. Photon-number distribution of a squeezed vacuum and the vacuum state (inset) reconstructed from the quadrature-componentdistributions according to eq. (91). Solid points refer to experimental data, iustograms to theory. The experimentally determined statistical error is 0.03. (After Schiller, Breitenbach, Pereira, Miiller and Mlynek [1996].)
radio-frequency local oscillator, so that the scheme is basically a heterodyne detector. Examples of reconstructed diagonal density-matrix elements are shown in fig. 13 (for reconstructed off-diagonal density-matrix elements, see, e.g., Schiller, Breitenbach, Pereira, Muller and Mlynek [ 19961, Breitenbach and Schiller [ 19971). Further, the method was used successfully to measure the time-resolved photon-number statistics of a 5 ns pulsed field with a sampling time of 330 fs, set by the duration of the local-oscillator pulse (Munroe, Boggavarapu, Anderson and Raymer [1995]). From eqs. (91) and (92) it is easily seen that the photonnumber probability distribution p n = &, can be given by
where p(x) = (2n)-' J dq,p(x, q) is the phase-averaged quadrature-component distribution. Equation (95) reveals that for sampling the photon-number statistics the phase need not be controlled - a situation that is typically realized when the signal and the local oscillator come from different sources. In the experiment, an argon-laser-pumped Tixapphire laser is used in combination with a chirped-pulse regenerative amplifier to generate ultrashort, transform limited local oscillator pulses (330 fs) at a wavelength of 830 nm and a
112
PI, § 3
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
.I0
-6
0
5
Quadrature amplitude
5
to
:::B, 0 a1
0 00 0
10
0
20
30
Pholon number n
Photon 10 number 20 30 n
Fig. 14. Measured phase-averaged quadrature-component distributions for (a) t = 4.0ns and (b) t = 6.0ns, and the resulting time-resolved photon-number distributions for ( c ) t = 4.011s and (d) t = 6.0ns, obtained from (a) and (b) by using eq. (95). P(E) andp(n), respectively, correspond to p ( x ) andp,. (After Munroe, Boggavarapu, Anderson and Raymer [1995].)
repetition rate of 4 kHz with approximately lo6 photons per pulse. The signal is from a laser diode of wavelength 830 nm and pulse width 5 ns. Figure 14 shows examples of the measured (with 65% overall quantum efficiency) quadrature-componentprobability distributions and the resulting photon-number probability distributions at two times in the signal pulse. In particular, the photonnumber statistics are seen to change from nearly Poissonian statistics (laser above threshold) to thermal-like statistics (laser below threshold) (for ultrafast homodyne detection of two-time photon-number correlations, see 8 3.8.1). So far in the formulas, perfect detection has been assumed. The problem of extending eq. (91) to imperfect detection such that p(x, q) and iU,,(x, cp), respectively, are replaced with p(x, QJ; r ] ) and a sampling function K,,(x, cp; r ] ) that compensates for the losses has also been considered37,
-
~ m n ( x ,QJ; r ] ) =
(4kQJ; r>l.),
(96)
where
37 Equation (96) follows
see footnote 7.
from eqs. (B.31) and (B.32), expressing Y ( z ,QJ) in terms of Y ( z ,rp; 1);
11,
0 31
QUANTUM-STATE RECONSTRUCTION
113
(D’Ariano [1995], D’Ariano, Leonhardt and Paul [1995], Leonhardt, Paul and D’Ariano [1995], D’Ariano [1997a], D’Ariano and Pans [1997a]). It has been shown that Kmn(x,cp; r ] ) is a well-behaved bounded function provided that r] > 1/2. It is worth noting that the reconstruction formula (91) also applies to other than harmonic-oscillator systems (Leonhardt and Raymer [19961, Richter and Wiinsche [ 1996a,b], Krahmer and Leonhardt [ 1997b,c], Leonhardt [ 1997a1, Leonhardt and Schneider [1997]). To be more specific, p(x, cp) in terms of pmn reads as38
where g m n ( x ) = VJ;(X>
(99)
%(XI,
and from eqs. (91) and (98) together with eqs. (92), (94) and (99) it follows that the functions nfmn(x),eq. (94), are orthonormal to products of energy eigenfunctions gmn(x),eq. (99),
n / d ~ j ~ , , ( x ) g ~ + , ~sn,mlsnnf ( x ) = for E , - E , =E,~-E,,~.
(100)
can be the regular and irregular solutions, It can be shown that VJm(x)and qm(X) respectively, which solve a Schrodinger equation,
for chosen energy Em, with U ( x ) being an arbitrary potential39. This offers the possibility of reconstruction of the density matrix (in the energy representation) of a particle in an arbitrary one-dimensional potential U ( x ) from the timedependent position distribution of the particle. The quadrature-component distribution in eq. (91) must be regarded as a time-dependent position distribution p(x,t) = (xl$(t)lx) and the phase integral converts into a time integral,
/ / +T/2
pmn= lim !! T-00
T
-T/2
dt
dx eiYmntfntn(x)p(x, t),
(102)
[pmn = pntn(t)1,-0; vntn,transition frequencies]. Obviously, the position distribution p ( x , t ) can also be used for reconstruction of other quantum-state Note that substituting in eq. (91) for p(x, q) the expression on the right-hand side of eq. (98), carrying out the tp integral and using the orthonormalizationrelation (100) just yields an identity. 39 For a proof, see, e.g., Leonhardt and Schneider [1997], Leonhardt [1997c].
38
114
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI, 5 3
representations, such as tomographic reconstruction of the Wigner function. Note that the time interval T , in which the position distribution can be measured is limited in general (for a data analysis scheme for determining the quantum state of a freely evolving one-dimensional wave packet; see Raymer [1997]; see also Q 4.4.1). In the numerical implementation of eq. (91) [or eq. (102)] the sampling function can be calculated on the basis of the analytical result given in eqs. (92) and (94), using appropriate numerical routines (Leonhardt, Munroe, Kiss, Richter and Raymer [1996]). An alternative way is the direct inversion of the underlying basic eq. (98) that expresses the quadrature-component distributions p(x, cp) in terms of the density-matrix elements pmn (Tan [1997])40. For any physical quantum state the density-matrix elements must eventually decrease to zero with increasing m(n). Therefore it follows that the expression on the right-hand side of eq. (98) can always be approximated to any desired degree of accuracy by setting ennrn M 0 for m(n) > urnax. To obtain the (finite) number of density-matrix elements from the measured quadrature-component distributions, the resulting equation can always be inverted numerically, using standard methods, such as least-squares inversion (Appendix D; see also Q 3.9.2). It should be pointed out that the method also applies when the detection efficiency is less than unity. Recalling eqs. (19) and (20), it is easily seen that p(x, q;q ) is related to en,,, according to eq. (98), with gmn(x)being replaced with
en,,?
gnn~(x;q ) = J’h’viz(x’)vn(x’)p(x
q).
(103)
When the characteristic function Y(z, q) of the quadrature-component distribution p(x, q) can be measured directly (Q 4.2. l), then the density matrix in the Fock basis can be obtained from Y ( z ,q) by replacing in eqs. (91) and (92) the x integral with the z integral over f (z)Y(z, q), where -nin
More explicitly, the result can be given by (Wallentowitz and Vogel [1996b])
en,l+k =
l7 1
rxi
d q eCikT
dz SAk’(z)
Re Y(z, q) if k even, Im Y ( z ,q ) if k odd,
(105)
The underlying basic equation for eq. (102) is given by eq. (98) with p(x. t ) and e?’~~~f*‘ in place of p ( x , q )and e-i(n4-n)$, respectively. Inverting it numerically, the time need not be infinitely large as it might be suggested from the analytical result given in eq. (102) (Opatm9, Welsch and Vogel [ 1997cl).
40
11,
8 31
115
QUANTUM-STATE RECONSTRUCTION
(k 3 0), with
(5)
$qz) =a ,/T ~
J-C
( n + k)!
k+l
~;k)(~2/2) e-z2/4
(-2)"2
if keven,
(-2)(k-1)'2 if k odd, (106)
[ ~ ; ~ ) (Laguerre x), polynomial]. So far, reconstruction of arbitrary quantum states has been considered, which can require measurement of the quadrature components at a large number of phases (see also $ 3.9.1), which reveals that the Pauli problem [i.e., reconstructing a quantum state from p(x, q) and p(x, q~ + n/2)] cannot be uniquely solved in general. However, if there is some a priori information on the quantum state to be reconstructed, then p(x, q) need not be known for all phases within a n interval. In particular when the state is known to be a pure state that is a finite superposition of Fock states,
n=O
then it can be reconstructed from two quadrature-component distributionsp(x, q) and p(x, q +n/2) 4 1 . In this case, the problem reduces to solving blocks of linear equations for the unknown coefficients in the Fock-state expansion of the state (Orlowski and Paul [ 19941)42. 3.3.2. Sampling of the displaced Fock-states on a circle
From $2.1.I we know that in unbalanced homodyning the photon-number distribution of the transmitted signal mode is, under certain conditions, the displaced photon-number distribution of the signal mode, pm(a), the displacement The Pauli problem of determining the quantum state of a particle from the position distribution and the momentum distribution has been studied widely, and it has turned out that it cannot be solved uniquely even if the particle moves in a one-dimensional potential and is prepared in a pure but arbitrary state (for the problem including finite-dimensional spin systems, see Pauli [1933], Feenberg [1933], Kemble [1937], Reichenbach [1946], Gale, Guth and Trammell [1968], Lamb [1969], Trammell [1969], Band and Park [1970, 19711, Park and Band [1971], d'Espagnat [1976], Kreinovitch [1977], Corbett and Hurst [1978], PrugoveEki [1977], Corbett and Hurst [1978], Vogt [1978], Band and Park [1979], Park, Band and Yourgrau [1980], IvanoviC [1981, 19831, Moroz [1983, 19841, Royer [1985, 19891, Friedman [1987], PaviEiC. [1987], Wiesbrock [1987], Busch and Lahti [1989], Wootters andFields [1989], Stulpe and Singer [1990], Bohn [1991], Weigert [1992]). 42 These coefficients can also be obtained from p ( x , cp) and a/dp p(x, q)IT=o Wchter [1996c]). More generally, it can be shown that a pure state can always be determined from p(x,cp) and Warp p(x, cp); i.e., from the position distribution and its time derivative in the case of a particle that moves in a one-dimensional potenial and is prepared in an arbitrary pure state (Feenberg [1933], for the problem, see also Gale, Guth and Trammell [1968], Royer [1989]).
41
116
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTlON
[II, § 3
parameter a = 1 ale'Qlbeing controlled by the local-oscillator complex amplitude [see eq. (51)]. Expanding the density operator in the Fock basis, p,(a) can be related to the density matrix of the signal mode as
where the expansion coefficients (nlm,a) can be taken from eq. (B.15) or eq. (B. 16). Equation (108) can always be inverted in order to obtain en,,, in terms O f p k ( a ) (Mancini, Tombesi and Man'ko [1997], Mancini, Man'ko and Tombesi [19971)43 :
where
is bounded for s E (-l,O], the 6 operator being given by eq. (B.19). This offers from the displaced Fock-state probability the possibility of direct sampling of pnnl,, distribution pm(a). Since pm(a)as a function of a for chosen m already determines the quantum state ($9 2.1.5 and 3.3.3), it is clear that when m is allowed to be varying, then in contrast to eq. (109) - p m ( a )need not be known for all complex values of a in order to reconstruct the density-matrix elements @kn fromp,(a). In particular, it is sufficient to knowp,(a) for all values of m and all phases 47, la\ being fixed (Leibfried, Meekhof, King, Monroe, Itano and Wineland [1996]44;Opatrn? and Welsch [ 19971, Opatrnl, Welsch, Wallentowitz and Vogel [ 19971). For chosen lal, we regardp,,(a) as a function of cp and introduce the Fourier coefficients,
Equation (109) follows directly from eqs. (B.20) and (129). Here the method was first used for reconstructingexperimentally the density matrix of the centerof-mass motion of a trapped ion (see also § 4.2.2).
43
44
K o 31
QUANTUM-STATE RECONSTRUCTION
117
(s = 0, 1,2,. . .), which are related to the density-matrix elements whose row and column indices differ by s as
where
with J = min(m, n + s) and L = min(m, n). Inverting eq. (1 12) for each value of s yields the density matrix sought. Since there has not been an analytical solution, setting ~ , ern,,M 0 for m(n) > nmax eq. (112) has been inverted n ~ m e r i c a l l y ~ (cf. the last paragraph but two of Q 3.3.1) and using least-squares inversion (Appendix D; Q 3.9.2). In this way, can be given by
where it is assumed thatpn is measured for n = 0,1,2,. . . ,N , with N < nmax,and the matrix F;,,(lal) is calculated numerically. Combining eqs. (1 14) and (1 1 l), can be given in a form suitable for statistical sampling. An extension of eq. (1 12) to nonperfect detection is straightforward. In this case, eq. (50) applies, and the measured probability distribution Pm(a;q) can be related to pnl(a)as shown in Appendix C. From eqs. (1 11) and (C.3), it can be seen that when in eq. (1 12) the Fourier component p;(lal) is replaced with the actually measured one, then the matrix G;,(lal) must be replaced with the matrix
Similarly, multichannel detection of the photon-number distribution can be taken into account. In particular, when a photon-chopping scheme as outlined in
45
For examples, see, e.g., Opatmy, Welsch, Wallentowitz and Vogel [1997].
118
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
tII, § 3
Q 2.1.7 is used, then the Fourier components of the measured N-coincidenceevent probability distribution can again be related to the density-matrix elements according to eq. (112), but now with Ghn(IaI>V,W
=C P m l k ( W k l l ( V ) k.1
GYn(IaI)
(1 16)
in place of Gkn(la1,v), where Pmlk(N)can be taken from eq. (53). 3.3.3. Reconstruction from propensities As already mentioned, the displaced-Fock-state probability p,(a), eq. (1 08), as a function of a delines for each m a propensity prob(a) = .n-‘pm(a)of the type given in eq. (40). The simplest example is the Q function (m = 0), which can be measured in perfect eight-port balanced homodyne detection, with the vacuum as quantum filter (0 2.1.4). Propensities contain all knowable information on the quantum state, and therefore all relevant properties of it can be obtained from them in principle. However, in order to obtain the “unfiltered” quantum state, the additional noise introduced by the filter must be “removed”, which may be an effort in practice. When the Q function is known, then the density matrix in the photon-number basis can be calculated straightfowardly, using the well-known relation46
which corresponds to
(Wiinsche [1991, 1996a1). Equation (118) can also be extended to other than vacuum filters, in order to obtain the density operator in terms of derivatives of more general propensities of the type given in eq. (40) (Wiinsche and Buiek [1997]). In order to obtain the density matrix from the measured homodyne data, derivatives on them must be carried out, which probably could be done with sufficient accuracy only for states whch contain very few photons. Writing .nela12Q(a) = eIal2(alfila) = Em,, e ~ a ~ ’ ( a ~ m ) ( m ~ ~ and ~ ~ ) recalling ( n ~ a ) , that (nla)= (n!)-1’2ane-la12’2, eq. (117) can be derived easily.
46
11,
o 31
QUANTUM-STATE RECONSTRUCTION
119
In practice, it may be more convenient to handle integrals rather than derivatives. It can be shown that4'
r
-
(119) where m 3 n and s 6 -1, L:(z) being the Laguerre polynomial (Paris [ 1996b1). In particular, when s = 1 -2q-', then P(a;s) is just the smoothed Q function measured in nonperfect detection (cf. 0 2.1.4). Unfortunately, eq. (1 19) is not suitable for statistical sampling, since the integral must be performed first and after that the summation can be carried out. Moreover, the inaccuracies of the measured P(a; s) together with the Laguerre polynomials can give rise to an error explosion in the reconstructed density matrix, so that an exact reconstruction of the density matrix from measured data may be expected to be possible only for states which contain finite (and not too large) numbers of photons. In th~scase, the p sum in eq. (1 19) can be truncated at p = N - rn, where the value N has to be chosen large enough to ensure that ( l i t i ; l j ) = 0 for i , j 2 N . Now the p sum can be performed first, and a (state-dependent) integral kernel for statistical sampling can be calculated (Paris [ 1996b,c]). 3.4. MULTIMODE DENSITY MATRICES
The extension of the methods outlined in 99 3.2 and 3.3 to the reconstruction of multimode density matrices from the corresponding multimode joint quadraturecomponent distributions or multimode joint propensities is straightforward. The situation is somewhat different when combined distributions, i.e., distributions that are related to linear combinations of the modes, are measured. Let us consider the two-mode detection schemes shown in fig. 4 in 0 2.1.3. When the sum quadrature-component distribution of two modes, ps(x, a , cpl , @), is known for all phases q11 and @ within JT intervals and all superposition parameters a E ( 0 , ~ / 2 ) ,then it can be shown, on recalling eq. (30), that the reconstruction of the two-mode density matrix in a quadrature-component basis
(B.22) to k = In)(ml and calculating the c-number fimction F(a;s) associated with fi in chosen order. Note that for s = -1 the integral form (119) corresponds to the differential form (117).
47 Equation (119) can be derived, applying eq.
120
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[K 5 3
can be accomplished with a three:fold Fourier integral (Opatrnq, Welsch and Vogel [ 1996, 1997b1):
where
and
(k = 1,2148. The generalization to the N-mode case is straightforward. Suppose that we can measure the probability distribution of a weighted sum of quadratures f = a[})&, where a/ are N - 1 parameters which can be controlled in the experiments, 1 = 1,2,. . . N - 1, and fk are real functions which satisfy C:=,f,2({a[}) = 1 identically for each set of parameters {a/}.Let gr be the inversions of fk, gr[{fk({am})}] = a[ (i.e., the set { a [ }parametrizes the surface of an N-dimensional sphere). From the measured sum quadraturecomponent probability distribution ps(x, {a[},{ qk}) its characteristic function Ys(z,{a,},{qk}) can be calculated as a Fourier transform. The characteristic function of the joint quadrature-component probability distribution can then be calculated as
zr=
(xf=
[z = z ~ ) ~ ” ]from , which the N-mode density matrix can be obtained by an N-fold Fourier transform; the whole N-mode density matrix reconstruction is thus accomplished by an N + 1-fold integration of the measured data.
48
For an extension to imperfect detection, see Opatrn?, Welsch and Vogel [1997b].
11,
P 31
QUANTUM-STATE RECONSTRUCTION
121
In the Fock basis the reconstruction of a two-mode density matrix from the combined quadrature-component distributionps(x, a, 971,@) can be accomplished with a four-fold integration (Raymer, McAlister and Leonhardt [ 19961, McAlister and Raymer [1997b], Richter [1997a]): (ml,mzlGlnI,nz) = S % = d a ~ = d ~ l ~ = d ~ R , : : ~ (cpl@)Ps(x,a, x,a, 971, Qb), ( 124) where the integral kernel R:$;(x, a, 971472) is suitable for application of statistical sampling. It can be given by
where
(126) ( y ) and &I ( y ) are harmonic-oscillator energy eigenfunctions, and Here, yml fm,n,(y) is given by eq. (94). An alternative integral expression for rz:,":(x,a) reads
with f-mn (z) being given by eq. (104). Since f-mn (z) is the kernel fbnction for reconstruction of the single-mode density matrix in the Fock basis from the quadrature-component characteristic function Y ( z , @),from eqs. (105) and (106) it is seen that it can be expressed in terms of the associated Laguerre polynomial. Using t h s in eq. (127), then the integral can be performed to obtain a representation of r:$,":(x,a ) as a finite sum over confluent hypergeometric hnctions (Richter [ 1997a1). Finally, the problem of reconstruction of the quantum state of unpolarized light was studied, by considering the (two-mode) density operator
(Lehner, Leonhardt and Paul [ 19961). Obviously, the probability distributionp n of finding n completely unpolarized photons in the signal field characterizes uniquely the quantum state (128). Two schemes for determiningp n were studied.
122
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[I& § 3
First, expressions for the sampling function were derived for the case when the two orthogonal polarization modes are delivered to two balanced homodyne detectors and the joint quadrature-component distributions are measured (Krahmer and Leonhardt [1997a]. Second, it was shown that measurement of the quadrature-component distributions of any linearly polarized component of the signal field is sufficient to determinepn,the corresponding sampling function being closely related to the single-mode hnction (Richter [1997b1). 3.5. LOCAL RECONSTRUCTION OF P(n;s)
The displaced photon-number statistics measurable in unbalanced homodyning (0 2.1.7) can be used for a pointwise reconstruction of s-parametrized phasespace functions (Wallentowitz and Vogel [ 1996a1, Banaszek and W6dkiewicz [1996])49.From eq. (B.21) together with eqs. (B.19) and (B.13) it is easily seen (cf. also Moya-Cessa and Knight [ 19931) that
Hence, all the phase-space functions P( a ; s ) , with s < 1, can be obtained from pol(@for each phase-space point a in a very direct way, without integral transformations. In particular, when s = -1, then eq. (129) reduces to the wellknown result that Q(a) = .n-'pO(a), with po(a) = (alela).Further, choosing s = 0 in eq. (129), we arrive at the Wigner function,
Equation (130) reflects nothing but the well-known fact that the Wigner function is proportional to the expectation value of the displaced parity operator (Royer [19771)5 0 . The method was first applied experimentally to the reconstruction of the Wigner function of the center-of-mass motion of a trapped ion (Leibfned, Meekhof, King, Monroe, Itano and Wineland [1996]; 0 4.2.2). From 0 2.1.7 we know that in unbalanced homodyning the quantum efficiency q = lukl l2qO is always less than unity, even if qD = 1, because of l ~ k I l < 1.
4y 50
For a squeezed coherent local oscillator, see Banaszek and Wodluewicz [1998]. For proposals of measuring the Wigner h c t i o n of a particle using this fact, see Royer [1985,
19891.
11, § 31
Q U A N m - S T A T E RECONSTRUCTION
123
Equation (129) can be extended, to imperfect detection in order to obtain (Wallentowitz and Vogel [ 1996a1, Banaszek and Wodkiewicz [ 1997bl)5 1
[a = -(UkZ/ukI)a~,, eq. (49)], where the measured photon-number distribution P , can be regarded as a smoothed displaced photon-number distributionp,(a; q ) of the signal mode. The method works very well and is we1 suited for statistical sampling if s < 1 - q-’ ;i.e., when the weighting factors improve the convergence of the series52. Hence reconstruction of the Q function with reasonable precision is always possible if q > 1/2 (for computer simulations of measurements, see Wallentowitz and Vogel [ 1996a1, Banaszek and Wodkiewicz [ 1997b1). As already mentioned in Q 2.3, measurement of the photon-number distribution of a linearly amplified signal can also be used - similarly to measurement of the displaced photon-number statistics in unbalanced homodyning - for reconstructing the quantum state of the signal mode. It can be shown that when the idler mode is prepared in a coherent state, then the phase-space funcion P(a;s) of the signal mode can be related to the measured photon-number distribution as 53
(Kim [ 1997a,b]), where a = -[(g - l)/g]”2a;. Note that the gain factor g and the quantum efficiency m of the detector enter separately into eq. (132) [in contrast to q = IUkl12rlrin , eq. (131)l. 3.6. RECONSTRUCTION FROM TEST ATOMS IN CAVITY QED
Let us now turn to the problem of reconstruction of the quantum state of a highQ cavity field from measurable properties of test atoms in detection schemes 5 1 This can be easily proved correct, applying eq. (C.4) and expressingp,(a) in eq. (129) in terms of Pm, eq. (50). 52 The feasibility of reconstruction of the Wigner function of truncated states was also demonstrated for 11 < 1 (Wallentowitz and Vogel [1996a]). 53 From eq. (C.5) it can be found that for a = 0, eq. (129) relates the measured distribution P , to the phase-space function of the detected mode at the origin of the phase space, Pdet(O,S). Equation (132) can then be proved correct, using the unitary tranformation (63) and expressing Pd,(a,s) in terms of phase-space functions of the signal and idler modes by convolution (see also Leonhardt [ 19941, Kim and Imoto [ 19951).
124
H O M O D m DETECTION AND QUANTUM-STATE RECONSTRUCTION
DL § 3
outlined in 0 2.4. Though at a first glance the schemes look quite different from the homodyne detection schemes, there are a number of remarkable similarities between them. 3.6.1. Quantum state endoscopy and related methods Let us first consider a two-level (test) atom that interacts resonantly with a singlemode cavity field according to a k-photon Jaynes-Cummings model, the atomfield interaction Hamiltonian being given by
which for k = 1 reduces to eq. (65). When the atoms are initially prepared in superposition states 1 6 )= 2-1(2(18) fe-iv 1 . ) and the excited-state occupation probabilities P:(t) are measured as functions of time, then the cavity-mode density-matrix elements @,,n+k can be determined (Vogel, Welsch and Leine [1987]). To be more specific, it can be shown that the difference P;(t) - P,'(t) reads as
i.e., the off-diagonal density-matrix elements enn+k can be obtained dlrectly from the coefficients aLk) for two phases I/J, such as I/J = 0 and )I = d 2 . Provided that the interaction time t can be varied in a sufficiently large interval (0, T ) , the Fourier transform of P;(t) - P,'(t) consists of sharp peaks, whose values yield the sought coefficients aik' as 54
If T is not large enough, then the peaks in the Fourier integral contain non-negligible contributions of the tails of the corresponding sinc functions. In this case, the coefficients aLk' can be calculated from a set of linear equations obtained from eq. (134) for different times. 54
11,
o 31
125
QUANTUM-STATE RECONSTRUCTION
enn,
(T -+ m). To measure the diagonal density-matrix elements it is sufficient to prepare the atom in the excited state, P , ( t ) l , = ~= 1, and observe (for arbitrary k) the atomic-state inversion AP = P, - Pg = 2Pe - 1,
en,,
from which can be obtained by Fourier transformations5. In cavity QED the 1-photon Jaynes-Cummings model is typically realized, so that the method - also called quantum state endoscopy - does not yield the off-diagonal density-matrix elements @,,,,+k with k > 1. When the quantum state is known a priori to be a pure state such that is given by = cc ,C , with C,C;+~ $ OVrn56,then eq. (134) [together with eq. (135)] can be taken at a sufficiently large number of time points (and at least at two phases) in order to obtain [after truncating the state at a sufficiently large photon number nmaxaccording to eq. (107)] a system of conditional equations for the expansion coefficients c,, whch can be solved numerically (Bardroff, Mayr and Schleich [ 19951, Bardroff, Mayr, Schleich, Domokos, Brune, Raimond and Haroche [1996]). The reconstruction problem for arbitrary quantum states can be solved by performkg a displacement of the initial state of the cavity field such that 8 is replaced with b t ( a ) $ b ( a )and , hence ( k = 1):
em,,
em,,
(8= 2 ~ K , = id1)), where &(a) = (nlbt(a)$b(a)ln)= (n,al$[n,a). Now Qnn(a) can again be obtained from U ( t ) by Fourier transformation, and fiom &(a) the quantum state of the cavity mode can be obtained, applying, e.g., the methods outlined in $8 3.3.2 and 3.5. Alternatively,the quantum state can also be reconstructed when the interaction time is left h e d and only a = (alei'J'is varied
55 For k = 1, very precise measurements of the Rabi oscillations have been performed recently (Brune, Schmidt-Kaler, Maali, Dreyer, Hagley, Raimond and Haroche [1996]), the peaked structure of the Fourier-transformed data being interpreted as a direct experimental verification of field quantization in a cavity. s6 This condition is not satisfied, e.g., for even and odd coherent states as typical examples of Schr6dinger-cat-like states. For even and odd coherent states, see Dodonov, Malkin and Man'ko [1974]. For a review of Schrodinger cats, see Buiek and Knight [1995].
126
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
[K 0 3
(Bodendorf, Antesberger, Kim and Walther [ 19981). Regarding the measured atomic occupation probabilities as functions of a = (a(eip,M ( t ) = AP(t,a), and introducing for chosen t and la1 the Fourier coefficients,
ixi2x
hP”(t,la[)= -
d q eiSqAP(t,a),
(139)
= 0,1,2,. . .), it can be shown that they are related to the density-matrix elements by equations of the form
(s
(for explicit expressions for Y;(t, la\), see Bodendorf, Antesberger, Kim and Walther [1998]). Inverting eq. (140), whch resembles, in a sense, eq. (112) in Q 3.3.2, for each value of s then yields the density matrix of the cavity mode. Similarly to eq. (112), the inversion can be carried out numerically; e.g., by means of least-squares inversion (Appendix D), choosing an appropriate set of values of lal. In the two-mode nonlinear atomic homodyne detection scheme (Wilkens and Meystre [ 19911) it is assumed that the signal cavity mode is mixed with a local oscillator cavity mode according to the interaction Hamiltonian
Equation (138) can then be found treating the effect of the local oscillator semiclassically (i.e., replacing the operator 6~ with a c number aL,i i ~ --+ aL).In this case, the scheme is obviously equivalent to an initial displacement a = -aL of the density operator of the cavity mode, so that the atomic-state inversion W ( t )is given exactly by eq. (138). In particular, when laLI is sufficiently large, then AP(t) can be rewritten as
In other words, for I aL I 4 ca the atomic occupation probabilities Pe(g)(t)can be related directly to the characteristic function @(p)of the Wigner function W ( p ) of the cavity mode. Varying the interaction time and the phase q~ of aL, the whole function @(p)can be scanned in principle. Knowing @(p),the
KQ
31
QUANTUM-STATE RECONSTRUCTION
127
Wigner function can then be obtained by Fourier transformation5'. Since for appropriately chosen arguments @( /3) is nothing but the characteristic function of the quadrature-component distribution p(x, 6) for all values of 6 within a n interval (see Appendix B.5), the density matrix in both a quadrature-component basis and the photon number basis can be reconstructed straightforwardly from eq. (142) (08 3.2 and 3.3.1). Later it was found that the semiclassical treatment of the local oscillator restricts the time scale to times less than a vacuum Rabi period, because of the quantum fluctuations in the local oscillator cavity mode (Zaugg, Wilkens and Meystre [ 19931, Dutra, Knight and Moya-Cessa [ 1993]), and it was shown that this difficulty can be overcome when the atoms are coupled weakly to the local oscillator but coupled strongly to the signal (Dutra and Knight [ 19941). A priori knowledge on the quantum state to be measured is required in magnetic tomography (Walser, Cirac and Zoller [1996]). Here, the idea of quantum state mapping between multilevel atoms and cavity modes prepared in truncated states (Parkins, Marte, Zoller, Carnal and Kimble [1995]) is combined with a tomography of atomic angular momentum states by SternGerlach measurements (Newton and Young [1968]). It is assumed that an angular-momentum degenerate two-level atom passes adiabatically through the spatial profile of a classical laser beam [Rabi frequency: Q(t)] and, with a spatioternporal displacement t > 0, through the profile of a quantized cavity mode [atom-cavity coupling: g(t - t)] such that the coupled atom-cavity system evolves according to the time-dependent Hamiltonian
1,Jg,Je
Co,m s ,
being Clebsch-Gordan coefficients.
57 Recall that in the dispersive regime the Wigner function can be measured directly (Lutterbach and Davidovich [1997]; see 5 2.4), without any reconstruction algorithm.
128
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
PI, § 3
It can be shown that if the time-dependent change of the Hamiltonian during the total interaction time is much less than the characteristic transition energies and if the delay and shape of the pulse sequences are chosen such that
then a coupled atom-cavity-field density operator $AF that can be factorized initially into a pure atomic state and a field state containing less than 2Jg photons will be mapped to a product of atomic ground state superpositions and the cavity vacuum
with
and 58
srn,snbeing possible sign changes. Hence, if the final atomic density matrix (Jg, Jg - rn - 1[$A lJg,Jg - n - 1) is known, then the original cavity-field density matrix (ml$F)Fln)is known. Atomic states of this type can be determined from magnetic dipole measurements using conventional Stern-Gerlach techniques (see 04.5). 3.6.2. Atomic beam defection
When a cavity mode is known to be in a pure state, then the expansion coefficients of the state in the photon-number basis can be inferred from the measured deflection of two-level probe atoms during their passage through the cavity (Freyberger and Herkommer [ 19941; for an application of related schemes to the reconstruction of the transverse motional quantum state of two-level atoms,
5 8 Note that with reverse adiabatic passage, an internal atomic state is prepared uniquely by reading out the cavity state.
a, o 31
QUANTUM-STATE RECONSTRUCTION
129
see 44.4.1). A narrow slit put in front of one node of the standing wave (see fig. 8) transmits the atoms only in a small region Ax << A centered around x = 0 (x, atomic center-of-mass position; A, wavelength of the mode). The dependence on 2 of the coupling strength K = ~ ( 2 =) K’ sin(k2) in eq. (65) can then be approximated by K = K‘ k2, k = 2jt/A, so that the interaction Hamiltonian of the combined system reads as
It is also assumed that the system is initially prepared in a state I Y )= I YF)I YA), where
The subscripts F and A label the states of the cavity field and the atom, respectively, v(x) being the initial wave function of the atomic center-of-mass motion. Assuming c = 1 and calculating the temporal evolution of the state vector in the Raman-Nath regime5’, it can be shown that the atomic momentum probability density for the interaction time t is given by 00
IW(p + f i h k ~ ’ t ) l ~+(eiqcn-l ~ , l2
P ( p , t ) = ahk n= I
W
+ ihk
IW(p - fihkK’t)121cn- eiqcn_ll2
+ $hklt,b(p)121co12, -
n= I
(151) where v ( p ) is the initial wave function of the atomic center-of-mass motion in the momentum basis. For an initial position distribution lv(x)I2, such as a Gaussian, with Ax << A, and sufficiently strong interaction, i.e., K’t CQ, the corresponding shifted momentum distribution Iv ( p f Ji;hkK’t)12 becomes sharply peaked at p = T f i f i k K ’ t , n = 1,2,.... In this case, Iv(p)12 and IY(p f fihkK’t)12 obviously select in the measured momentum distribution
in this approximation it is assumed that the transverse displacement of the atoms during the interaction time t is small compared with the wavelength of the field, so that the kinetic energy of the atomic center-of-mass motion can be disregarded in the Hamiltonian of the combined system. For times longer than the interaction time, the now deflected atoms move freely, and the spatial distribution of the atoms on a screen put up far away from the resonator is a picture of their momentum distribution at the moment when they leave the interaction zone. 59
130
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II,
P3
P ( p , t ) a sequence of peaks ,whose heights are proportional to lc0l2 and (c,, f e1pc,,-112,respectively, for n = 0 and TI > 0. In this way (c0l2 and if the measurement is performed at two phases cp - the sequence of products c$cI,cTc2,.. . can be obtained, from which (after truncating the state at a sufficiently large photon number nma) the expansion coefficients c,, can be calculated easily (up to an unimportant overall phase factor) 60. To allow the cavity mode to be prepared in a mixed state, a running reference field aligned along the y-axis (i.e., perpendicular with respect to the standingwave cavity field aligned along the x-axis and the z-direction of the moving atoms) can be included in the scheme such that the atoms initially prepared in the electronic ground state interact simultaneously with the two fields (Schneider, Herkommer, Leonhardt and Schleich [1997]). In the dispersive regime the effective interaction Hamiltonian for the system can then [on extending eq. (66) to two modes] be given by
Here the index L refers to the running reference field, which has the same frequency as the cavity field and can play a similar role as the local oscillator in a homodyne detector. In the calculations it is assumed that the atoms are initially prepared in a state I YA)of the form given in eq. (150) with c = ' 0, and the reference field is prepared in a strong coherent state and can be treated classically ( 2 ~ -+ aL = laL[ ei'J'L, 1 aL1 -+ m) 6 1 . The cavity field can be prepared in an arbitrary mixed state described by a density operator 0. Again making the approximation ~ ( 2M) K'k,? and asuming that the initial atomic wave function v ( x , y )factorizes into a very narrow y-wave function 6(y -yo) and a Gaussian in the x-direction, it can be shown that the transverse momentum distribution P ( p ,t ) of the atoms for an interaction time t is a (scaled) smeared quadraturecomponent distribution p(x, cp; q) of the cavity mode,
analysis of a diffuse peak structure, see Freyberger and Herkommer [1994]. Note that an atom that is initially in the electronic ground state (c = 0) stays in the electronic ground state, because of the effective Hamiltonian (152). 6o For an
K P 31
QUANTUM-STATE RECONSTRUCTION
131
k = QLK’hklaLlt/6. Here, the quantum efficiency 17 with which the quadraturecomponent distribution is measured is given by
and the phase ~1 is determined by the reference-field phase and the position yo, cp = - c p ~ -kyo. The result proves that the atomic probe carries the (smeared) quadrature-component information on the cavity field that corresponds to the reference angle 47. Hence, after having measured the transverse momentum probability distribution P ( p , t ) of the deflected atoms for a sequence of phases cpL of the (local-oscillator) reference field, the phase-space function P(q,p;s = 1 - up’) of the cavity mode can be reconstructed from eq. (75) by means of tomographic imaging ( Q 3.1). Note that the larger the effective coupling constant Fi is, the closer is P ( p ,q;s) to the Wigner function. However the initial position uncertainty must not exceed a fraction of the wavelength of the cavity mode, because of the approximations made. Needless to say, the interaction time t must be small compared with the decay time of the cavity mode. 3.1. ALTERNATIVE PROPOSALS
Since the quantum state of a radiation field mode is known when its densitymatrix elements emnin the photon-number basis are known, it was proposed to measure them directly (Steuernagel and Vaccaro [ 19951). The diagonal elements can be measured by direct photodetection in principle. In order to measure the off-diagonal elements, it was proposed to prepare a probe field in a superposition of two photon-number states, Jamn)= N d l m ) + a In)),
(155)
N, = (1 + ~ U I ~ ) - ~ ’ ~ and , combine it with the signal mode at a beam splitter. The observed joint photon-number probabiltity in the two output channels of the beam splitter then reads as (m > n ) pq,k+m-q (bkllolbkl),
( 156)
1 - k = m - n, where Ibkl) is again a state of the type given in eq. (159, so that
When for chosen difference, k - I = n - m, the diagonal elements e k k are - known from a direct photon-number measurement, then the offhagonal elements p k k+m-n, k = 0,1,2,. . ., can be obtained from two (ensemble)
and
132
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II,
03
measurements for different superposition parameters. The whole density matrix can then be obtained by means of a succession of (ensemble) measurements, varying the difference m - n of photon numbers in the reference state (155) from measurement to measurement. The presently unresolved problem, however, consists in building an apparatus that prepares travelling waves in superpositions of two photon-number states Im) and In) for arbitrary difference m--12 in a controllable way. For a radiation-field mode that is known to be prepared in a pure state I Y) and that contains only a finite number of photons, eq. (107), it was proposed to extend the method of photon chopping outlined in 0 2.1.7 to a complete determination of the expansion coefficients c, (Paul, Torma, Kiss and Jex [ 1996a, 1997a1)62. First the photon-number distribution pn = (c,t2 is determined according to eqs. (52)(54). Then the balanced 2N-port apparatus (N 2 nmax)is used for mixing the signal mode with N - 2 reference modes in the vacuum state and one reference mode prepared in a coherent state la) (in place of the N vacuum reference inputs in 6 2.1.7). Using the selected statistics when only photons in the first N/2 output channels are observed, it can be shown that the probability p,(N) of detecting n photons with at most one photon in each channel is given by (Paul, Torma, Kiss and Jex [1996a]) p,(NI=
I
I
I Lm)
- >,
d(n- m)!a"'cn-m .
(158)
m=O
Determining the coincidence probability distribution p,(N) for two coherent states with different phases, the phases cp, of the expansion coefficients c, = Ic,Jel" can be calculated step by step (for known amplitudes Ic,( and up to an overall unimportant phase) from the two measured coincidence-event distributions.More involved formulas are obtained in the case when the complete coincidence statistics are included in the analysis; i.e., photons are allowed to be observed in any output channel and realistic photodetection is considered (Paul, Torma, Kiss and Jex [ 1997a1). It turns out that the reconstruction scheme with two coherent states requires photodetectors which can discriminate between 0,1,2,. , . photons; i.e., detectors which have rather low quantum efficiency. Using avalanche photodiodes, which can have high quantum efficiency, the reconstruction might become rather involved,because additional reference beams
For a proposal to extend the method to a two-mode signal field, see Paul, Torma, Kiss and Jex [ 1997bl.
62
11,
8 31
QUANTUM-STATERECONSTRUCTION
133
with different phase properties Must be used. Similarly, when a mixed quantum state is tried to be reconstructed, then extra reference beams (reference phases) must be used, the maximum number of reference beams being limited by the cutoff in the photon number. It can also be shown (Bialynicka-Birula and Bialyniclu-Birula [ 19941, Vaccaro and Barnett [ 19951) that when a radiation-field mode is prepared in a pure state I Y) that is a finite superposition of photon-number states, eq. (107), then it can be reconstructed from the photon-number distribution p n = /cnl2and the (PeggBarnett) 63 truncated canonical phase distribution Pps(@)= I q(@)12, with64
Taking eq. (159) at 2(n,,, + 1) values of @, such as 01 = h/(nmaX + l), the resulting equations can be regarded as conditional equations for the unknown phases qn,provided that all absolute values I~I(@I)I and Ic,I are known. Apart from the fact that the quantum state must be known to be a pure state, the question remains of how to obtain the phase statistics (see also 5 3.8.3). 3.8. RECONSTRUCTION OF SPECIFIC QUANTITIES
Since the density matrix in any basis contains the full information about the quantum state of the system under consideration, all quantum-statistical properties can be inferred from it. Let fi be an operator whose expectation value,
mn
is desired to be determined. One may be tempted to calculate it fiom the reconstructed density matrix (or another measurable quantum-state representation). However, an experimentally determined density matrix always suffers from various inaccuracies which can propagate (and increase) in the calculation process (cf. 53.9.1). Therefore it may be advantageous to determine directly the quantities of interest from the measured data, without reconstructing the
See Pegg and Barnett [1988]. Note that @(@) = ( @ p ~ ( @ ) l Ywhere ), IY) is given by eq. (107), and the truncated phase state J@ppg($))reads as l @ p ~ ( @ = ) ) (nma, + l)-"* C"""" n = O e&@\n) (cf. 8 3.8.3).
63
134
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, § 3
ern,,
whole quantum state. In particular, substituting in eq. (160) for the integral representation (91) one can try to obtain an integral representation,
with
suited for direct sampling of (k) from the quadrature component distributions p(x, q) [provided that both the sampling function KF(x,q) and the integral in eq. (161) exist]. 3.8.1. Normally ordered photonic moments It is often sufficient to know some moments of the photon creation and destruction operators of a radiation-field mode rather than its overall quantum state. Let us again consider the measurement of the quadrature-component probability distributions p(x, q) in balanced homodyning (0 2.1.2). Then the normally ordered moments of the creation and destruction operators, (;ltniirn) can be related to p(x, q) as6'
where H,(x) is the Hermite polynomial (Richter [1996b])66.Equation (163) is just of the type given in eq. (161) and offers the possibility of direct sampling of normally ordered moments (;lt";l") from the homodyne data. It is worth noting that knowledge of p(x, q) at all phases within a n interval is not necessary to reconstruct (;lt";l") exactly, and therefore the q integral in eq. (163) can be replaced with a sum. It was shown that any normally ordered moment (;lt";l") can already be obtained from p(x, q) at N = n + m + 1 discrete different phases Qlk (Wiinsche [1996b, 19971).
65 Equation (163) can be proven correct if both sides are expressed in terms of the density matrix in the photon-number basis and standard summation rules for the Hermite polynomials are used. 66 A more involved transformation was suggested by Bialymcka-Birula and Bialymcki-Birula [1995]; see footnote 77.
11,
8 31
QUANTUM-STATE RECONSTRUCTION
135
The method is especially useful, e.g., for a determination of the moments of photon number. Note that for finding the mean number of photons (2) from the Fock-basis density matrix a relatively large number of measured diagonal elements must be included into the calculation, in general, each of which being determined with some error. After calculation of the sum ( 6 ) = Enne,, the error of the result can be too large to be acceptable, and for higher-order moments severe error amplification may be expected. Equation (163) can be extended easily to nonperfect detection (q < l), since replacing p(x, cp) with p(x, cp; q ) simply yields q("+m)'2(iitniim) 67. Provided that the whole manifold of moments (2t"iim)has been determined, then the quantum state is known in principle (Wiinsche [1990, 1996b], Lee [1992], Herzog [1996b])6*. To be more specific, the density operator can be expanded as [cf. eq. (69)]
en,,
M
k.l= 0
where
[{k,Z} = min(k,Z)] provided that (iitkii') exists for all values of k and Z and the series (164) (in chosen basis) exists as well. Since all the moments do not necessarily exist for any quantum state, and the series need not necessarily converge for existing moments, the quantum-state description in terms of density matrices is more universal than that in terms of normally ordered moments69
Recall that an imperfect detector can be regarded as a perfect detector with a beam splitter in front of it, so that the destruction operator 2 of the mode that is originally desired to be detected i,s transformed according to a beam-splitter transformation (4 2.1. l), Cr( 7) = JiiCr + f i b , where b is the photon destruction operator of a reference mode prepared in the vacuum state. This is also true for other than normally ordered moments, such as symmetrically ordered moments (Band and Park [1979], Park, Band and Yourgrau [1980]). 6y An example of nonexisting normally ordered moments is realized by a quantum state whose photon-number distribution behaves like p n n-3 for n + co. Even though it is a normalizable state with finite energy, its moments (itk) do not exist for k 2 2. For a thermal state the relation (iit"~3~) = d,,n! A" is valid ( A , mean photon number), which implies that the series (164) for pmmn does not exist. Note that the problem of nonconvergence of the series (164) may be overcome by analytic continuation.
67
-
136
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[K5 3
The extension of the method to the, reconstruction of normally-ordered moments of multimode fields from joint quadrature-component distributions is straightforward. Moreover, normally-ordered moments of multimode fields can also be reconstructed from combined distributions considered in Q 2.1.3 (Opatrn-, Welsch and Vogel [1996, 1997a1, McAlister and Raymer [1997a,b], Richter [1997a]). For simplicity, let us restrict our attention here to two-mode (the subscripts 1 and 2 refer to the two modes) moments (i?pli?pi?yliiy) and assume that the weighted sum i = SZl(q1)cosa + iz(@)sina, eq. (29), is measured in a homodyne detection scheme outlined in 0 2.1.3 (see fig. 4). Measuring the moment (2") for n + 1 values of a, a set of n + 1 linear algebraic equations can be obtained whose solution yields the two-mode quadrahuecomponent moments ( i ; - k ( q l ) i $ ( @(k) )= 0,1,2,. . . ,n). Varying ql and @, the procedure can be repeated to obtain the dependence on q~ and cp2 of the twomode quadrature-component moments. These can then be used to determine step by step the normally ordered moments of the photon creation and destruction can be obtained operators of the two modes. In a closed form, (ii~'i?inzi?yli?~) from the combined quadrature-component distribution ps(x, a, ql,912) as
ei(nl-ml) nl+nz+m1+m2)(~)
x ~n,+n,+m1+m2(~)~~2+m2
e l i(n2-ml)
(166) [C,, = n!m!/(n2("+"Y2(n + m)!)].The functions F f ) ( a )form a biorthonormal system to the fimctions G t ) ( a )= (:) cosl-ka sinks in some Q interval, so that daFf)(a) Gg)(a)= S k p . Note that since for given 1 there is a finite number of functions Gf)(a),the system of functions F f ) ( a )is not determined uniquely and can be chosen in different ways. In particular, when only the phase difference A q = (p2 - ql is controlled and the overall phase ql is averaged out (this is the case when the signal and the local oscillator stem from different sources), then the moments (i?In1i?pi?yiiiT2)can be reconstructed for nl -ml = m2 -n2. If both phases ql and @ are averaged out, then those moments ( i i ~ n 1 i ? ~ n 2 ican i ~ 1still ii~) be obtained, which carry the information about the photon-number correlation in the two modes. The method was used to demonstrate experimentally the determination of the ultrafast two-time photon number correlation of a 4 ns optical pulse (McAlister and Raymer [1997a]). In the experiment, the local oscillator pulses are derived from a Ti:sapphire-based laser system that generates ultrashort, near transformlimited pulses (150 fs) at a wavelength of 830 nm and a repetition rate of
s,
11,
o 31
137
QUANTUM-STATE RECONSTRUCTION
-1 0
-5
0
5
10
7 (PS)
Fig. 15. The second-order coherence, eq. (167),,experimentally determined via balanced four-port homodyne detection (dots) and from the measured optical spectrum (solid line). The value of tl is set to occur near the maximum of the signal pulse. (After McAlister and Raymer [1997a].)
1 kHz. The signal is fi-om a single-spatial-mode superluminescent diode. The broadband emission at 830 nm is filtered spectrally to produce a 4 ns pulse having a 0.22 nm spectral width. For each value of the relative delay At = tl - t2 between two local-oscillator pulses the phase-averaged combined quadrature-component distribution p ~ ( xa) , = ( 2 . 7 ~J) dqp, ~ ~ J dcp2ps(x, a, q1, cp2) is measured for three different values of a: 0, n/2 and x/4. From these the normalized second-order coherence function,
is computed (6, and 62 being the photon destruction operators of the spatialtemporal modes defined by the local-oscillator pulses centered at times tl and t2). This experiment represents an extension of measurements of the HanburyBrown-Twiss correlations70 to a sub-picosecond region. Results are shown in fig. 15. 3.8.2. Quantities admitting normal-order expansion
The basic relation (163) can also be used [similar to eqs. (160)-(162)] to find sampling formulas for the mean values of quantities that can be given in terms 70 See Hanbury Brown and Twiss [1956a,b, 1957a,b]; for more discussion of the Hanbury Brown and Twiss experiments, see Pefina [I9851 and Mandel and Wolf [1995].
138
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
tII, § 3
of normally ordered moments of photon creation and destruction operators (D'Ariano [1997b]). Let us consider an operator h and assume that it can be given by a normal-order expansion as
n.m = 0
The mean value
(h)can then be obtained from p ( x , q;r ] ) according to 71
where the integral kernel, KF(x, fp; r ] )
p(% 1111
= Tr
%
>
[I?(x, q ;r ] ) being defined in eq. (97)] can be given by
with
(n + m )
-I
00
GF(z,q) =
fnm
(-i.z)"+mei(m-nf9,
n,m = 0
Here it is assumed that the integral kernel K&, q;q) exists and the x integral in eq. (169) exists as well72. The integral kernel (171) exists if for z + 00 the hnction G&, q) grows slower than eqz2'2,so that the integral in eq. (171) converges (for examples, see D'Ariano [ 1997b], D'Ariano and Paris [ 1997b])73. Recall that when the c-number function F ( a ; s )that is associated with an operator k in s order exists for s < -1, then (b)can be obtained from the phase-space function P( a ; s)(measurable, e.g., in balanced eight-port homodyning; see 52.1.4) according to eq. (B.22), provided that the integral exists. 72 Equations (169)-(172) can be obtained by taking the average of eq. (168), substitutingfor (ht"6") the right-hand side (multiplied by q-(n+"y2) of eq. (163), and using the generating function of the Hermite polynomials with the argument (i/m)(d/dz). 73 Since the integral kernel K&, q; q ) does not depend on rp if k is a function of the photonnumber operator, this case can be regarded as the first realization of Helstrom's quantum roulette wheel (D'Ariano and Paris [1997b]).
11. Q 31
QUANTUM-STATE RECONSTRUCTION
139
The integral kernel (17 1) is therefore applicable to such quantum states whose (smeared) quadrature-component distributionsp(x, cp; r ] ) tend to zero sufficiently fast as 1x1 goes to infinity such that eq. (169) converges even if KF(x,cp; q) increases with 1x1. At this point it should be noted that K&, q;r ] ) is determined only up to a function O(x,cp); i.e., K&, q;q) in eq. (169) can by replaced with Kk(x, cp; q) = &(x, cp; r ] ) + O(x,cp) such that
for any normalizable quantum state. Hence, if the integral kernel KF(x,cp; r ] ) that is obtained from eq. (171) is unbounded for 1x1 -+ cm such that the x integral in eq. (169) does not exist for any normalizable quantum state, it cannot be concluded that fi cannot be sampled from the quadrature-component distributions of any normalizable quantum state, since a different, bounded kernel may exist.
3.8.3. Canonical phase statistics The quantum-mechanical description of the phase and its measurement has turned out ta.be troublesome and is still a matter of discussion. Many papers have dealt with the problem and an extensive literature is available (for reviews, see Lukl and Peiinova [1994], Lynch [1995], Royer [1996], Pegg and Barnett [1997]). Here we confine ourselves to the canonical phase that is obtained in the attempt - in close analogy to the classical description - to decompose the photon destruction operator into amplitude and phase such that 2 = hfi. The (nonorthogonal and unnormalizable) phase states,
fl=O
(London [1926, 1927]), which are the right-hand eigenstates of the one-sided unitary operator k , &I$) = ei@l$),can then be used to define - in the sense of a POM - the canonical phase distribution P($) = (A($)),with A($)= I$)($\ 74. Note that the Hermitian operators 2 = (k+k 9 / 2 and 3 =.(A - kt)/(2i) are
For a two-mode orthogonal projector realization in the relative-photon-numberbasis, see Ban [1991a-d, 1992, 19931, Hradil [1993].
74
140
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI, 5 3
the Susskind-Glogower sine and coshe operators (Susskind and Glogower [ 19641)75. Various proposals have been made to measure P ( @ )using homodyne detection or related schemes 76. So the Wigner function or the density matrix reconstructed from the homodyne data could be used to infer the phase statistics in principle (Beck, Smithey and Raymer [1993], Smithey, Beck, Cooper and Raymer [1993], Breitenbach and Schiller [1997]). Further it was proposed to obtain P(@) = (I?(@))from a homodyne measurement of (all) normally ordered moments (ij+m+"ij"),because I?(@) has a series expansion of the type given in eq. (168),
(Bialynicka-Birula and Bialynicki-Birula [ 19951)77. Unfortunately, (I?(@)) cannot be given (even for = 1) by an integral relation of the form (169) suited to statistical sampling, because the integral kernel does not exist (see footnote 79). It was therefore proposed to introduce parametrized phase distributions P(@,E) such that for any E > 0, P(@,E) can be obtained by direct sampling fromp(x, cp), and P ( ~ , E-+) P(@)for E 0 (Dakna, Knoll and Welsch [1997a,b]). The sampling functions can then be obtained as convergent sums of the sampling functions for the density-matrix elements in the Fock-basis (see 5 3.3). Since the value of the parameter E for which P(@,E) M P(@)is determined by the number of photons at which the quantum state under study can be effectively truncated, the method is state-dependent. Further, it was proposed to measure the canonical phase by projection synthesis, mixing the signal mode with a reference mode that is prepared in a quantum state such that, for appropriately chosen parameters, the joint-photon-number probabilities in the two output channels of the beam splitter directly correspond to the canonical phase statistics of the signal mode (Barnett and Pegg [1996], Pegg, Barnett and Phillips [1997]). Apart from the --f
75 For constructing the operators corresponding to classical phase-dependent quantities, see Bergou and Englert [1991]. 76 The proposal was made to regard the quadrature-component distribution at x = 0 as (unnormalized) phase distribution, P(4) p(x = O,$) (Vogel and Schleich [1991]), which yields phase statistics which are quite different from the canonical phase statistics in general (for an improvement, see B&ek and Hillery [1996]). 77 Here it was proposed to obtain (dtm+"dn)from the homodyne data via the (2n + m)th derivative of dq e-imrPe-1z'2(exp[i~(rp- n/2)]).
-
s,"'
K 5 31
QUANTUM-STATE RECONSTRUCTION
141
direct photon-number measurement needed and the fact that the quantum state under study must again be truncated, so that the method is state-dependent, the difficult problem remains to design an apparatus that produces the reciprocal binomial states needed. The problem of homodyne measurement of the canonical phase can be solved when the exponential phase moments yk;[i.e., the Fourier components of P(4)] are considered and not the phase distribution itself,
where 'Yk= (kk)if k > 0, and ' Y k shown that ( k > 0)
= y?k
if k < 0 (YO= 1). Then it can be
(Opatrny, Dakna and Welsch [1997, 19981, Dakna, Opatrny and Welsch [1998]). The integral kernel Kk(x)(fig. 16) can be used for sampling the exponential phase moments from the homodyne data for any normalizable state '*. In particular, Kk (x) rapidly approaches the classical limit 79
K(c/) k
()=
i(-l)(k-')'2k signx
if k odd,
(2n)-'(-l)(kf2Y2klnx if k even
as 1x1 increases, and it differs from the classical limit only in a small interval around the origin. It is worth noting that the method applies to quantum and classical systems in a unified way and bridges the gap between quantum and classical phase. In particular, the integral kernel Ky)(x)as given in eq. (178) is nothing but the integral kernel for determining the radially integrated Wigner function, which reveals that in the classical limit the canonical phase distribution is simply the radially integrated probability density for the complex amplitude a. Note that any radially integrated propensity prob(a), such as the
''
Note that when Kk(x) is calculated according to eq. (171), then it does not apply to all normalizable states (for analytical expressions and properties of &(X), including nonperfect detection, see Dakna, Opatmy and Welsch [1998]). 79 Already from the classical kernel (178) it is seen that eik(q-@)Kk(x)does not exist, and hence P(Q) cannot be obtained from p ( x , rp) by means of an integral transformation of the form given in eq. (177).
xi??-,
142
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
1.5 -
k=5
0.5 -
Kkb)
0
k=l
-
-0.5
-
-1.5
-
k=3
,
-6
-4
-2
0
2
4
6
X
Fig. 16. The x-dependent part &(x) of the sampling function &(x, rp) = eikpKk(x) for the determination of the exponential moments 'Yk = ( E k ) of the canonical phase from the quadraturecomponent distributions p(x, rp) according to eq. (177) for various (a) odd k and (b) even k. (After Dakna, Opatm? and Welsch [1998].)
11,
8 31
143
QUANTUM-STATE RECONSTRUCTION
radially integrated Q function measurable, e.g., in balanced eight-port homodyne detection, can be used to define an operational phase probability density (Noh, Fougkres and Mandel [1991, 1992a,b, 1993a,b,c]),which in the quantum regime differs from the canonical phase distribution in general (for hrther readings, see Turski [1972], Paul [1974], Shapiro and Wagner [1984], Hradil [1992, 19931, Vogel and Welsch [1994], Leonhardt, Vaccaro, Bohmer and Paul [1995], Leonhardt [ 1997~1). 3.8.4. Hamiltonian and Liouvillian So far, measurement and reconstruction of quantum-state representations and averages of particular quantities at certain time have been considered. The quantum state of an object at chosen time t is, of course, a result of state evolution from an initial time to, the state evolution being governed by the Hamiltonian H of the system or a Liouvillian i in the more general case of the system being open. Since the Liouvillian of an object expresses in most concentrated form the dynamics of the object, knowledge of the Liouvillian is essential for understanding the behavior of the object, and the question may arise of how to experimentally determine its form. To answer the question, it was proposed to appropriately apply quantum-state reconstruction routines, such as direct sampling of the density matrix in balanced homodyning (D'Ariano and Maccone [ 19971). Let us consider a system that is initially prepared in some known state Gin = $(to) and assume that in the further course of time it evolves to a state Gout = g ( t ) at time t ,
where for a system that is homogeneous in time the superoperator 6 = G(t,to) can be given by the exponential of the (time-independent) Liouvillian of the system,
t
The superoperators 6 and i can then be obtained by measuring (reconstructing) Gout for various probe inputs G i n . A typical example may be a radiation-field mode whch (at time to) is fed into nonlinear resonator-like equipment giving rise to amplification and damping. The quantum state of the outgoing mode (at
144
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
P, 5 3
time t ) can then be related to that ofthe incoming mode according to eq. (179). In the Fock basis, eq. (179) reads as w
Now let us assume that in a succession of measurements the quantum states of the outgoing mode are reconstructed for an appropriately chosen (overcomplete) set of input quantum states Qyn, p = 1,2,. . ., such that for chosen m and n the block of equations, w
can be used as conditional equations for the matrix elements GI,,. Repeating the procedure for all values of rn and n, the G matrix and [by using eq. (180)] the L matrix can be calculated in principle. Apart from a truncation of the Hilbert space in numerical implementations, the main problem that remains in practice is of course the (controlled) initial-state preparation such that all (relevant) matrix elements of the Liouvillian can be probed. In parficular, when the Liouvillian is phase-insensitive, then mixtures of Fock states evolve into mixtures of Fock states. For this case a setup was proposed which uses correlated twin beams produced by a nondegenerate parametric amplifier in combination with conditional photon-number measurement on one beam in order to prepare the other beam in (random) photon-number states. These states can then be used as inital states for probing the Liouvillian. To realize the scheme with available techniques, it should be noted that the reconstruction algorithm can be extented such that the quantum efficiency of the photon-number measurement need not be unity (D' Ariano [ 1997~1). 3.9. PROCESSING OF SMEARED AND INCOMPLETE DATA
In practice, there are always experimental inaccuracies which limit the precision with which the quantum state in a chosen representation can be determined. Typical examples of inaccuracies are data smearing owing to nonperfect
For limits on the measurement of state vectors expressed in terms of channcl capacities for the transmission of information by finite numbers of identical copies of statc vectors, see Jones [1994].
11, § 31
145
QUANTUM-STATE RECONSTRUCTION
detection, finite number of measkement events, and discretization of continuous parameters, such as the quadrature-component phase in balanced homodyning. In particular, the latter is an example of leaving out observables in the expansion (69) of the density operator @.Whereas in balanced homodyning the distance between neighboring phases can be diminished such that the systematic error is reduced, in principle, below any desired level, there are also cases in which a principally incomplete set of observables is available. Then either additional knowledge of the quantum state is necessary to compensate for the lack of observables or other principles must be used to reconstruct the density operator according to the actual observation level.
ai
3.9.1. Experimental inaccuracies
Since in a realistic experiment the quantum efficiency q is always less than unity, the probability distributions of the measured quantities are always more or less smeared, because of losses. The problem of compensating for losses in direct sampling of the quantum state from the quadrature-component distributions measurable in balanced homodyning ( Q 2.1.2) has been studied widely. As mentioned in Q 2.3, active loss compensation may be realized, in principle, by mwns of a squeezer, such as a degenerate parametric amplifier (Leonhardt and Paul [1994a]; see also Leonhardt and Paul [1995]). When the signal mode is preamplified before detection by means of a squeezer [eq. (63) with ei@;liin place of ;[I7 then it can be shown that for appropriately phase matching the (scaled) quadrature-component distribution of the preamplified signal measured with quantum efficiency q reads as8’ P’(&x,
q; rl) =
/
h ’ P ( x ’ , q)P[&(x
-X%
rll,
(183)
(g, amplification factor), with p(x; q) being given by eq. (20). From eq. (183)
together with eq. (20) it is seen that q is effectively replaced with gq/ (1 - g + gg), and hence &p’(&x, Q?; q) tends to p(x7cp) as the amplification becomes sufficiently strongp2.Note that the degree of improving the quantum
’’
Note that squeezing the local oscillator has the same effect as antisqueezing the signal field under the assumption that the coherent component is large (Kim and Sanders [1996]; see also Kim [199-3). 82 For details, see also Vogel and Welsch [1994], and for a discussion of eqs. (183) and (20) in terms of moments of the measured quadrature components and those of i (rp), see Marchiolli, Mizrahi and Dodonov [1997]).
146
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[XI, 5 3
efficiency is limited by the realizable mode-matching in degenerate parametric amplification, which deteriorates with increasing pump strength. Without active manipulation of the signal, it has been well established that above a lower bound for q, it is always possible to compensate for detection losses, introducing a modified sampling function that depends on q such that performing the sampling algorithm on the really measurable (i.e., the smeared) quadrature-component distributions p(x, cp; r]) yields the correct quantum state (D’Ariano, Leonhardt and Paul [ 19951). This bound depends on the chosen state representation in general. In particular, reconstruction of the density matrix in the photon-number basis is always possible if q > 1/2. In this case,
where the now q-dependent sampling function Knln(x,cp; q ) is defined by eq. (96) [together with eq. (97)]. Another approach to the problem of loss compensation is that the sampling function is left unchanged [i.e., K,,(x, cp; q) -+ K,,(x, cp) = K,,(x, cp; q = 1) in q ) of the quantum state eq. (184)] and, at the first stage, the density matrix that corresponds to the smeared quadrature-component distributions p(x, q;q) is reconstyucted. After that, at the second stage, the true density matrix = @(; = 1) is calculated from emn(q) (Kiss, Herzog and Leonhardt [1995], Herzog [ 1996a1). Extending the Bernoulli transformation (C.3) and the inverse Bernoulli transformation (C.4) to off-diagonal density-matrix elements, can be calculated from emn(q) asg3
emn(
emmn
entn
It is worth noting that eq. (185) is exact. In other words, when (for precisely given overall detection efficiency) q) is known exactly, then can always be calculated precisely, irrespective of the value of q.
ern(
Replacing in eq. (185) q-‘ with q yields @,,,,,(q) in terms of em,,,which corresponds to a generalization of the Bernoulli transformation (C.3) to off-diagonal density-matrix elements. Since an imperfect detector can be regarded as a perfect detector with a beam splitter in front of it, h(q) and $ can be related to each other applying the beam-splitter transformation (7), h(q) = 2(01e’/’i2$e-iPi2J0)2, with i 2 according to eq. ( 6 ) and c0s2(p/2) = 8 (the mode indices 1 and 2, respectively, refer to the signal and vacuum inputs). It can then be proved easily that $[a) = +$, where i = t ( i t 2 $ +hiti - 22@t), which corresponds to the time evolution of a damped harmonic oscillator at zero temperature (8 -+ e+). 83
11, § 31
QUANTUM-STATE RECONSTRUCTION
147
In practice however, q is not @own precisely in general, and the experimenalways differ from the exact ones. In particular, tally determined values of emn(r) when 11 6 1/2 and the error of does not vanish with increasing m and n, and when there is no a priori information about the quantum state, such as the photon number at which it can be truncated, then cannot be obtained for chosen number of measurements - from the measured @ k / ( v ) , because of error explosion. A typical example of such an error is the statistical error with q) is sampled from the quadrature-component distributions measured which in balanced homodyning. In this case loss compensation (for arbitrary quantum states) is possible only if Q > 1/2 (D’Ariano and Macchiavello [ 19981)84. Clearly when the quantum state to be reconstructed can be truncated at some maximum photon number nmaxrthen the sum in eq. (185) is finite and the density-matrix elements encan also be obtained for q < 1/2, the accuracy being determined by that of Q m n ( V ) * Limitations on variables in real experiments always give rise to systematic errors. This is the case, e.g., when the quadrature components i ( q )are measured at discrete phases Q?k or when only some part of the n interval can be scanned experimentally (for an example, see Q 4.4.1). In particular, in balanced homodyning the quadrature-component distribution p(x, q) is measured at a finite number of phases q k within a n interval and finite x-resolution. When p(x, q) is measured precisely at N equidistant phases qk = ( n N ) k , then the reconstructed density-matrix elements are given by
emn(r])
(186)
in place of eq. (91). A measure of the systematic error is the difference Aemn= emn(N)which reads ass5
em,,,
A e m n = ~ ~ G k m ; e kwith l r k-l=m-nf2jN j=l
k,l
84 When the density matrix at initial time t = 0 of a signal mode that undergoes phase-insensitive damping or amplification is tried to be reconstructed from the quadrature-component distributions p(x, rp, t; TJ) measured at time t > 0, then the additional (phase-insensitive) noise gives rise to a modified overall quantum efficiency q*, so that qs > 1/2 must be valid in order to compensate for the losses (D’Ariano [1997b], D’Ariano and Sterpi [1997]). 85 Equation (187) can be derived by substituting in eq. (186) for p(x, rp) the result of eq. (98) and recalling eqs. (92) and (100). Note that G T = 6,,,k6,,~only holds for k - I = m - n.
148
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[K § 3
[G:r = n J dxfm,(x)gk~(x)](Leonhardt,and Munroe [1996], Leonhardt [1997b]). Equation (187) reveals that when the quantum state can be truncated such that
then the density matrix-elements en*,,for 1m - nl < N can be reconstructed precisely from p ( x , q ) at N phases qk according to eq. (186). In particular, one phase is required to reconstruct a completely dephased quantum state that contains only diagonal density-matrix elements. Since any quantum state can be approximated to any desired degree of accuracy by setting emnM 0 for m(n) > nmax,there is always an N M nmax for which the condition (188) can be assumed to be satisfied, so that all (essentially nonvanishing) densitymatrix elements can be reconstructed precisely. If N is not known a priori, the quantum state can be reconstructed in an iterative way, increasing N in a sequence of (ensemble) measurements until I is sufficiently small. Note that any normally ordered moment (6tn6")can be reconstructed exactly from p(x, q) at N = n + m + 1 phases (Wunsche [1996b]; see Q 3.8.1). The methods for quantum-state reconstruction are based on ensemble measurements; i.e., on a sequence of individual measurements carried out on identically prepared systems8 6 . Since the number of individual measurements is principally finite, the measured quantities are always estimates of the true ones. Hence all the quantities which can be inferred from the measured quantities are also estimates, and the statistical error with which the original quantities are measured propagates to the quantities inferred from them. In particular, when the quantities which are desired to be determined can be sampled directly from the measured data, then the statistical error can be estimated straightforwardly by also using the sampling method. The problem of statistical error in quantum-state reconstruction has been studied in a number of papers, with special emphasis on balanced four-port homodyne detection (Leonhardt, Munroe, Kiss, Richter and Raymer [1996], D'Ariano [1997b], D'Ariano and Paris [1997a], D'Ariano, Macchiavello and Sterpi [ 19971, Leonhardt [ 1997~1).Let us consider a quantity and assume that it can be sampled directly from the quadrature-component statistics according to eq. (161). When in an experiment n ( q k ) individual
For the problem of measuring the state of single quantum systems, see Uecla and Kitagawa [1992], Aharonov and Vaidman [1993], Aharonov, Anandan and Vaidman [1993], Imamoglu [1993], Royer [1994, 19951, Alter and Yamamoto 119951, D'Ariano and Yuen [1996]. 86
11, I 31
QUANTUM-STATE RECONSTRUCTION
measurements are performed at phase q k , k estimated as
=
149
0,1,. . . ,N - 1, then (& can be
where x n ( c p k ) is the result of the nth individual measurement at phase cpk. Taking the average over all estimates ($eSt)(N))yields ( E ( N ) ) ;i.e., the desired quantity within the systematic error owing to phase discretization,
where
) terms of the averaged Accordingly, the statistical fluctuation of ( p ( e s t ) ( N )in can be given by, on taking into account estimates of the variance of (k;cest)(N)) that the individual measurements are statistically independent of each other ”,
where
is the variance of the sampling function, and N(cpk) is the number of measurements per phase interval, N(cpk) = n(cpk)N/n. Let us mention that when the kernel KF(x, cp) is a strongly varying function of x in regions where p ( x , cp) is non-negligible, then the first term in eq. (193) is much larger than the second one. In this case the second term can be neglected and the statistical error can be approximated by averaging the square of the kernel. Note that the same result is obtained if one assumes that the numbers of
*’For simplicity, in eq. (192) it is assumed that F is a real quantity.
150
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, § 3
events yielding the values of x(cpk) in given intervals of x (bins) are independent Poissonian variables (Leonhardt, Munroe, Kiss, Richter and Raymer [19961, Leonhardt [ 1997~1,McAlister and Raymer [ 1997bl). However, these variables are neither strictly independent [their sum is always N(cpk)] nor Poissonian Ethe probability that there is more than N ( cpk) events in a bin is zero]. Therefore, care must be taken before one decides to use the simplified error estimation. Whereas for the Fock-basis density matrix elements the simplified estimation is very good (the kernels are strongly oscillating), for the exponential moments of canonical phase one must take into account both terms in eq. (193), otherwise the error would be overestimated (the kernels are slowly varying functions). From eq. (192) it can be expected that the statistical error depends sensitively on the analytical form of the sampling function. To give an example, let us consider the integral kernels Kmn(x, cp), eq. (92), needed for sampling the densitymatrix elements in the photon-number basis, Qmn. For chosen cp, Kmn(x,cp) is an oscillating function of x, and with increasing distance d = ( m- nl from the diagonal the oscillations become faster and the oscillation range slowly increases. It turns out that with increasing m the diagonal-element variance oimm becomes independent of m ; i.e., saturation of the statistical error for sufficiently large m is observed, oimm< 2/R (R, mean number of measurements per phase interval). On the contrary, the off-diagonal statistical error increases with d, without saturation. The influence of the quantum efficiency q on the statistical error of the density-matrix elements is very strong in general. For chosen m and n the oscillation range of Km,(x, cp; q), eq. (96), increases very rapidly as r] approaches the lower bound r] = 1/2, and the statistical error increases rapidly as well (for numerical examples, see D’Ariano, Macchiavello and Sterpi [ 19971)88. Using the experimentally sampled density-matrix elements for calculating the expectation values of other quantities, such as ( p ) according to eq. (160), the resulting statistical error is determined by the law of error propagation. It is worth noting that error propagation can lead to additional noise which is not observed if the quantities are also directly sampled from the measured data (provided that the sampling method applies).
**
For the error of the exponential phase moments sampled from the quadrature-component distributions in balanced homodyning, see Dakna, Opatrn? and Welsch [1998], and for the error in quantum-statemeasurement via unbalanced homodyning and direct photocounting, see Wallentowik and Vogel [1996a], Banaszek and Wbdkiewicz [1997b], Opatrn?, Welsch, Wall&owitz and Vogel [1997].
11,
5 31
QUANTUM-STATE RECONSTRUCTION
151
3.9.2. Least-squares method In the quantum-state reconstruction problems outlined in the foregoing sections a set of measurable quantities (in the following also referred to as data vector) is related linearly to a set of quantities (state vector) that can be used to characterize the quantum state of the system under study. Both sets of quantities can be discrete or continuous or of mixed type. Typical examples are the relations (70) and (98), respectively, between the Wigner hnction and the density-matrix elements in the Fock basis, and the relations (112) and (140) between the Fourier components of the displaced Fock-state distributions and the atomic-state inversion, respectively, and the density-matrix elements in the Fock basis. A powerful method for inversion of such relations has been leastsquares inversion89. The method has been used for quantum-state reconstruction in balanced optical homodyning (Tan" 1997]), unbalanced homodyning (Opatrny and Welsch [ 19971, Opatrny, Welsch, Wallentowitz and Vogel [ 1997]), cavity QED (Bardroff, Mayr, Schleich, Domokos, Brune, Raimond and Haroche [1996], Bodendorf, Antesberger, Kim and Walther [ 19981) and for orbital electronic motion (Cline, van der Burgt, Westerveld and Risley [1994]). It has been used further to reconstruct the quantum state of the center-of-mass motion of trapped ions (Leibfried, Meekhof, King, Monroe, Itano and Wineland [ 1996]), the quantum state of a particle in an anharmonic potential and the quantum state of a particle that undergoes a damped motion in a harmonic potential (Opatrny, Welsch and Vogel [ 1997~1). An advantage of least-squares inversion is that it is a linear method - the density matrix elements can be reconstructed in real time together with an estimation of the statistical error. Moreover, it allows for an easy incorporation in the reconstruction of various experimental peculiarities, such as nonunity quantum efficiency, finite resolution or discretization of the data, finite observation time, dissipative decay of the system, etc.. These aspects can hardly be treated on the basis of analytically determined (and existing) sampling functions. On the other hand, the method does not guarantee (similarly as any other linear method) that a reconstructed density matrix is exactly positivedefinite (cf. § 3.9.3). To illustrate the method, let us assume that a distributionp(x, cp) of the type of a quadrature-component distribution is measured, and that p(x, cp) can be given
The method of least squares was discovered by Legendre [1805] and Gauss [1809, 18211 for solving the problem of reconstruction of orbits of planetoids from measured data. 89
152
PI, 5 3
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
en,,!
by a linear combination of all density-matrix elements of the quantum state (x, cp), to be reconstructed, with linearly independent coefficient functions &,,I
where Snn,(x,cp) need not be of the form used in eq. (98)90.Since the density matrix of any physical state can be truncated at some value nmax,the sum in eq. (194) is effectively finite. Direct application of least-squares inversion (Appendix D) yields the reconstructed density-matrix elements @:$)as
where P ( ~ ) ( xcp), is the experimentallymeasured distribution, X and @ being the intervals accessible to measurement. The integral kernel Knnt(x,cp) is given by
and F = G-', with the matrix G being defined by
It can be proved by direct substitution that if the data correspond to the exact quantities p ( x , cp), i.e., P ( ~ ) ( xcp) , = p(x, cp), then the reconstructed density matrix equals the correct one, i.e., = On the other hand, if the experimental data suffer from some inaccuracies, then the reconstructed density matrix has the property that it reproduces the data as truly as possible (in the sense of least squares). In the above given formulas we have assumed that x and ~1 are continuous variables, and that n is discrete. The formulas for other combinations of discrete andor continuous arguments can be obtained in a quite similar way. An essential point of the method is the inversion of the matrix G, which requires the matrix to be sufficiently far from singularity; i.e., the data must carry enough information about all the density-matrix elements which are desired to be reconstructed. Otherwise regularized inversion must be applied (Appendix D). @,,,,I.
90 In particular, when x is the position of a moving particle and p corresponds to the time t , and the particle undergoes damping, then the quantum state evolves according to a master equation whose solution then determines Snn,(x, t ) .
11, § 31
153
QUANTUM-STATE RECONSTRUCTION
Regularized inversion usually decreases the statistical error of the reconstructed density-matrix elements, but on the other hand they are biased. Therefore, in practice such a degree of regularization should be used for which the introduced bias is just below the statistical noise. 3.9.3. Maximum-entropy principle As already mentioned, it is principally impossible to measure the exact expectation values of an infinite number of operators 2;in an expansion of the density operator of the type given in eq. (69), because any realistic experiment can only run for a finite time. So far, the exact formulas have been applied to the analysis of the incomplete measurements including an estimation of the error made. However, the question may arise of how to obtain an optimum result of has been reconstruction of a quantum state when only a finite number of (2;) measured in the experiment. An answer can be given using the Jaynes principle of maximum entropy9' (Buiek, Adam and Drobny [ 1996a,b], Buiek, Drobny, Adam, Derka and Knight [ 19971). of n quantities i = 1,. . . n Let us assume that the expectation values (ji) are determined experimentally9'. The set of measured quantities can be regarded as a measure of the realized observation level. Certainly, there is a number of potential density operators 8, Tr6 = 1, which are compatible with the experimental results; i.e.,
a;,
Tr($;)
=
(&),
i = 1, . . .n.
Among them, that density operator is chosen that maximizes the von Neumann entropy 93 ,S[6]=-Tr(61nQ).
( 199)
See Jaynes [1957a,b]. Probability distributions (or density operators) describe our stage of knowledge about physical systems. If we do not know anythmg, we usually assign uniform distributions to the quantities (or a multiple of the unity operator to the density operators). If we have partial knowledge, we choose such distributions which are as broad as possible and still reflect our stage of knowledge. A suitable measure of the breadth is the entropy; one therefore seeks for such distributions (density operators) which maximize the entropy under the condition that known quantities are reproduced. 92 Note that the determination of already requires an infinite number of individual measurements which cannot be realized during a finite measurement time. 93 See von Neumann [1932]. 9'
(A,)
154
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
Introducing Lagrange multipliers, the resulting density operator @ the familiar (grand-canonical ensemble) form
tII, = @S
03
takes
where
and represents a partially reconstructed (estimated) density operator on the chosen observation level. Substituting in eq. (198) for @ the density operator @ S from eq. (200), a set of n nonlinear equations is obtained for the calculation of the n Lagrange multipliers A; from the measured expectation values Any incomplete observational level cad be extended to a more complete observational level, in principle, by including additional observables in the scheme, which is usually associated with a decrease of the entropy. However, since rather involved calculations are required to be performed, the method has been studied for reconstructing the quantum state of a radiation-fieldmode on particular (not very high) observational levels (Buiek, Adam and Drobn9 [ 1996a,b], Buiek, Drobnq, Adam, Derka and Knight [ 19971) and/or low-dimensional systems, such as spin states Wuiek, Drobny, Adam, Derka and Knight [ 19971). Since the expectation values cannot be measured with infmite precision, only estimates can be inserted into eq. (198), and therefore it can happen that the solution does not exist; i.e., the non-precisely measured averages are not compatible with any density operator. To overcome this problem, the method can be combined with least-squares minimization (Wiedemann [ 1996194); i.e., the sum of squares of differences
(ai).
(a;)
as a function of the parameters Ai is tried to be minimized, non-precisely measured averages.
(ij)(M) being the
94 In this paper, which is unfortunately unpublished, the operators d are identified with the photon number h and quadrature-componentprojectors 1x1,q k ) (XI, rpk I, where the subscript I labels a finite subset of the continuous quadrature-components at chosen phase rpk. Due to computational limits, 4 phases and 13 values of x at each phase are considered. The partial reconstruction of the quantum state (in phase space) from computer-simulated homodyne data is performed for various states and yields results which reflect typical properties of the states sufficiently well.
n, Q 31
155
QUANTUM-STATE RECONSTRUCTION
3.9.4. Bayesian inference The statistical fluctuations of the data are taken into account in the Bayesian inference scheme (Helstrom [1976], Holevo [1982], Jones [1991, 19941, Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [1996]). Let us assume that the system under consideration is prepared in a pure state that belongs to a continuous manifold of states in a state space 0 and i = 1 , . . . ,n, with consider a repeated N-trial measurement of observables eigenvalues AJ,. The determination of the quantum state of the measured system is then performed in a repeated three-step procedure: (i) As a result of a (single) measurement of 8, a conditional probabilityp(A,, I@) is defined which specifies the result AJ, if the measured system is in state
a,,
0 = IW(W,
where !‘A,, = \Ajt)(Aj,1. (ii) A probability distribution PO(@) defined on the space 0 is specified such that it describes the a priori knowledge of the state to be reconstructed. The joint probability distribution p(AjL,@)is then given by
When no initial information about the measured system is available, then the prior probability distributionPO(@) is chosen to be constant. (iii) Finally, the Bayes rule95 is used to obtain the probability p(@lAj,)of the system being in state @ under the condition that Aj, is measured,
Now the procedure can be repeated, making a second measurement of some observable & , (& = or & # ,&) and using p(@IAji)obtained from the first measurement as the prior probability distribution for the second measurement.
ai
156
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
tII, § 3
Proceeding in the way outlined, the Ndrial conditional probability distribution ~ ( $ 1 {Aj, 1) is given by
where
is the likelihood function (regarded as a function of 6; the measured values {A,} playing the role of parameters). Note that C(@)is the probability distribution of finding the result { A ] , } in the sequence of N measurements under the condition that the state of the system is 6. The partially reconstructed density operator @LJ is then taken as the average over all the possible states @96,
To apply the method, the state space 9 of the measured system and the corresponding integration measure dS2 must be defined and the prior probability PO(@)must be specified. In particular, the integration measure must be invariant under unitary transformations in the space 9. T h s requirement uniquely determines the form of the measure. It should be pointed out that t h s is no longer valid when S2 is tried to be extended to mixed states. Since a system which is in a mixed state can always be considered as a subsystem of a composite system that is in a pure state, the Bayesian reconstruction can be applied to the composite system and tracing the resulting density operator over the degrees of freedom of the other subsystem (Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [ 19961). As already mentioned, the prior probability PO(@) can be chosen constant if there is no a priori information about the state to be r e c o n ~ t r u c t e d It ~ ~turns . out that with increasing number of measurements the method becomes rather insensitive to the prior probability. In particular,
96 Note that & can correspond to a mixed state even though it is assumed that the system is prepared in a pure state. 97 Note that when the integration measure d B is not defined uniquely, then a prior probability that is constant with respect to a chosen integration measure need not be constant with respect to another one.
11,
D 41
QUANTUM STATES OF MATTER SYSTEMS
157
when the number of measurements approaches infinity, N -+ m, eq. (208) corresponds, on chosen observation level, to the principle of maximum entropy on a microcanonical ensemble98. Contrary to least-squares inversion, Bayesian inference always yields a (partially) reconstructed density operator which is (formally) positive definite and normalized to unity. However, the price to pay is a rather involved procedure that has to be carried out in practice. For this reason, the method has been mostly considered - similar to the method of maximum entropy - for spin systems; i.e., for systems with small dimension of the state space (Jones [1991], Derka, Buiek and Adam [1996], Derka, Buiek, Adam and Knight [ 19961). Another statistical method related closely to the Bayesian reconstruction method is the quantum state estimation based on the maximization of the likelihood function C@), eq. (207). Here, that state 0 in the state space is selected for which C(6) attains its maximum. Because of the difficulty of finding the maximum of L(0)in higher-dimensional state spaces, a procedure was proposed which is based on a sequence of general inequalities satisfied by the likelihood function (Hradil [ 19971).In this way, the problem can be tranformed to that of the diagonalization of an operator given by a linear combination of the projectors where the expansion coefficients must solve a set of nonlinear algebraic equations. The method simplifies the Bayesian treatment but still guarantees the positive definiteness. Significantly, all the solutions based on the deterministic relation (69) between counted data (frequencies) and the desired density matrix are involved. Whenever such solution exists as a positively defined density matrix, then it should maximize the likelihood function as well. The method was applied successfully to the reconstruction of (low-dimensional) density matrices in the photon-number basis of a radiation-field mode from computer-simulated homodyne data, and a comparison with direct inversion of the linear basic relation between the measurable quantities and the density-matrix elements was given (Mogilevtsev, Hradil and PeEna [ 19971).
8 4.
Quantum States of Matter Systems
In the preceding sections we have considered phase sensitive measurements of radiation fields and methods for reconstructing the quantum state of the
98 Note that on a chosen (incomplete)observational level the two methods yield different fluctuations
of the observables in general.
158
HOMODYNE. DETECTION AND QUANTUM-STATE RECONSTRUCTION
“1,
04
fields from the measured data. The problem of quantum-state measurement and reconstruction has also been studied for various matter systems. Different matter systems require, in general, different detection schemes for measuring specific quantities that carry the full information on the quantum state of the system. Although these methods may be, at first glance, quite different from the methods outlined in Q 2 for phase-sensitive measurements of light, there have been a number of analogies between the reconstruction concepts for radiation and matter.
4.1.
MOLECULAR VIBRATIONS
It was shown and demonstrated experimentally that the quantum state of molecular vibrations can be determined using a tomographic method (Dunn, Walmsley and Mukamel [1995]) which resembles the one for a light mode outlined in 9 3.1. The method, called molecular emission tomography, is based on the fact that the time resolved emission spectrum of a molecule allows one to visualise the time dependence of a vibrational wave packet withn the excited electronic state from whch the emission originates (Kowalczyk, Radzewicz, Mostowsh and Walmsley [19901). Alternatively, the desired information on the wave-packet dynamics can be obtained by photoelectron spectroscopy (Assion, Geisler, Helbing, Seyfried and Baumert [1996]). Let us assume that the molecule is prepared in a given vibrational quantum state in the excited electronic state. As can be seen from fig. 17, for appropriately displaced potential energy surfaces of the molecule the position of the vibrational wave packet can be effectively mapped onto the frequency of the emitted light. This fact is used for the tomographic reconstruction of the vibrational wave packet by measuring the time-resolved emission spectrum with a time resolution that is fast compared with the characteristic time period of the molecular state to be studied. The experimental realization has been performed as follows (Dunn, Walmsley and Mukamel [ 19951). A sample of Na2 molecules is illuminated by a 4 k H i train of laser pulses of 60 fs duration and mean wavelength of 630 nm. The laser pulses generate vibrational wave packets in the A’Z;+ state of the sodium dimer, which evolve with a time period of 310 fs. A fraction of the pulses is split off and plays the role of a time-gate shutter. The light emitted from the molecular sample is collected and focused synchronously with the split-off part of the exciting pulse onto a nonlinear crystal. A prism monochromator is used to filter the resulting sum-frequency and the field is recorded by a photon-counting photomultiplier. The resulting temporal resolution of the device is about 65 fs.
QUANTUM STATES OF MATTER SYSTEMS
159
15,000
10,000
5,000
c
Fig. 17. The vibronic energies for the A'ZG ---t X'Z; transition of Na2 clearly show the possibility to display the vibrational motion (in the excited state) in the time-resolved emission spectrum. (After Kowalczyk, Radzewicz, Mostowski and Walmsley [ 19901.)
4.1.1. Harmonic regime
The first reconstruction of the quantum state of molecular vibrations from a time-resolved emission spectrum was based on the assumption that only low vibrational quantum states are excited such that the relevant potentials can be approximated by harmonic ones. Furthermore, it was assumed that the vibrational frequencies in the two electronic quantum states, which contribute to the emission spectrum, are nearly equal. In this case, the vibronic coupling is caused solely by the displacement of the equilibrium positions of the potentials in the two electronic states. When these approximations are justified, then the time-gated spectrum S(Q, T ) can be related to the s-parametrized phase-space distribution P(q,p;s) 3 2-'P[a = 2-"2(q + ip);s] as (Dunn, Walmsley and Mukamel [ 19951)
S(Q, T ) =
s
dyP[x(Q)cos(vT)+ysin(vT),ycos(vT)-x(Q) sin(vT);s]. (209)
160
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
111,
54
28 7926 15. 247122 6820 6418 61-
14.544
... .,........
-6.2
-4.6
-3
-1.4
0.2
1.8
3.4
Momentum (Ground state width).' Fig. 18. Tomographic reconstruction of an s-parametrized phase-space function for a vibrational wave packet in sodium from a measured set of emission spectra. The time gate duration was 65 fs, which implies s = -0.8. (After Dunn, Walmsley and Mukamel [199S].)
Here,& and T are the setting frequency of the spectral filter and the setting time of the time gate, respectively ( Y , vibrational frequency). The function x(Q) = (D- K ~ Y ) / ~ Kdescribes V the mapping of the position of the wave packet onto the emitted frequency, K being the ratio of the displacement of the potentials to the width of the vibrational ground state. The ordering parameter s in eq. (209) is related to the temporal width r-' of the time gate as s = -(T/KY)2.
Obviously, the relation (209) between the phase-space distribution P(q,p ; s) of the molecular vibration and the measured spectrum S(Q,T) is very close to the basic relation (70) of optical homodyne tomography. Thus P ( q , p ; s )can be obtained from S(Q, T ) by means of inverse Radon transform (see 9 3.1). A typical experimental result is shown in fig. 18. 4.I .2. Anharmonic vibrations
In general, molecular vibrations are known to be significantly anharmonic when their excitations are not restricted to very small numbers of vibrational quanta. In such cases one cannot apply the approximate reconstruction procedure based on eq. (209). Due to the anharmonicity effects it is no longer possible to reconstruct the quantum state from only one half of a vibrational period.
11,
P 41
QUANTUM STATES OF MATTER SYSTEMS
161
It was proposed (Shapiro [1995]) to reconstruct the vibrational quantum state from the time-resolved (spectrally integrated) intensity Z(T) = 1d S S ( S , T ) of the light emitted by the molecular sample, which can be related to the densitymatrix elements of the vibrational mode in the excited electronic state as
emn
Here, vmnare the vibrational transition frequencies in the excited electronic state, and k h (o‘;‘:)~ 1(kl.2)2 is determined by the Franck-Condon overlap l ( k l . 2 ) ~of the vibrational wave functions in the two electronic states and the vibronic transition frequency w?!. Equation (210) reveals that, as long as the transition frequencies vmnare nondegenerate, the corresponding density-matrix elements pmncan be obtained, in principle, from an analysis of I ( T) as a function of T. However, the separation of the density-matrix elements from each other may require a rather long time series. Further, the dimension of the set of equations to be inverted can be large, because of the large number of densitymatrix elements that may contribute to the intensity of the emitted light. In the degenerate case, which is observed for the diagonal elements of the density matrix and for some off-diagonal elements due to the anharmonicities, adhtional information is needed. It was proposed to use the stationary spectrum of the light, whose determination requires an additional measurement. Alternatively, the two measurements can be combined such that the density-matrix elements are reconstructed from the time-resolved spectrum (Trippenbach and Band [ 19961)99. It is worth noting that the dimension of the sets of equations that must be inverted numerically can be reduced substantially by employing the 1 1 1 information inherent in the time-resolved spectrum:
-
(Waxer, Walmsley and Vogel [ 19971). In eq. (2 1l), the blurring function g ( w ) = exp(-w2/4r2) is determined by the resolution time r-’ of the time gate (see 54.1.1). In practice, the time-dependent spectrum is available only in a finite time interval of size z, which can be taken into account by multiplying S ( a , T)
99 Trippenbach and Band [1996] also discussed the inclusion of molecular rotations in the reconstruction.
162
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II,
54
by the corresponding samplingswindow function G ( T ,z) in order to obtain S ' ( 0 , T ) = S ( 0 , T ) G ( T ,z). The Fourier transform of S ' ( 0 , T ) with respect to T reads
with G(Y- v,,,,, z) being the Fourier transform of the sampling window. Fixing the frequency Y of the time-series spectrum, the structure of the window function ensures that only some of the density matrix elements contribute at this frequency, depending on the chosen size z of the sampling window. Let us assume that the density matrix elements @,,,(i = 1,2,3,. . . ,N ) contribute to the spectrum at chosen frequency Y. By choosing N frequencies 0 = Qi of the emission spectrum, one gets from eq. (212) a linear set of N equations of rather low dimension that can be inverted numerically loo. In this way, the method allows one to reconstruct rather complicated quantum states on a time scale that can be shorter than the fractional revival time, as was demonstrated in a computer simulation of measurements (Waxer, Walmsley and Vogel [1997]). Note that a reduction of the dimension of the set of equations to be inverted improves the robustness of the method with respect to the noise in the experimental data. It was also proposed to reconstruct the density matrix from the time-dependent position distribution according to eq. (102) (Leonhardt and Rayrner [ 19961, Richter and Wunsche [ 1996a,b], Leonhardt [ 1997a1, Leonhardt and Schneider [1997]). This requires a scheme suitable for measuring either the position distribution of molecular vibrations or another set of quantities that can be mapped onto the position at different times. Note that this is hardly possible in molecular emission tomography in general. Finally it was proposed to use a wave-packet interference technique (Chen and Yeazell [1997], $4.5.1) for reconstructing pure vibrational states in the excited electronic state (Leichtle, Schleich, Averbukh and Shapiro [ 19981).
loo Note that the emission frequencies Sri are determined by the two vibrational potentials involved in the vibronic emission, whereas the degenerate values of the vibrational frequencies v,,,, are determined solely by the vibrational potential in the excited electronic state. The (within the resolution of the sampling window) degenerate density-matrix elements usually contribute to the emission spectrum at distinct frequencies Sri.
11,s 41
QUANTUM STATES OF MATTER SYSTEMS
163
4.2. TRAPPED-ATOM MOTION
Since the first observation of a single ion in a Paul traplo' (Neuhauser, Hohenstatt, Toschek and Dehmelt [ 1980]), much progress has been achieved with respect to laser manipulation of the quantized motion of single atoms in trap potentials. Such systems are of particular interest since the quantized lowfrequency (- MHz) motion is very stable, and laser manipulations allow one to prepare very interesting nonclassical states. Until now, motional Fock states and squeezed states (Meekhof, Monroe, King, Itano and Wineland [1996]) as well as Schrodinger-cat type superposition states (Monroe, Meekhof, King and Wineland [19961) have been realized experimentally. One might expect that the reconstruction of the quantum state of the centerof-mass motion of a trapped atom may be very similar to the reconstruction of molecular vibrations. However, the vibronic couplings in the two systems are basically different. In the case of a molecule, the vibrating atoms are close together within atomic dimensions and the change of the electronic state substantially alters the potential of the nuclear motion. Ln the case of an atom in a Paul trap, the potential of the center-of-mass motion is given externally by the trap. In this case, electronic transitions can hardly affect the potential. The vibronic interaction in this system is induced by the interaction with radiation. Therefore one may expect that appropriate interactions of a trapped atom with laser fields may open various possibilities for measuring the motional quantum state. 4.2.1. Quadrature measurement
The first proposals to reconstruct the motional quantum state of a trapped atom were based on measuring the quadrature components of the atomic center-ofmass motion in the (harmonic) trap potential. In the scheme in fig. 19 a weak electronic transition 18) = 11) H le) = 12) of the atom is proposed to be driven simultaneously by two (classical) laser beams whose frequencies orand cob, respectively, are tuned to the first motional sidebands, w, = 021 - Y and o.$, = 021 + Y of the electronic transition of frequency ~1 (Wallentowitz and Vogel [1995, 1996b1). Since the linewidth of the transition is very small, the motional sidebands can be well resolved. For a long-lived transition and in the
lo'
For the trap, see Paul, Osberghaus and Fischer [1958].
164
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
/7
v
...............
14 '
.............
Fig. 19. Scheme of a trapped ion with a we& transition 11) ++ 12) and a strong transition 11) ++ 13). Two incident lasers of frequencies y = y~- v and = ~1 + v are. detuned from the electronic transition by the vibrational frequency Y to the red and blue respectively. The laser driving the strong transition is used for testing the ground state occupation probability by means of resonance fluorescence. (After Wallentowitz and Vogel [ 19951.)
LamkDicke regime lo2 these interactions are well described by Hamiltonians of the Japes-Cummings (and anti Japes-Cummings) type '03. Assuming that the Rabi'fiequencies of the two (classically treated) laser fields are equal, the two Jaynes-Cummings interactions can be combined to an electron-vibration coupling that in the interaction picture reads as
A' = ;hQLJZ (6- + G+)qq), where QL is the vibronic Rabi frequency of the driven transitions, and q is the phase difference of the two lasers which can be controlled precisely. Assuming
Io2 In this regime, the (one-dimensional) spread of the motional wave packet, Ax, is small compared with the laser wavelength AL over 2n, A x << A ~ / 2 n .The Lamb-Dicke parameter, rl~o= 2n(Ar)o/A~,is a measure of the spread of the motional wave packet in the ground state of the trap potential relative to the wavelength. '03 For the description of the dynamics in the Lamb-Dicke regime by a Jaynes-Cummings type Hamiltonian, see Blockley, Walls and Risken [ 19921, Wineland, Bollinger, Itano, Moore and Heinzen [1992], Cirac, Blatt, Parkins and Zoller [1994]. Note that the fact that the method is based on the Lamb-Dicke regime is not a serious limitation. In recent experiments (Meekhof, Monroe, King, Itano and Wineland [1996]) the frequency of the weak transition is in the GHz range and it is driven by two lasers in a Raman configuration. In this case the LambDicke parameter can be changed easily by the laser beam geometry.
11,
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QUANTUM STATES OF MATTER SYSTEMS
165
that the atom is initially prepared in the electronic ground state, P,(t)l,=o = 1 (incoherent preparation), it can be shown that the atomic-state inversion at time t is nothing but the real part of the (scaled) characteristic function Y ( x , 9)) of the quadrature-component distribution p(x, 9)) at phase Q? of the center-of-mass motion, AP(i"c)(t) = -Re
Y ( d k & . t ,9)) ,
(214)
[AP= P, - Pg]. Similarly, the imaginary part of the characteristic function can be obtained by preparing the atom initially in an appropriate coherent superposition of the two electronic quantum states such that
Hence, varyhg the phase difference between the two driving laser beams, the complete information about the motional quantum state can be determined from the time evolution of the atomic-state inversion of the weak transition. This occupation probability can be probed by testing an auxiliary, strong transition 11) H 13) for the appearance of fluorescence (cf. fig. 19). The existence and the absence of fluorescence detects the weak transition in state Il} and 12), respectively. This method has an almost ideal quantum efficiency,whlch is a great advantage for the present purpose Io4. Applying eqs. (85H87), the reconstruction of density matrix of the center-of-mass motion in a quadrature-component basis from the measured signal can be accomplished with a single Fourier integral, and application of eqs. (105) and (106) offers the possibility of direct sampling of the density matrix in the Fock basis from the fluorescence signal. Further, using eqs. (B.24) and (B.27), s-parametrized phase-space functions can be inferred from the signal by performing a two-fold integral. The latter was demonstrated in another proposal suitable for measuring Y ( z ,Q?)(D'Helon and Milburn [1996]). Here, the 11) c--) 12) transition (fig. 19) is driven by a standing-wave laser pulse tuned to whose duration z p is much shorter than the vibrational period, YZP << 1, and it is assumed that the center of
Io4 For the experimental detection of a weak transition via fluorescence from another, strong transition, see Nagoumey, Sandberg and Dehmelt [1986], Sauter, Neuhauser, Blatt and Toschek [1986], Bergquist, Hulet, Itano and Wineland [1986].
166
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, 0 4
the trap potential coincides with a node of the standing wave. In the interaction picture, the interaction Hamiltonian in the LambDicke regime then reads as lo'
Ei'
=
$QL(t)JZ (2- + 3+)qf).
(216)
Here, i ( t ) corresponds to the freely evolving position operator of the centerof-mass motion, which for the harmonic motion considered is simply given by the quadrature-component operator at phase Q, = vt, and the time-dependent (slowly varying) Rabi frequency Q L ( ~ ) includes the shape of the pulse. Owing to the shortness of the driving pulse it can be assumed that its action at time t gives rise to a unitary kick described by the operator
k = Texp
[ I 1 -ih-l
dt'H'(t')
M
exp[-iiO(2-
+ 2+)i(f)],
where 0 = Jdt'Q(t') is the pulse area. Obviously, its action on the state is formally the same as the action of the unitary operator that corresponds to the interaction Hamiltonian (213). Hence, Achc)P(t)and A(c"h)P(t)measured immediately after application of the pulse at time t are again related to Y ( z , Q,) according to eq. (215), but now with 8 and vt in place of QL and q, respectively ' 0 6 . It is well known that a squeezed coherent state approaches a quadraturecomponent eigenstate as the strength of squeezing goes to infiruty; i.e.,
with &a) the coherent displacement operator, and 3(5)= exp[(E*i2- E;It2)/2] the squeeze operator Io7. Hence, the quadrature-component distribution p(x, q) can be given asymptotically by
Io5 Note that the Hamiltonian (216) is not based on a vibrational rotating wave approximation as it is the case for the Hamiltonian (213). In eq. (216) the position operator results from the Lamb-Dicke approximation of the (operator-valued)mode b c t i o n of the standing wave, sin(k& M k ~ f . Io6 Whereas in the scheme of Wallentowitz and Vogel [1995] the phase difference between the two laser beams must be controlled, the scheme of D'Helon and Milburn [1996] requires control of short-pulse areas. Io7 For a proof, see, e.g., Vogel and Welsch [1994].
11, Q 41
QUANTUM STATES OF MATTER SYSTEMS
167
where 6'is a coherently displaced mmd.squeezed version of the density operator 6 to be determined,
Equations (219) and (220) reveal that when the quantum state to be measured can be squeezed strongly and displaced coherently, then the quadrature-component distributions can be obtained, in principle, from the occupation probability of the motional ground state that is measured after these manipulations, the phase being controlled by the free evolution of the vibrating system (Poyatos, Walser, Cirac, Zoller and Blatt [ 19961). To realize the scheme, the following procedure was proposed: (i) Wait for a time t such that q = vt. (ii) Perform a sudden displacement of the center of the trap to the right for a distance d such that x = m being the mass of the atom. (iii) Change the trap frequency instantaneously from Y to (lower) v ' I o 8 . (iv) Determine the population of the motional ground state Io9. Knowingp(x, y ) for all phases q in a n interval, the complete information on the motional quantum state is available and the (tomographic) methods of quantumstate reconstruction outlined in $ 3 apply ' I o .
d w d ,
4.2.2. Measurement of the Jaynes-Cummings dynamics
Let us consider the dynamics of a trapped ion that is driven by a single (classical) laser beam in a resolved sideband regime and assume that the laser frequency is tuned to the (red) kth-order motional sideband of the weak electronic transition 11) cs 12), =u l - kv. In this case the resulting coupling (in the interaction picture) between the electronic transition and the center-of-mass motion has
The sudden change of the trap frequency leads to squeezing with the squeeze parameter being (Janszky and Yushin [1986]). Note that a significant change of the trap frequency is needed since the measurement scheme requires strong squeezing. log It was demonstrated experimentally that the motional number statistics can be determined from the measured Jaynes-Cummings revivals (Meekhof, Monroe, King, Itano and Wineland [1996]; 5 4.2.2). 'lo The steps (i), (ii) and (iv) were proposed in order to measure the Q function. In the practical realization the single laser may also be replaced with two lasers driving a dipoleforbidden transition in a Raman configuration, which essentially yields the same basic Hamiltonian. lo8
161 = ln(v/v')/2
'''
168
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI,
54
the form of a nonlinear multiquantum Jaynes-Cummings interaction (Vogel and de Matos Filho [1995]),
H
=
~hSZL6+]"k'(2t2)iik+ H.c.,
(22 1 )
where the operator functionh reads in normally ordered form as
Lamb-Dicke parameter). It describes the influence of the interference between the driving laser wave and the (extended) atomic wave function on the dynamics of the system. The nonlinear Jaynes-Cummings model can be solved exactly. In particular, when the atom is initially prepared in the excited electronic state, then the atomic-state inversion is given by an equation of the type of eq. (137)"*, (QLD,
where now G n n + k = C2~(nI](~)h~(n + k ) , and are the motional density-matrix elements at the initial time t = 0. The motional quantum state can be reconstructed from the measured (nonlinear) Jaynes-Cummings dynamics by applying typical methods outlined in Q 3. When the motional quantum state which is desired to be reconstructed is shifted coherently [G .+ bt(a)Gb(a)]before M(t) is measured, then the Fock-state probability distributionp, = Qnn in eq. (223) must be replaced with the displaced Fock-state probability distribution p,(a) = &(a) = (n,aloln, a ) [cf. eq. (138)], and inversion of eq. (223) yieldsp,(a). Changing a in a succession of (ensemble) measurements, the complete information about the original (unshifted) motional quantum state can be inferred from the measured data. In particular, applying the method outlined in § 3.5, pointwise reconstruction of phase-space functions is feasible. This was demonstrated successfully by reconstructing the Wigner function (eq. 130) of a single 'Be+ ion that is stored in a rf Paul trap (Leibfried,
' I 2 The dynamics were realized experimentally and used to determine the (initial) excitation statistics pnn of the motional state by inverting eq. (223) (Meekhof, Monroe, King, Itano and Wmeland [ 19961). The same dynamics can be realized by tuning the laser onto the corresponding r + k v , and preparing the electronic subsystem initially in the ground state. blue sideband, ql = o
QUANTUM STATES OF MATTER SYSTEMS
169
Fig. 20. Surface and contour plots of the reconstructed Wigner function W ( a )of the In = 1) motional number state. The plotted points are the result of fitting a linear interpolation between the actual data points to a 0.1 by 0.1 grid. The octagonal shape is an artifact of the eight measured phases per radius. (After Leibfiied, Meekhof, King, Monroe, Itano and Wineland [1996].)
Meekhof, King, Monroe, Itano and Wineland [1996]). In the experiment, the relevant oscillation frequency in the trap potential is v/2n z 11.2 MHz, and the transition between the states 18) and le) is a stimulated Raman transition between the hyperfine ground states 2S1/2 ( F = 2, mF = -2) and 2S1/2 ( F = 1, mF = -l), respectively, which are separated by about 1.25 GHz. The coherent displacement of the initially prepared motional quantum state is realized by applying a classical, spatially uniform rf field. In fig. 20 an example of a reconstructed Wigner hnction of a motional number state is shown. The density matrix in the Fock basis, can be obtained from the displaced Fock-state probability on a circle, applying the method outlined in 53.3.2. This was also demonstrated in the above mentioned experiment (Leibfned, Meekhof, King, Monroe, Itano and Wineland [1996]). In fig. 21 an example of a reconstructed density matrix of a superposition of two Fock states is shown. The method of coherently displacing the motional quantum state to be detected can be extended in order to measure the Wigner hnction rather than reconstructing it from the measured data (Lutterbach and Davidovich [1997]) ' I 3 .
emn,
' I 3 For comparison, see the approach to direct measurement of the Wigner function in cavity QED as outlined in 5 2.4.
170
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI,
54
Fig. 21. Reconstructed density-matrix amplitudes IpJ= ]@Irnn of an approximate l/&l(ln = 0) iln = 2)) motional state. The state was displa%edby JaJ= 0.79 for eight phases on the circle. (After Leibfried, Meekhof, King, Monroe, Itano and Wineland [1996].)
For this purpose the laser driving the Jaynes-Cummings dynamics is tuned on resonance to the electronic transition, so that the Hamiltonian (221) for k = 0 applies. For sufficiently small Lamb-Dicke parameter, VLD << 1, the nonlinear function f c O ) in eq. (222) can be approximated by f c O ) M 1 - &(2+2 + 1/2). Consequently,the vibronic Rabi frequenciesread as Qnn M QL[ 1 - &, (n+ 1/2)]. Inserting &is result into eq. (223), with &(a) in place of en,,,and choosing the interaction time as t = t w = n/(Qr$,)yields
n=O
which reveals that under the conditions given the measured electronic-state occupation probability directly gives the Wigner function at the phase-space point a. Further it was proposed to reconstruct the density matrix in the Fock basis following a line similar to that given by eqs. (134H136) for cavity QED (Bardroff, Leichtle, Schrade and Schleich [1996]). In cavity QED the k-photon transitions needed to calculate the off-diagonal density matrix elements n+k from the measured atomic-state inversion (provided that initially a coherent superposition of the two states is prepared) can hardly be realized for larger values of k. In the case of a trapped ion the situation is improved significantly. The k-quantum interaction of Jaynes-Cummings type as given by eq. (221) can be realized as outlined above, by tuning the laser on resonance to the kth motional sideband. This enables one to reconstruct the density matrix elements
en
11,
o 41
QUANTUM STATES OF MATTER SYSTEMS
171
@,,,,+k for all k in principle, by successively tuning the laser on all sidebands. Needless to say, this requires sufficiently large values of the Lamb-Dicke parameter, since otherwise the coupling of the laser to high-order sidebands is very small. Formally, the difference to cavity QED only consists in the frequencies Q,, ,+,+, which are different for the two systems 14. It was also proposed to reconstruct the density matrix in the coherentstate basis by combining coherent displacements with motional ground state measurements (Freyberger [ 19971). The scheme allows a pure motional quantum state to be reconstructed by coherent displacements on a circle. Reconstruction of the density matrix of a mixed state requires additional variation of the displacement amplitude. A first dsplacement is introduced before a filtering measurement is performed in order to decide whether or not the atom is in the motional ground state. A second displacement is performed which is followed by detecting the probability that the atom is in the motional ground state, which may be obtained from the Jaynes-Cummings dynamics as outlined above.
4.2.3. Entangled vibronic states Let us consider the situation where the motional state is entangled with the electronic state of the system and ask for measuring the complete vibronic state. Such 8 measurement may be realized by the interaction Hamiltonian (221) for k = 0 (Wallentowitz, de Matos Filho and Vogel [1997]). In the scheme the initially prepared vibronic state @ is first displaced coherently in the phase space of the motional subsystem, @ -+ @(a)= bt(a)@h(a). Next the dnving laser causing the electron-motional coupling is switched on for an interaction time Z. This procedure is followed by probing the atom for fluorescence on the strong, auxiliary transition. Provided the atom is detected in the excited electronic state ) . 1 (no fluorescence), the density operator @(t) of the system reduces to
At this point it should be noted that in a Paul trap the applied rf field may modulate the motion of a trapped atom, which gives rise to the so-called micromofion. The rf frequency is usually large compared with the (effective) motional frequency, and its effect may consist in a modification of the excitation-dependent functionf(k) in the interaction Hamiltonian (221) (Bardroff, Leichtle, Schrade and Schleich [1996]). Apart from such modifications, the structure of the Hamiltonian is preserved. For details of the treatment of the micromotion, see the original article.
172
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[K0 4
-
where @,&) ( e [ @ ( t > l eis) the corresponding density operator of the motional subsystem. Its diagonal elements in the Fock basis read
Equation (226) reveals that appropriately chosen interaction times t allow the displaced vibronic density-matrix elements @:!(a) to be mapped onto the reduced motional number statistics [ @ ~ (Itt is )] worth ~ ~noting . that the underlying vibronic coupling is not only suited for this mapping but also for a QND measurement of the reduced motional number statistics, since the Hamiltonian commutes with the motional number operator 'I5. Note that the reduced motional number statistics can also be determined using other methods, such as the methods outlined in 3 4.2.2. Knowing the displaced vibronic density-matrix elements @::(a)as functions of a, the density matrix: ;@ of the original vibronic quantum state can be reconstructed, following the lines given in 5 3. In particular, the elements @$(a) can be summed up to obtain the vibronic quantum state in terms of the Wigner function matrix Wub((r),
(Wallentowitz, de Matos Filho and Vogel [19971). The Wigner-function matrix WcZb(u)has the following properties. Its trace with respect to the electronic subsystem is the (reduced) motional Wigner function. Integrating Wub(a)with respect to the motional phase-space amplitude CT yields the (reduced) electronic density matrix. A typical quantum state for which such a measurement scheme would be of interest is an entangled state of the type I q) (12) 10) f I 1) I - a ) ) 1 6 . Clearly, the nonclassical features of such a state are lost completely when one measures only the reduced motional quantum state.
-
I " For the application of this type of vibronic coupling to QND measurement of the motional excitation of a trapped atom, see de Matos Filho and Vogel [1996], Davidovich, Orszag and Zagury [1996]. States of this type have been realized experimentally (Monroe, Meekhof, King and Wineland [1996]). For a reconstruction of W,h(a) from a computer simulation of measurements, see Wallentowitz, de Matos Filho and Vogel [1997]).
n. o 41
QUANTUM STATES OF MATTER SYSTEMS
173
4.3. BOSE-EINSTEIN CONDENSATES
The recent progress in evaporative cooling of an atomic gas has rendered it possible to realize experimentally Bose-Einstein condensation (Anderson, Ensher, Matthews, Wieman and Cornell [ 19951, Bradley, Sackett, Tollett and Hulet [1995], Davies, Mewes, Andrews, van Druten, Durfee, Kurn and Ketterle [ 19951). The Bose-Einstein condensate (BEC) represents a macroscopic occupation of the ground state of the gas, which is an important signature of quantum-statisticalmechanics. Moreover, coherence effects, such as interference fringes between two condensates, could be demonstrated (Mewes, Andrews, Kurn, Durfee, Townsend and Ketterle [ 19971, Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997]; for a theoretical interpretation, see Wallis, Rohrl, Naraschewski and Schenzle [1997]). In view of these feasibilities it is interesting to get more insight in the exact nature of the quantum state of BEC. Let us consider a two-mode BEC consisting of N atoms ' 1 7 , where the two modes may correspond to different hyperfine states. To reconstruct the quantum state, it was proposed to introduce a controllable phase shift 4 between the modes and to mix them by applying a (lossless) beam-splitter transformation. The numbers of atoms in the two modes are counted, and the two-mode quantum state is inferred from joint counting statistics (Bolda, Tan and Walls [1997], Mancini and Tombesi [ 1997b], Walser [19971) ' I 8 . Neglecting collisions between the atoms, the two-mode density operator 6 is transformed as
with
[cf. eq. (7)], where the mode subscripts 1 and 2 label the corresponding hyperfine states. Raman transitions between these states can be made by optical pulses which are off-resonance from an excited state 3. An rf field may be used to couple state 1 to another state 4, and atoms in state 4 are repelled
I" The two-mode BEC is considered in view of the lack of a coherent reference state within a condensate of a fixed number of atoms. ' I s The three proposals are basically similar. Concerning the practical realization of such a measurement, we briefly outline the rather detailed scheme proposed by Bolda, Tan and Walls [1997]. For the interaction with radiation of an atomic Bose gas in an isotropic harmonic oscillator potential, see also Javanainen [1994].
174
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[n, § 4
from the trap. The beam splitter transformation can be realized by applying two copropagating light pulses which are detuned from the 1 - 3 and 2 - 3 transitions. The phase 4 is controlled by the phase difference of the lasers. The pulses are followed by an rf n-pulse that transfers the state 1 t0 4. Consequently, the atoms in state 4 fall freely and those in state 2 remain trapped. Eventually, the separate groups of trapped and untrapped atoms are counted (for the experimental setup, see also Mewes, Andrews, Kurn,Durfee, Townsend and Ketterle [1997], Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997]). The probability Pm(@)= P n , ~ - n ( 4of) counting n and N - n atoms, respectively, in the modes 1 and 2 for a phase shift setting of 4 can be given by119
where T,"(@) is defined by the corresponding matrix elements of the beam splitter transformation in the basis of the states Im). The density matrix of the two-mode state can be obtained by numerically inverting eq. (230) (Bolda, Tan and Walls [ 19971, Mancini and Tombesi [ 1997b1) by means of the least-squares method (Appendix D; see also Q 3.9.2). A further simplification can be achieved by talung the Fourier transform of Pm(@)with respect to 4 (Bolda, Tan and Walls [1997]). Since the sth Fourier component P; = (2n)-' s,'" d$ eis@Pm(4) is related only to the matrix elements whose row and column indices differ by s, the corresponding blocks of equations can be inverted separately (cf. Q 3.3.2). Alternatively, the inversion can also be performed analytically by applying orthogonality relations for Clebsch-Gordan coefficients (Walser [19971). The complex amplitude of a BEC field, q ( r , t ) = ( ~ ) ( r , t ) )is, frequently assumed to satisfy a nonlinear Schrodinger equation:
(m, particle mass; U , potential; g , atom-atom interaction constant). The density of particles lq12can be measured by phase-contrast imaging (Andrews, Mewes, van Druten, Durfee, Kurn and Ketterle [1996]). In order to determine v ( r ,t ) for given Iv(r, t)12, it was proposed to use a nonlinear propagation for a trial function
' I 9 For simplicity, the abbreviating notation lm) for the two-mode states In), 8 1 N - n ) ~is used.
11,
41
QUANTUM STATES OF MATTER SYSTEMS
175
q,so that the correct function can be found by adaptive modification of the trial function (Leonhardt and Bardroff [ 19971). 4.4. ATOMIC MATTER WAVES
Stimulated by the successful experimental demonstration of atom interferometry (Carnal and Mlynek [ 19911, Keith, Ekstrom, Twchette and Pritchard [ 19911) the study of coherence properties of atomic matter waves has been of increasing interest. In particular, several methods have been considered for determining the (single-particle) quantum state of both the transverse and the longitudinal centerof-mass motion of the atoms, and the first experiments have been performed. 4.4.1. Transverse motion
The density matrix of the (with respect to the direction of propagation) transverse motion of a single particle can be determined by methods of refractive (atom) optics (Raymer, Beck and McAlister [ 19941, Janicke and Wilkens [ 19951). Combining the effect of lenses and propagation, it is possible to reconstruct the quantum state from measured position distributions by tomographic methods (0 3.1). The desired rotation of the quadrature components can be realized by appropriately changing the parameters of the experimental set-up, such as the focal lengths of the lenses, their positions and the positions of the detectors (Raymer, Beck and McAlister [1994]) IZo. In the fkst experimental reconstruction of the transverse quantum state of an atomic matter wave (Kurtsiefer, Pfau and Mlynek [1997], Pfau and Kurtsiefer [ 19971) the scheme is simplified by avoiding the use of lenses. Metastable helium atoms are used whose quantum state of the transverse motion is prepared by a combination of an entrance slit and a double slit, which realizes a nonclassical two-peaked atom-wave state. The atoms then propagate for some distance to a time- and space-resolving detector, which allows one to measure the transverse atomic distribution of the atoms with a spatial resolution down to 500 nm, and the arrival time of the atoms with an accuracy of 100 ns. This renders it possible to record spatial atomic distributions for different free-evolution times of the atomwave packet after the double slit (fig. 22), which are suited for reconstructing
I2O
For possible realizations of optical elements in atom optics, see, e.g., Adams, Sigel and Mlynek
[ 19941.
176
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
[II,
54
40 1000
600
200 400
30
I
20 10
1
'
o -7-Y2-
--r
0
F
25 50 75 100 125 150 175 x l p
0
Fig. 22. Measured time-resolved spatial atomic distribution in the Fraunhofer diffraction limit. The two wave packets emerging from the individual Slits overlap almost completely, and a maximal visibility of the interference fringes is observed. (After Pfau and Kurtsiefer [1997].)
the quantum state. The measured distributions are position distributions of the sheared Wigner function, W(X,P,t)= w ( x - P t,P,O), m
(232)
(m, mass of the atom) of the freely evolving atom-wave packet after the double slit. Knowing the position distributions for a sufficiently large set of evolution times, the original quantum state can be obtained (6 3.3.1; for reconstruction of an original Wigner function from marginals of the sheared Wigner finction, see also Lohmann [19931). In particular, the measured position distributions correspond to the quadrature-component distributions whose phase parameter is given by cp = arctan
(3) ,
(233)
(xo,scaling length). Obviously, the phase cp can be scanned only between 0 and n/2, and in a real experiment this interval is further reduced in general. Since a precise reconstruction of the quantum state requires a phase interval of size n, some a priori information about the state should be available (cf. 0 3.9.1). In the present case it may be possible to assume symmetry in position space for the Wigner function, because of the preparation of the quantum state by a double slit. The Wigner function reconstructed (with a limited range of phases cp) from the measured diffraction pattern in fig. 22 by applying inverse Radon transform (9 3.1) is shown in fig. 23.
11,
0 41
177
QUANTUM STATES OF MATTER SYSTEMS
\
Q -1
-8
-4
b x/pm-
4
Fig. 23. Wigner function of the atomic motional state immediately behind the double slit, reconstructed from the measured time-resolved spatial atomic distribution in fig. 22. (After Pfau and Kurtsiefer [ 19971.)
Although the shape of the Wigner function and especially the oscillating interference part are well reproduced, there is a systematic error owing to the limited range of phases accessible in the experiment. For example, the reconstructed Wigner function appears sheared. This shear could be avoided by forced symmetrization of the Wigner function in position. Using the mttukerShannon sampling theorem (see, e.g., Marks [1991]), it can be proved which features of the reconstructed Wigner function of an atom-wave packet are obtained correctly when the set of data is not tomographically complete (Raymer [1997]). The analysis shows that the density matrix in the momentum basis ni,)&( around is correct everywhere except in an excluded region of width the diagonal, with being inversely proportional to the time interval over which the diffraction pattern is recorded. Transferring this result to the reconstructed Wigner function, it turns out that there is a low-frequency error in W ( x , p )as a function of x, whereas the high-frequency behavior is essentially correct. In another tomographic scheme for reconstructing the Wigner function of an initially prepared transverse motional state, a set-up is considered in which a classical standing light field that is strongly detuned from the atomic transition serves as a thick gradient-index lens for two-level atoms which are prepared ~
178
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
PI,
54
in the ground state and cross the'light field close to a node (Kienle, Fischer, Schleich, Yakovlev and Freyberger [1997]1; for a set-up of this type, see also fig. 8). It was found that here the phase cp of the quadrature-component distribution associated with the measured time-resolved position distribution can be tuned over a complete n interval. In a modified (interferometric) scheme it is assumed that the phase c p ~of the mode hnction of the standing-wave laser field can be controlled via a movable mirror, which is used to create the standing wave (Freyberger, Kienle and Yakovlev [ 19971, Kienle, Fischer, Schleich, Yakovlev and Freyberger 119971). It is shown that when a pure motional state is realized, then the phase of the freeevolving wave function can be inferred from the atomic position distributions for phases of the laser field c p ~= -n/4, 0, n/4 and n/2 at chosen time. Knowing from a pre-measurement the absolute value of the wave function for t h s time, the initial wave function can then be reconstructed by inverting the free time evolution.
4.4.2. Longitudinal motion
In order to determine the quantum state of the longitudinal center-of-mass motion of atoms in an atomic beam, it was proposed to tomographically reconstruct the Wigner function via state-selective time-dependent measurement of longitudinal position distributions (Kokorowski and Pritchard [ 19971). In the scheme, twolevel atoms that are initially prepared in the ground state are considered. Located at an adjustable position in the apparatus is an electromagnetic excitation region that is assumed to be effectively a n pulse; i.e., any particle exposed to the radiation in the region is excited completely. Downstream, a state-selective detector counts the number of particles which have made a transition into the excited state owing to the n pulse. Time dependence may be introduced either by operating the excitation region in short pulses at definite times, or by using an excited state whose lifetime is much shorter than the desired time resolution. In the latter case, detection consists of observing spontaneously emitted radiation, and the time is simply the time of detection. Measurement of the longitudinal position distribution as a function of time then yields, similar to eq. (233), the quadrature-component distributions for phase parameters in a n/2 interval. Again, symmetry assumptions may be made in order to compensate for the lack of accessible phases. It was also proposed to reconstruct the quantum state in terms of the density matrix in the energy basis by using an interferometric method that is based on
K 8 41
QUANTUM STATES OF MATTER SYSTEMS
179
a generalization of Ramsey’s classic separated oscillatory fields technique 12’ and the observation that the longitudinal momentum of an atom can be shifted coherently via interaction with off-resonant radiation (Dhirani, Kokorowski, Rubenstein, Hammond, Rohwedder, Smith, Roberts and Pritchard [ 19971). In the interferometer an atom-wave in the internal (electronic) ground state is split into ground-state and excited-state components by an electromagneticfield (detuning 61) at position 21. The excited-state component receives a momentum shift hAk1 ( x 61/u if the kinetic energy mu2/2 is much larger than h61). The remaining ground-state component is split again by a second field (detuning &) at x2, where the excited-state component is momentum-shifted by hAk2. Both field regions are assumed to have different detunings (61 # &). The excited state detected at the interferometeroutput contains a coherent superposition of two distinct energy components, each with a different corresponding longitudinal momentum. It can be shown that the probability of detecting an atom in the excited state at location z (output port) and time t is given by 122
P&, t ) =
/ / d52’
x
d52” @(52’,52”) exp [-i(52’ - Q”) ( t - z h ’ ) ]
{ f [(Sl - &)t - AkI(z - 21) + A k i ( ~- ~ 2 ) ]} ,
(234)
COS~
where g(Q’,Q”) is the motional density matrix in the energy basis (strictly speaking, SZ = EA). Equation (234) reveals that the measured excited-state probability P&, t ) is determined by a convolution of the density-matrix elements @(a’, 52”) with two factors. The exponential represents the free evolution of the density matrix, and the cosine squared contains the phase Qfference accumulated by the two paths of the interferometer. Eventually, eq. (234) can be inverted in order to obtain @(a’, Q”) in terms of P,(z, t ) by Fourier transforming twice after appropriate change of the variables (for details, see Dhirani, Kokorowski, Rubenstein, Hammond, Rohwedder, Smith, Roberts and Pritchard [19971). 4.5. ELECTRON MOTION
So far, knowledge of electronic quantum states has been typically required in order to reconstruct motional and vibrational quantum states of atoms and
For details, see Ramsey [1956]. Here it is assumed that En‘ - En” << En’, En”; for the relation without this restriction, see Dhirani, Kokorowski, Rubenstein, Hammond Rohwedder, Smith, Roberts and Pritchard [ 19971. 12’ 122
180
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI, 5 4
molecules. This may have brought ,the reconstruction of the motional quantum state of electrons into question (for the spin, see 9 4.6). 4.5.1. Electronic Rydberg wave packets
Highly excited Rydberg states have offered novel possibilities of preparing electron-wave packets. Since the early experiments at the end of the 1980s (Yeazell and Stroud [1988], ten Wolde, Noordam, Lagendjk and van Linden van den Heuvell [1988]), various schemes have been used in order to produce Rydberg wave packets (for a brief review of generation and detection of Rydberg wave packets, see Noordam and Jones [1997]). A Rydberg wave packet is produced whenever several Rydberg states are excited coherently. In particular, it has also been possible to prepare an atomic electron in a coherent superposition of two wave packets which are localized spatially at the opposite extremes of a Kepler orbit (Noel and Stroud [1996]). On a sufficiently short time scale these nonclassical states are close to the evedodd coherent states of the harmonic oscillator. To demonstrate important features of these states, two kinds of measurements have been used. State selective ionization allows one to verify that only every other atomic level is populated, and a Ramsey fringe measurement verifies the coherence of the superposition. A wave-packet interference technique was proposed to measure engineered atomic Rydberg wave functions (Chen and Yeazell [ 19971) 123. It is assumed that the Rydberg wave packet is produced by (a sequence of) short optical (Gaussian) pulses whose interaction with the atomic system can be treated in lowest-order perturbation theory, Describing the atomic system by the radial wave function,
where a,, and w,, are the Schrodinger amplitudes and eigenenergies for the Rydberg states and ug and wg are these for the ground state, the excited-state amplitudes after the last excitation pulse, a,,, are given by a,, = -
where z, = 1 + Ciexp[i(w + A,).s;] (Q?,,Rabi frequency of the excitation between the ground state and a given Rydberg state n; At?= w,, - w, detuning;
The method was also proposed for reconstructing pure vibrational states in a molecule (Leichtle, Schleich, Averbukh and Shapiro [1998]).
11, 0 41
QUANTUM STATES OF MATTER SYSTEMS
181
0,pulse width; ti,time delay for the ith individual pulse). When a probe pulse (identical to the previous pulses) interacts with the atom, then the new excitedstate amplitudes, 2 2
a'n = -liQne-*nu 2
'2
{zn + exp[i(w + An)z]},
(237)
are produced (z, delay time for the probe pulse). By measuring the population in each eigenstate, lak12, for different phases between excitation and probe pulses (on applying high-resolution technique, such as state-selective field ionization), the complex quantities z,, can be obtained; i.e., the (radial) wave function created by the excitation pulse may be inferred. Note that the pump and probe pulses need not be identical, but they must be phase coherent with each other. Using optical pulses, Rydberg wave packets can usually be detected when they are near to an ionic core. This limitation can be overcome by using subpicosecond,unipolar electromagnetic^ field (half-cycle) pulses 124, whch have been a powerful tool for studying dynamics in weakly bound systems (Jones, You and Bucksbaum [1993]). Such pulses can track the wave packet throughout its orbit and detect wave-packet motion anywhere in the atom, as was demonstrated experimentally (Raman, Conover, Sukenik and Bucksbawn [19961). In particular, it has been possible to monitor the momentum-space probability of the wave packet as a function of time (Jones [1996]; for details, see also Noordam and Jones [1997]). In the experimental demonstration of the method, also called impulsive momentum retrieval, Na atoms are first excited by a tunable nanosecond laser to the 25d Rydberg state. Subsequently, two half-cycle pulses are applied. Kicking the electron with the first pulse, the Rydberg population is redistributed and a complicated, dynamically evolving wave packet is created. The second pulse is used to measure the ionization as a function of its peak field and delay relative to the first pulse. If the duration of the probe pulse is negligible compared with the time scale for variations in the position and momentum of the electron, then the energy gained (or lost) by the electron depends only on its initial momentum (i.e., the momentum immediately before the application of the pulse) and the time-integrated field. By measuring the ionization probability (at threshold) as a function of field, the momentum distribution of the initial state along the field axis can be obtained.
'24 For these pulses, commonly referred to as HCP's, see also Greene, Federici, Dykaar, Jones and Bucksbaum [1991] and You, Jones, Bucksbaum and Dykaar [1993].
182
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
PI,
04
4.5.2. Cyclotron state of a trapped electrpn
Let us consider an electron in a Penning trap and assume a uniform magnetic field along the positive z axis. The cyclotron and the axial motions are well separated in their frequency ranges (for a review, see Brown and Gabrielse [1986]). It was proposed to reconstruct the cyclotron state by mapping it onto the axial degree of freedom (Mancini and Tombesi [1997a]). By applying appropriate fields on the trapped electron, a coupling between the two degrees of freedom may be realized such that the Hamiltonian in the (cyclotron) interaction picture is of the form
fi = hv,
+ $)+ hg2,(cp)i.
(6$,
Here, 6; (6,) is the creation (annihilation) operator for the axial motion of frequency vr,and 2 and iC(cp), respectively, are the axial positon operator and the quadrature operator of the cyclotron motion, g being the (field induced) coupling constant. For an interaction time that is short compared with the axial period, z << 2n/v,, the Hamiltonian (238) gives rise to a unitary kick and the time evolution of the mean axial momentum yields the mean of the quadrature operator,
+ Q) = (M>) + figt(Mcp))cos(vzt).
(239)
By measuring the current due to the induced charge variation on the cap electrodes of the trap, one may obtain the axial momentum and then the cyclotron quadrature component. Knowing the cyclotron quadrature-component statistics, the quantum state can be derived; e.g., by applying the methods of $83.1, 3.2 and 3.3.1. To determine the cyclotron state from the displaced number statistics ($0 3.3.2 and 3.5), it was proposed to perform a QND measurement of the cyclotron excitation number (Mancini and Tombesi [1997a]). Using a magnetic bottle configuration (Brown and Gabrielse [ 1986]), an interaction Hamiltonian of the type
fi' = h#$;c
c
22
(240)
may be realized. It gives rise to a dependence of the axial angular frequency on the number of cyclotron excitations, which may be obtained by probing
'25
For the trap, see Penning [1936].
11, P 41
QUANTUM STATES OF MATTER SYSTEMS
183
the resonance frequency of the output electric signal. The displacement of the cyclotron state may be realized by a driving field acting immediately before the measurement process induced by the Hamiltonian (240). 4.5.3. Electron beam A method was also proposed for determining the quantum state of the (onedimensional) transverse motion of a beam of identically prepared electrons (Tegmark [1996]). Originating from the electron source, the beam crosses a magnetic field and in conjunction with collimators a highly monochromatic beam is prepared. This beam enters a shielded box, wherein a harmonic potential acts in the x-direction. The needed potential is obtained by subdividing the walls of the box into a large number of metal plates which are insulated from one another, and by fixing the potentials of the plates appropriately. A slidable detector (along the z-direction of longitudinal motion of the beam) allows one to measure the particle density p(x) along the x-direction. Performing these measurements for a series of locations of the slidable detector, the density matrix characterizing the transverse motion of the electrons of the beam may be reconstructed. 4.6. SPIN AND ANGULAR MOMENTUM SYSTEMS
In the history of quantum-state measurement, spin (angular momentum) systems have played a particular role, because of the finite-dimensional Hilbert space (see, e.g., Fano [1957] and the references in footnote 41 for the Pauli problem). Spin-quantum-state reconstruction is usually based on data obtained from SternGerlach experiments, in which particles with magnetic momentum are deflected in a magnetic-field gradient and the spin (or angular momentum) projection onto the direction of the field gradient is measured (Stem [1921], Gerlach and Stern [1921, 19221, Feynman, Leighton and Sands [1965]; for the optical SternGerlach effect, i.e., state-dependent deflection of particles in optical fields, see Kazantsev [1975, 19781, Cook [ 19781, Tanguy, Reynaud and Cohen-Tannoudji [ 19841, and for its experimental realization, see Sleator, Pfau, Balykin, Carnal and Mlynek [ 19921). Interacting the particles with fields of magnetic multipoles, multipole moments of the particles can be measured (Bohn [1991]). In a Stern-Gerlach analyzer as described by Feynman, Leighton and Sands [1965] a beam of particles is split in an inhomogeneous magnetic field into different paths provided with gates that may be opened or closed. The diagonal density-matrix elements pmm are then given by the probability that the particle is deflected into the corresponding mth path. Measurement of the off diagonal
184
HOMODY NE DETECTION AND QUANTUM-STATE RECONSTRUCTION
tK 5 4
element @kn requires two analyzers. In the first analyzer, the nth and kth paths are opened while the remaining paths are closed. In the second analyzer the field gradient is in the plane that is perpendicular to the quantization axis defined by the first analyzer. Let the orientation of the field gradient of the second analyzer. be QI. The probability pm(q)of the particle being deflected into the mth path in the second analyzer is then given by (Gale, Guth and Trammel1 [1968]) 2
pm(QI)= N{ pt2(n/2)]
@kk
+ [d$;(n/2)]
+ 2l@knld2(n/2)d$;(X/2)
@nn
(241) cos[(k - n ) q + QIh]
where @h= l@knl exp(iqh). The matrix d$i(t9) is the &dependent part of the Measuring Wigner rotation matrix (see Wigner [1959]), and N = (@& + @J1. (for known @kk and &Inn) the probabilities for two different orientations QI, the off diagonal-density matrix element @An can be determined. The quantum state can also be determined by using a single Stern-Gerlach apparatus with variable orientation (Newton and Young [1968]). The probability pm(z9,QI) that the projection of the spin on the quantization axis (0, QI)is m can be expressed in terms of the density matrix elements @kn as
which can be inverted to obtain
where
.
25
e,:,
cp, = 2 n d ( 4J + l), and are Clebsch-Gordan coefficients. It is seen that in order to reconstruct the density matrix, it is sufficient to keep 19 fixed and take ~1 at 4 J + 1 values such that d$(6) does not vanish for any combinationj and w. It should be noted that the Stern-Gerlach apparatus can also be allowed to take all possible orientations (Dodonov and Man'ko [1997]). In this case, one need
11,
o 41
185
QUANTUM STATES OF MATTER SYSTEMS
not search for an optimum choice of orientations, since all possible orientations are considered on an equal footing. As already outlined, the problem of reconstruction of spin density matrices has also been treated by means of the principle of maximum entropy (Buiek, Drobny, Adam, Derka and Knight [1997]; see 0 3.9.3) and Bayesian inference (Jones [ 19911, Derka, Buiek and Adam [ 19961, Derka, Buiek, Adam and Knight [1996]; see 93.9.4). It was also proposed to tomographically reconstruct the discrete Wigner function of a spin system in analogy to the inverse Radon transform for the continuous Wigner function (Leonhardt [1995, 19961). However, the scheme does not only require measurement of probability distributions of spin (angular momentum) components but also those of discrete phase states. With regard to pure states, it can be shown that the spin wave function can be reconstructed from the populations of the spin projections on two directions deviated from each other by- an infinitesimal angle (Weigert [1992]). Particular attention was paid to the determination of the class of pure states which are eigenstates of rotated spin projection operators J,, = n,J, + nyJy + nzJz, where n = (n,,n,,n,) is a direction vector of unit length. As shown, measurement of spin projections on three directions is sufficient to determine the unknown direction n and the corresponding spin component eigenvalue (IvanoviC [ 19931). Finally, experiments for reconstructing the angular momentum density matrix of electrons in hydrogen atoms from measurements of the Stark structure of the emitted light have been reported (Havener, Rouze, Westerveld and Risley [1986], Ashburn, Cline, Stone, van der Burgt, Westerveld and Risley [1989], Ashburn, Cline, van der Burgt, Westerveld and Risley [1990], Cline, van der Burgt, Westerveld and Risley [ 19941, Renwick, Martell, Weaver and Risley [1993], Seifert, Gibson and Risley [1995]). In the experiments, the density matrix is considered for the manifold of states of principal quantum number n = 3 and n = 2, the matrix elements being parametrized by the orbital and magnetic quantum numbers 1 and m respectively. The excited hydrogen atoms are produced by collisions of protons with noble gas atoms. Measuring the (n = 2) and coherence parameters of the emitted light [transitions (n = 3) (n = 2) t ( n = l)] in dependence on the external electric field, one can infer the density-matrix elements (for theoretical results, see Jain, Lin and Fritsch [1987a,b, 19881). --f
4.1. CRYSTAL LATTICES
More than ten years before the first demonstration of reconstruction of the quantum state of light, experiments were already performed for reconstructing
186
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
P,0 4
off-diagonal elements of the single-particle density matrix of crystal lattices (Golovchenko, Kaplan, Kincaid, Levesque, Meixner, Robbins and Felsteiner [1981], Schiilke, Bonse and Mourikis [198l]). The methods are based on Compton scattering or X-ray scattering. Before giving the basic idea of these methods, let us outline briefly the information available in standard X-ray and Compton scattering experiments. In conventional X-ray diffraction, the Bragg diffracted intensity corresponding to a reciprocal lattice vector g is proportional to the squared modulus of the form factor F(g). Provided that the phase of the form factor can be determined, some limited insight into the one-particle density matrix in momentum space, @(p,p+ g), can be gained. To be more specific, the form factor yields the momentum average of the density matrix according to FW
dP Q(P,P+g).
=
(245)
In conventional Compton scattering one may obtain the position average of the density matrix in position space, @(r, r’).The one-dimensional Fourier transform of the Compton profile, the so-called reciprocal form factor B(r), can be given bY B(r) =
/
dr’ Q(r’,r’ + r).
(246)
Obviously, conventional methods do not provide the full density matrix, neither in position nor in momentum space. The basic idea to overcome thls problem consists of the use of coherent scattering methods (Golovchenko, Kaplan, Lncaid, Levesque, Meixner, Robbins and Felsteiner [1981], Schiilke, Bonse and Mour~kis [1981], Schiilke and Mourikis [1986], Schiilke [1988]). Let us consider the situation in Compton scattering. In conventional experiments a propagating plane wave is used, so that each point in position space is excited equally. Thls explains the fact that one can only measure position-space averages of the density matrix as given in eq. (246). To avoid this limitation, one may introduce a weighting of certain positions within the elementary cell. This can be done by using a standing-wave field with a spatial periodicity of the nodes that is commensurable with the lattice periodicity. The spatial distribution of the corresponding intensity is given by i ( r ) = I , + 12
+ 2J17?; cos(g . r + cp),
(247)
where the phase cp defines the position of the nodes with respect to the atomic planes of the lattice. One may perform two measurements of the Compton
11, App. A]
RADIATION FIELD QUANTIZATION
187
profiles by choosing the values of the phase to be Q, = 0,n. The difference of the corresponding form factors, AB(r;g), is then given by AB(r;g)
-
1
dr’ &’, r’ + r ) cos@ . r’).
That is, coherent Compton scattering allows to determine spatial Fourier transforms of the one-particle density matrix.
Acknowledgments
We thank all the colleagues who supported us during the preparation of this article. We are indebted to them for, reading foregoing drafts and giving valuable comments which improved the final version, for bringing particular details of the subject to our attention, for providing us with experimental data and for the help in preparing figures. In particular, we are grateful to E.L. Bolda, T. Coudreau, G.M. D’Ariano, E. Giacobino, Z. Hradil, W.M. Itano, L. Knoll, D. Leibfkied, U. Leonhardt, M.G.A. Paris, J. Pefina, Th. Richter, S. Wallentowitz, D.F. Walls, I.A. Walmsley, L. Waxer, D.J. Wineland and A. Zucchetti. Finally, we wish to emphasize that our research in the field of phase-sensitive measurements and quantum-state reconstruction has been supported by the Deutsche Forschungsgemeinschaft.
Appendix A. Radiation Field Quantization
From Maxwell’s theory it is well known that the vector potential of the radiation field in free space can be given by an expansion in transverse travelling waves. For dealing with radiation inside resonators, such as cavities bounded by perfectly reflecting mirrors, a standing-wave expansion is appropriate, where the spatial structure of the waves depends sensitively on the resonator geometry through the boundary conditions. An expansion of the vector potential,
in orthogonal transverse waves,
188
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[n,App. A
is usually called mode expansion. The mode functions Ak(r) satisfy the Helmholtz equation,
4
V x V x A k - -A* C2
=O,
(A-3)
where the condition of transversality, V . A A = 0, implies that V x V x An = -MA.The mode amplitudes a ~ ( t evolve ) according to harmonic-oscillator equations of motion, cik = -iwi
(-4.4)
uk.
Since harmonic oscillators can be associated with the modes, quantization of radiation reduces to quantization of harmonic oscillators. In eq. (A.l), the complex amplitues ak and u; are regarded as non-Hermitian destruction and creation operators irk and iii, respectively, which satisfy the well-known (equaltime) bosonic commutation relations:
[&4,] &*,, I,&[ 0. =
=
(A.5)
The bosonic excitations introduced in this way are called photons. It should be pointed out that the expansion in monochromatic modes of the (Heisenberg) operator of the vector potential,
A(r, t ) =A(+)@, t ) + A"(r, A(+)(r, t)=
t),
(A4
A*(r)e-iwAt &A,
(A.7)
c i
can be used to introduce photons which are associated with other than monochromatic waves; viz., I
A(r,t ) =
[Ai,(r,t )
&it + A';(r, t )271 ,
(-4.9)
11
where CItnare elements of a unitary matrix, so that the &it and photon destruction and creation operators, respectively, and
&'jare again (A. 10)
A
(for details, see Titulaer and Glauber [ 19661). The spatial-temporal (nonmonochromatic) modes can be used advantageously for describing and analysing pulse-like radiation.
11, APP. Bl
QUANTUM-STATE REPRESENTATIONS
189
The mirrors used in resonator experiments are of course not perfectly reflecting. Moreover, fractionally transparent mirrors are deliberately used to open input and output ports in resonator equipments. Another typical example is the use of beam splitters in various interference and correlation experiments. In many cases these and related (passive) optical instruments can be regarded as macroscopic dielectric bodies which respond linearly to radiation and whose action can be included phenomenologically in the Maxwell theory through a space-dependent permittivity E (Knoll, Vogel and Welsch [1987], Glauber and Lewenstein [1991]). When the radiation under consideration is in a frequency interval where the effect of dispersion is sufficiently small, then the dependence of the permittivity on frequency can be disregarded. Performing a mode expansion (A.2), the mode functions are now solutions of a modified Helmholtz equation (A.3), with u = u(r) = c/n(r) in place of c, where n(r) = &@ is the real refractive index. The mode functions satisfy a generalized condition of transversality, V . (&An)= 0, and the relation of orthogonality requires inclusion of E in the space integral (for details, see Vogel and Welsch [1994]), because of the dependence of the permittivity on r 126. Appendix B. Quantum-State Representations Among the wide variety of possible radiation-field states, there are some hdamental states which, with regard to representation and measurement, play a special role in quantum optics. Let us consider a (free) radiation field whose vector potential is given by a mode decomposition [eq. (A.7) or eq. (A.9)] and restrict our attention to single-mode states. The single-mode states can then be used to build up multimode states as (direct-)product states. B.I. FOCK STATES
A mode (with photon destruction and creation operators 2 and 2t, respectively) is said to be in a Fock state (photon-number state) In), when the state is an eigenstate of the photon-number operator 2 = 2t2,
;In)
= nln),
n
.. .
= 0,1,2,.
(B.1)
For extensions to dispersive and absorbing media, see Gruner and Welsch [1996] and references therein. In this case, the permittivity is a complex function of frequency, the real and imaginary parts being related to each other according the Kramers-Kronig relations, and a mode decomposition outlined here fails.
190
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[n, App. B
Each number state In) can be obtained from the vacuum state (0) according to 1 In) = -fit"l0).
fi
Since the photon-number states form an orthogonal and complete set of states, the density operator of any (single-mode) quantum state can be expanded in the photon-number basis, n, n'
the diagonal elements p n photons.
=
(n@ln) being the probabilities for observing n
B.2. QUADRATURE-COMPONENT STATES
Next let us consider a (single-mode) field-strength operator of the type of P(q)= lFl(fie-ip + fiteiq). Depending on the choice of the (scaling) factor IF[, the operator P may represent the vector potential or the electric or magnetic fields associated with the mode. In what follows we set P = 21'21FJP(cp),where qcp)
= 2-1/2
(fie-@
+ fiteiq
(B.4) is called quadrature-component operator. The quadrature components P( cp) and E(cp') are observables which cannot be measured simultaneously for cp f cp' + k ~ , k = 0, f1,. . ., because of [2(cp),P(cp')] f 0. For chosen cp quadrature-component states Ix, cp) can be introduced (Schubert and Vogel[1978a,b], see also Vogel and Welsch [ 1994]), solving the eigenvalue problem -m
P(cp) 1x7 cp) = x IX, cp), 6 x < 00. Obviously, (x,cp) is related to) . 1
(B.5) =
Ix, cp = 0) as Ix, cp) = eiirqlx),
and can be given by
-
Jd-1/4e-x2/2
exp[-+ (eiViztl2+ 21/2xeiqfit] lo),
[Hn(x), Hermite polynomial]. The states lx, cp) define an orthogonal Hilbert space basis for each value of cp, so that the density operator can be expanded as
a
=
ss dx
h' 1x7 cp)(., cpl@lx',cp)(x', cpl.
03.7)
In particular, p(x, cp) = (x, cpI@Ix,cp) is the probability density for observing the value x of the quadrature component 2(cp). Note that the symmetry relation P(X,~ 1 X) + =P(-x, v) holds.
11, APP. Bl
QUANTUM-STATEREPRESENTATIONS
191
B.3. COHERE” STATES
The coherent states introduced by Schrodinger [ 19261 to simulate the motion of a (near)classical particle in a harmonic potential are usually defined by the eigenvalue problem
’*’
(a, complex). Equivalently,
where b(a ) is the coherent displacement operator,
&a)
= exp(a&t-
[note that b(a)i?bt(a)
a*&), =& -
(B.10)
a]. The coherent states are not orthogonal,
(a’la)= exp(-iIa- a’lz) exp[i(aa’* - a*a’)],
(B. 1 1)
and they are overcomplete (Cahill [1965], Bacry, Grossmann and Zak [1975]), so that there are various possibilities of representing the density operator. In particular, it can be expanded as (B.12) (d2a = d Re a d Im a). Equation (B.9) can be extended in order to define generalized coherent states by the action of the displacement operator on arbitrary states of the Hilbert space (see, e.g., Perelomov [1986]). In particular, the action of b ( a ) for all values of a on a set of states IYA)which resolve the unity introduces a foliation of the Hilbert space into orbits such that each state belongs exactly to one orbit of
‘27 The coherent states were recovered in the 1960’s (Klauder [1960, 1963a,b], Glauber [1963a], Sudarshan [1963], Klauder and Sudarshan [1968]).
192
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[n, App. B
generalized coherent states. For example, the Fock states can be used to define the generalized coherent states,
1% 4 = &oln),
(B.13)
which are also called displaced Fock states 12*. Their expansion in the ordmary Fock basis reads as
(B.14) m
where the expansion coefficients (rnln,a ) can be given by
(B.15) [{m,n}
= min(m, n),
a = Iale'Q)]or
(B.16) [L,"(x),Laguerre polynomial]. B.4. .+PARAMETRIZED PHASE-SPACE FUNCTIONS
Representations of the density operator in terms of phase-space functions whch are formally similar to classical probability distributions are used frequently. Defining an s-parametrized displacement operator by
the following expansion of the density operator can be proved correct: d2aTr{(3&-a;s)}b(a;-s),
(B.18)
Similar to the coherent states, these states correspond to wave packets of a particle in a harmonic potential which keep their shapes and follow the classical motion (Husimi [1953], Senitzky [1954], Plebanski [1954, 1955, 19561, Epstein [1959]).
U, APP. Bl
193
QUANTUM-STATE REPRESENTATIONS
(for details, see, e.g., Cahill and Glauber [1969a,b], Agarwal and Wolf [1970], Pe%a [1991]) 129. Introducing the Fourier transform of b ( a ;s),
‘J
&a-2;s) = n 2
d2pfi(p;s)exp[a/3*-a*p] (B.19)
the expansion (B.18) can be rewritten as13o
$=n
s
d2aP(a;s)&a-2;-s),
(B.20)
where
P(a;s)= Tr{@&a-Z;s)}
(B.21)
is the s-parametrized phase-space function, which is normalized to unity 13’ . Casespf particular interest are the Glauber-Sudarshan P function P(a) = P(a; 1) (Glauber [1963b,c], Sudarshan [1963]), the Wigner function W ( a ) = P (a ;0) (Wigner 119321) and the Q function Q(a) = P(a;-1) (Husimi [1940]). Note that 8((x - 2, -1) = n-’la)(al, which implies that Q(a) = n-’(aI$la). The expectation value of an operator h can then be given by
(h) =
/
d2aP(a;s)F(a;s),
where F ( a ; s ) / n is defined according to eq. (B.21) with h in place of
(B.22)
0.
For s = 0 the expansion (B.18) is known from Weyl’s quantization method (Weyl [1927]). Note that eqs. (B.20) and (B.21) also apply to other than density operators. Let fi be an operator function of 6 and d t . Then it can be shown that F = j- d 2 a F ( a ;s) 6(a - 6;s), where F(a;s) = nTr{fi &a - 6;-s)} is the associated c-number function in s order. 13’ Frequently the definition P ( q , p ; s )= 2-’P[a = 2-’”(q + i p ) ; s ] is used, so that P ( q , p ; s )can be regarded as a function of “position” and “momentum”, with j” dq dp P(q,p;s) = 1. IZ9
130
194
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
From eqs. (B.19) and (B.21) we see that
[n, App. B
,
@(a;s) = Tr { 0 b(a;s)}
(B.23)
can be regarded as the characteristic function of the phase-space function
P(a;s), 1
P(a;s) = n2
/
d2/?exp(a/?*- a*/?) @(/?; s).
(B.24)
Using eqs. (B.17), (B.23) and (B.24), the phase-space hnctions P(a;s) can be related to each other as
It should be noted that the formalism can be extended to finite-dmensional Hilbert spaces (Opatm?, Welsch and Buiek [ 19961; for the Wigner function, see Wootters [I9871 and Leonhardt [1995,1996]; for the Q function, see Opatrn9, Buiek, Bajer and Drobn? [1995] and Galetti and Marchiolli [1996]) 132. B.5. QUANTUM STATE AND QUADRATURE COMPONENTS
It is worth noting that there is a one-to-one correspondence between the phasespace function P(a;s) and the set of quadrature-component distributionsp(x, QI) for all values of QI within a n interval (Vogel and Risken [1989]). Introducing the characteristic function Y(z, QI) of the quadrature-component distribution P(% QI),
(B.27)
132 For an approach to phase-space functions for systems with finite-dimensional Hilbert spaces that uses a continuous phase space, see Varilly and Gracia-Bondia [1989] and Dowling, Aganval and Schleich [1994].
D,APP. Cl
195
PHOTODETECTION
Equations (B.24), (B.26) and (B.27) reyeal that p(x, cp)
=
'J x
dz exp(-izx- $2) (B.28)
J d2a exp[iz(alt(cp)la)] P(a;s)
and
In other words, knowledge of p(x, cp),for all values of q, within a JG interval is equivalent to knowledge of the quantum state 1 3 3 . In particular, the expansion of the density operator as given in eq. (B.18) can be rewritten, on using eqs. (B.23) and (B.27), as
0=
& 1" 1 dcp
dz IzI esZ2I4 Y(-z, c p ) b ( i ~ 2 - ' / ~ ,-s) e'~.
7
(B.30)
which offers the possibility of relating 0 to p(x, cp) as (B.3 1) provided that the operator kernel 1
cp)
=
J dz
IZI
exp{iz N c p )
-XI)
9
(B.32)
exists (D'Ariano [1995], D'Ariano, Leonhardt and Paul [1995]).
Appendix C. Photodetection
In order to give an introduction into the quantum theory of photoelectric detection of light (Mandel [1958, 19631, Kelley and Kleiner [1964], Glauber
133 Note that the
rp integral in eq. (B.29) can be performed over any n interval.
196
HOMODYNE DETECTION AND QUANTUM-STATERECONSTRUCTION
[n, App. c
[ 1965]), let us consider, as a simple example of a photodetection device, a large
sample of atomic systems which are capable of absorbing light through the photoemission of electrons in a certain time interval t , t + At. Next suppose that the photoemission of the actual number m of photoelectrons is dominated by the process in which exactly in atomic systems are involved, so that each emits just one electron. We further assume that the total number N of atomic systems is much larger compared with the mean number of emitted electrons, so that for any (relevant) actual value m the inequality m < N may be assumed. Under these assumptions, the main features of the theory may be developed by applying Dirac's perturbation theory to the basic process of light absorption and combining the corresponding results with methods of classical statistics with respect to the ensemble of photoelectrons generated by the absorption processes (see, e.g., Vogel and Welsch [1994]). Here, we will give a more intuitive anhlysis rather than an exact derivation. When each photon that falls - in the chosen time interval - on the detector gives rise to exactly one emitted electron, then the number of counted electrons agrees exactly with the number of photons, and the statistics of the counted electrons reflect exactly the photon-number statistics; i.e., the probability Pm of detecting m photoelectrons is equal to the probability p m of m photons being in the field, Pm =Pm = (ml&+ (C.1) However owing to losses, the probability r] of converting a photon to an electron is less than unity in general (0 r] 1). This probability is also called detection (or quantum) efficiency. Since (under the assumptions made) the individual events of emission of a photoelectron can be regarded as being independent of each other, the probability Pmln(r])of observing m photoelectrons under the condition that n photons are present corresponds to a Bernoulli process,
< <
and Pmln(q)= 0 if m > n. The joint probability that n photons are present and m photoelectrons are counted is then Pmln(r])pn, and hence the prior probability of detecting m photoelectrons is given by
which for r] -+ 1 (perfect detection) reduces to eq. (C.1). Note that eq. (C.3) can be inverted in order to obtain p n from Pm by simply replacing r] with q-l, m
m=n
11, APP. Dl
ELEMENTS OF LEAST-SQUARES lNVERSlON
Equation (C.3) can be rewritten as 134 P , = ( : - (rl2)" m!
e-l)i
.), .
and the characteristic function
can be given by
where the symbol : : indicates normal ordering, with the operator it to the left of the operator 2. So far, we have assumed that the photons are effectively associated with a single (nonmonochromatic) mode. The extension of the above given formulae to multimode fields is straightforward. In particular, the multimode version of the single-mode photocounting formula (C.5) reads as
where h k is the photon-number operator of the kth mode, and V k is the quantum efficiency which with the photons of this mode are detected.
Appendix D. Elements of Least-Squares Inversion 135 Let f be a (possibly unknown) no-dimensional "state" vector and consider a stochastic linear transform,
yielding an rno-dimensional (mo 3 no) "data" vector y available from measurements. Here, A is a given mo x no matrix, and n is an mo-dimensional
'34 Note that the identity In)(nI =: (i?n/n!)e-i : is valid, wiuch easily can be proved correct from the associated c-number function l(aln)I2of In)(nl in normal order. '35 For details, see, e.g., Golub and van Loan [1989], Robinson [1991], Press, Teukolsky, Vetterling and Flannery [19951.
198
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[II, App. D
vector whose random elements,with zero means and covariance matrix W-' describe the noise associated with realistic measurements. The probability of a state vector f being realized under the condition that there is a data vector y can be given by
where for Gaussian noise the conditional probability P(ylf) of observing the data vector y is
The probability P(f) is a measure of a priori knowledge of the state and can be set constant if no state is preferred. The most probable state? can then be found by minimization
from which AtWAf
= AtWy,
(D.5)
and when At W A is not singular, then
Otherwise, the inversion of eq. (D.5)is not unique and further criteria must be used to select a solution. when the matrix W is not known, then it may be set a multiple of a unity matrix, so that eq. (D.6) reduces to
provided that AtA is not singular. Equation (D.7) still gives the correct averaged inversion, but the statistical fluctuation of the result may be (slightly) enhanced.
136 If W is diagonal (i.e., the noise is uncorrelated), then eq. (D.4) represents a s u m of weighted squares of the differences between the components of the data vector y and the components of the transformed vector Af,each term of the sum being multiplied by a weight given by the corresponding element of W.
11, APP. El
ELEMENTS OF LEAST-SQUARES INVERSION
199
If the data are not sensitive, enough to some state-vector components, then these components can hardly be determined with reasonable accuracy. Mathematically, At W A becomes (quasi-)singular and regularizations, such as Tikhonov regularization and singular-value decomposition, are required to solve approximately eq. (D.5). For simplicity let us set W = I, with I the unity matrix. Using Tikhonov regularization, it is assumed that some components of the state vector can be preferred by a properly chosen a priori probability P ( f ) , such as
the parameter A. (A 2 0) being a measure of the strength of regularization. Maximization of P(fly)then yields, on recalling eqs. (D.2) and (D.3),
Note that A21 + AtA has only positive eigenvalues and is thus always invertible. A possible choice of A. is based on the so-called L curve, which is a log-log plot of I If1 I versus I lAyl 1, where Ay = y - Af,for different values of d. The points on the horizontal branch correspond to large noise, whereas the points on the vertical branch correspond to large data misfit. Optimum choice of A. corresponds to points near the comer of the L curve. Applying singular-value decomposition, the inversion of the matrix At A is performed such that their eigenvalues whose absolute values are smaller than the (positive) parameter of regularization 00 are treated as zeros, but the inversions are set to zero (instead of to infinity). This operation is called “pseudoinverse” of a matrix,
f = Pseudoinverse(A A; 00) fty.
(D.lO)
For 00 close to zero, the result of eq. (D.lO) is similar to that of eq. (D.7). With increasing 00, smaller absolute values of components off are preferred. The effect of the regularization parameters h and 00 is similar. The statistical error of the reconstructed state vector? is decreased, but bias towards zero is produced simultaneously. Hence, optimum parameters are those for which the bias is just below the statistical fluctuation. The bias can be estimated, e.g., by Monte Car10 generating new sets of “synthetic” data from the reconstructed state. From these sets one can again reconstruct new sets off. The difference between the mean value of the states reconstructed from the synthetic data and the originally reconstructed state estimates the bias.
200
HOMODYNE DETECTION AND QUANTUM-STATE RECONSTRUCTION
[n
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E. WOLF, PROGRESS IN OPTICS XXXIX 0 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
111 SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION BY
S.K. SHARMA S.N. Bose National Centre for Basic Sciences, Block-JD, Sector-3, Salt Lake City, Calcutta-700091, India
AND
D.J. SOMERFORD Department of Physics and Ashonomy, University of Wales, College of C a r d 8 Card18 CF2 3YB, UK
213
CONTENTS
PAGE
6 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .
3 2 . THE EIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING. . . . . . . . . . . . . . . 9 3. EIKONAL APPROXIMATION IN OPTICAL SCATTERING . . 6 4. APPLICATIONS OF THE EIKONAL APPROXIMATION . . . 0 5. CONCLUSIONS AND DISCUSSIONS . . . . . . . . . . . mFERENCES . . . . . . . . . . . . . . . . . . . . . . .
214
215 218 230 268 282 285
8
1. Introduction
The eikonal approximation (EA), also referred to as the high energy approximation, was first introduced in problems connected with high energy potential and nuclear scattering in the late 1940s and early 1950s. The earliest development of this approximation appears to be due to Molikre [1947] in his work on the elastic scattering of fast charged particles. However, according to Schiff [1956], Montroll and Greenberg [1954] attribute it also to Glauber [1953] whilst Goldman and Migdal [1954] attribute it to Landau and Lifshitz [1948]. In the Soviet literature, this approximation has also been referred to as the SitenkoGlauber approximation after the work of Sitenko [ 19591. An approximation similar to the EA was obtained by Raman and Nath [1935, 19361 in the context of the scattering of light by acoustic waves. A detailed account of the early development of the EA in non-relativistic potential and nuclear scattering has been given by Glauber [1959] in his famous Boulder lectures on high energy collision theory. Despite its extensive use in nuclear scattering problems, it was quite some time before applications of the EA to new disciplines began to appear. It was extended to relativistic potential scattering of spin 1/2 particles by Hunziker [1963] and Baker [1964]. Franco [1968] demonstrated its usefulness in calculations of e-H( 1s) elastic scattering amplitude. Applications to electron-molecule and atomatom scattering soon followed (Byron, Krotkov and Medeiros [ 19701, Yates and Tenney [1972a,b], Chang, Poe and Ray [1973]). At about the same time, it was used successfully to sum the high energy behavior of generalized ladder diagrams in quantum field theory in a compact and useful manner (Abarbanel and Itzykson [1969], LBvy and Sucher [1969], Englert, Nicoletopoulos and Truffin [ 19691). Drawing an analogy between a potential and a refractive medium, the EA was then adapted to scattering problems in optics. Although this suggestion first came quite early (Greenberg [ 1960]), there seems to have been no activity until Borovoi and Krutikov [ 19761 applied it to the scattering of light by an ensemble of particles and later Sharma and Debi [1978] suggested its use in the analysis of light scattered by biological cells. In the 1970s and later, the EA has been applied to many other scattering problems. Gersten and Mittleman [ 19751 applied it to the scattering of charged 215
216
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
tIIk
81
particles in the presence of an electromagnetic field. The process is of central importance in the study of plasma heating by electromagnetic waves. In the same year it was applied to the atomic scattering from corrugated surfaces (Garibaldi, Levi, Spadacini and Tommei [ 19751). Group theoretical techniques in eikonal physics were initiated by Harnad [1975] and by Leon, Quiros and Mittelbrunn [1977]. Stepanov and Shelagin [1986] studied the scattering of very cold neutrons from heterogeneities in condensed matter while Schulp [19891 used the EA for inversion of atomic and molecular beam scattering data. G6mez and Castaiio [19881 showed that the multi slice approximation used in the context of transmission electron microscopy (Cowley and Moodie [ 19571) was nothing but the EA, while the real space method (Van Dyck and Coene [1984]) is in fact a modified form of the EA. The inverse scattering problem in the EA has been discussed by Varsimashvili, Demkov and Ostrovskii [1980]. Recently, it has also been used to investigate the elastic electron-ion collision processes in strongly coupled plasmas (Jung [1496]). Currently, the EA is being used extensively in the context of a variety of scattering problems in various disciplines. A search ofthe literature based on “eikonal” as the key word from the data base of the Science Citation Index yielded about 90 papers per year on the subject since 1994. Although the EA made its debut as a high energy approximation, studies on its validity domain in various disciplines show that this need not always be the case. For-example, it fails as a high energy approximation in field theories describing the interaction of two scalar particles via a scalar meson exchange (Banerjee and Mallik [1974]). In these theories, it must be viewed as a long range approximation (Banerjee, Mallik and Sharma [ 19771, Banerjee and Sharma [ 19781); i.e., the interpretation of the validity domain of the EA is dependent on the nature of the interaction involved. Thus despite the use of the term “high energy approximation”, a more appropriate choice of name seems to be the “eikonal approximation”. The term EA is also used at times to describe a class of approximations which approximate wave propagation in a medium which varies slowly in space compared with the wavelength of the wave. It is in this-sense that the semi-classical approximations are at times also referred to as the eikonal approximation. Historically, the term eikonal first appeared in optics when Bruns [1895] used it to describe some functions similar to the characteristic functions of Hamilton [1828] (see, e.g., Born and Wolf [1970]). Sommerfeld and Runge [191I] showed that a convenient starting point for obtaining the eikonal equation, which characterizes the eikonal function, is the scalar wave equation of optics,
V2$(r)+ m2(r)k2$(r) = 0,
(1.1)
111, 8 11
217
INTRODUCTION
where k = (2n/A) is the wave nymber in vacuum, A is the wavelength in vacuum and m(r) is the refractive index of the medium. The complex amplitude 4 may be written in the form
4(r) = A exp(ikS),
(1.2)
where A ( x , y , z ) is the real amplitude and kS(x,y,z) is the phase. The function S is called the eikonal. Substitution of eq. (1.2) in eq. (1.1) leads to 1 2i i -V2A+-VA.VS-(VS)2+-V2S+m2 k2A kA k
=O.
(1.3)
In the geometrical optics limit A + 0, the above equation redqces to
(vs)*= m2.
( 1.4)
Equation (1.4) is known as the eikonal equation and determines the wave propagation in the geometrical optics approximation. The surfaces S = constant are the wavefronts and the normals to these surfaces represent ray directions. The solution of eq. (1.4), in a scattering problem, is: S ( r ) = k .r +k
lor
rn(r’)ds’,
where ro is an arbitrary point outside the region of interaction and the integration is along the actual ray path. A particular case of this approximation is obtained if the curvature of ray paths is neglected and if A is assumed to be unity. The approximate $(r),
#(r) = exp(ikz) + ik
I
[m(r’)- 11 dz’,
(1.6)
is then nothing but the WKB interior phase (Saxon [1955]) or the interior phase function in the anomalous diffraction approximation (ADA) (Van de Hulst [1957]). The assumption that A is unity holds only if the relative refractive index of the scatterer is close to unity. Under these circumstances, one may also cast eq. (1.6) in the form
It is the solution (1.7) of eq. (1.1) that has come to be known as the EA and which forms the main theme of this article. Many reviews on various aspects of
218
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[IK § 2
the EA have been published. In relation to the optical scattering, however, only a short review was published some time ago (Sharma and Somerford [ 19901). Many more developments have taken place since then. The purpose of this article is to present an up to date account of the EA in the context of optical scattering.
0
2. The Eikonal Approximation in Non-Relativistic Potential Scattering
Much of the work on the development of the theory of the EA in scattering of light by a particle has been adapted from the work on quantum mechanical potential scattering by drawing an analogy between a potential and the refractive index of the particle. It is, therefore, essential to first review the work on the EA in relation to the potential scattering. This section is devoted to this preliminary but essential exercise. Some earlier qrticles wherein the EA has been reviewed in the context of potential scattering are Glauber [1959], Abarbanel [1972], Joachain and Quigg [ 19741, Gerjuoy and Thomas [ 19741, Joachain [19751, Byron and Joachain [1977] and Gien [1988]. 2.1. PRELIMINARIEiS OF THE PROBLEM
We begin by considering the non-relativistic elastic scattering of a spinless particle of mass rn by a local potential V(r) of range a. Let ki and kf be the initial and h a 1 wave vectors associated with the particle and let 0 be the scattering angle between them. The particle energy E is h2k2/2m and (ki(= lkfl = k. The problem to be considered is the solution of the Schroedinger equation,
or alternatively, the solution of the Lippman-Schwinger equation: v(r) = exp (iki . r) -
/
G(r - r’) U(r‘) v(r’),
(2.2)
where G(r - r’) =
exp(ik(r- r’l) 4x(r-r’( ‘
The scattering amplitude f (8) is then given by the expression
f(0) =
-& /
exp (ik.r) U ( r )v(r) dr,
where U(r) = (2m/h2)V ( r ) is the reduced potential.
(2.4)
111, 8 21
THE EIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING
219
2.2. THE EIKONAL APPROXIMATION
The EA for the wave function V ( r )can be obtained in a number of ways. Each derivation brings out more clearly a different feature of the approximation. Our presentation in Q 2.2 and 2.3 follow closely that of Glauber [ 19591. 2.2.1. Approximation from the Schroedinger equation
Consider a trial solution of eq. (2.1) in the form
q ( r ) = exp(iki . r)@(r).
(2.5)
This, when substituted in eq. (2.1), gives
[.’ +
2 i k a ] @(r)= U ( r )@(r).
(2.6)
In arriving at eq. (2.6), the z-axis has been chosen along the direction of the incident momentum ki. Neglecting V2 in eq. (2.6), we obtain
a
2ik-@(r) = U ( r )(b(r), & which, with the boundary condition @(-oo)
where ;y(b,z)=
2 2k
(2.7) =
1, gives the solution
/--
U(b,z’)dz’
The solution (2.5) along with eqs. (2.8) and (2.9) is the EA in its simplest form. The assumptions made in arriving at eq. (2.7) can be checked a posteriori. ) U(r)@(r)and V 2 @ ( r<< ) 2ikd@(r)/& reduce to the These are that V 2 @ ( r<< following inequality (Alvarez-Estrada, Calvo and Juncos del Egido [ 19801):
(2.10) which essentially means that the validity criteria for the EA are
9
<
and
k a > 1,
where I Uol is the ‘strength’ of the potential.
(2.1 la,b)
220
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[HI, 5 2
2.2.2. Approximationfrom the integral equation
Substitution of the trial wavefunction (2.5) in the Lippman-Schwinger equation (2.2) gives
‘S
@(r)= 1 - 4n
eiklr-r‘ I-&.(r-r’) U(r’)q5(r’)dr’. Ir - r‘I
Defining a new variable, r” eq. (2.12) can be recast as @(r)= 1 -
4n
/
r” dr”
=
/
(2.12)
r - r’, and employing spherical coordinates,
dq5 d p eikr”(‘-p)U ( r - r”) q5(r - r”),
(2.13)
with p = cos(k, r”). The largest contribution to the integral in eq. (2.13) comes when p = 1; i.e., for those values of r” which lie close in direction with ki. Otherwise, the exponential oscillates rapidly and if the function Uq5 is slowly varying, the contribution of the integral becomes negligible. If it is assumed that the product U @varies significantly over a distance d, which is such that kd >> 1, the integration over p gives
‘s
@(r)= 1 - 4nik
dr” dq5 eikr’’(’-!‘)V ( r - r”)qb(r- r’’)$I!,
+ O(1M).
Ignoring terms of order (lfid), one arrives at 2k
dr” U(r - r”)$(r
-
r”)lr,qk,.
(2.14)
The contribution from the limit p = -1 is also of the order (l/kd). Since the z-axis lies in the direction of propagation, eq. (2.14) can be recast in the simpler form (2.15) Differentiation of eq. (2.15) with respect to z yields eq. (2.7) whose solution, as we have seen, gives the EA. This derivation shows that the EA is a good approximation when kd >> 1, where d is a distance in which the product Uq5 varies appreciably. From eq. (2.8), it can be seen that q5 varies appreciably in a
111, 3 21
22 1
THE EIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING
distance WlUol. The potential varies appreciably in a distance a. The distance d is therefore the smaller of the two quantities. That is, for
k a < -, IUOl
or
~
lUola < 1 : k
M >> 1 implies ka >> 1, (2.16a)
and k for a > -, IUOI
or
‘Uola
-> 1 :
k
kd >> 1 implies
lU0l/k2 << 1.
(2.16b) In either case both the conditions, ka >> 1 and lU0l/k2 << 1, must be satisfied. It may be noted that no restriction has been placed on the parameter I UOla/%.In contrast, the Born approximation requires this parameter to be much less than unity and the WKB approximation requires this product to be much greater than unity. The EA is, therefore, particularly useful for intermediate values of the parameter I Uola/k.
2.2.3. Propagator approximation At high energies, i.e., ka >> 1 and lU0l/k2 << 1, it is reasonable to assume that the particle propagates nearly undeflected from its initial direction. Consequently, the only values of the intermediate momentum, p , in the Green’s function
SdP
G(r - r’) = -(2.7~)~
exp [ip . (r - r’)] p2 -k2-i€ ’
(2.17)
which are significant, are those which are near k;. Introducing a new variable, Q = p - ki, the propagator (2.17) can be expressed as
Since the main contribution comes from values lQl << k, the denominator of the propagator can be linearized by neglecting the Q2 term. The linearized (or eikonalized) propagator &A, afier performing simple integrations, becomes
Here O is the step function and [d/&]O(x) = 6(x) is the Dirac delta function. Employing the trial wavefunction (2.5) and the linearized propagator (2.18),
222
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
Fig. 1. Scattering of a plane wave by a potential in the eikonal approximation (EA).
the Lippman-Schwinger equation can be solved without further approximation, leading to the solution (2.8). The eikonal propagator (2.18) clearly exhibits straight line propagation in the forward direction. It is, therefore, expected that the EA will be justified only for $mall angle collisions. The limitation on the scattering angles is 8’kd CK 1 (Glauber [1959]). 2.2.4. Physical picture of propagation in the EA
The physical picture of the scattering process that emerges from the wavefunction (2.19)
may be viewed as follows. A high energy particle is assumed to pass through the scattering potential at an impact parameter b in a straight line trajectory (fig. 1). The presence of the potential introduces a change in the phase of the wavefunction of the incident particle. Its amplitude remains unaffected. The change in the phase of the wavefunction of the incident particle is a linear function of the potential. Consequently, the phases in the EA are additive. This property of additivity has allowed extensive use of the EA in the context of scattering by composite systems. 2.3. SCATTERING AMPLITUDE
2.3.1. Eikonal amplitude Substituting the approximate wavefimction , be written as amplitude, ~ ( O ) E A may
f(O),,
=
J’ei(q~-b+’1/sin2(i)/2i)
U ( b m)e: 7”
into eq. (2.4), the eikonal
r:=.=
U(b,z’)dz’
db&,
(2.20)
In, § 21
THE EIKONAL APPROXIMATION IN NON-RELATIVISTIC POTENTIAL SCATTERING
223
where q1 . b = kb sin 8. For small pngle scattering, exp[2ikz sin2(8/2)] may be approximated by unity. The error involved is of the same order as the order of error that already underlies the eikonal wavefunction; i.e., of the order 02kd (Glauber [1959]). With this small angle approximation, the f (O)EA becomes (2.2 1) The z-integration in eq. (2.21) can be performed easily, leading to a twodimensional impact parameter representation of the scattering amplitude,
f(%A
=
-2Jt
ik
J’
eiql.b[exp (iXo(b)) - 11 db,
(2.22)
where the phase function ~ o ( bis) roo
(2.23) The amplitude (2.22) is referred to as the eikonal amplitude. Strictly speaking though, it is a combination of the EA and an additional small angle approximation. For a potential that possesses azimuthal symmetry, the two-dimensional representation of the scattering amplitude reduces further to the one-dimensional integral
f (O)EA
= -ik
/
b db Jo(kb sin 8) [eiXo(’)- 11 ,
(2.24)
where JOis the Bessel function of order zero. The angular range of the EA is governed by the relation tI2kd << 1, which means: (2.25a) and (2.25 b) These inequalities follow directly from eqs. (2.16). 2.3.2. Glauber variant of the EA In the foregoing discussion, because of the special role assigned to the incident momentum ki, the eikonal amplitude is not invariant under time reversal.
224
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
fm 8 2
A simple way to introduce the time reversal symmetry is to choose the z-axis along the average momentum direction k,, = (ki + kf)/2. The momentum transfer vector q then lies entirely in the &pact parameter plane. The scattering amplitude (2.23) then becomes *m
(2.26) where 41 . b = 2kb sin(8/2). T h s choice of z-axis was suggested by Glauber [1959]. Therefore, the resulting amplitude is known as the Glauber amplitude. Here, the only change fi-om eq. (2.22) is replacement of kb sin 8 by 2kb sin(W2). In what follows, it is this amplitude which will be referred to as the EA unless stated otherwise. At first sight, it may appear that by choosing the z-axis along the average momentum direction the Glauber formula has been able to avoid the small angle approximation. But if the z-axis is chosen along the average momentum direction right from the beginning, the scattering amplitude obtained is
1
00
fA,(8)= -ik,
JO (2kb sin(8/2))
(2.27) where k, = kcos(8/2). The subscript AI stands for Abarbanel and Itzykson [19691, Numerically, however, this apparently more systematic approximation turns out to be inferior to the amplitude in eq. (2.26), suggesting that the terms neglected in making the small angle approximation in fact correct certain approximations made in arriving at eq. (2.19). That t h s is indeed the case has been verified by Wallace [1971,1973a], Weiss [1974], Swift [1974] and by Banerjee, Dutta-Roy and Sharma [1975]. More recently, in an interesting derivation, Williams [I9881 formulated the EA utilizing the Feynman path integral method. By requiring conservation of energy, the direction of travel the particle takes in the EA has been shown to be determined uniquely to be that of the average momentum. 2.4. RELATIONSHIP WITH PARTIAL WAVE EXPANSION
It has been shown by Glauber [ 19591that the EA can be obtained from the partial wave expansion (2.28) k J db, 261 -+ ~ o ( band ) by making the transformations I + 112 + kb, X I P,(cos 8) -+ J0[2(1+ 1/2) sin(8/2)]. Further investigations have been conducted --f
HI, 0 21
THE EIKONAL APPROXIMATION IN NON-RELATIVISTICPOTENTIAL SCATTERING
225
by Wallace [1973b], who showed that the partial wave sum can be converted, without any approximation, to the integral
f(6)= -ik
/
00
b dbJo (2kb sin(W2)) [SF(b)- 11,
(2.29)
0
where &(b)
= exp[2i6(b)] W ( 6 ) ,
(2.30)
W ( 6 ) is an expansion in powers of k-2, with unity as its leading-order term, and 6(b) is the phase function. In the high energy limit (k + m), S(b) can be approximated by its Born approximation value,
With this approximation the amplitude in eq. (2.29) is nothing but the eikonal amplitude. 2.5. COMPARISON WITH THE BORN SERIES
A detailed investigation of the relationship between the Born series and the EA was conducted by Byron, Joachain and Mund (see Byron and Joachain [1977] and references therein). The Born series is an infinite series in powers of the interaction potential, and may be expressed as
(2.31) To compare the EA with the Born series it is convenient to define an analogous elkonal series by expanding eq. (2.26): (2.32) A comparison of eqs. (2.31) and (2.32) shows thatfEA 1 =fsl for all interaction potentials, energies and momentum transfers. But no such simple relationship exists among higher order terms. Nevertheless, a detailed analysis of the series
226
[IK§ 2
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
for n < 3 for Yukawa type potentials (Byron and Joachain [1973]) further suggests that the following relations, lim R
+
? !& !-
RefEAn
=
1 (n = odd)
and
lim k ---*
ImfB n
___ ImfEA n
1 (n = even),
hold for all n and all momentum transfers. For other potentials, such as Gaussian, square well, or polarization potential, the above relationships hold only for small angle scattering. Important studies in this connection were undertaken by Moore [19701, Weiss [1974] and Swift [1974], who examined each term of the Born series in the limit k 4 00 and q fixed, and showed that when the resulting series is summed, the EA is the unique result. 2.6. INTERPRETATION OF THE EA AS A LONG RANGE APPROXIMATION
High incident momentum is one way of realizing the basic assumption ka >> 1. The same condition may be realized if the interaction is long range. Indeed, for a Coulomb potential Ze2 V ( r ) = --F(r), r
(2.33)
where Ze2/r is the unscreened Coulomb potential, the EA gives an exact scattering intensity. The screening is necessary because the phase function ~o(b) diverges otherwise. Subsequently, the screen is removed to a large distance from the scattering center. If the screen F(r) is chosen as a step function, F(r) = 1 for r < a and F(r) = 0 for r > a, the scattering amplitude in the EA can be shown to be (Glauber [1959]) -Ze2/mhk f(e)EA = 2k sin2(8/2)
Ze2
ln[2ka sin(W2)] + 2iv (-2i mhk
is nothing where q = argr(1 + i(Ze2/mhk)). The scattered intensity lf(8)~~I~ but the Rutherford formula. That is, the EA is exact for the Coulomb potential. 2.7. NUMERICAL COMPARISONS AND POTENTIAL DEPENDENCE OF THE EA
Since the particle can be imagined to propagate through the interaction region in an infinite number of ways, many variants of the EA appeared on these
m, 8 21
THE EIKONAL APPROXIMATION IN NON-RELATIVISTICPOTENTIAL SCA7TERING
227
lines. Notable among them are those by Schiff [1956], Saxon and Schiff [1957], Abarbanel and Itzykson [1969], Sugar and Blankenbecler [1969], LBvy and Sucher [ 19691. Numerical comparisons of these variants with exact results have been made by Hahn [1970], Wallace [1971, 1973a1, Kujawaski [1971, 19721, Bemman and Castillejo [1973], Byron, Joachain and Mund [1973], and Weiss [ 19741, among others. The Yukawa, exponential, square well, Wood-Saxon, Gaussian and polarization potentials have been considered in these comparisons. The following features emerge from these comparisons: (1) When the conditions I UOl/k2 << 1 and ka >> 1 are satisfied, the EA compares very well to the exact results. The angular domain is generally found to be larger than that expected from theoretical considerations. For certain potentials, such as Yukawa and exponential, the EA holds even at large angles. For these potentials, results are reasonably good even if I UOl/k2 is not much less than unity and ka not much greater than unity. At small angles the EA appears to work well even if I UOl/k2 >> 1, provided the condition ka >> 1 is satisfied (Byron and Joachain [ 19731, Byron, Joachain and Mund [ 19731). (2) The validity of the EA has also been noted to be potential dependent. But it is not clear how to delineate the class of potentials for which the EA is valid at large angles. Nevertheless, a detailed study of the EA amplitude (Swift [1974]) and corrections to it in the limit k + 00, q -+ 00 for Yukawa, Gaussian and polarization potentials suggests that the EA amplitude is a good approximation to exact amplitude at all angles when ~ o ( bhas ) ) analytic at b = 0, the correction term a singularity at b = 0. If ~ o ( b is dominates at large q. (3) The potential dependence of the EA has been discussed by Sugar and Blankenbecler [1969] as well. It is concluded that the EA is a good approximation at high energy and large momentum transfers, if the potential is such that its Fourier transform does not fall off too rapidly for large momentum transfers. For example, the EA will certainly not reflect the large momentum transfer behavior for the case of a Gaussian potential. However, for potentials such as Yukawa or exponential, which fall off like a power in momentum space, the EA is expected to be an accurate approximation for a wide range of momentum transfers. 2.8. MODIFLED EIKONAL AF'PROXIMATIONS: CORRECTIONS TO TKE EA
2.8.1. The eikonal expansion
Wallace [1971, 1973a1 developed an eikonal expansion in powers of k-'
228
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[III, 5 2
for infinitely often differentiable potentials in which the EA appears as the zeroth-order term. For smooth potentials corrections systematically improve the EA results. The eikonal amplitude corrected to the first two orders for a spherically symmetric potential may be written as
f (4)= -2k
1
db eiq'b k(byi: - 11 ,
(2.35)
where fiAand f;; denote first and second order corrected functions given by (2.36a) (2.36b) where (2.37a) 1 z2(b) = -- (1 + k5
+ fa)
00
dz U 3 ( r ) -
b (x;(b))3 24k2 '
(2.37b) (2.37~)
and Pn = b"d"/db". Wallace [1971, 1973al has obtained third order corrections, too. These corrections have been rederived by Weiss [1974], Swift [1974], Wallace [ 1973bl and Baker [ 1972, 19731 employing alternative techniques. Fourth order corrections have been obtained by Sarkar [1980]. All the phase corrections vanish for the Coulomb potential. This is as it should be, because the EA is exact for the Coulomb potential. Numerical evidence of systematic improvement with each increasing order (up to third order) of the eikonal expansion has been presented by Wallace [1971, 1973al and Weiss :[ 19741. 2.8.2. The eikonal-Born series
The eikonal-Born series interpolates between second order Born and the eikonal approximations. For Yukawa type potentials, the exact scattering amplitude in the limit x >> 1 can be expressed as (Byron, Joachain and Mund [1973]) fex
=h,+ [ki+Lk] A . B + [ki+lk'] C .D + . . . (2.38)
111,
5 21
THE EIKONAL APPROXIMATION M NON-RELATIVISTIC POTENTIAL SCATTERING
229
In contrast, the eikonal series yields:
In the above equations, A and B are of second order in potential strength and C and D are third order in potential strength. A comparison of eq. (2.39)with (2.38) shows that neither & 1 +f ~ 2nor & 1 +fB2 is correct to order l/k2. But because A and B are second order in potential strength while C and D are third order in potential strength,f =fB1 +fB2 is more accurate than the eikonal amplitude for the weak coupling case. The addition of the real part of the second Born term to the Glauber amplitude thus results in impressive improvements for Yukawa type potentials. Two alternative amplitudes corrected in this way are (Byron and Joachain [19771) fEBS
=fBl +fB2 +fE3,
and
f&S
=fE + Ref ~ 2 ,
(2.40a,b)
which are correct up to order (1/k2)and have been referred to in the literature as the eikonal-Born amplitudes. 2.8.3. The generalized eikonal approximation An alternative approach to rectify the defect of the missing Refs2 term in the EA is to write a generalized linear propagator of the form (Chen [ 19841)
1 Gl(r) = -b2(b) 21a
Q ( z )e'p',
(2.41)
where the arbitrary parameters a and are determined in such a way that the dominant real part as well as the dominant imaginary part of the second Born amplitude are correctly reproduced. The resulting formula, termed as the generalized EA(GEA), has been found to work very well for Yukawa and Gaussian potentials even at large angles. 2.9. RELATIONSHIP WITH RYTOV APPROXIMATION
The Rytov approximation (Rytov [1937],Nayfeh [1973])assumes a solution of the Schroedinger equation of the form
230
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[a8 3
where R(r, p) = C,"=, R,(r) is an expansion in powers of coupling constant (R,(r) is of order 5;"). The scattering amplitude may then be expressed as (Car, Cicuta, Zanon and Riva [19771) f ( q ) = -2k
1
eiq'bd2b [ex 7 Rn - 11 ,
(2.43)
where
The lowest nontrivial order of the Rytov expansion is same as the EA. The first order correction differs from the first order Wallace correction. This is because while the eikonal expansion is an expansion in powers of k-' ,the Rytov expansion is an expansion in powers of potential strength.
5
3. Eikonal Approximation in Optical Scattering
Light scattered by an obstacle is related to its physical properties and hence in principle it is possible to obtain information about the scatterer from an analysis of the scattered IigM. Thus, for many years, the light scattering technique has been used to infer the size, shape and refractive index of particles in various scientific disciplines such as biophysics, colloid physics, molecular physics, optical fibers, plasma diagnostics, atmospheric physics and astrophysics, etc. The fact that it is a non-destructive technique and can be used for atmospheric and astrophysical particles, which are not easily accessible otherwise, makes it a very attractive diagnostic tool. Light scattering experiments generally involve measurements either from one particle at a time or measurements from an ensemble of particles. In either case, a theory for predicting the scattering pattern from a single particle is necessary. Unfortunately, the problems involving the scattering of light are very complex. The exact solutions are unknown except in the simplest and most idealized cases of a sphere (Mie [1908]), an infinitely long cylinder (Wait [ 19551) and a spheroid (Asano and Yamamoto [1975]). Simple inhomogeneous objects such as a concentric sphere (Aden and Kerker [1951]) and an infinitely long cylinder (Tang [1957], Kerker and Matijevi6 [1961]) can also be treated exactly. For particles of other shapes one may resort to numerical procedures. The complexity of numerical solutions has become of less consequence with the advent of faster and faster computers. However, in many applications a numerical
111,
8 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
23 1
approach still proves to be tedious, impractical or even impossible, and one must resort to approximation methods. That apart, approximation methods give simple expressions for quick and easy use and also provide a deeper physical and mathematical insight into the general scattering problem. Thus, one is aware of important factors involved in the use of any approximation. A brief review of some analytic approximation methods for spherical particles has been recently given by Kokhanovsky and Zege [ 19971. The EA has proved to be a very good approximation for the analysis of the near forward scattering pattern for particles whose refractive index is close to that of the surrounding medium. It has been applied to the scattering of light by homogeneous spheres, coated spheres, infinitely long cylinders, spheroids and rough particles. A large variety of particles can be modeled using these shapes. It was introduced in the context of optical scattering by drawing an analogy between the Schroedinger equation and the scalar wave equation of optics. Consequently, it is expected to be applicable in situations where the scalar description of the scattering process is sufficiently accurate to allow the vector nature of light to be ignored. Attempts have been made to include the vector character of light in the analysis of the scattering process and a reasonable degree of success has been achieved. 3.1. _ANALOGY WITH POTENTIAL SCATTEFUNG
Consider a scalar wave characterized by the field v(r) propagating through a medium of spatially varying relative refractive index m(r). The field v(r) then satisfies the wave equation V 2 q ( r ) +k2m2(r)v(r) = 0.
(3.1)
A comparison of eq. (3.1) with the Schroedinger eq. (2.1) shows that
U(r) = [1 - m2(r)] k2.
(3 .a
With this relation between potential and refractive index, one can easily write the scattering function S(~)E,(=-i&"(O)E*) as
S(0),, where
= -k2
/
db eiq" [eixo@) - 13 ,
(3.3)
232
[IK8 3
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
The conditions (2.11) for the validity of the EA in potential scattering now translate to
Im(r)- 11 << 1,
and x >> 1,
(3.5aYb)
where the dimensionless parameter x = ka is the size parameter and is essentially a measure of the size of the scatterer in terms of the wavelength of the scattering radiation. The notation x , has also been used in the coordinate system r = ( x , y , z ) . However, this notation does not result in any confusion and has been used in most books on the subject. The requirement (3.5a) ensures that at boundaries there is no deviation of the incident ray and that the energy reflected is negligible. The second requirement ensures that the ray travels undeviated through the scatterer as the refractive index varies slowly in a wavelength. The angular range given by,eqs. (2.25a) and (2.25b), now translate to
and
Here, m may be taken as the maximum value of the refractive index. For complex m(r) = n(r) + in’@), the condition (3.5a) is equivalent to In@) - 11 << 1 and n’(r) K 1. 3.2. VALIDITY OF SCALAR SCATTERING APPROXIMATION
for the scattering Table 1 shows the values of the ratio i(O),,,,,/i(O),,,~, of unpolarized light by a homogeneous sphere and transverse electric wave Table 1 The ratio i(0)exact/i(O)sca~ar for m = 1.15
1
2
5
10
20
Sphere
0.791
0.918
0.953
0.975
0.982
Cylinder (TEWS)
0.769
0.890
0.956
0.976
0.970
111, § 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
233
scattering (TEWS) by an infinitely long homogeneous cylinder, where i(0) is the forward scattered intensity. It may be concluded fiom the results in the table that for near forward scattering the approximation of light by a scalar wave is a reasonably good approximation for x 3 3.0. 3.3. SCATTERING BY A HOMOGENEOUS SPHERE
3.3.1. The eikonal approximation Spherical or near spherical scatterers are of most common occurrence in natural media. Not surprisingly, the model of scattering by a homogeneous sphere has been used extensively to examine the validity domains of various approximation methods. For a sphere of radius a and refractive index m, the scattering function (3.3) can be expressed easily as S(&A
= -k2
la
b db Jo (2kb sin(W2)) [exp (iXo(b)) - 1 1 ,
where
(3.9) The distance travelled by the ray in the medium at impact parameter b is 2 d m (see fig. 2). The quantity piAis also expressed as p( 1 + i tan&*), where the phase shift suffered by the central ray in passing through a diameter is
and 2nn'x
PEA = -
(3.11) P is that part of the phase shift function which describes the decay of the amplitude due to the absorption. The scattering function given by eq. (3.8)
-
Fig. 2. Scattering geometry for a homogeneous sphere.
234
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[UI,
P3
is formally the same as the expression for S ( 6 ) in the anomalous diffraction approximation (ADA) (Van de Hulst [ 19571). The inequalities (3.5) define the validity domain of this approximation, too. The difference in the two approximations is that in place of p$Aone has (3.12)
piDA = 2x(m - 1).
Clearly, as n -+ 1 and n‘ 4 0 the two approximations tend to the same limit. In optical scattering an alternative expression used in place of eq. (3.8) is lr/2 S ( 6 ) E A = x2
Jo(z sin y ) [l - ew*‘OS y ] cos y sin y d y ,
(3.13)
where z = 2xsin(6/2) and w* = k ( m 2 - 1). In arriving at eq. (3.13), a change of variable, b = a sin y, has been made in eq. (3.8). For a completely absorbing sphere (n’ 4 GO), exp(-ptan&A) = 0, and S(O),, becomes (3.14) which is nothing but the scattering function in the Fraunhofer diffraction. On the other hand, for x(m2- 11 << 1 , one may approximate exp[w* cosy] N- 1 + w* cos y. The-scattering function is then (3.15) whxh is nothing but the scattering function for a homogeneous sphere in the Rayleigh-Gans approximation (RGA) (see, e.g., Van de Hulst [ 1957]), which is known to be a good approximation for wavelength size or smaller particles. That is, despite the original premise x >> 1, the EA is, in practice, a good approximation for arbitrary x for scattered intensity. Clearly there is a cancellation between errors arising from the scalar approximation (0 3.2) and errors inherent in the EA. A difference in sign between the exponent in eq. (3.13) and the corresponding expression presented by Van de Hulst [I9571 may be noted. It is due to the choice of the sign in the incoming plane wave. In potential scattering, the usual choice for the incoming plane wave is exp[i(kz - w t ) ] .Here, we have adopted this convention. The same convention has been adopted in books by Newton [ 19661, Bayvel and Jones [ 198 I] and by Bohren and H u m a n [ 19831. On the other hand, Van de Hulst [ 19571 and Kerker [ 19691 use the convention exp[-i(kz - wt)] for
n1, o 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
235
the incoming wave. The implications of choosing a particular convention have been discussed by Shifrin and Zolotov [19931. The fact that s(@EA is formally identical to the s ( e ) A D A , allows many a result of the ADA to be used mutatis mutandis for the EA. For forward scattering the integration in eq. (3.8) can be evaluated in a closed form. T h s gives
(3.16)
where p' = 4ReS(0)/x2 is the extinction efficiency. For pgA + 03, eq. (3.16) gives Q,"A = 2, which is the correct asymptotic value of the extinction. For a dielectric sphere, sinhA = 0, and the extinction is given by (3.17) where &A = x(n2 - 1). For non-forward scattering, it is not possible to analytically evaluate S(O)EA in a clesed form. However, it is possible to express the scattering function in terms of known functions by means of a series expansion for real values of refractive index. Following Van de Hulst [1957], the imaginary part of the scattering function can be expressed as
- -
(3.18) where y2 = piA + z2. The real part of the scattering function can be expressed as ReS(o)EA=X2 [ ( ? ~ J Z ( Z ) - P--J3(Z)+-:A 1.3 z3 for small values of PEA,and as
J4(z)+ . . . 1.3.5z5
for PEA > z. The expansion (3.20) is obtained by making a doubly infinite expansion in powers of kA and z.
236
SCATTERING OF LIGHT P 4 THE EIKONAL APPROXIMATION
[IIL § 3
3.3.2. Derivation of the EA scattering func,tion from the Mie solutions
The exact scattering functions Sl(I9) and S,(I9), corresponding to perpendicular and parallel polarizations, respectively, for the scattering of light by a homogeneous sphere are (3.21) and 21+1
[aiq(cos 0) + bln/(cos 691,
(3.22)
I= 1
where ni(cos
P/(COS 0) e) = dd(cos 0) ’
(3.23)
and ~ ( C O 19) S = cos
BTC/(COS0) - sin I9
[
dni(COS 6) d(cos0) ’
]
(3.24)
with P/(cos 0) as the Legendre polynomial. The scattering coefficients a/ and b/ can be expressed as (see, for instance, Newton [ 19661)
where (3.26a) and (3.26b) with u/(z) = zj&) functions.
and u / ( z ) = zvl(z) as the Riccati-Bessel and Riccati-Neumann
111, 8 31
231
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
Bourrely, Lemaire and Chiappetta [1991] have shown that in the limit x -+ 00, (nxI+ 00 and nI2x/n << 1, the coefficients a6 and @I may be approximated as
where f=cosy-[y-;n]siny,
f’=COSy’-[y’-;n]siny’,
(3.28)
and x’ = nx. The reflection coefficients rl and r 2 are given by cos y - n cos y‘ (3.29) cos y + n cos y” and the angle of incidence y, (fig. 2), is related to y’ through the relation rI
=
ncosy-cosy’. n cos y + cos y’ ’
2x sin y = 2x’ sin y’
= (21
r2 =
+ 1).
(3.30)
For large x and small scattering angles (6’ << Ux), one can introduce the approximations 1(1+ 1) ~ ( C O 6’) S N ~(CO 6’)SN -Jo [ (1 + f ) 6’1 , (3.31a) 2 and (3.31b) In addition, if n -+ 1, Y ] and r2 also go to zero. The leading term in the expansion of xf - xlfl in powers of n2 - 1 is -(n2 - 1) cos y/2 (see next section). It is then straightforward to see that eq. (3.22) reduces to S(~’),A. Expressions corresponding to eq. (3.27) for a homogeneous dielectric sphere (n’ = 0) have been derived in most of the books cited here. 3.3.3. Relationship with the anomalous difraction approximation An expansion of xf -xlf‘ in powers of (n2 - 1) to order (n2 - 1)3 gives (Sharma
[19921)
-2i(xf-x’f’)=i(n2-l)xcosy
L
1-$(n2-1)(1-tan2y)
+$(n2-1)2(1-2tan2y)-$(n2-1)2tan4y
1
(3.32)
.
The first term on the right hand side of eq. (3.32) is the EA phase. Clearly, the phase in the EA should be viewed as an n --+ 1 approximation for fixed (n2 - 1)x.
238
SCATTERING OF LlGHT IN THE EIKONAL APPROXIMATION
[IK 5 3
The corrections to the EA phase given ,in eq. (3.32) may now be compared with the phase obtained by translating Wallace results to optical scattering. The Wallace phase, to order (n2- 1)3, then reads (it does not involve contributions o f w type) (n2 (n2 - i)3 (1-tan2y)cosy+ix(1 - 2 tan2 y) cos y. i(n2- 1)xcosy-ix8 4 (3.33) It is gratifj4ng to note that the first order correction in eq. (3.32) is identical to the first order correction of Wallace. The second order correction, however, agrees only partially. Regrouping the terms, we can rewrite eq. (3.32) as [i(n2-1)-$(n2-1)2+t(n2-~)3]~~~~y -2i(xf-x’f’)=2i (3.34) - ;x(n2 - 1>2tan y sin y - $x(n2- i)3tan y - &x(n2 - 1 p sin tan3 y.
If the term in square brackets is approximated as [[l + (n2 - 1)]1’2 - 11 = (n - I), eq. (3.34) becomes -2i(xf - x’f’) = 2ix(n - 1) cos y + ;x(n2 - 1)2tan y sin y (3.35) - ;x(n2 - 1i3 tan y sin y - $x(n2 - 1)3 tan3 y sin y.
The first tern- on the right hand side of this equation is the ADA phase. Note that corrections to the EA phase as well as corrections to the ADA phase are of the order (n2 - 1). For large PADA, the important contributions of the refraction term to forward scattering come from y values close to zero. The corrections to the ADA phase are then small even when (n- 1I is not much less than unity. T h s is an important observation and explains why the ADA is a reasonably good approximation even for n as large as 2. 3.3.4. Corrections to the EA When translated to optical scattering, the first order corrected scattering function of Wallace can be written as (Sharma, Somerford and Sharma [1982]) n/2
S ( 0 ) = x2
JO(Zsin y) [1 - eixo+irL] cos y sin y dy,
(3.36)
where T I = --fx(m2- 112 cos y(1-
tan2 y) cos y.
in,
However, a difficulty arises here. At y = the correction to the eikonal phase diverges. This is essentially a consequence of the sharp cut-off at the
UI,
P
31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
239
boundary of the scatterer. The eikonal expansion of Wallace, derived for infinitely often differentiable potentials, thus need not hold even at the zeroth order level for optical scattering. But, because the numerical comparisons confirm that the EA is a good approximation, it is customary to ignore the mathematical problems associated with the discontinuous behavior at the boundaries and consider that part of the correction which can be evaluated in an analytic form. An approximate form which gives an analytic expression for S(0) in the domain $xlm2- 112 << 1 may thus be written as S ( 0 ) F C I = x2
/
Jo(z sin y) [l - eiX”( 1 + irl)] sin y cos y dy.
(3.37)
The abbreviation FCI in the subscript refers to the first order correction. Another similar correction can be based oq eq. (3.35):
ix’2
s(e)FCII = x2
[
JO(Zsin y)
1 -e2ix(m-l)cosy
(1 + $ix(m2 - 1)sin2 y
>I
(3.38) sin ycos ydy.
This is essentially a modified form of the ADA. For forward scattering, eq. (3.37) as well as eq. (3.38) can be evaluated analytically to yield
+ $x2(m2 - 1) [epir(m2-l)
s(o)F~~ =~ ~ s ( o ) E A
(3.39)
and
with C ( m ) = 1 + 0.25(m + 1)’(m - 1)(2 - m2). The correction given by eq. (3.40) clearly preserves the simplicity of unmodified approximations. The GEA and a modified form of the GEA (MGEA) have been employed by Chen [1989], and Chen and Smith [1992] for the problem of scattering of light by a homogeneous sphere. The scattering function in the GEA utilizes a parametrized propagator. In potential scattering, the parameters of thls propagator are determined by requiring that the amplitude reproduces the real and imaginary part of the second Born term correctly. In optical scattering parameter determination includes two effects: (i) The phase change in propagation through the medium is given by 2k(m - 1 ) d m (in the case of the GEA) and by 2k(m - cos W 2 ) m (in the case of the MGEA). The
240
rm § 3
SCATTERMG OF LIGHT N THE EIKONAL APPROXIMATION
phase change in the MGEA is assumed to be angle dependent. (ii) The edge effects due to the sharp boundary are recovered. The scattering function in the MGEA then takes the following form: 2 ik S(~)MGEA = -( 1 - 6)s~ - (m2 - 1)6 xk I, 4n where
a2
/
n/2
Jo(z sin y) [eipGEA ‘OS Y PMGEA o the first Born term, S,, is given by
I=
~
&(O) 6=
= -4x2(m2 - l)a
m2 - 1 2a[n - cos(0/2)]’
m-tl- 3i [A - ___ 2 2
8 x
-
11 cos y sin y d y,
r2
and =
(3.41)
sin2 y cos yJo(z sin y) dy,
(5- $11,
PMGEA
with QI = 2 + 2.4i, a2 = 2 + 6i and PMGEA = 2x(m - cos 0). For forward scattering, eq. (3.41) can be evaluated analytically to give
where a0 = a k . Equation (3.42) reduces to the EA for 6 = 1 and a. = 1. For large spheres, that is for PGEA >> 1, the integral in eq. (3.41) can be evaluated analytically, leading to (Chen [ 19931)
s(e)&, = (n2 - 1)x3 where y = [z2
jl(z) z
6)-
+
6Jl(z) ~
ZPMCEA
+ -6
y2
(.ie
iy
+---1 ;ei.”)]
, (3.43)
and a is redefined as
The superscript LS refers to large scatterers. The formula (3.43) clearly exhibits diffraction, refraction and other terms which are due to interference between diffraction and refraction.
o 31
24 1
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
From the point of view of extending the angular domain of the validity of the EA, Perrin and Chiappetta [1985] proposed a modification of the EA which is referred to as the eikonal picture (EP). Explicit small angle approximation made for arriving at the two-dimensional scattering amplitude is not made here. The EP is simply a translation of eq. (2.20) to optical scattering. Thus, the scattering function in the EP is S ( 0 ) ~ p=
--
ik3 2
s
dbdzeiq”[m2(b,z)- 11
(3.44) For 8 = 0, exp(2ik sin W2) = 1, the EP is then identical with the EA. For a homogeneous sphere, z integration in S(O),p can be performed. This yields ei[q,+k(mZ-l)lv‘FP
S(~)E =P -k3(m2 - 1) L
- e-iq,v‘FP
2qz + k(m2 - 1)
(3.45) It can be seen that when Imlm2- 1I is negligible and 4 sin2(8/2) + Relm2- 1I = 0, resonances are produced. But these resonances are spurious. A variant similar to this was suggested by Berlad [1971] in the context of scattering of spin $ particles. An improved version of the EP for an absorbing sphere (Imlml x > 1; ka >> 1) has been obtained recently by Bourrely, Chiappetta and Lemaire [1996]. Scattering process in an absorbing sphere physically implies that the difiactive and the first order reflective parts of the scattering function give a reasonable description of the scattering phenomenon. It is then easy to see that the modified scattering function can be written as (3.46) where r I is the Fresnel reflection coefficient for the perpendicular component of the electric field. 3.3.5. Numerical comparisons To delineate the validity domains of the EA and the ADA, numerical comparisons for scattered intensity have been performed by Debi and Sharma [1979]
242
PI, § 3
SCATTERING OF LIGHT rN THE EIKONAL APPROXIMATION
Table 2 Percent error in various approximations of i ( 0 ) for a homogeneous sphere of refractive index 1.05 ~~
X
FCI
EA
pEA(m2 - 1)/4 ~~~
~~
ADA
FCII
~~~
1.o
2.63 x
-3.07
-8.43
1.88
-7.59
3.0
7.88 x 10-3
1.37
-3.79
6.09
-2.97
5.0
1.31 x
2.58
-2.56
7.20
-1.76
10.0
2.63 x
3.67
-1.65
8.04
-0.84
20.0
5.25 x
4.90
-1.37
8.35
-0.50
30.0
7.88 x
6.60
-1.33
8.45
-0.39
40.0
1.05 x lo-'
9.11
-1.31
8.55
-0.28
50.0
1.31 x lo-'
12.65
-1.19
8.65
-0.17
60.0
1.58 x lo-'
16.65
-0.89
8.67
-0.14
70.0
1.84 x lo-'
15.17
-0.67
8.10
-0.77
80.0
2.10 x 10-1
8.20
-0.67
6.80
-2.20
90.0
2.35 x lo-'
-2.60
-3.44
6.65
-2.36
100.0
2.63 x lo-'
2.98
-3.73
7.04
-1.93
for a dielectric sphere. Since both of these approximations are known to be small angle approximations,percent errors in the forward scattered intensity were examined. A typical set of results is shown in table 2. The percent error has been defined as (3.47) The comparison shows that the EA is superior to the ADA in the domain PEA < 4.0. This observation is important because it relates to intermediate size particles. For higher values of PEA the ADA gives more consistent results. On the other hand, for small size particles, Rayleigh or Rayleigh-Gans approximations are more useful. It is interesting to note that the value of PEA _N 4 corresponds to the fist maxima in the extinction curve. The significance of this observation, if any, is not clear. As expected, the EA improves as n 4 1. But, contrary to expectation, one finds that as x increases for a fixed n, the EA results do not improve continuously. The errors oscillate around the correct value. The reasons suggested to explain t h s behavior are: (i) Because of the energy dependence of the effective potential, the condition Uo/k2 << 1 becomes In2 - 11 << 1, which is k independent.
x 3 1.0 and
IU,
o 31
243
EIKONAL APPROXIMATION IN OPTICAL SCATTERTNG
Table 3 Percent error in various approximations of Q" for a homogeneous sphere of refractive index 1.05 X
pEA(m2 - 1)/4
EA
FCI
ADA
FCII
1.o
2.63 x
-155.26
-155.26
-143.03
-154.48
3.0
7.88 x
-23.41
-23.44
-17.48
-23.02
5.0
1.31 x
-9.47
-9.55
-4.32
-9.23
10.0
2.63 x
-2.47
-2.77
2.19
-2.42
20.0
5.25 x lo-'
0.34
-0.88
4.04
-0.48
30.0
7.88 x
2.15
-0.62
4.36
-0.15
40.0
1.05 x lo-'
4.33
-0.60
4.42
-0.09
50.0
1.31 x lo-'
6.85
-0.66
4.34
-0.17
60.0
1.58 x
lo-'
8.65
-0.66
4.13
-0.34
70.0
1.84 x lo-'
6.69
-0.65
3.79
-0.75
80.0
2.10 x 10-1
0.89
-0.87
3.57
-1.00
90.0
2.35 x lo-'
-1.19
-1.33
3.64
-0.90
100.0
2.63 x lo-'
-1.22
-1.68
3.71
-0.83
(ii) The condition x >> 1 is essentially a consequence of the requirement that the refractive index varies slowly over a wavelength. For a homogeneous sphere, the refractive index is constant. Thus, increasing x need not result in increased accuracy. (iii) The requirement of the slow variation of the refractive index is not satisfied at the boundary where there is a sharp cut-off. The validity of the EA has also been examined for extinction efficiency (Sharma and Somerford [1989], Sharma [1993]). The EA no longer performs better than the ADA in the domain x 2 1 and PEA < 4. Table 3 shows the percent error in various approximations in ptfor a homogeneous dielectric sphere of n = 1.05. It is clear from tables 2 and 3 that in comparison with i(O), all approximations work better for p'when x 3 5.0. For smaller values of x , the performance of the EA as well as the ADA is poor for p'This . is because, for small values of x neither the EA nor the ADA reproduces accurately the real part of the scattering function. Figures 3 and 4 show the effect of a small absorptive part (n' = 0.1) on i(0) and pt, respectively (Chen [1988]). As a consequence of the presence of the absorption, the forward scattering results are now better approximated. The oscillations in percent errors now begin to die down.
244
83
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
OO
1.0
1.2
1.4
1.6
1.8
2.0
REAL INDEX Fig. 3. Loglo i(0) against the real part of refractive index at x = 10. The imaginary part of the index is fixed. (a) n’ = 0, (b) nr = 0.10. Solid line, exact results; dash-single dot, EA; dashdouble dot, ADA. (From Chen [1988].)
(b)
52
-,,..
..-.-__.
-
O 1
0 1.0
1.2
1.4 1-6 REAL INDEX
1.0
2.0
Fig. 4. Extinction efficiency, Qext, against the real part of the refractive index at x = 10.0. The imaginary part of the index is fixed. (a) nr = 0, (b) n’ = 0.10. Solid line, Mie theory; dashsingle dot, EA, dash-double dot, ADA. (From Chen [1988].)
Inclusion of corrections improve considerably the results. The modified approximations are found to work extremely well in the domain x 6 5.0 and $pEAln2 - 1I < 1 (Sharma [1993]). Significant improvement is achieved for i(0) as well as for pt. In particular, S(i3)~crris found to give very good results. It was noted in the context of potential scattering that the EA is expected to
111,
o 31
245
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
lo7
%
5
105
lo6
C
aJ
4
.r
104
aJ
.-> c
d
d
Q,
a lo3 10'
0
10
20
30
40 50 Scattering Angle in degrees
60
Fig. 5. i(0) versus 0 for perpendicularly polarized light by a dielectric sphere with rn = 1.33 + iO.001 and x = 30,60,90. Solid curve, Mie theory; dashed curves, MGEA, dash-dotted curve at x = 30, GEA. (From Chen and Smith 119921.)
be a good approximation at all angles if ~ o ( bis) singular at b = 0 (see Q 2.8). For a homogeneous sphere ~ o ( bis) analytic at b = 0. Hence the approximation is expected to be valid only at small scattering angles. Not many numerical studies of angular variation of i(0)have been performed. Whilst Chen [1988, 19891 and Chen and Smith [1992] have compared the angular scattering patterns of the EA, the ADA and the GEA with exact results, Perrin and Chiappetta [ 19851 and Sharma, Roy and Somerford [1988a] have examined the EP and the EA against exact results. The following conclusions emerge from these studies: (i) The EA works to within 25 percent error in the domain x 3 1, 1 n 6 1.2 and 8 < 10.0. (ii) The GEA method works very well for light scattering by a dielectric sphere. It greatly improves the EA results. More importantly, it appears to work very well for y1 as large as 4.0.However, the success is only for scattering angles up to 5". Its improved variant, MGEA, is found to work well for the scattering of light with perpendicular polarization. It predicts accurately the positions of minima and maxima for 0 up to 60", x 2 5.0, n < 4 and n' < 0.5. Figure 5 shows a comparison of GEA and MGEA with exact i(0) versus 8 curves for x = 30, 60 and 90. The refractive index of the scatterer is m = 1.33 + iO.001. (iii) The simplified version of the GEA, given in eq. (3.43), is found to work as well as the GEA for x > 10. (iv) For a dielectric sphere the EA as well as the EP agree well with the exact results for small angle scattering. In the forward lobe the two approximations are very similar, but the EA has some advantage over the
<
-
246
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
8 , (degree) Fig. 6. Scattered intensity i(0) as a function of 0 for a perfect homogeneous sphere with n and x = 10.0. Solid line, Mie theory; dashed line, EA; crosses, El?
=
1.10
EP. Outside the forward lobe the positions of maxima and minima are determined more accurately in the EP but the errors in the intensity at the minima in this approximation are much larger compared with those obtained when the EA is used. These points are clearly demonstrated in fig. 6. (v) As the imaginary part of the refractive index increases, the oscillations in the scattering pattern decrease at large angles. The accuracy of the EP then increases and it appears to qualify as an all angle approximation. The scattered intensity IS(@g{d12for absorbing particles has been compared with the Mie calculations for a variety of particle sizes and refractive indices. Figure 7 shows a typical comparison between IS(O)EPI2 and IS(0)Ep)2 with Mie intensity for x = 20.0 and m = 1.5 + iO.1. The modified formula is in much better agreement with the Mie theory for scattering angles greater than 40". The model is valid for x 2 10.0.
IK
P 31
247
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
” ” , ” ” , ” ”
j -I
1
l
0
n
50
‘
,
a
l
,
,
,
100
,
l
,
,
150
scattering angle (degree) Fig. 7. Scattered intensity for a sphere of x = 20.0 and m = 1.5 + iO.1. Solid line, Mie theory; dashed line, modified ekonal model; dotted line, original elkonal model. (From Bourrely, Chiappetta and Lemaire [1996].)
3.3.6. One-dimensional models
Further insight into the validity domain of the EA may be gained by examining one-dimensional models. Clearly, a one-dimensional model constitutes a tremendous simplification of what one encounters in reality, but it has the compensating advantage of being soluble exactly. One-dimensional models have been used for a long time to study the validity domain of the Glauber multiple scattering series and the eikonal type approximations in the context of potential scattering (Tobocman and Pauli [1972], Chen [1974], Banejee, Dutta-Roy and Sharma [ 19751) and in examining optical scattering approximations (Alvarez-Estrada and Calvo [1981], Sharma, Ghosh and Roy [1988], Lin and Fiddy [1992]) and have served successfully as a useful guide to more realistic three-dimensional calculations. For a one-dimensional homogeneous scatterer the transmission amplitude is (Sharma, Ghosh and Roy [1988]) 4m 4m
(3.48)
It is clear from eq. (3.48) that the domain of the applicability of the ADA is determined by the condition ( m - 11 << Im + 11. That is, the ADA is a good
248
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
w,5 3
approximation even for particles with m as large as 2. The condition under which
fgA = exp[ix(m2 - l)] would be a good approximation to the exact f + , can be obtained easily by rewriting eq. (3.48) as (3.49) Clearly,A;f + f' as Iml + 1 for fixed x)m2- 11. It may be noted that as x increases for a fixed M , one would expect oscillations in the errors in the EA due to the factor exp[-ix(m Oscillations will be smaller as n' increases. This is exactly what one observes in actual numerical calculations. Next let us consider a continuouslyvarying refractive index profile of the form n2(2) = 1 +
A(A - 1) X2
sech2az,
( 3.50)
where A(A - 1)/x2 s (n2(0)- 1) = (n2 - 1). l / a determines the characteristic length over which n2(z)- 1 drops by an order of magnitude, that is, it essentially determines the size of the scatterer. The transmission amplitude for such a problem can be written as (Flugge [1971]) f+ =
$ [e2i44e
-e 2 ~ ~ ~ ~ 1
7
where
and
r(ix)exp(ix In 2)
r(?+$) r ( 1 - + + $ ) To examine the condition for whichA;f is a good approximation to the exact and $0 may be expanded in powers of (n2 - 1) and l/x. To this end, ' f can be written correctly up to order (n2 and I/! as (Sharma, Ghosh and Roy [1988])
f',ge
f += exp[i
(PEA
&)I.
-EL 6x + 30x
(3.51)
The first term in the exponential is nothing but the eikonal phase. Clearly the EA improves as x + 00 for fixed PEA; i.e., the EA must be viewed as a n ---t 1 approximation for continuously varying index profiles as well.
UI,
5 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
249
In another one-dimensional study, Alvarez-Estrada and Calvo [ 19811 have considered the transmission amplitude for an inhomogeneous dielectric slab where the dielectric permittivity E is assumed to be dependent on frequency u at each point z through the Kramers-Kronig dispersion relation. The authors find that the EA can be regarded as k + cy) approximation here because approaches unity in this limit. But if E(Z) the permittivity E(Z,U) = 1 is independent of u,then the EA can no longer be viewed as a k 4 00 approximation. 3.3.7. Backscattering in the EA
Although the EA is a forward scattering approximation, it can also serve as a useful basis to describe the backward scattering if the conditions (3.5a) and (3.5b) are satisfied. This has led to theories in which a large angle scattering is achieved by a few hard scatterings rather than the accumulation of a number of small angle scatterings. The Saxon and Schiff [1957] approximation assumes that the backscattering is due to a single hard scattering event. When translated to the optical scattering, this gives
The result is depicted schematically in fig. 8a. There are two reflections, both at impact parameter zero, and with one at the front surface and the other at the back surface. A comparison of i(n)ss with the exact result for a dielectric sphere of m = 1.05 is shown in fig. 9a for x ranging from 5.0 to 50.0. Approximate results are in good agreement with exact results except at those values of the size parameter where the scattered intensity has a minimum. Interestingly, the positions of minima are reproduced quite accurately. The contributions of the two and three hard scattering events, shown in figs. 8b and 8c, respectively, have been calculated for a three-dimensional square well potential by Reading and Bassichis [1972]. In the context of the scattering of
eD0 (a)
fb)
(C)
Fig. 8. Description of backscattering process of a wave by (a) single hard scattering, (b) double scattering and (c) triple scattering.
250
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
5.0
18.0 32.0 Size Parameter
45.5
5.0
18.5
4 5.5
32.0
Size Parameter Fig. 9. A comparison of (a) Logi(n)ss (dotted line) and (b) Log IS(rr)ss + S(n)ks with i(n)MIE (solid line) for m = 1.05.
I2 (dotted line)
light by a sphere, these additional contributions are (Sharma and Somerford [19941):
and
If the medium is highly absorbing, the main contribution comes from the first = l/x(n" - 1)2, reflection. In contrast, for dielectric spheres (S(n)~sl~/(S(n)',,(~ and the contribution from two hard scattering events becomes important and may even dominate if x > 1/Jn2- 112. Figure 9b shows the effect of two hard
111,
0 31
25 1
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
Table 4 A comparison of the average actual separation between two successive minima with prediction of eq. (3.56) m
1.01
1.02
1.03
1.04
1.05
1.06
1.07
1.08
1.09
1.10
Mie
1.562
1.544
1.526
1.508
1.490
1.474
1.458
1.426
1.418
1.410
Approx.
1.555
1.540
1.524
1.509
1.494
1.479
1.465
1.450
1.435
1.421
scattering events. The deep minima predicted by i(n)ss now get filled up to give values closer to the exact values. The scattered intensity corresponding to eq. (3.52) for a dielectric sphere is
The positions of minima in the scattering pattern are then given by the relation x(n2 + 1) = p n , where p is an integer. The separation between two successive minima for a given n is thus related to refractive index through the relation A x -
n n2 + 1’
(3.56)
Table 4 shows a comparison between true minima and those predicted by eq. (3.56) for various values of n. The results are based on the average separation obtained from the fist 11 minima beginning at x = 5.0. It may be seen that the predicted separation is in good agreement with the actual separation. Equation (3.56) therefore is a potentially useful relation for diagnostic purposes. A backscattering formula based on a single hard scattering event has also been given by Chen and Hoock [1975]. This differs from the Saxon and Schiff formula in the way it takes into account the rescattering effects (Chen [1981]). Sharma and Somerford [ 19941 included this formalism also in the comparisons. It shows improvement over the Saxon and Schiff results, but is less accurate in comparison with S(n)ss + S ( n ) i s . 3.3.8. Vector description
Attempts to incorporate vector nature in the EA description to get access to polarization have been made by Perrin and Lamy [1986] and by Bourrely, Lemaire and Chiappetta [1991]. Perrin and Lamy [1986] identify the scalar eikonal solution with the amplitude of the perpendicular component Sl(6) of
252
[III, 5 3
SCATTERING OF LlGHT IN THE EIKONAL APPROXIMATION
the scattered field. The parallel component is obtained by multiplying SI(8) by the ratio of reflectivities which are obtained from single and double reflections (Wolf [1980, 19811). Therefore, the polarization is due entirely to reflection. Bourrely, Lemaire and Chiappetta [1991], on the other hand, perform separately the eikonalization of the two amplitudes. The vector eikonal function so obtained is
la (
Sl(0) = S;"ff(0)H(6m,x -6) + k db e2ja(@= cos e JI(kb sin 8)
where Sdiff ( 8)
= - [(1-cos6)-[l-Jo(xsinH)]+xJl(xsin~) cos 8
sin 8
sin 8
1
(3.58)
is the difiactive component which is zero for 8 > em,, = 18O/x radians and a(b) and P(b) are as defined in 33.3.2. The scattering function for the other polarization component $(8) is obtained from eq. (3.57) by permuting a(b) and P(b), where a(b) and P(b) are the Mie coefficients a/and 01 in the impact parameter representation. For a sphere, b is related to z and Z' (fig. 2) via the relations
r = tan-'
(g
1/2 -
.I)
,
t l =
tan-'
(b2
1/2
n2a2 -
1)
.
(3.59)
The approach above has been tested numerically against Mie predictions for
+
+
x = 150, rn = 1.3 iO.0 1 and for x = 500 and rn = 1.10 i0.0 1. The comparison is
shown in figs. 10a and 10b for scattered intensity and the degree of polarization for the first set of parameters. The agreement with the Mie theory is satisfactory for i(8) except for 8 > 160°, where the eikonal model predicts oscillations not present in Mie calculations. The degree of polarization is in perfect agreement with the Mie theory predictions for 8 > Om,,. Agreement is better for particles of larger sizes. Bourrely, Lemaire and Chiappetta [ 199I] have also examined the possibility of applying the above formalism to ellipsoidal particles. For a perfectly
n ~o, 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
253
I *
102 101
1 8 (degree)
s
1
.c N ._ L
0
0
a 0.5
r
W
Em W
U
0 0
50
100
150
Fig. 10. (a) The perpendicular component of the scattered intensity and (b) the degree of polarization as a ftmction of scattering angle 0 for a perfect homogeneous sphere of radius a = 150pm and index of refraction rn = 1.3 + iO.01. Here 1 = 6.28 pm. (From Bourrely, Lemaire and Chiappetta [1991I).
homogeneous ellipsoidal particle whose axis of symmetry is parallel to the direction of incident ray, cos z is given by cos z =
bb/a [ 1+ (pi - l)b2/u2]”* ’
where u is the size of the ellipsoid along its minor axis and ,HQ is the ratio of the major axis to the minor axis. This approach, however, is valid when (b- 1) is not very large because the Mie coefficients’are specific to spherical shape. The scattered intensity exhibits a behavior similar to the scalar eikonal solution for a perfect sphere. Numerical comparisons with the exact results have not been performed. 3.4. SCATTERING BY AN INFINITELY LONG CYLJNDER
Scattering of light by an infinitely long homogeneous circular cylinder is another situation where the EA has been examined in detail. This model of scattering
254
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
§3
lends itself to many practical ,situations and is perhaps the second most widely employed model in light scattering applications. 3.4.1. The scattering function for normal incidence
Any electromagnetic wave incident normally on an infinite long cylinder can be considered a superposition of transverse electric and transverse magnetic waves. If the electric vector is perpendicular to the axis of the cylinder, the wave is termed transverse electric. On the other hand, if the magnetic vector is perpendicular to the cylinder axis the wave is termed transverse magnetic. In either case, both the electric as well as the magnetic vector are perpendicular to
I
HY TEWS
TMWS
Fig. 1 1. Scattering geometry for an infinitely long cylinder.
the direction of propagation. The scattering geometry is shown in fig. 11. The plane ( x , y ) is the scattering plane and the direction of the incident beam is taken to be the z-axis. Since E,(x,y) is independent of z, it can be shown easily that for transverse magnetic wave scattering (TMWS), the Maxwell equations reduce to the following two-dimensional equation:
[V: + k2m2(x,y)]&(x,y)
= 0,
& &
(3.60)
: = + and m ( x , y ) is the relative refractive index inside the where 0 cylinder. For transverse electric wave scattering (TEWS) however, Maxwell's equations take the form eq. (3.60) only if ka >> 1. Using standard techniques, the scattering function for this problem is written as (Alvarez-Estrada, Calvo and Juncos del Egido [1980]) T(ki,/if)=
-ik2 4 ~
J
db e-ik'b[m2(b)-
n ~g, 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
255
where b is a two-dimensional vector in the (x,y) plane. For eikonalization of E,(b), as in the Mie case, a trial solution E;(b) = exp[ikx] $(b) is chosen. This in eq. (3.61) gives
[.:+
$(b) = -k2 [m2(b)- 11 $(b).
Zik;]
(3.62)
Ignoring 0 : in the above equation, the resulting equation with boundary condition ( ~ ( 6=) 1 at x = -00, yields the solution
(3.63) with a as the radius of the cylinder. Theoretically, the inequalities lm2(b)- 1 I << 1 and x >> 1 govern the validity domain here, too. The solution (3.63), when substituted in eq. (3.61), gives
where q = ki - kf is the momentum transfer. For small angle scattering, q is nearly perpendicular to the x-axis and one may approximate q . b by qy, where q = -k sin 8. The EA then gives
(3.64) where
(3.65) is the phase shift suffered by the ray in travelling undeviated across the cylinder. 3.4.2. Scattering by a homogeneous cylinder
For a homogeneous cylinder, m(r) = m, and the scattering function becomes
(3.66) Making the change of variable y form,
1
=
a sin y, it can also be cast in an alternative
n/2
T(o>,A = x
dy cos y cos(z sin y ) [I - exp(u* cos y)] ,
(3.67)
where z = x sin 8 and as before U * = (i& - pgAtanB). T ( 6 ) A D A can be obtained from T(~ ) E Aby replacing pEAby piDA.For a thin soft cylinder, such
256
SCATTERMG OF LIGHT IN THE EIKONAL APPROXIMATION
that x/m2- 1I << I, one may approximate exp(w* cos y ) gives
1
Id2
T(@EAM
-W*X
dy
COS'
y cos(z sin y)
xXW* = --Ji(z),
22
M
[a0 3
1 + o*CQS y. T h s
(3.68)
which may be recognized as the Rayleigh-Gans approximation for the TMWS case. For Im - 1I + 0 this is also RGA for the TEWS case. This means that the EA applies to the entire x region despite the original premise that x >> 1. For a completely absorbing cylinder, it can be verified easily that T(@EA
sin(2x sin 0) 2sin8
(3.69)
which is nothing but the scattering function for an infinitely long cylinder in the Fraunhofer diffraction approximation. For forward scattering, integration in eq. (3.67) may be carried out analytically. This yields
for an absorbing cylinder (Stephens [1984]), and
for a dielectric cylinder. In eq. (3.70), L(o*) = I ~ ( w-*Ll(w*) ) is a combination of the modified Bessel function II(w*) and the modified Struve fimction L I( w * ) , while J1( PEA) and S1(PEA) are, respectively, Bessel and Struve functions of order 1. The extinction efficiency, related to T(0) via the relation p'= (2/x)Re T(O), gives
QFL = xRe [L(o*)].
(3.72)
For w* --+ m, QrL approaches asymptotically its correct value 2, whle for very small o*,
QFL
x
M -- p
2
tan /? M xxnn',
(3.73)
is the limiting form of extinction efficiency for a thin absorbing cylinder. For nonforward scattering, as in the case of a homogeneous sphere, the scattering. function can be expressed in an analytic form in terms of known
UI,
8 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
251
functions only for a dielectric,cylinder (n’ = 0), and then only by means of appropriate series expansions (Sharma, Somerford and Roy [1997]). The imaginary part of the T(O)EAcan be integrated readily. This yields (3.74) where y is as defied earlier. The integration of the real part of eq. (3.67) gives (3.75) for PEA >> 1 and (3.76) for PEA << 1. Interesting formulae have been obtained by truncating these series appropriately (Sharma, Somerford and Roy [ 19971). 3.4.3. The EAJi-om exact solutions For perpendicular incidence, the exact scattering functions TI(@ and Tz(0) for TMWS and TEWS are given by (see, e.g., Bohren and Huffman [1983]) m
.
m
(3.77a) and W
.
m
(3.77b) respectively. The phase angles PI and a/ and the scattering coefficients a/ and b/ are given by (3.78a) (3.78b) (3.78~) (3.78d) where J/, NI and HI’) are Bessel, Neumann and First Hankel functions, respectively, of order 1. Primes denote derivatives with respect to the argument.
258
SCATTERlNG OF LIGHT IN THE ElKONAL APPROXIMATION
[m§ 3
Eikonalization of the exact scattering functions for a dielectric cylinder was carried out by Sharma, Roy and Somerford [1988b]. The procedure is analogous to what has been followed for the Mie case. Even the discussion pertaining to the relationship between the EA phase and the ADA phase remains unaltered. 3.4.4. Corrections to the EA
Corrections similar to those described in connection with the scattering of light by a homogeneous sphere have been studied for an infinitely long homogeneous dielectric cylinder by Sharma, Powers and Somerford [1981] and by Sharma [1993]:
1
dy cos y cos(z sin y) [1 - (1 + izl) e'"] ,
1
dy cos y cos(z sin y ) [ 1 - (1 + ir;) eixADA] ,
x/2
T(~)FCI =x
(3.79)
and n/2
T(@FCU =x
(3.80)
where x(n2 - 1)(1- 2 sin2 y) 7 4 cos y
=-
7;= -
x(m2 - 1)(2- m2)sin2 y 4 cos y
For forward scattering, integrals can be evaluated analytically, leading to
(3.81)
W~FCII =Wm) W~ADA,
(3.82)
with D(m) = [l + 0.125 x (m + 1)2(m - 1)(2 - m2)]. In eq. (3.81), SO(PEA) is the Struve function of order zero and 1F2 is the generalized hypergeometric function. Analytic solutions of expression (3.79) without approximating e"' = 1 + izl have been obtained by Di Marzio and Szajman [1992] in terms of rapidly converging infinite sums, and the numerical results were shown to be quite accurate.
a P 31
259
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
Table 5 Percent error in various approximation methods in I ' ( 0 ) ~ w sfor an infinitely long homogeneous cylinder of refractive index 1.05 ~
X
0.2
&A(rn2
- 1)/4
5.25 x lo4
EA
FCI
ADA
FCII
0.79
0.79
5.58
1.08
0.6
1.58 x
2.62
2.62
7.31
2.89
1 .o
2.63 x
2.05
2.05
6.77
2.32
2.0
5.25 x
0.45
0.44
5.24
0.72
3.0
7.88 x 10-3
0.10
0.05
4.88
0.35
5.0
1.31 x
0.11
0.01
4.83
0.30
10.0
2.63 x
0.22
-0.17
4.69
0.14
15.0
3.94 x
0.58
-0.32
4.59
0.04
20.0
5.25 x
1.25
-0.37
4.59
0.04
25.0
6.57 x
2.12
-0.46
4.58
0.03
3.4.5. Numerical comparisons Numerical comparisons of i(0)EA against the exact results have been performed by Sharma, Powers and Somerford [1981], Sharma and Somerford [1982, 1983al and by Sharma [1993] in intermediate and small size domains for m = 1.05 to m = 1.50. Typical percent errors in forward scattered intensities I'(O)TEWS = I T(o)TEWS l2 and i(O)TMWS = I T ( o ) T ~ S l2 for a homogeneous cylinder of relative refractive index m = 1.05 are shown in tables 5 and 6, respectively. The percent error is defined through the relation (3.47). It is clear from these comparisons that the EA is superior to the ADA in the domain Q 3.5 for TMWS. For TEWS, it is superior to the ADA in the domain x 2 3 and p Q 3.5. Again the value of PEA M 3.5 corresponds to the fist maximum in the extinction curve. For decreasing In - 1I and increasing x the general trend is similar to that observed for a homogeneous sphere. For x < 1, as in the case of a sphere, the scattered intensity in the EA is found to be a good approximation for TMWS despite the original premise x >> 1. Its performance is comparatively poor for TEWS in this domain. Further, a comparison of tables 5 and 6 with table 2 shows that the EA and its variants for an infinite cylinder are more accurate in comparison with the case for a sphere. Comparisons of i ( 6 ) E A with exact results (Sharma and Somerford [1983a]) conlirm it to be a small angle approximation. It has been found to work fairly
260
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[IIL
03
Table 6 Percent error in various approximation methods in i(O)TEWS for an infinitely long homogeneous cylinder of refractive index 1.05 X
PEA(m2 - 1)/4
0.2
5.25 x 1 0 - ~
0.6
1.58
10-~
EA
FCI
ADA
FCII
-10.04
-10.04
-4.70
-9.73
-8.56
-8.56
-3.33
-8.26
1.o
2.63 x 10-3
-7.63
-7.63
-2.45
-7.33
2.0
5.25 x 1 0 - ~
-4.05
-4.06
-0.96
-3.77
3.0
7.88
10-3
-2.97
-3.01
1.96
-2.72
5.0
1.31 x lo-*
-1.64
-1.74
3.17
-1.45
10.0
2.63 x lo-*
-0.67
-1.06
3.84
-0.74
15.0
3.94 x 10-2
-0.03
-0.93
4.01
-0.57
20.0
5.25 x lo-'
0.76
-0.87
4.12
-0.45
25.0
6.57 x lo-'
1.70
-0.90
4.16
-0.41
accurately at least up to about 30" for intermediate sizes. As the imaginary part of the relative refractive index increases the approximation appears to improve (Sharma and Somerford [1983b]). For very high absorption, the EA as well as the ADA<end to the Fraunhofer diffraction. Inclusion of corrections improves the results and the approximations are then found to work extremely well in the domain PEA In2 - 11/4 < 1. However, in contrast to a sphere, but as expected, TMWS also shows improvement for small values of x because the EA for TMWS does not involve any scalar approximation. No extra advantage is achieved by the more complicated form of the correction FCI. 3.4.6. l%e EA as Im - 1I + 0 approximation
In $3.3.6, in a one-dimensional analysis it was noted that the EA should be viewed as a Im - 11 + 0 approximation for fixed pgA.A partially successful attempt was made by Sharma, Roy and Somerford [1988a,b] to extend this analysis to two dimensions. For lm2 - 11 << 1, various Bessel and Hankel functions can be expanded as (Gradshteyn and Ryzhik [ 19801) (3.83)
111,
o 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
26 1
It is then possible to write b/ as
(3.84)
which may be rearranged as a series in powers of pgA as
b/ = Y p i A +S&;
+ t& + . . . ,
(3.85)
where the coefficients r , s and t in the limit x
4
00
are
Thus, bl can be expressed as bl =
i [l - is/ -
-
-
. . .] M (1 -eir'/),
(3.86)
Neglected terms are of the type ( pgA/x,pi:/., . . .) where 61 = pEA-/., and (pgA/x2,pg:/x2, . . .), etc. That is, the approximation improves as x increases for fixed pEA. Strictly speaking, the above result is valid only in the domain pEA < 1. Nevertheless, it shows that the EA, or at least its phase approximation, should be viewed as a m + 1 approximation for fixed p. 3.4.7. Vectorformalism Equations (3.78a) and (3.78b) in the limit PEA + 0 can be written.as (Van de Hulst [ 19571, Sharma, Roy and Somerford [1988b]) (3.87a) (3.87b) The scattering functions (3.77) with and cq given by eqs. (3.87a) and (3.87b), respectively, are then good approximations if pEA << 1. However, in the limit x , l + 00 with x > I, these yield eikonal phase Xo(b), when 1 is expressed
262
[IK § 3
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
- h g lo2
.-
X X X
0
30
60
90
120
150
180
8 (degrees)
rk
"lo' loo-
30
60
90
12 0
150 18C 6 (degrees)
Fig. 12. (a) Comparison of i(@)TMWS with i(0) corresponding to eq. (3.87a). (b) Comparison of i(6)TEWS with i(0) corresponding to eq. (3.87b). In both (a) and (b), M = 1.05 and x = 10.0.
in impact parameter representation. Thus, the formulae (3.77) with PI and a1 given by eq. (3.87) may be regarded as vector eikonal scattering functions for an infinitely long cylinder. Scattering fimctions given by eqs. (3.77) and (3.87) have been examined numerically (Sharma, Roy and Somerford [1988b]). A comparison of approximate
n ~9, 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
263
intensities with exact intensities for m = 1.05 and x = 10.0 is shown in figs. 12a and 12b for TMWS and TEWS, respectively. As expected, the approximate results reproduce the exact results reasonabIy well at small angles.
3.4.8. Scattering at oblique incidence The scattering function for the scattering of light by a tilted cylinder has been studied by Stephens [1984] in the framework of the ADA. It can be translated easily to the EA. Let us denote by Y the angle the incident wave makes with the z-axis of the cylinder. That is, Y = z / 2 for perpendicular incidence. The phase shift suffered by the central ray when the wave is inclined at an angle Y is then w*/ sin Y . The Scattering function in the EA thus becomes
1
;r/2
T(&A
=x
dy cos y cos(z sin 7) [ 1 - exp(w* cos y / sin Y ) ].
(3.88)
The above expression has been rederived by Sharma [1989] from the exact solution of the problem. No numerical estimates of the error have been made for this case. However, comparisons with exact results have been made for the ADA (Stephens [1984]). As the obliqueness of the incidence increases, the maxima and minima in-the extinction curves are out of phase with the exact theory. 3.4.9. Scattering by an anisotropic cylinder
The above discussions for a homogeneous cylinder are also valid for a normally illuminated cylinder composed of uniaxial material, where the cylinder axis coincides with the optic axis. For such a cylinder, the permittivity is characterized by a dyadic of the form
The Maxwell equations can be solved without any approximation in this case. It is found that, if the incident light is polarized parallel to the cylinder axis (TMWS), the cylinder scatters and absorbs light as if it' were isotropic with permittivity €11. On the other hand, if the incident light is polarized perpendicular to the cylinder axis (TEWS), it scatters light as if it were isotropic with permittivity E 1.
264
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[III, § 3
3.5. SCATTERING BY A COATED SPHERE
The coated sphere model has proved to be very useful in problems relating to the scattering of light by biological, atmospheric and other environmental and astrophysical particles. If m l and a1 denote the refractive index and radius, respectively, of the core and m2 and a2 denote the corresponding quantities of the coating, then the equivalent potential for the problem is (Chen [1987]) U ( r )= -k2 [(m:- 1) @(a1- Irl) + k2(m; - 1) @(a2 - Irl) @(lrl - a l ) ] , (3.89) where 0 is the step function. The scattering function in the EA may then be written easily as S(O),,
= k2
[ iu'
b db Jo(qb) (1 - eip
+
(&Z- &3)
&Z+ipz
l:
(3.90)
b db Jo(qb) (1 - e1p2
where PI = k(m: - 1) and p2 = k(m; - 1). Employing the change of variable, b = a1 sin y1 = a2 sin y2, the scattering function for a coated sphere may also be cast in the following form:
+x;
dy2 cos y2 sin y2Jo( 4a2 sin y2) [ 1 - eipzaz 'OS
.
sin-' (ul/'2)
(3.91) Unlike the scattering by a homogeneous sphere, the s(&A for a coated sphere can not be evaluated analytically even for forward scattering. But simple expressions are possible in special cases corresponding to a thinly or a thickly coated dielectric sphere. These models represent a number of realistic situations. As a result, many approximation schemes have been developed to desciibe the scattering of light by such particles (Morris and Jennings [1977], Arag6n and Elwenspoek [ 19821, Bhandari [ 19861). For a thinly coated dielectric sphere (a1 N a2; cosy1 N C O S Y ~following ), Morris and Jennings [1977], one can write n/2
S(O)EA = x:
[
]
dyl cos y1 sin y1 Jo(zl sin yl) 1 - e i ( ~ i + ~ 2 ) c 0 s+RI,
(3.92) where PI = xl(n: - l), p2 = x&$ - l), ZI= 2x1sin 812, X I = kal and x2 = k a ~ . The first term on the right hand side can be treated exactly in the same manner
111, Q 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
265
as the scattering function for ,a homogeneous sphere. The residue term, R I , is given by
where z2 = 2x2 sin(W2). As expected, R I + 1 as al/az sphere, a2 >> a t , S(O)EAmay be approximated as
+
1. For a thickly coated
where the residue R2 is approximately given by
and goes to zero as al/a2
+ 0.
Since
the integrals in eq. (3.93) are seen to be similar to those occurring in expressions for homogeneous spheres and may be treated in the same manner. Numerical comparisons of the EA and the ADA against the exact results have been performed for i(0) and p'for the scattering of unpolarized light. Chen [1987] made the comparisons in the context of atmospheric particles of core-mantle structure. The ratio of core radius to the total radius in his study was taken to be 0.8. Absorptive as well as non-absorptive coated spheres were examined. Salient features emerging from this study are essentially the same as those which emerge from the studies on a homogeneous sphere. Sharma and Somerford [1991] examined the accuracy of the EA for p'over the whole range of (= a1/a2) values. The values nl = 1.16 and n2 = 1.02 are typical of refractive indices of sand grains and biological materials relative to water. The performance of the EA as well as the ADA is fairly accurate over the entire range of x2 3 10.0. The maximum percent error in this domain in the EA and the ADA are 20 percent and 13 percent, respectively. For x > 500, the percent error in both the approximations is less than 6 percent.
266
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
SII,
53
Y X
Fig. 13. Scattering geometry for a spheroid. (From Chen and Yang [1996].) 3.6. SCATTERING BY A SPHEROID
It has been shown by Chen [1995] that within the framework of the EA, the scattering by a spheroid can be effectively replaced by a sphere of radus a’ and refractive index m‘ given by a‘
where
=
Po -a,
iuo m’ = 1 + --(m
clo
clop0
= a/b
a.
(3.94)
- 11,
is the aspect ratio of the spheroid (fig. 13), =
J,-
,
p0
= Ja2 cos2 (9
+ sin* (9,
and 0 and @ are related to 80, $0 and 8: cos(812)cos 80 cos $0 - sin(012) cos 0, > sin 0 cos 0 = cos(812) cos 00+ sin(012) sin 00cos $0. cos @ =
The scattering amplitude Soid(a,b, m) is then related to the scattering amplitude by a sphere Sere(a’,m‘) in the following way: soid(&
clo m,8) = -sere(a’,
Po2
m’,0).
(3.95)
The relationship (3.95) has been shown to be valid for a spheroid whose size parameter at smallest radius of curvature is at least 4. The scattering hnction
o 31
EIKONAL APPROXIMATION IN OPTICAL SCATTERING
267
for a spheroid can thus be fqund by calculating Sereemploying Mie theory for a sphere of radius a' and refractive index n'. Chen and Yang [1996] have examined the validity of using the GEA on the right hand side of eq. (3.95) for a water droplet with ka = 20 and a h = 1.25.The angular range considered was 0" to 40". They concluded that: (i) The GEA results agree well with T-matrix calculations for small scattering angles up to about 10". The agreement continues to be fairly good for larger scattering angles up to about 30". (ii) For a prolate spheroid the GEA works better when the direction of the incident light is normal to the major axis. For an oblate spheroid, the GEA formula works better when the direction of incident light is parallel to the major axis. Similar observations have been reported by Holt and Shepherd [1979] while using ADA for the analysis of the electromagnetic scattering in forward and backward directions by a dielectric spheroid. 3.7. SCATTERING OF LIGHT BY NEIGHBORING DIELECTRIC SPHERES
The potential function U ( r )for the system of N spheres of radii ai and refractive indices ni and centered at Rican be expressed as (Chen [1990])
c(n' N
U ( r )= -k2
- 1) @(ai
-
Ir - Ril)
i= 1
(3.96)
N
N
-2k2
C(ni
-
1) O(ai - Ir - Ri I).
i= I
The EA with this potential function has been tested numerically and is found to work fairly well when the gaps between spheres are small. A particular case where the EA greatly simplifies the problem is the case of closely packed dielectric spheres of the same size (but not necessarily the same refractive index) which are lined up along a common axis. If this axis is chosen to be the z-direction, the potential function simplifies to N
U ( r )= -2k2 x ( n i - 1)
@(a- b),
(3.97)
i= 1
whxh is also the potential function for a single sphere with the effective refractive index given by N
nett- =
1 + c ( n i - 1). i= I
(3.98)
268
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
-.-
Q
I
102
0
5
10
15
20 25 ka
30 35 40 45
Fig. 14. i(0) versus x. Set A is for m = 1.4, set B for m = 1.33 and set C for m = 1.33 + iO.01. The magnitudes in sets A and B are multiplied by factors of lo4 and lo2, respectively. Solid line, Mie theory for a single sphere with effective refractive index (3.99); dashed line, results of multiple Scattering calculation; dash-dotted line, results of noninteracting spheres only. (From Chen [ 19901).
The scattering function for the multiple scattering by N spheres ( S N )in this case is therefore nothing but the scattering function by a single sphere of effective refractive index (3.98). This gives the sum rule
Results of the sum rule are found to be good for n up to about 1.5. The single sphere calculations with neffup to about 2 yield accurate positions of maxima and minima of multiple scattering calculations except at ripples and sharp resonances which are not expected to be reproduced by the EA. A typical comparison is shown in fig. 14.
5
4. Applications of the eikonal approximation
4.1. PARTICLE SIZING
4.I . 1. One particle at a time
It is clear from the examination of validity domain of the EA and its modified variants that the EA is best suited for analysis of light scattered by intermediate and large size particles. For large particles, the Fraunhofer diffraction can be employed more conveniently, particularly for absorbing particles. A comparison of Fraunhofer diffraction approximation and the exact solution for scattered
III, o 41
APPLICATIONS OF THE EIKONAL APPROXIMATION
269
intensity of an infinitely long cylindrical particle of circular cross-section has been performed by Powers and Somerford [1979] for fiber size determination based on positions of the maxima or minima. The Fraunhofer approximation is found to agree well with the exact theory for absorbing cylinders o f x 2 20, whilst for transparent fibers having x 6 100 the agreement is poor. However, the apparent diameter of the transparent fiber is still a useful quantity since the true diameter may be obtained by applying a suitable correction factor (Powers and Somerford [ 19821). For particles with x < 20, an obvious choice is to employ the EA or one of its modified forms for particle sizing. However, since it is a small angle approximation, a signature of the scattering pattern is needed that depends on small scattering angles. Hodgkinson [ 19661 has noted that a measurement of the ratio between the scattered intensities at a pair of convenient angles within the forward lobe can give a useful estimate of the size of the scattering particle. Theoretical errors in spherical particle sizing employing this technique and the Fraunhofer diffraction were quantified by Boron and Waldie [ 19781. Following this work, Sharma and Somerford [1983a] examined the variation of the ratio of the scattered intensities R(81,h) = i(O,)/i(&) versus x for the EA, the FCI and the RGA against the exact R(81, &) for TMWS at selected angle pairs. The EA and the FCI were found to be in close agreement with the exact graphs. A typical comparison is shown in fig. 15. Measurement of R(@,8 2 ) for a known m allows the size parameter to be determined from theoretical graphs of the type shown in fig. 15. The errors involved in using the FCI in place of exact theory for fiber size determination have been assessed by Sharma and Somerford [1983a]. The percent error as a function of size parameter for m = 1.05 and for angle pairs (So,2.5"); (lo', 5') and (20", 10") is shown in table 7. The errors are indeed small. Similar estimates have also been made for spherical particles (Sharma, Sharma and Somerford [1984]). For a given size parameter, R ( @ ,02) values at two appropriate angles in the forward lobe were computed using the exact theory. Any divergence between the exact theory and the FCI predictions was obtained by applying the intensity ratio computed by the exact theory to the FCI graph of the type shown in fig. 15. For an accurate fiber size determination for x < 20.0 and m < 1.15, a suitable angle pair was noted to be (5",2.5'). The percent error in x determination has been defined as (xex- XFCJ) x 1O0/xex. The method applies neither to small nor to large values of x. For smaller x values the error in using a graphical technique becomes significant. For larger particles, the forward lobe shrinks. The choice of 81 and 6 must then be restricted to small angles. But at small angles, errors in practical measurements
270
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
Size parameter Fig. 15. Scattered intensity ratio R(81, &) versus size parameter for an infinitely long homogeneous cylinder of m = 1.05 and for angle pair (5",2.5"). Solid line, exact theory; triangles, RGA; crosses, EA; diamonds, FCI.
may become large. The situation may be improved to some extent by using a radiation of longer wavelength. Alternatively, one may also choose to use an index matching liquid to bring the relative refractive index of the fiber close to unity. Chen [1994] has deduced relations between the positions of minima in the scattering pattern and the size of a homogeneous dielectric spherical scatterer in the framework of the GEA. Since the EA and its variants are small angle approximations, the position of the first few minima only will yield accurate results. Starting from the formula (3.43), the locations of the first two minima in the scattering pattern are:
(n - l)/(n + 1) + 15.87sinyl/yl n(1.667 - 12.43 Sinyl/yl)
1 1
and
(4.1)
(n - l)/(n + 1) - 50.0 siny2/y2 n(2.15 +21.50siny2/y2)
where
d n 2 + 1 - 2 n d 1 - (1.916/q2,
YI
=k
y2
= 2xdn2
+ 1 - 2nJ1
- (3.508/~)~.
(4.3)
111,
8 41
27 1
APPLICATIONS OF THE EIKONAL APPROXIMATION
Table 7 Percent error in size parameter determination for an infinitely long cylinder of rn FCI
2.5-5
5-10
=
1.05 in using
10-20
1.0
2.05
1.65
3.0
0.38
0.01
-1.38
5.0
0.27
-0.13
-1.60
7.0
0.19
-0.17
-1.71
0.2 1
9.0
0.17
-0.23
-1.91
10.0
0.17
-0.22
-3.4 1
11.0
0.15
-0.24
12.0
0.13
-0.24
13.0
0.14
-0.25
14.0
0.13
-0.27
15.0
0.12
-0.15
16.0
0.08
-0.21
17.0
0.12
-0.30
1-8.0
0.10
-0.45 -0.76
19.0
0.10
20.0
0.10
21.0
0.08
22.0
0.08
23.0
0.20
24.0
0.14
25.0
0.10
Figure 16 shows results from eqs. (4.1) and (4.2). The agreement with exact results is excellent, especially for the first minimum and for size parameters above about 50. Equations (4.1) and (4.2) provide a good way of obtaining information about the scatterer from the position of the minima in the scattering pattern. However, the accuracy of this method does not seem to have been investigated for particle size determination. It is known that the positions of the minima do not determine uniquely the size of the scatterer. That is, it is possible to have a minimum at the same angle for different size parameters. However,
272
SCATTERING OF LIGHT
IN THE EIKONAL APPROXIMATION
second minimum
Fig. 16. Locations of the first and second maxima versus size parameter for light scattering by a water droplet. Solid line, Mie theory; dotted lines, eqs. (4.1) and (4.2); dash-dotted line, Fraunhofer approximation. (From Chen [ 19941.)
this difficuhy may be overcome by choosing an x which satisfies simultaneously eqs. (4.1) and (4.2). It may also be pointed out here that the scattered intensity i( 0)Ep reproduces very accurately the positions of minima. This fact, therefore, must be exploited to obtain relationships between the positions of minima and the size parameter of the scatterer for accurate size parameter determination. 4.1.2. Suspension of particles
For a monodisperse system of spherical particles the experimentally observed quantity Z(0)/Zoc is related to the scattering function i(0) for a single particle through the relations (Heller, Nakagaki and Wallach [ 19591)
and (4.5) where f(0)/focis the specific intensity of light scattered at an angle 0, and p2 and p12 are the densities of the particles and the total system. Particle weight
111,
P 41
273
APPLICATIONS OF THE EIKONAL APPROXIMATION
and diameter are denoted by w and .d, respectively, and c is the concentration of the spheres. Since the EA gives a simple analytic expression for the i(O), one can easily obtain particle size from eq. (4.4) by checking the value of x that satisfies eq. (4.4). The value of w then follows from eq. (4.5). The percent error introduced in weight and size determination in using I'(O)EA in place of exact i(0) has been estimated and compared with the percent error introduced in using the RGA and the ADA (Sharma and Debi [1980]). The maximum percent error in using the EA for weight and size determination in the domain 1 x 6 25 and n < 1.05 is of the order 9 and 3 percent, respectively. Borovoi and Krutikov [ 19761have calculated, within the framework of the EA, the statistical characteristics of the wave field propagating in the polydispersion of weakly refractive homogeneous spheres and have shown that the measured statistical characteristics can be used to determine the average characteristics of individual particles. The simple form of the EA for forward scattering can also be used to obtain size distribution of particles by analytic inversion methods. Analytic inversion methods based on the ADA have been derived for spheres (Fymat [1978, 19791, Fymat and Smith [1979], Box and McKeller [1978, 19791, Wang and Hallett [ 1996]), for infinitely long cylinders (McKeller [ 1982]), for particles with variable complex refractive index (Smith [1982]) and for absorbing spheres (Klett [1984]). The EA scattering function can be used in exactly the same manner to determine the particle size distribution in the domain where it is more accurate in comparison with the ADA.
<
4 . 2 . SCATTERING BY ROUGH
SURFACES
The effect of surface roughness was incorporated by introducing a Fermi distribution in the phase shift function by Chiappetta [1980]. Such a procedure is based physically on the fact that the refractive index decreases continuously in a region close to the surface and is not cut-off sharply. The refractive index of such a particle may be written as
The predictions of this model are in good agreement with the microwave scattering measurements of Giese, Weiss, Zerull and On0 [1978] for irregular compact and fluffy particles. In these calculations the EA was used to parametrize the forward scattering function. For the backward direction, the
274
SCATTERING OF LIGHT
IN THE EIKONAL APPROXIMATION
1n1, 0 4
Table 8 Characteristics of particles for whch the model of Perrin and Lamy has been examined Material
Average diameter (pm)
Montmorillonite Murchison meteorite
Type
110
dielectric
42
dielectric
Allende meteorite
28
dielectric
Magnetite
20
metallic
reflective model and shadow function were used (Wolf [1975], Giese, Weiss, Zerull and Ono [1978]). The above model neglects the effects of roughness along the incident direction and does not take into account the local fluctuations at the surface of the particle, the so called asperities. Thus, Perrin and Lamy [I9831 modified the refractive index model (4.6) to 1 + exp m(r) - 1 = (m - 1)
zL
da2- b2Q(a - b) n
1 + exp[(b - a)/d]
1 J
(4.7)
Here Q is the step function and d is the mean amplitude of asperities. For large angle scattering, these authors adapt the work of Wolf [1975, 1980, 19811 on single and double reflections by the microstructures of the surface and non-polarized transmission and extend it to particles, t a h g into account the curvature of grains. This model was tested against the experimental measurements of Weiss [ 19811 on four dlfferent indlvidual particles. The material type and average diameters are given in table 8. Apart from magnetite, good fits were possible for all other particles. For magnetite the angle of maximum polarization always remains too small. But this is not surprising because the EA is not expected to be good for metallic particles. Perrin and Lamy [1986] and Bourrely, Lemaire and Chiappetta [1991] have refined the above model f!urther by introducing the vector description in the EA. Figure 17a shows parallel and perpendicular components of scattered intensity calculated from the vector eikonal model (see 6 3.3.8) for a rough sphere of radius 20.5prn, index of refraction m = 1.55 + iO.01 and mean size of asperities (dh) = 3 x lo3. The incident wavelength is 0.633pm. The perpendicular component of the scattered intensity has roughly the same magnitude as for a perfect sphere. The degree of polarization is shown in fig. 17b. When the size
n ~P, 41
275
APPLICATIONS OF THE EIKONAL APPROXIMATION
107
lo6
.-
c v)
C
aJ
c
.-C 01 .-CL aJ c
0
V
ul
1 o5
lo4 lo3 lo2
1 0' 1
0
50
100
150
0 (degree)
C
0 .c
a
.-
L
4 0 Q
r
0 aJ
L0, aJ
'0
0
50
100
150
0 (degree) Fig. 17. (a) The perpendicular (triangles) and parallel (squares) components of the scattering intensity from eikonal vectorial model for a rough surface of radius u = 20.5wm,rn = 1.55 + iO.01 and The wavelength is A = 0.633 pm. Solid line, perpendicular mean size of asperities d/u = 3 x component of the M e theory. (b) The degree of polarization. Solid line, Mie theory; squares, eikonal vectorial model. (From Bourrely, Lemaire and Chiappetta [1991].)
216
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
tIK 8 4
of asperities is small, variations from a perfect sphere are small except in the backward direction. An alternative way of introducing surface roughness has been investigated by Bourrely, Chlappetta and Torrksani [1986] and by Bourrely, Torrksani and Chiappetta [1986]. Their formulation, based on the EP, accounts exactly for the shape of the surface of the particle. To see how this is done, let us rewrite eq. (3.44) for an axially symmetric scatterer as s( O)EP = -ink3
s
b db Jdkb sin 6) G(b) db,
(4.8)
where
(4.9) and Z(b) denotes the height of the boundary of the scatterer for a fixed impact parameter. To account exactly for the shape of the scatterer, one can first perform integration over the z-variable for a fixed value of b = bo. The problem is then to find the intersection points of line b = bo and the curve b = S(z), where S(z) is the expression of the boundary of the scattering object. This may be done analytically for simple geometries. For instance, for a sphere the intersection gives Z(b) = d n . For complex scatterers, this may be done numerically. The scattered intensities predict backward scattering enhancement. This is in qualitative agreement with experimental measurements of light scattered by a collection of particles (Weiss-Wrana [ 19831). The model also predicts strong oscillations in the scattering pattern. But these strong oscillations are not seen in experimental results because the scattering particles are not all identical in shape and hence the oscillations are averaged out. Also, the angular resolution of the scattered intensity is much greater than the period of predicted oscillations. Tests have been made in the microwave range. The presence of strong oscillations and backward scattering enhancement is observed. The angular domain 160°-1800 is most sensitive to the degree of roughness of the particle. 4.3. PLASMA DENSITY PROFILING
If the wavelength of the incident radiation is much greater than the Debye length, the plasma column in the presence of an axial static magnetic field along the z-
111,
P 41
APPLICATIONS OF THE EIKONAL APPROXIMATION
277
axis behaves like a dielectric rod with dielectric tensor given by (Platzmann and Ozaki [ 19601)
where €1 =
1 - Q(l + iv) , (1 + iv)* - s2
€3=l--
Q 1+ i v y
€1
=
Qs 5 (1 + iv)2 - s2
(4.11)
with Q = ( o , / o ) ~s , = ( u , / o )and ~ Y = ( Y J w ) ~Here . wp is the electron plasma frequency, wc is the electron cyclotron frequency, vc is the collision frequency and w is the frequency of the incident wave. Under such conditions, Maxwell equations for TMWS reduce to the scalar equation for the scattering by a homogeneous cylinder with m2 = € 3 . The EA is, therefore, expected to be a useful tool for the analysis of the light scattered by a cylindrical plasma column (Sharma [1986], Sharma and Dasgupta [1987]). Ignoring v,/w in comparison with unity, €3 can be approximated as €3 =
1 - [n(x,y)/n,] .
(4.12)
Here n, is the cut-off density that forms the upper limit beyond which the radiation of wavelength A will not propagate through the plasma and n(x,y) is the plasma density. The eikonal phase shift for the scattering of electromagnetic waves from a cylindrical plasma can therefore be written as
Je
~EA(Y = )n(x,y ) dx. 2% -@I7
(4.13)
Further, for a radially symmetric distribution, the phase shift function can be expressed as 2n
~ E A ( Y v= )
n,
bdbn(b)
dzq’
(4.14)
whose Abel inversion leads to (4.15) Here, @kA(y)= d @ ~ ~ ( y ) / dAy .simple method of obtaining $ E A ( ~ ) is suggested by eq. (3.64). The quantity exp[i#EA( y ) ] is the field behind the plasma column
278
SCATTERING OF LIGHT IN THE ElKONAL APPROXIMATION
PI, 5 4
and e xp[ i $~~( y)-]1 is the, added field which determines the scattered wave. This added field may be obtained by using a small stop in the spatial frequency plane that just shields the focus of the beam. The amplitude u ( y ) in the image plane may then be written as (4.16a) (4.16b) where F and F-' stand for Fourier and inverse Fourier transforms,fU is the spatial frequency and t u j ) is the transmittance of the spatial filter. Assuming that only a negligible part of the spectral components of [exp[i&~(y)]- I] is shielded by the stop, one can write
and hence one can obtain the I$EA(Y)by scanning the intensity pattern in the image plane, I(Y) = I exp[i@EA(y)l- 1l2
= 2 11 - cos[#EA(y)II.
(4.18)
A simple integration program then yields the electron density profile. This method is analogous to the method proposed by Brinkmeyer [ 19781 in the context of refractive index profiling of optical fibers. 4.4. LIGHT SCATTERING BY CLADDED FIBERS
A typical cladded optical fiber is characterized by a &electric constant (Calvo and Juncos del Egido [ 19791)
and
where R1 is the radius of the core, R2 is the radius of the cladding, and 6 is limited to 0 < 6 << 1. The external medium is supposed to be air with dielectric constant unity. E I , €2 and 6 are all real, and 0 6 a << 1. For an optical fiber, typical physical characteristics may be taken as R1 = 2 pm, R2 = 50 ym,
UI,
0 41
APPLICATIONS OF THE EIKONAL APPROXIMATION
279
nl = 1.52, n2 = 1.50, 6 = lo-’, a = 6. If the incoming radiation has a wavelength corresponding to a He-Ne laser, it may then be shown easily that the quantity K defined in eq. (2.10) is much less than unity, so that approximate eikonal solutions are reliable. 4.5. DIFFRACTION BY A VOLUME HOLOGRAM
Calvo and Juncos del Egido [1982] have considered the diffraction of electromagnetic waves by a volume hologram in the framework of the EA. For a medmm that is periodic along some plane, the dielectric permittivity of the medium, &(b,z),is a function of the two-dimensional vector b = (x,y). The hologram occupies a finite region xo in the z direction and the periods of the grating are given by (ul,u2,O). The scalar wave equation for the electric field inside the hologram can then be written as
[V:
+ k’]
E(r) = -k2 [ ~ ( r-)11E(r),
and the corresponding integral equation, assuming z direction as the direction of propagation, as
E(r) = Eo exp(ikz) +
1
dz’
db’ G(r - r‘) k’ [ 1- ~ ( r E(r’), )]
(4.20)
where SZ is the “periodicity shell” determined by the origin and the two vectors a1 and 112. In the EA, the Green’s function can be linearized to yield
T h s approximate Green’s function in eq. (4.20) leads to the following integral equation: & E A ( ~Z) , = Eo
+ii
-
1”’
dz’ [ 1 - ~ ( bz’)] , O(Z -z’)&EA(~, z’),
(4.21)
where &EA(b, Z) = E E A ( ~ , Z) e-ikz,
The solution of eq. (4.21), as has been noted earlier, is (4.22) which turns out to coincide with the Raman-Nath approximation (Calvo and Juncos del Egido [1982], Raman and Nath [1935, 19361).
280
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
w, 0 4
Corrections to the EA have also been obtained. For this purpose, the exact scalar field may be expressed as the sum of two contributions,
where Ec is the corrective term. Writing
it can be shown that &(b,z) satisfies the following integral equation: &(b, z) = R(b,z)e-ik
(4.24) with
Successive iterations of eq. (4.24) yield the corrections to the EA. If it is assumed that EEA >> Ec, the first correction dominates and & and R(b,z) may be approximated as
and
R(b,Z) M
lza
dz’ k d b ’ k2[1 - ~ ( b z’)] ’ , [G(b- b ’ , ~- z’) - G E A (-~b’, z - z’)]
(4.27) Numerical evaluation of corrections to several orders of diffraction has been carried out by Calvo and Juncos del Egido [1982]. Following observations regarding the behavior of the corrections have been made: (i) The correction to zero-order vanishes exactly. (ii) The most important correction for the physical case considered occurs for the first order of diffraction, and (iii) in higher orders, the importance of the correction decreases as the order of diffraction increases.
ID,
P 41
APPLICATIONS OF THE EIKONAL APPROXIMATION
28 1
4.6. MISCELLANEOUS APPLICATIONS
There are many light scattering problems where the ADA has been employed. The EA as well as the FCII can be used in exactly the same way in these applications. One only needs to substitute 2x1~1 - 11 by xlm2 - 11 for the EA in case the and C(m)-'QyL for a sphere and D(wI)-~Q?;for a cylinder for of FCII. In this section, we mention some of these applications. The ADA is an active field of research in the context of light scattering by spherical as well as non-spherical particles. It has been used to study the scattering of light by a cube (Napper and Ottewill 119631, Napper 119661, Maslowska, Flatau and Stephens [1994]), a hexagon in particular orientations (Flatau, Stephens and Draine [ 1988]), a hexagon in arbitrary orientations (Kuznetsov and Pavlova [1988]), a disk and an ellipsoid of revolution (Bryant and Latimer [ 1969]), ellipsoids (Latimer [ 19751, Holt and Shepherd [ 19791, Streekstra, Hoekstra, Nijhof and Heethaar [ 1993, 1994]), layered ellipsoids (Kokhanovsky [1991a]), anisotropic and optically active particles (Zumer [ 19881, Kokhanovsky [ 1991b]), crystals (Heffels, Heitzmann, Hirleman and Scarlett [ 1995]), prismatic particles in particular orientations (Chylek and Klett [ 1991]), arbitrarily oriented finite cylinders (Liu, Arnott and Hallett [1998]), coated spheres (Zege and Kokhanovsky [ 19891) and nonconcentric spheres (Lopatin [1986]). The ADA has also been used in the context of light propagation in random media (Borovoi [1988]). Accuracy of the ADA for column-like ice crystals has been examined recently by Liu, Jonas and Saunders [1996]. A fast algorithm to get efficiencies for most common shapes and in arbitrary orientations has been developed recently by Flatau and Cotton [19921. Numerical comparisons of the ADA for non-spherical particles with an exact solution are also presented by him. An extensive review of the theory and applications of the ADA has been given by Mahood [1987]. The ADA has been used extensively to study the colloidal solutions. Champion, Meeten, Moon and Gate [1979] applied the ADA to the case of platelike kaolinite particles to explain the particle size dependence of the turbidity of a dispersion at rest, where the particles are oriented randomly. Their study also revealed why the transmitted intensity on shear increases or decreases depending on the particle size and the wavelength. Expressions for the refractive index increment of dilute colloidal dispersions were obtained by Champion, Meeten and Senior [1978] employing the ADA. Meeten [1979, 1980a,b] generalized the ADA to obtain linear birefringence and dichroism of dispersion of colloidal particles. References to its use in the context of inversion of extinction data have already been given in 6 4.1.
282
[In, 5 5
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
The ADA has also been applied to a non-linear medium (Orenstein, Speiser and Katriel [1984, 19851, Orenstein, Katriel and Speiser [1986, 19871) in the context of optically multistable devices.
9 5.
Conclusions and discussions
The approximation methods play an important role in the analysis of light scattering problems. The arguments in favor of the desirability for studying approximate solutions, despite the availability of fast computational facilities, were given in 6 3 and are not repeated here. The type of approximation to be used depends mainly on three parameters: the size parameter x, the refractive index relative to the surrounding medium, m, and the phase shift parameter xlm2 - 11. Approximation methods generally employed have been listed in table 9. Table 9 Analytic approximations and their domains of validity Approximation
Domain of validity
~~~~
~
~
Rayleigh
x<
SlU&in
polarizability << 1
Rayleigh-Gans
.
1; lml< 1
Im - 11 < 1; 2x Irn - 11 << 1
Modified Rayleigh-Gans (Shimizu [1983], Gordon [1985])
Im-ll
Geometric optics
x>>
WKB
same
1
Two wave WKB (Klett and Sutherland [1992])
same
NW (Nussenzveig and Wiscombe [1980])
x>l
ADA (anomalous diffraction)
Im- 1
EA (eikonal or high energy)
same
S-approximation
same
Discrete Dipole (Draine and Flatau [1994])
Iml 5 2
HM (Hart and Montroll [1951])
1 < m < 1.5; 1/2 < x < 6
EF (Evans and Fournier [1990])
Im- 11 < 1; anyx
Fraunhofer dlffraction
x
Penndorf (Penndorf [ 19621)
Re S(0) >> Im S(0)
>> 1; H’
SP (Shlfnn and k i n a [ 1%8])
same
FM (Fymat and Mease [1981])
x>l
+ large
111, § 51
CONCLUSIONS AND DISCUSSIONS
283
In the present contribution, we have reviewed in detail the status of one of the soft particle approximations, namely the eikonal approximation. Theoretically, the domain of validity of the EA is x >> 1 and lm2 - 1 I << 1 . In practice, however, the EA has been found to be a good approximation for arbitrary x values for many cases. It is clear from table 9 that the domain of validity of the EA not only has an overlap with other soft particle approximations, namely the ADA and the S-approximation, it also has an overlap with other approximations. It has been noted that for x 2 30, asymptotic methods such as the geometrical theory of diffraction or some of its variants provide accurate answers (Chiappetta and Torrksani [1997]). For particles less than or equal to the size of the scatterers, approximations like the Rayleigh or the Rayleigh-Gans provide simple and accurate analysis. Thus, it is the intermediate size domain where the EA and the ADA are most useful. The EA is of maximum interest in the domain xlm2 - 1I < 4 because ithisin this domain where it has an advantage over the ADA. Corrected variants, namely FCI and FCII, further expand this domain to pgAlm2 - 1\/4 < 1. The size range over whch the EA is better than the ADA can be widened by the use of an index matching liquid. By a judicious choice of wavelength further adjustments can be made. The longer the wavelength the smaller the size parameter. The EA has been employed profitably in the large particle size domain as well, where many other approximation methods are also available. The reason for this is two fold: (i) It can take into account the shape of the particle in cases where the boundary of the particle can be described in terms of an equation. (ii) Simple and accurate modified variants of the approximation are now available in the form of the FCI, FCII, GEA and the EP. The EA and its variants have been compared numerically with the ADA, the HMA (Sharma [1994]) and the SA (Sharma and Roy [1996]), but have not been compared with other large particle approximations. The existing evaluation of the EA, uis a uis other large particle approximations, is therefore not really complete. Clearly there is a need for an exhaustive comparison of various approximation methods for completeness of relative accuracy studies. The corrections FCI and FCII are first order corrections in expansions where the leading approximations are the EA and the ADA, respectively. The second expansion is a realization of what was suggested long ago by Van de Hulst [1957] (p. 196); i.e., an expansion where the ADA appears as the leading term. The FCII considers only the first order correction to the ADA. The role of higher order corrections remains to be examined in future. As expected, the EA improves as 1m2 - 11 + 0. But as x increases for a fixed 1m - 11, the approximation improves in an oscillating manner. Thus, in contrast
284
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
[IIL 5
5
to potential scattering where the EA is viewed as a high energy approximation, the EA in optical scattering must be viewed as an Im - 11 + 0 approximation. As for angular range, the EA is a small angle approximation. This range is governed by the inequality 8 < 1/&. Interestingly, the EA, although developed as a small angle approximation, has served as a useful basis for many backward scattering approximations. T h s is because the backscattering by a soft scatterer is dominated by one or two hard scattering events rather than many small angle scattering events. As the EA underlies these approximations, theoretically all of these approximations have the same domain of validity as the EA. For a long time, it has been observed that the ADA is a good approximation for Jmlas large as 2 despite the original premise lm - 1I << 1. The analysis of the relationshp between the EA and the ADA presented here demonstrates clearly why t h s is so. The EA phase and part of the corrections to the EA are found to constitute the ADA phase. The terms neglected in the ADA phase are small for the near central incidence. For a large particle, it is precisely this region from where the dominant contribution to the near forward scattering arises. Again, despite the requirement x >> 1, the EA is noted to be a good approximation for scattered intensity for all x for a homogeneous sphere and for TMWS for an infinitely long cylinder. For a sphere this has been attributed to the cancellations between the errors due to the scalar approximation and the errors inherent in the EA. FQr an infinitely long cylinder, on the other hand, there is no scalar approximation involved for TMWS. But the EA for scattered intensity is still good approximation for x < 1. For extinction, however, the EA does not give good results for x < 1. Clearly, the EA performs well for i(0) for small x because ImS(0) >> ReS(0) for small x and the EA accurately reproduces ImS(0). The main shortcoming of the EA is that being a scalar model, it can not describe the polarization effects. For large particles, although the polarization effect is small, nevertheless it is non zero and must be taken into account. Attempts have been made to incorporate the vector nature of light into the treatment of the EA. However, these attempts are limited to exactly soluble models only and to particles whose shapes do not deviate much from spherical geometry. Of the three soft particle approximations listed in table 9, two (namely, the EA and the ADA) have been compared in details and their “relative domains” of validty delineated. Another soft particle approximation in table 9 is the S-approximation. This approximation is due to Perelman [19781. Recently there has been an increase in activity relating to this approximation (Perelman [1991], Granovskii and Stbn [1994a,b], Sharma and Roy [1996], Sharma, Ghosh and Somerford [1997]). The approximation takes into account the vector nature of
1111
REFERENCES
285
light and is extremely accurate for particles whch are large when compared with the wavelength of the light. In fact, for a homogeneous sphere of rn 6 1.06 and x > 2.0, the error in using this approximation for extinction efficiency is less than 5 percent in the entire domain (Perelman [1991]). For an infinitely long cylinder this error is 2.27 percent for m 6 1.05 and x 2 2.0 (Sharma, Ghosh and Somerford [19971). The approximation is very accurate for scattered intensities as well. The maximum percent error for an infinitely long cylinder in the domain 1.0 x 500 is less than 5 percent. But, then it is not clear how to employ it for particles where the problem does not have exact solutions. In this sense its use is still limited. Similarly, the GEA has also been applied only to spheres and spheroids (of small eccentricity). The EA, on the other hand, can be applied to particles of a variety of shapes including rough particles, where it has already found. applications. In this article only those applications have been included where the EA has been employed explicitly. These relate to the scattering of light by soft scatterers of various shapes, plasma density profiling, scattering by particles of astrophysical interest, diffraction by volume holograms, etc. However, the mathematical structure of this theory closely resembles the mathematical structure of the ADA, which has been applied to a much larger variety of problems. Thus, the number of possible applications of the EA is very large. Clearly the EA, together with its variants, constitutes a powerful and versatile tool for the analysis of light scattered by soft particles.
< <
References Abarbanel, H.D.I., 1972, in: Cargese Lectures in Physics, Vol. 6, ed. D. Bessis (Gordon and Breach, New York) p. 519. Abarbanel, H.D.I., and C. Itzykson, 1969, Phys. Rev. Lett. 23, 53. Aden, A.L., and M. Kerker, 1951, J. Appl. Phys. 22, 1242. Alvarez-Estrada, R.F., and M.L. Calvo, 1981, Opt. Acta 28, 1253. Alvarez-Estrada, R.F., M.L. Calvo and P. Juncos del Egido, 1980, Opt. Acta 27, 1367. Aragh, S.R., and M. Elwenspoek, 1982, J. Chem. Phys. 77, 3406. Asano, S., and G . Yamamoto, 1975, Appl. Opt. 14, 29. Baker, A,, 1964, Phys. Rev. 134, B240. Baker, A,, 1972, Phys. Rev. D 6, 3462. Baker, A., 1973, Phys. Rev. D 8, 1937. Banejee, H., B. Dutta-Roy and S.K. Sharma, 1975, Ann. Phys. NY 95, 127. Banejee, H., and S. Mallik, 1974, Phys. Rev. D 9, 956. Banejee, H., S. Mallik and S.K. Sharma, 1977, Phys. Lett. B 66, 239. Banejee, H., and S.K. Sharma, 1978, Ann. Phys. NY 117,447. Bayvel, L.P., and A.R. Jones, 1981, Electromagnetic Scattering and its Applications(Applied Science, London).
286
SCATTERING OF LIGHT IN THE EIKONAL APPROXIMATION
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[111
E. WOLF, PROGRESS IN OPTICS XXXE 0 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
IV THE ORBITAL ANGULAR MOMENTUM OF LIGHT BY
L. ALL EN^,^, M.J. PADGETT~ AND M. BABIKER~ a
Department of Physics, University of Essex, Colchestel: C04 3SQ. UK; Department of Physics and Astronomy, University of St. Andraos. St. Andrews, Ffe, KYI6 9SS, Scotland, UK
29 I
CONTENTS
PAGE
§ 1. INTRODUCTION
. . . . . . . . . . . . . . . . . . .
294
. . . . . . . . . . .
296
. . . . . . . . . . . .
302
. . .
306
. . . .
309
§ 2. THE PARAXIAL APPROXIMATION § 3. NONPARAXIAL LIGHT BEAMS .
§ 4 . EIGENOPERATOR DESCRIPTION OF LASER BEAMS § 5. GENERATION OF LAGUERRE-GAUSSIAN MODES
§ 6. OTHER GAUSSIAN LIGHT BEAMS POSSESSING ORBITAL
ANGULAR MOMENTUM . . . . . . . . . . . . . . .
319
§ 7 . SECOND-HARMONIC GENERATION AND ORBITAL
ANGULAR MOMENTUM . . . . . . . . . . . . . . .
322
§ 8 . MECHANICAL EQUIVALENCE OF SPIN AND ORBITAL
ANGULAR MOMENTUM: OPTICAL SPANNERS . . . . . § 9. ROTATIONAL FREQUENCY SHIFT
. . . . . . . . . . .
324 326
9 10. ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT. . . . . . . . . . . . . . . . . . . . . . . . $ 11. ATOMS AND MULTIPLE LAGUERRI-GAUSSIAN BEAM CONFIGURATIONS . . . . . . . . . . . . . . . . .
328
.
342
fj 12. MOTION OF Mg+ IN MULTIPLE BEAM CONFIGURATIONS
345
. . . . .
356
. . . . . . . . . . .
363
9
13. ATOMS AND CIRCULARLY POLARIZED LIGHT
Q 14. SPIN-ORBIT COUPLING OF LIGHT 292
5
15 . CONCLUSIONS
. . . . . . . . . . . . . . . . . . . . .
ACKNOWLEDGEMENTS
. . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .
293
366 369 369
0
1. Introduction
Electromagnetic radiation carries both energy and momentum. As well as involving the exchange of energy, any interaction between radiation and matter inevitably involves the exchange of momentum, which often has a mechanical consequence. Such an exchange can involve either linear momentum or angular momentum. However, the large body of work associated with radiation pressure (Brevik [1979]) or the mechanical influence of light on atoms (Letokhov and Minogin [ 19871, Kazantsev, Surdutovitch and Yakovlev [ 19901, Phillips [ 1992]), has been concerned almost exclusively with linear momentum. A seminal work is that of Poynting [1909], who showed by use of a mechanical analogy, that circularly polarized light should exert a torque per unit area on a quarter-wave birefringent plate, equal to AI2n times the light energy per unit volume. This is consistent with what can now be trivially written in photon terms as the ratio of angular to linear momentum, hlAk = i112n. Poynting proposed a subsequently unperformed experiment to measure the torque by means of quarter-wave plates. An elegant variation of Poynting’s proposed experiment was performed by Beth [1936], in which the torsional resonances of a fiber were exploited to detect the torque on a single suspended half-wave plate and so became the fist definitive measurement of h. Such work is concerned exclusively with polarized light and as such is associated with its intrinsic angular momentum. The component of intrinsic angular momentum in the direction of propagation, o,is known as the helicity. The eigenvalues of the helicity, f l , are known as the components of the spin of the photon (Jauch and Rohrlich [1976], Mandel and Wolf [ 19951). Although the most commonly observed light comes from dipole transitions, it is well known that multipolar processes (Rose [1955], Bouwkamp and Casimir [ 19541) can produce radiation which possesses orbital angular momentum as well as intrinsic spin. The concept of a field whch possesses an orbital angular momentum has an extensive literature (see, e.g., Gottfned [1966], Jackson [1962] and Simmonds and Guttmann [ 19701). While polarized beams, which depend on spin angular momentum, can be readily prepared by the use of birefnngent and other optical components, the concept of a beam of light possessing orbital angular momentum was until very recently quite unfamiliar. It is this concept 294
rv,§ 11
INTRODUCTION
295
and its practical realization,in a predicable laboratory light beam which is the focus of attention in this review. We begin the review with an outline of the theoretical basis for the orbital angular momentum of beams within the paraxial approximation and then indicate the unapproximated theory, based on the full set of Maxwell equations. We discuss the problems associated with the separation and identification of spin and orbital contributions to the angular momentum properties of a field. Beams with an exp(iZ4) azimuthal phase structure are shown to have an orbital angular momentum of Zh per photon. We describe the properties of LaguerreGaussian beams which have this phase and are physically realizable in the laboratory, together with the ways in which the beams may be generated. It is shown that elliptic beams, although not having an azimuthal phase of exp(iZ@), may also have orbital angular momentum. We then review the phenomenological behavior of beams possessing orbital angular momentum and their interaction with matter in bulk. The passage of such beams through nonlinear materials leads to second harmonic generation, to mode transformation and to conservation of orbital angular momentum within the light. The process of generation of beams with orbital angular momentum from beams possessing none, requires that angular momentum be transferred to the optical elements involved in the conversion process. The subsequent torque presents a way that the angular momentum transfer can be measured, just as in Beth’s experiment. The mechanical equivalence of spin and orbital angular momentum is discussed with particular reference to the context of an optical spanner. Finally, we describe the measurement of the rotational Doppler shift which arises when beams possessing orbital and spin angular momenta are rotated. The dipole-interaction of atoms with the orbital angular momentum of light beams is considered next. The results include changes to the gross motion as well as the internal motion of the interacting atoms, which depend on the orbital angular momentum of the beams. The detailed behavior of the atoms is examined within the context of Doppler cooling and trapping. Spin-orbit coupling, well known in the context of atomic electrons, is also shown to occur within the light beam and t h s coupling is manifest in the interaction of such light with atoms. The roles of spin and orbital angular momentum are compared and contrasted throughout. In optical interactions, analogues are shown to exist between asymmetric optical elements and birefnngent materials and close comparison is shown to be possible between the momenta in the context of the rotational Doppler shift (Courtial, Robertson, Dholakia, Allen and Padgett [ 19981). Yet, in
296
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
IN§2
their interaction with atoms the two contributions appear to play quite different roles. The implications of this are discussed in the conclusions. The structure of this review is straightforward. The concept of orbital angular momentum is outlined in $9 1-4. The production of Laguerre-Gaussian beamsand their phenomenological behavior and interaction with matter in bulk is similarly reviewed in $0 5-9. Sections 10-14, concerning the interaction of light with atoms, present all the recent theoretical work on this topic within a single unified approach. The interaction of light and atoms is written in full generality and is valid for a beam with an arbitrary amplitude distribution; the now well known quasi-plane wave results are shown to follow, and only subsequently is the field taken expressly to be that of a Laguerre-Gaussian mode and specific results shown for a magnesium ion.
0
2. The Paraxial Approximation
The orbital angular momentum of a realizable light beam has been investigated within the paraxial approximation (Allen, Beijersbergen, Spreeuw and Woerdman [1992]). It is well known that this approximation provides a satisfactory description of beam propagation properties which arise in the study of real lasers (see, e.g., Siegman [1986]). The change in the transverse beam profile of a reasonably well collimated beam occurs only slowly with position z, along the direction of propagation. For an amplitude distribution u(xy,z), the paraxial approximation is made by ignoring d2u/dz2 compared with k(du/dz), where k is the wavenumber, and ignoring Idu/bz( compared with u, in the scalar wave equation. This gives the paraxial wave equation,
This is not without some inconsistencies; Lax, Louisell and McKnight [1975] pointed out that the approximation implies that one can have a plane polarized electromagnetic wave whose electric vector depends on transverse position. However, they showed that t h s result is a consistent solution of Maxwell’s equations to lowest order, and they identified the lugher-order correction terms. It was shown (Davis [1979]) that if the vector potential is assumed to have just one component, the non-vanishing component of the vector potential obeys a scalar wave equation and produces the same order transverse and longitudinal field components for a Gaussian field as those derived by Lax, Louisell and
rv, 0 21
THE PARAXIAL APPROXIMATION
291
McKnight [1975]. It is qntirely consistent to assume that the vector potential of a paraxial light beam is confined to a single direction, whereas h s is not the case for the electric or magnetic fields. This has been exploited by Haus [1984], and his approach was used by Allen, Beijersbergen, Spreeuw and Woerdman [1992], an outline of whose argument follows. For a vector potential A polarized along P,
A
= Pu(x,y, z )
exp(ikz),
(2.2)
the paraxial approximation, after use of the Lorentz gauge for convenience, yields:
and
We should note that there is an axial, z, component to both the electric and magnetic fields, as there is for the TEM modes of a real laser. It is readily possible to use these fields to evaluate the real part of the time averaged Poynting vector, EOEx B . This is: [(E* x B ) + ( E x B*)] = iw-EO (uVu' - u*Vu)+ wk h 1u1* 2. 2 (2.5) It should be noted that this beautifully simple result only occurs provided a term of the order zero, namely &/&, has been retained, allowing the grad term to be written concisely. Clearly the result applies also for cylindrical coordinates, where u is written as u(r,@, z). The cycle averaged momentum density in eq. (2.5) is the linear momentum density in the beam. The angular momentum density along the beam axis, $, may be found by taking the cross product of the @-componentof expression (2.5) with the radius vector, r . In the case of any field of the form Q,
(E x B ) =
which obeys the conditions of the paraxial approximation, it is easy to see from eq. (2.5) that the ~1 component of the linear momentum density becomes
298
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!$ 2
simply EO ( E x B)# = &ow2 lu12/r, and the cross product with r gives an angular momentum density of magnitude j z= &owlI u I 2. The energy density in such a 2 beam is w = ceo ( E x B); = c ~ w IuI k = EOO’ luI2; i.e., the speed of light multiplied by the linear momentum density, and so:
w
(2.7)
w
When the angular momentum and energy densities are integrated over the x-y plane, the ratio of angular momentum to energy per unit length of beam
is obtained. This is:
The ratio of angular and linear momentum densities may be seen trivially to be wllwk = 1A12x,and shows that the beam possesses an orbital angular momentum of 1 per photon; the beam is not polarized and the angular momentum can not be due to spin. Clearly, eq. (2.8) is the orbital angular momentum equivalent of the ratio of spin angular momentum to energy, f h l h w = f l l w , for circularly polarized light and arises from the azimuthal phase term. It is necessary to generalize the approach before the full story of angular momentum in a light beam becomes clear. The light may also be circularly polarized and have components of electric field in both the x- and y-directions. That is, the vector potential of the field is defined by A = (a3+ +Pj)u(x,y,z) exp(ikz), when, again following Haus [ 19841, we find
(2.9)
iwa&+iw+Puj-c (2.10) and B = -/3
[
e+iku f+a
L
)
--+I&
y+
I):
+P--a-
i exp(ikz).
(2.1 1) This time, the calculation of EO ( E x B ) is more tedious but more rewarding, and it is found that @
( E x B ) = 5 [(E* x B) + ( E x B*)] 2
2
(uVu’ - u*Vu) + 2wk@ IuI2 1
(2.12)
n! 8 21
THE PARAXIAL APPROXIMATION
299
But, (2.13) and because 1uI2 is @-independent, (2.14) The term i (@* - a*B) arising from the relative amplitude and phase of the xand y-components of the electric field is readily identified with the spin in the z-direction, o.Thus, (E x B) has a polarization-independent part and a spin or polarization part. It is easy to show that j,
=
(r x ~o (E x B)), = r ~ (oE x I#)@ = awl IuI --am2 &..
(2.15)
The paraxial approximation ensures that Eo (E x B ) , = EoWk IuI29
(2.16)
and so the local ratio of angular momentum to energy becomes (2.17) Clearly, the effect of the light beam being other than linearly polarized is to introduce an additional term to the result given in eq. (2.7). This is the spindependent part of the ratio; it depends on the local gradient while the spin independent contribution is constant and independent of its position in the beam. As before, integration across the beam gives the ratio of total angular momentum to energy per unit length as (2.18) and we see that in the absence of a phase term exp(iZ@)the result would be the usual ratio for spin angular momentum divided by energy; viz., hohw. Consequently the other term, Z/w = hZ/hw, is also an angular momentum divided by energy and it again follows that the orbital angular momentum in the beam
300
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
In! 5 2
is given by Iii per photon. The ratio of angular and linear momentum densities may be seen trivially to be w(l + a ) / w k = (I + o)AJ2n, and shows that the beam possesses an orbital angular momentum of h per photon as well as spin. The expressions (2.17) and (2.18) are interesting. When integrate& over the beam it is found that the ratio of spin angular momentum to energy depends only on 0,which for left- and right-handed circularly polarized light is f1. However, the local value depends on the intensity gradient in the beam. Consequently, in a beam that satisfies the paraxial condition, this means inevitably that the ratio changes from place to place. It is not clear that the Allen, Beijersbergen, Spreeuw and Woerdman [19921 expression (2.17) had been derived previously, but the problem of the way the polarization of the beam depends on its finite extent has been the subject of detailed examination (Simmonds and Guttmann [1970]). The relations (2.17) and (2.18) are important because they demonstrate that even for unpolarized light, a beam with an azimuthal phase term exp(iZ4) possesses angular momentum. This may well prove not to be a necessary condition (see $6) for orbital angular momentum, but within the paraxial approximation it appears to be sufficient. What has to be investigated is whether this result is an artefact of the approximation or whether it has more general validity. The detailed form of u has not been invoked, other than that it is a normalizable amplitude distribution which leads to finite energy in the beam and satisfies the conditions of the paraxial approximation. In the original paper by Allen, Beijersbergen, Spreeuw and Woerdman [ 19921, emphasis was put on a mode distribution known to be physically realizable and to satisfy the conditions of the paraxial equation. An experimentally realizable field of this form is that of the Laguerre-Gaussian mode (see $ 5), whose distribution is an allowed solution of the paraxial equation. The amplitude distribution of such a mode is
where Cp’;G is the normalization constant, Lk (2r2/w2( 2 ) ) Laguerre polynomial, the radius of the beam at position z is
(2.19) is a generalized
w(2) = w(O)d(zi +Z’)/Zi,
where w(0) is the width at the beam waist, and (2p + I + 1) arctan(z/zR) is the Gouy phase where ZR is the Rayleigh range.
n! 8 21
THE PARAXIAL APPROXIMATION
301
For such an amplitude distributi~nthe linear momentum density is readily calculated, and for the purpose of this discussion it suffices simply to consider linearly polarized light. In evaluating uVu* - u*Vu in eq. (2.5), only the imaginary part of the expression contributes and we find, for the linear momentum density, now ignoring terms in &/I%, (2.20) while the cross product with r gives the angular momentum density, (2.21) We may note that the magnitude of the z-component is again simply EOIW1uI2 and is locally a constant for all z. The form of the mode amplitude distribution, u, for a Laguerre-Gaussian mode, eq. (2.19), may be substituted into the expression for E , eq. (2.4), to give the electric field components. These will be used in 9 7. It has been shown (Padgett and Allen [1995a]) using expression (2.20), that at a constant radius r, the Poynting vectorp follows a spiral path with a welldefined pitch, z,,, given by:
zp =
2nkr2 ~
I
’
(2.22)
They showed that the angle of rotation 8 of the Pohting vector from the beam waist at z = 0, for a point in the relative intensity distribution chosen such that r(z)/w(z) is a constant, is given by (2.23) Clearly, the rotation of the Poynting vector is proportional to the change in the Gouy phase from that at the beam waist. For a p=O mode, the radius corresponding to maximum amplitude in the mode is given by (2.24) When this is substituted into eq. (2.23), the angle of rotation of the Poynting vector associated with the radius corresponding to the maximum field of
3 02
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!5 3
a p = O , 1 # 0 mode is seen to. be O=arctan(z/z~)which is, surprisingly, independent of I. Far from describing a multi-turn spiral, the Poynting vector in this case rotates by exactly n/2 either side of the beam waist as the light propagates from and to the far field. For p f 0, Padgett and Allen [ 1995al show that the situation is more complex.
0 3. Nonparaxial Light Beams The separation of spin and orbital angular momentum found for paraxial beams is, at fist sight, somewhat surprising. There has long been an argument that to separate them in this way is against gauge invariance and is therefore not permissible (Biedenharn and Louck [ 19801). It is necessary to consider whether the result is an artefact of the paraxial approximation and to what extent a welldefined orbital angular momentum is actually possible. A full theory requires the solution of Maxwell’s equations without approximation. The absence of the frailties of the paraxial approximation mean that it is not necessary to make use of the vector potential A ; it may be written in the usual way in terms of the electric field components and their relative phase. The formulation is otherwise consistent with that in 9 2. The linear momentum and angular momentum densities of an electromagnetic field may be written as
p
= q~(E
x B);
j
=
&o(rx ( E x B ) ) ,
(3.1)
where E G E ( r , t ) and B = B(r,t). These densities may be integrated over all space to give the total linear momentum and angdar momentum associated with the field. For monochromatic fields we use the complex notation
E
=
(8e-iul
+pe+iut)
B
=
4 (Be-iuf
+
gre+iuf) ,
L
(3.2)
Then, following van Enk and Nienhuis [1992], the magnetic field may be eliminated by use of the Maxwell equation i o B = V x 8,which gives
“S
d3r (E* x (V x E ) ) ; J = 2io
d3r(r x {E* x (V x E ) } ) .
(3.3) On partial integration and use of the transversality of E , it is found for fields which vanish sufficiently quickly as r -+ 00 that J= A/d3r 2io
ET(r x j=x,y,z
2io
(3.4)
rv, 5 31
NONPARAXIAL LIGHT BEAMS
303
The total energy is given by
wr = y / d ' r E * . E .
(3.5)
With the use of eqs. (3.3)-(3.5), Barnett and Allen [1994] investigated the role of angular momentum in nonparaxial beams. The linear momentum, angular momentum and energy per unit length were examined for a beam propagating, in the z-direction. These quantities were found by integration of the respective densities over the x-y plane. The quantities are not in general time-independent and must be cycle-averaged; this gives P=//dxdy(E* J=A 2 i /o/ d x d y ( r
x (V x E)),
x (E* x (D x E))),
(3.6) (3.7)
F = s2/ / d x d y E * . E . These expressions do not depend on the paraxial approximation and, within the defined limits, are general results whch follow from Maxwell's equations. From the knowledge gained from the study of paraxial beams, the nonparaxial beams chosen were such that the electric field was of the form
E = (a2+ Pjj) E(z,p) exp(iZ@)+ E,?.
(3.9)
Note that the choice of x- and y-components of the field again allows polarization to be taken into account while the term exp(iZ@)embraces the feature of the field found previously to be sufficient for the presence of orbital angular momentum in the beam. Note, too, that the z-component of the field has been included to ensure it is a transverse electromagnetic field of the kind associated with a laser. The relation (3.9) represents a general transverse field which includes an arbitrary superposition of z-polarized plane waves propagating in the x-y plane. When the field is made to satisfy the time-independent wave equation, the Helmholtz equation, as well as V . E = 0, the field is found to be of the form
E - I ' dKE(K) exp(il@)exp(i-)
3 04
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[Iv, § 3
We may note that the z-component,has a different azimuthal dependence to that which was superimposed on the x - and y-components of the field. The calculation is long and complex, but the important result that follows is that (3.11) Clearly this is not the simple result of the paraxial beams, where J, - l + o w w .
(3.12)
Indeed, it shows that the orbital and spin angular momenta cannot be separated even classically, but that there is an additional correction term. Nevertheless, the result is important in a number of ways. The most important is that the additional term depends only on B and not on 1. It follows, therefore, that the concept of a light beam possessing only orbital angular momentum is a valid one, because for linearly polarized light when B = 0, the correction term is zero and the paraxial relation between J , and W is exact. In the paraxial limit, the expression (3.11) may be shown to yield eq. (3.12) as expected, showing that the paraxial formulation following Davis [ 19791 and Haus [1984] agrees with the full Maxwell equation approach in the paraxial limit for a beam with an azimuthal phase of the form exp(il4). This corresponds to neglecting terms of order ( ~ / k in ) ~the integrand in eq. (3.11). Thus for practical purposes, namely those situations where the paraxial approximation applies satisfactorily, the simple result (3.12) is permissible. This corresponds physically for laser beams to those areas where the beams are not strongly focused. Great care must be taken in interpreting eq. (3.11) further. Any choice of E ( K ) will be specific but will not necessarily correspond to a field realizable in the laboratory. It is tempting to interpret the well-known result (3.4) for J as the sum of orbital and spin contributions, J=L+S,
(3.13)
but there is a considerable literature which warns against such a separation (see Jauch and Rohrlich [1976], Biedenharn and Louck [1980]). Biedenham and Louck write “It is, indeed, not possible to separate the total angular momentum
n! P
31
NONPARAXIAL LIGHT BEAMS
305
of the photon field into an “orbital” and a “spin” part (this would contradict gauge invariance); the best that can be done is to d e h e the helicity operators . . . which is an observable (Beth)”. Indeed, for a massless particle, the gauge conditions introduce an interdependence between the vectorial aspects and spatial dependence of the field and spin and orbital rotations of the field can not be defined separately. Nevertheless, it is possible for the electromagnetic field J to be separated into two gauge-invariant parts which may be termed spin and orbital. This orbital term is similar to external angular momentum, as it is defined relative to a reference point or axis. The spin term is called the intrinsic part, as it is independent of the choice of this reference. Their significance has been considered (van Enk and Nienhuis [ 1994a1) by beginning with the usual definition of angular momentum (Jackson [1962]) in a classical electromagnetic field. Namely,
J = EO
s
r x [E x B] d3r,
(3.14)
which is just the integrated angular momentum density (2.15). The transverse component of the electric field creates a contribution J,d which may be shown to separate into two parts (Cohen-Tannoudji,Dupont-Roc and Grynberg [1989]): Jrad = Lrad
+ &ad,
(3.15)
where
Here E and A are purely transverse components of the field. These fields are gauge invariant and for a free field s m d and L a d are both conserved. However, the separation has in general been considered to be $physical. Van Enk and Nienhuis [ 1994al argue that this is not so, by investigathg the quantummechanical form of these operators. They show that S m d and Lrad do not commute with each other, unlike the corresponding operators for the internal and external angular momentum of particles. As believed at the outset, neither S f a d nor Lrad is an angular momentum operator because they do not generate orbital rotations. However, the three commuting components of &ad can be diagonalized simultaneously and both it and Lrad are separately measurable. The physical significance of the operators can only be investigated seriously by considering an interaction of radiation with matter (van Enk and Nienhuis
306
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[IY 5 4
[ 1994b1) when it becomes clear that only the components along the propagation direction can be measured by examining the change in the internal and external angular momentum of an atom (see Q 10). We show in 4 5 , in transfwning from a Hennite-Gaussian mode with no orbital angular momentum to a Laguerre-Gaussian mode, that a torque is introduced on the lenses but the polarization state is left unaltered. In the Beth experiment, the state of polarization is changed by the birefringent plate but the spatial distribution of the field is left unchanged. Classical theory allows these effects and the separation of contributions discussed in the beginning of t h s section to be understood. The apparent identification of terms as spin or orbital is, however, by no means unique. Barnett and Allen [1994] show that the z-component of angular momentum density, (3.7), may be expanded vectorially as
[r x {E* x (V x E ) ) ] =~
d C E*-E~ .' a@ + (E;E, -E;E,)
j=x,y, 2
' d E * - (xE,-yEx).
j =x, y , z
&j
(3.17)
.
It is tempting to ascribe the orbital angular momentum and spin to the first and second terms as in the paraxial theory. However, examination of the terms for the full Maxwell theory shows that the first depends on both o and I, while the second depends only on a; there is no simple separation into an I-dependent and o-dependent component of angular momentum. Yet it is this combination of terms which yields the simple summation contribution, (1 + o)/o,to the ratio of the z-component of angular momentum per unit length to energy per unit length in eq. (3.12). The o-dependent correction term of eq. (3.12) arises directly from the final term in the expansion (3.17). There remains the intriguing possibility that the z-components of spin and orbital angular momentum are not simply observable at the same time, but that a true interaction of the form 10 might occur within the light beam analogous to 1 . s for electrons (see 9 14).
0
4. Eigenoperator Description of Laser Beams
Certain aspects of the paraxial approximation calculation look familiar. This is not entirely unexpected, as the paraxial equation is well known to be analogous to the Schrodinger equation (Marcuse [ 19721) with z replacing the time t. The
n5 P 41
EIGENOPERATOR DESCRIPTION OF LASER BEAMS
307
term u*Vu - uVu* closely resembles the quantum-mechanical expression for the probability current of a wave function u. The paraxial calculation ($2) looks very much as though the beams had-been treated as if they were eigenmodes of the angular momentum operator of a particle; i.e.,
However, u is not a particle wave function; it is simply the purely classical distribution function of amplitude and phase. Nevertheless, the formal analogy between quantum mechanics and paraxial wave optics has been investigated (van Enk and Nienhuis [1992]) with the emphasis on orbital angular momentum. Instead of giving analytical expressions for the electromagnetic field, van Enk and Nienhuis [1992] specify it via Hermitian operators of which the field is an eigenfimction and by the Corresponding eigenvalues. The propagation of the laser light through an optical system is shown to be described by the evolution of the eigenoperators of the light field. They were able to show that the change of orbital angular momentum of a light field (see Q 6) by a system of lenses is describable by the evolution of the operator i,. For an electric field E =F(x,y,z) exp(ikz), the paraxial wave equation becomes simply d 2ik-F(x,y,z)
dz
=-
The field F(x,y,z) may be represented by the ket vector IF(z)), and operators acting on the field F are represented by operators acting on this ket vector. The evolution of a field propagating from Z=ZOto z = z l through a lossless optical system is determined by a unitary operator 6 as
where the coordinate z plays the same role as time in conventional quantum mechanics. Thus the quantity Q(z) is defined by
The transverse momentum operators px and p y obey the usual commutation relations with the position operators P and 9. The operator i, may be readily introduced together with the spin operator & whose eigenvalues are, of course, f l .
308
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
In! 5 4
Stoler [1981] has shown that the evolution of a light beam propagating in vacuum is found from the formal solution of the paraxial wave eq. (4.2). It can be written in the form (4.3) with 6, the free propagation operator describing the evolution, given by
where j 2 = j: +j,:.When a field evolves according to eq. (4.3), the evolution of a quantity Q(z) is given by Q@i> =
(F(zo) !6+QG/W o ) ) .
(4.6)
As in the Heisenberg picture, we can interpret the operator 6+06 as giving the evolution of the quantity corresponding to the operator Q, where the field is described by the state vector IF(z0)) at a single value zo of z. In this way the components of both orbital and spin angular momentum are shown (van Enk and Nienhuis [1992]) to be conserved during free propagation, as the operators i, and & both commute with j 2 .The formalism is used to investigate the propagation of light through lenses, and the results appear in 0 6. Something of the same spirit has also been applied in operator form (Nienhuis and Allen [ 19931) to the description of the Gaussian modes of a laser beam. It is well known that the analytic form of such modes resembles the wave functions of the stationary states of a two-dimensional quantum harmonic oscillator. It might, therefore, be expected that the orbital angular momentum of a light beam would be analogous to the angular momentum of the harmonic oscillator. Nienhuis and Allen [1993] investigated Gaussian modes in the presence of ideal, although possibly astigmatic, lenses. Raising and lowering operators were introduced which generate all higher-order modes from the fundamental, where the fundamental mode is an eigenvector of the lowering operator with eigenvalue zero. Once again the formalism of quantum mechanics with linear operators representing observables is carried over to a classical light beam in the paraxial approximation. As with the harmonic oscillator, the higher-order modes are obtained from the fundamental by applying the raising operator AT(z) lu/Y(z)) = Jn+iIUn+l(Z)),
The effect of a thin lens can be expressed by adding a phase factor to the field. This local phase jump is described by means of a propagation operator that does
n! Q 51
GENERATION OF LAGUERRE-CAUSSIAN MODES
309
not affect the commutation relations. This approach allows the description of both free propagation and refraction by ideal astigmatic lenses. The paraxial optics analogue of a coherent state is-shownto be equivalent to a ray in geometric optics. Within this framework, the expectation of the orbital angular momentum of a paraxial beam is found to be expressible in terms of a contribution analogous to the angular momentum of the oscillator plus contributions arising from the astigmatism of the beam. This helps to clarify the process by which transfer between the orbital angular momentum of the light beam and lenses or apertures can take place. The torque of a beam exerted on an astigmatic lens is found to be expressed in terms of a single expectation value. It is also confirmed that after passing through an angular aperture, a beam may transfer more angular momentum per photon to a lens than the apparent angular momentum per photon in the beam. The operator algebra approach allows a derivation of the algebraic transformation between Laguerre-Gaussian and Hermite-Gaussian modes, each of which forms a complete set. This agrees with that found analytically by Allen, Beijersbergen, Spreeuw and Woerdman [ 19921 and Beijersbergen, Allen, van der Veen and Woerdman [1993], and in the context of hydrogen wave functions, by group theoretic methods by Danakis and Aravind [ 19921.
9
5. Generation of Laguerre-Gaussian Modes
The rectangular symmetry associated with most laser cavities means that the transverse field distribution of a laser beam is nearly always best described in terms of a superposition of HermiteGaussian (HG) modes. These form a complete basis set with which any arbitrary field distribution can be described by an appropriate superposition of modes with indices n and m. The field amplitude of the individual modes, u::, is given by: =
unm
(
cexp(-ik we) x exp(-(n
( 2 + y2) z 2(Zk + 22)
)) exp(--) 2 +
y2 W*(.)
+ m + 1) q ) H n
where w2(2) =
2(Zi + 22) b R
'
q(z) = arctan
(9) ,
'
3 10
[n!0 5
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
Fig. 1 . The intensity distribution of HG and LG modes.
C,"," is a normalization constant, H,(x)is the Hermite polynomial of order n, k is the wavenumber, ZR is the Rayleigh range, ( n + m + 1) 2y is the Gouy phase,
and-w is the radius at which the Gaussian term falls to l/e of its on-axis value. These modes are paraxial solutions to Maxwell's equations and are structurally stable, which means that, ignoring radial scaling, their intensity profile remains the same on propagation. Laguerrffiaussian (LG) modes also form a complete basis set. Their field amplitude, uf;l.G,is given by eq. (2.19); viz.,
x exp
Cy
(2(z*-ikr2z+ z;) ) exp(-il@) exp (i(2p + I +
1) arctan
(3) ,
(5.2)
is the normalization constant and L$x) is the generalized Laguerre where polynomial, Z=m-n, and p=min(n,m). Rather than use n and m when describing LG modes it is more usual to use the mode indices I and p. The azimuthal phase term, exp(-iZ@), gives rise to I intertwined helical wavefronts and is the origin of the orbital angular momentum within the beam. Figure 1 shows the intensity structure of various HG and LG modes.
n! o 51
GENERATION OF LAGLJERRE-GAUSSIAN MODES
31 1
As the HG and LG modes each,form complete basis sets, linear superpositions of one set can be used to describe the other (Beijersbergen, Allen, van der Veen and Woerdman [1993], Danakis and Aiavind [1992]; see also Abramochkin and Volostnikov [1991]). Any LG mode can be expressed in terms of HG modes as follows: m+n
with
b(n,m,k ) =
x -- ((1 - t)"(l 2n+nln!m! k! dtk
+ t)"lt=o.
(5.4)
Note that the ik term corresponds to a nl2 phase shift between successive terms. It is perhaps more surprising that an HG mode rotated by 45" can be expanded into the same set of non-rotated HG modes:
In this case there is no ik term and therefore all the constituent modes are in phase. This will be shown to be important when we examine a particular method of transforming HG into LG modes. Figure 2 shows a number of such transformations. The LG; mode is a phase quadrature superposition of HG1,o and HGo,l modes oscillating with the same frequency (Rigrod [ 19631). Within most laser cavities, astigmatism breaks the frequency degeneracy of the HG1,o and HGo.1 modes, which explains why a true LG output is rarely observed. The frequently observed "doughnut", or hybrid, mode is a superposition of HGl,o and HGOJ modes but without any fixed phase or frequency relationshp. Progressive compensation for the astigmatism within the laser cavity allows the two modes to approach degeneracy and ultimately frequency-lock together ( T a m [ 19881). This can be facilitated by using a laser with a narrow gain bandwidth (Tamm and Weiss [199Ob]). Inclusion of an on-axis circular absorber favors the HGl,o and HGo,I modes loclung together in a phase quadrature, forming an LG; mode with zero on-axis intensity. The sense of the helical wavefronts is either left to chance (Harris, Hill, Tapster and Vaughan [ 19941) or can be controlled using selective feedback into the laser cavity (Tamm and Weiss [1990a]).
312
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n! 5 5
1
+2
1
G Fig. 2. The transformationof HG modes into LG modes.
- Computer-generated holograms or diffractive optics are used widely for generating monochromatic beams with specific phase/amplitude distributions. They can also be used for the generation of LG modes. In one form, the hologram is the computer-calculated interference pattern formed between a plane wave and the desired beam propagating at a small angle with respect to each other. When the hologram is illuminated subsequently with a plane wave, the firstorder diffracted beam has the required structure. For a beam with I intertwined helical wavefronts, such as an LG mode, the interference pattern with a plane wave resembles a diffraction grating with 1 additional lines forming a multi-pronged fork on the beam axis (Heckenberg, McDuff, Smith, RubinszteinDunlop and Wegener [ 19921); see fig. 3. These holograms can be made as binary amplitude holograms giving multiple, positive and negative diffraction orders. The boundaries between the light and dark lines of the hologram are given by the equation (Brand [1997])
.@2r I- -n+-cos#, 7t
D
where n is an integer and is the period of the grating far from the fork. When the hologram is illuminated subsequently by a plane wave such as a fundamental
N , P 51
GENERATION OF LAGUERRE-GAUSSIAN MODES
313
Fig. 3. A computer-generated hologram which, when illuminated by a beam with a plane wavefront, gives diffracted beams with helical wavefronts.
Gaussian beam from a conventional laser, the first-order diffracted beams have helical wavefronts with an exp(il@)phase structure. Alternatively, the amplitude hologram can be replaced with phase holograms which give a diffraction efficiency as high as 60% into the chosen order (He, Heckenberg and Rubinsztein-Dunlop [1995]). The exact structure of the diffracted beam depends on the form of the illuminating beam. For a hologram with a single additional line, that is, Z = 1, illuminated with a fundamental Gaussian mode, approximately 80% of the first-order diffracted beam is contained in the “single-ringed” LG; mode. The remainder of the energy lies in higher-order LG modes which also have an azimuthal mode index of unity. The holographic technique has been extended recently by the inclusion of a phase reversal at a fixed radius from the center of the hologram. This efficiently produces “multi-ringed” LG modes with a higher radial mode index (Arlt, Dholakia, Allen and Padgett [19981). Computer-generated holograms allow the conversion of a Gaussian beam from a conventional laser into a beam with helical wavefronts and associated orbital angular momentum. Rather than use a hologram, it is possible to use a transparent plate with a thickness that increases with azimuthal angle. These spiral phase plates, of refractive index n, are designed such that the radial step height, s, corresponds to a whole number of wavelengths, A,of the incident beam. The ideal step height is given by
A spiral phase plate of this type superimposes an exp(iZ@)phase structure on the transmitted beam.
3 14
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
tIv, § 5
Step Height
Incident Ray
t
Fig. 4. The form of the spiral phak plate showing the azimuthal refraction of a light ray at a radius Y.
It is difficult to make a phase plate with the required step height at optical frequencies. The best results have been obtained by suspending a spiral phase plate made from glass in a near index-matched fluid. Changing the temperature of the fluid modifies its refractive index, which controls the effective step height [Beijersbergen, Coeminkel, Kristensen and Woerdman [ 19941). The spiral phase plate technique has been extended to the mm-wave region of the spectrum where the longer wavelength makes the machming of a phase plate with the correct step height somewhat simpler (Turnbull, Robertson, Smith, Allen and Padgett [ 19961). Simple geometrical optics analysis of the spiral phase plate is sufficient to show that the orbital angular momentum content of the transmitted beam is Ih per photon. Consider a ring of radius r, projected on the spiral surface of the phase plate. Away from the axis of the phase plate where the angles are small, the angle 8 of the local azimuthal slope of the surface is given by 8 M d2nr (see fig. 4). A ray at a distance r from the optical axis will be refracted as it emerges from the spiral surface. The deflection angle, a, may be found using Snell’s law for small angles and is (8+ a) M no. Before refraction, the beam has a linear momentum of hlA per photon. After refraction, there is a component of linear momentum in the azimuthal direction, p @ ,given bype M halA. This corresponds to a transfer of angular momentum, L, between the spiral phase plate and the beam of light of L
=
h rp$ = r 1 a
n5 z 51
GENERATION OF LAGUERRE-GAUSSIAN MODES
315
per photon in the beam. Combining the three previous equations with that for the optimum step height gives s(n - 1) Lxh-xlh
(5.9)
A
per photon in the beam. This surprisingly simple result emphasizes the point that the orbital angular momentum is dependent solely on the azimuthal phase structure within the beam. The Zh per photon result is true for all beams with an exp(il4) azimuthal phase structure, as shown in 9 2. The methods discussed so far do not, in general, produce pure LG modes, but rather a superposition of LG modes each with the same azimuthal mode index. However, pure LG modes can be obtained fiom high-order HG modes using a mode converter based on cylindrical lenses (Tamm and Weiss [1990a], Beijersbergen, Allen, van der Veen and Woerdman [19931). The 100% efficient conversion between HG1,o and LGA modes has been used with such a converter to confirm the helical nature of a laser output (Tamm and Weiss [ 199Obl). However, it transpires that any HG mode with indices m and n is, on transmission, converted into the corresponding LG mode. The indices of this new LG mode are 1 = m - n and p = min(m,n). As discussed at the beginning of this section, an LG mode and an HG mode rotated by 45" can both be described m terms of the same superposition of HG modes. To convert from one to the other it is necessary to find a method of introducing a n/2 phase shift between successive mode components. When an HG beam is focussed by a cylindrical lens, the resulting Gouy phase shift depends both on the mode indices and the degree of astigmatism. The astigmatism can be characterized in term of differing Rayleigh ranges z k and ZR,, governing the divergence of the beam in the x-z and y-z planes. The resulting Gouy phase factor is given by: exp{ -i [(n + :)arctan(
5)+ ( m + ;)arctan(
ZRr
.}I)&
(5.10)
If two identical cylindrical lenses of focal lengthf are separated by f i f ,then the transmitted constituent HG modes of a beam with a Rayleigh range of (1 + l/fi)f undergo a relative phase shift due to the Gouy phase of (n - m)n/4. As (n- m) differs by 2 for each of the constituent modes, an incident HGl,o mode rotated at 45" is converted to a pure LGA mode. The only limitation on the range of LG modes that can be generated in this way, is the range of HG modes that can be generated from a conventional laser.
316
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
cylindrical
rn! 5
5
m
Input mode
In phase Fig. 5. The cylindrical lens mode converter for the conversion of an HG n = 1, m =O mode into the corresponding LG mode with 1 = 1 and p = 0.
The n12 phase difference introduced between the constituent modes leads such a device to be called a n12 mode converter. Beams with quasi-plane wavefronts are converted into beams with helical wavefronts possessing orbital angular momentum (see fig. 5). The cylindncal lens system is analogous to a birehngent quarter-wave plate that transforms h e a d y polarized light into circularly polarized light possessing spin angular momentum (Beijersbergen, Allen, van der Veen and Woerdman [ 19931). When the lens separation is increased to 2f ,the phase difference between the constituent HG modes is increased to n. These n mode converters reverse the handedness of any transmitted LG mode and are analogous to birefringent halfwave plates which reverse the sense of transmitted circularly polarized light (see fig. 6). Any optical device that reverses the handedness of the mode is effectively a n mode converter. Rather than make a converter from two cylindrical lenses, it is often easier to use an optical component such as a Dove prism. A short length of near single-mode optical fiber has been used to convert an input HGl,o mode into a beam with an azimuthal phase structure of exp(*il@) and corresponding helical wavefronts (McGloin, Simpson and Padgett [ 19981). Stress applied to the fiber breaks its circular symmetry. The fiber modes which
IV,
5 51
GENERATION OF LAGUERRE-GAUSSIAN MODES
317
Fig. 6. Quarter or half-wave plates and n/2 or rc mode converters play equivalent roles for spin and orbital angular momentum.
closely resemble HGl,o and HGo,l modes propagate with different velocities. By controlling the stress on the fiber, their difference in velocity can be made such that the modes emerge from the fiber in phase quadrature. As with the cylindrical lens mode converter, an input HG1,o mode aligned at 45' to the stress axis of the fiber is converted into an LG; mode.
318
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[rv, 5 5
Fig. 7. The intensity and phase structure of LG modes displayed by their interference with a plane wave.
A number of experimental configurations have been demonstrated for the determination of the phase structure of LG modes. This is particularly important as examination of the intensity distribution alone can not distinguish between helical wavefronts of opposite handedness. One approach is to interfere the LG mode with its own mirror image. A biprism placed in the LG beam produces an additional LG beam of opposite handedness that interferes with the original beam to give straight-line hnges with a 21 + 1 fork at the center of the overlapping beams (Harris, Hill, Tapster and Vaughan [1994]). If the handedness of the original beam is reversed, the direction of the fork also reverses. A Mach-Zehnder interferometer containing a Dove prism to invert the beam in one arm has also been used to the same effect (Courtial, Dholakia, Robertson, Allen and Padgett [1998]). A discrete series of beam splitters have also been used (Harris, Hill and Vaughan [1994]), which allows the two beams to be superimposed with different wavefront curvatures, resulting in spiral interference patterns with 21 arms. When the LG mode is generated external to the laser, it is possible to interfere the converted LG mode with the original laser output. By placing the mode converter in one arm and a beam expansion telescope in the other arm of a Mach-Zehnder interferometer, it is possible to observe the interference pattern between the LG mode and a plane wave of the same frequency (Padgett, Ark, Simpson and Allen [ 19961). Careful alignment produces the spiral interference patterns with the number of arms equal to I, as shown in fig. 7.
N,8 61
OTHER GAUSSIAN LIGHT BEAMS POSSESSING ORBITAL ANGULAR MOMENTUM
319
As an alternative to these interference experiments, a cylindrical lens mode converter can be used to transform the LG mode into the corresponding HG mode. The readily deducible m and n indices of the resulting HG mode can be used to identify the Z and p indices of the original LG mode (Dholakia, Simpson, Padgett and Allen [ 19961).
Q 6. Other Gaussian Light Beams Possessing Orbital Angular Momentum So far, we have considered only beams with an azimuthal phase structure exp(iZ4) which have an orbital angular momentum equivalent to ZA per photon. The origin of the angular momentum transfer to light beams may be clearly seen for the case of the spiral phase plate, where refraction gives rise to an azimuthal component of the ray vector. Alternatively, the torque exerted on the cylindrical lens mode converter can be seen to arise from the force experienced by the dielectric lens material within the electric field gradient that arises from the non-circularly symmetric light beam (Beijersbergen, Allen, van der Veen and Woerdman [ 19931). If a cylindrical lens is to exchange angular momentum with a light beam, it is essential that the incident light beam is not circularly symmetric. One obvious example is an elliptical Gaussian beam. Although the focusing of elliptical Gaussian beams with cylindrical lenses had been analyzed previously (Arnaud and Kogelnik [ 1969]), the angular momentum properties of such beams were not recognized until recently. When a light beam with field amplitude described by u(x,y, z) is transmitted through a thin lens, it acquires an additional phase factor x(x,y). The amplitude immediately after the lens, d(x,y,z), is given by: UI(X,Y,Z) = U(X,Y,Z) exP[ix(~lY)l.
(6.1)
The resulting change in the z-component of the orbital angular momentum, SL,, for light of an angular frequency o is given by (van Enk and Nienhuis [19921):
For a cylindrical lens of focal lengthf, inserted at an angle a to the x-axis, the phase factor introduced for light of a wavenumber k is given by: x(x,y)= n sin a + y 2 cos2 a + 2 ~ sin y a cos a , -2fk ( 2 ' 2
)
320
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[IV, 5 6
so that
6L,= -w/e Sa km ~ ~ d x d y [ ( x 2 - y 2 ) s i n 2 a + 2 r y c o s 2 a ]Iu(x,~,z)~~. 2f -w An elliptical Gaussian beam aligned with the x- and y-axes has an intensity distribution
where w, and wv are the distances at which the Gaussian term drops to lle of its on-axis value in the x- and y-directions, respectively. T h s field distribution is symmetric about the x and y-axes and 6L, simplifies to
It follows that the transfer of orbital angular momentum between the elliptical beam and cylindrical lens is given by (Courtial, Dholalua, Allen and Padgett [ 1997a1) kW,-W;
6L, = h-f
.
4
sin(2a)
per photon. Clearly this is not in general an integer multiple of h. For highly elliptical beams a few millimeters in diameter, the transfer can exceed 10OOOh per photon. The amount of orbital angular momentum that can be transferred is limited only by the f-number and aperture of the cylindrical lens. The ratio between the orbital angular momentum density, j,, and the energy density, w = EWJ* (uI2, of a beam with a local phase I,!@,$) is given by (Beijersbergen, Allen, van der Veen and Woerdman [19931):
For an LG mode, V ( r ,4) = 14 and, as shown in 5 2, the ratio of orbital angular momentum density to energy density is independent of the position within the
m, 5 61
OTHER GAUSSIAN LIGHT BEAMS POSSESSING ORBITAL ANGULAR MOMENTUM
32 1
beam and equal to l/4. In contrast, for an astigmatically focused elliptical Gaussian beam, the derivative of I$ with respect to I$ varies across the beam. The ratio of orbital angular momentum density to energy density can be shown easily to be (Courtial, Dholakia, Allen and Padgett [ 1997a1)
where x' and y' are the coordinate axes of the cylindrical focusing lens. The phase structure of astigmatically focused, elliptical Gaussian beams has been studied by interfering them with a plane wave (Courtial, Dholakia, Allen and Padgett [ 1997a1). In the region of the beam focus, their intensity and phase structure resembles that of a two-blade propeller (see fig. 8).
2%\
I
Intensity Fig. 8 The phase and mtensity structure of the astlgmatically focussed elliptical Gaussian beam III the focal plane of the cyllndncal lens.
A possible application of these astigmatically focused elliptical Gaussian beams is the optically induced rotation of micron-sized particles. In this case, the large size of the particle compared with that of the beam ensures that the variation in angular momentum density within the beam is not important. The significantly higher angular momentum content of the elliptical beams may allow
322
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[rv, 0 7
greater torques to be applied and correspondmgly greater rotational velocities to be achieved.
0
7. Second-Harmonic Generation and Orbital Angular Momentum
In nonlinear optical materials, the polarization is a nonlinear function of the electric field. For a second-order nonlinearity, the polarization has a term that is proportional to the square of the electric field. Thls leads to the wellknown phenomenon of sum-frequency mixing, in which two fields of optical frequencies 01 and 02 combine to produce a third field of frequency wg, where 01 + 02 = 1 3 3 . When 1 3 1 equals 02 the process is called second-harmonic generation. The efficiency of these second-order nonlinear interactions is determined by phase matching. Phase matching is simply the conservation of linear momentum, namely the condition kl + k2 = k3. When the beams are colinear, as in the case of second-harmonic generation with an HGo,o mode, this vector relationship is a simple scalar one. However, for a LaguerreGaussian mode, where the Poynting vector follows a spiral trajectory, one might expect the situation to be more complicated. Surprisingly, perhaps, it is found that $he optimum phase matching conditions for high-order LaguerreGaussian modes are the same as for an HGo,o mode. This implies that the wavevectors and Poynting vectors for the fundamental LG mode and its second harmonic are also co-linear (Dholakia, Simpson, Padgett and Allen [ 19961). As shown in 9 2, the Poynting vector for a Laguerre-Gaussian beam spirals around the beam axis at a rate given by (Padgett and Allen [1995a]): ~
Because the Poynting vectors are co-linear, their rotation rates in the fundamental and second-harmonic beams must be the same. Hence, when k doubles then I must also double (Dholalua, Simpson, Padgett and Allen [ 19961). Earlier work (Basistiy, Bazhenov, Soskin and Vasnestsov [1993]) made brief mention of the frequency doubling of a laser beam possessing a phase singularity centered on the axis of the beam. Two dislocations were observed in the second-harmonic beam, but no explanation as to the physical basis underlying this observation was suggested.
N,5 71
SECOND-HARMONIC GENERATION AND ORBITAL ANGULAR MOMENTUM
323
An alternative view of frequency doubling is that the field of the generated wave is proportional to the square of the input field. In the plane of the beam waist, the expression for a Laguerre-Gaussian mode simplifies to
For a p = 0 mode, the second-harmonic beam is also an LG mode with the following transformations:
k
+ 2k
wo + wo/& p = 0 -+ p = 0 I -+ 21
frequency doubling; reduction of the beam waist; the amplitude distribution remains single-ringed; the angular momentum per photon is doubled.
In the second-harmonic beam, the number of photons is halved. Consequently the doubling of I corresponds to a conservation of the orbital angular momentum within the light field. The predicted doubling of the azimuthal mode index has been confirmed by examining the phase structure of the fundamental and second harmonic beams. A Mach-Zehnder interferometer with a Dove prism in one arm forms theinterference pattern between a beam and its own mirror (Courtial, Dholakia, Allen and Padgett [1997b1). Figure 9 shows the forked interferograms obtained for a variety of fundamental Laguerre-Gaussian beams and their secondharmonic counterparts. The azimuthal index of the beam can be inferred directly from the number of fringes on either side of the fork, by dividing the number of additional fringes by two. These results confirm that the azimuthal index of the beam is doubled in the second-harmonic process and are consistent with the conservation of orbital angular momentum within the beams. We have seen that a p = 0 mode frequency doubles to give another p = 0 mode that propagates in a structurally stable fashion. However, for p > 0 the situation is more complicated. When p > 0, the associated Laguerre polynomial is a function of r, the square of which can no longer be described by a single Laguerre polynomial. In general, the square of a Laguerreaaussian mode with mode indices I and p can be expressed as the sum of LaguerreGaussian modes with indices 21 and p =0,2,. . . 2p (Courtial, Dholakia, Allen and Padgett [1997b]). Although these modes all have the same Rayleigh range, their Gouy phases are not the same. Consequently, the constituent modes interfere to give a beam distribution that changes with propagation. It is only in the far field, where the Gouy phase shifts of the constituent
324
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
Fundamental
I
Second Harmonic
e = i , p=o
e =I, p=i
e=2, p=i
e=i, p=2
Fig. 9. The forked interferograms formed between a beani and its mirror image, for a variety of LG beams and their second harmonic counterparts.
modes differ by multiples of 2n, that the distribution at the beam waist is reproduced.
0
8. Mechanical Equivalence of Spin and Orbital Angular Momentum: Optical Spanners Following the calculations wluch predicted that LG modes should possess an orbital angular momentum along the beam axis, attempts were made to demonstrate the transfer of this orbital angular momentum to matter. Initial experiments attempted unsuccessfully to measure the torque exerted on a cylindrical lens mode converter suspended from a quartz fiber (see Beijersbergen [1996]). Experiments of this kind are extremely challenging, as a slight misalignment between the optical beam and the suspension generates unwanted torque on the lens that may be many times larger than the torque of interest. As an alternative, it was suggested that the physical suspension mechanism could be
Iv, 0 81
OPTICAL SPANNERS
325
removed and the optical beam itself used to “suspend” the object to be rotated (Padgett and Allen [ 1995a1). Any dielectric material falling within an electric field gradient experiences a force towards the region of highest field. In the vicinity of a tightly focused laser beam the gradient force is strong enough to trap a micron-sized dielectric particle in three dimensions (Ashkin, Dziedzic, Bjorkholm and Chu [1986]). This technique is now commonly referred to as optical tweezers and is widely used in many biological applications, such as measuring the compliance of bacterial tails (Block, Blair and Berg [1989]), the measurement of the forces exerted by single muscle fibers (Finer, Simmons and Spudich [1994]) and the stretching of single strands of DNA (Wang, Yin, Landick, Gelles and Block [ 19971). The trapped object is held on the beam axis by use of an LG mode within optical tweezers and any bbserved rotation is due purely to the transfer of orbital angular momentum. The first observation of the transfer of orbital angular momentum to a particle was made using an 1 = 3 helical beam within optical tweezers to trap absorbing ceramic powder suspended in kerosene (He, Friese, Heckenberg and Rubinsztein-Dunlop [ 19951). However, the 100% absorption of those particles resulted in a force in the propagation direction of the laser beam and instead of being held at the beam focus, these particles were trapped against the microscope slide. The rotation of the ceramic particles was attributed to the absorption of the orbital angular momentum from the light beam. With particles which absorb only a few percent of the incident laser light, the gradient force can be sufficient to overcome the force associated with the radiation pressure. For partially absorbing teflon particles suspended in alcohol, it is possible to observe rotation while still forming a genuine 3-dimensional optical trap (Padgett and Allen [ 1995b], Simpson, Dholakia, Allen and Padgett [ 19971). The trapped Teflon particles were observed to rotate at several Hertz; hence the term “optical spanner”. Not only does this eliminate any doubts over the origin of the observed rotation, but it also creates a potentially useful manipulative tool. In principle, the rotation speed could be compared with predictions made on the basis of the estimated absorption of the light and the viscous drag of the surrounding fluid. However, the errors inherent in these measurements make quantification of the orbital angular momentum extremely difficult. However, the orbital angular momentum can be compared directly to the spin angular momentum of h per photon. With a circularly polarized, I = 1, LG mode the handedness of the polarization can be set to give a total angular momentum
326
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
Orbitalspin
Orbital
,
[n! 5 9
Orbital t spin
,
0 ms
40 ms
80 ms
v
120ms No rotation
Rotation
I
Fast
rotation
Fig. 10. Successive frames of a video showing that the spin and orbital angular momentum terms can be added to give faster rotation, or subtracted to give no rotation, in an optical spanner.
of h+-h=2h or h - h = O per photon. The observed start/stop nature of the rotation, which is shown in fig. 10, confirms that the orbital angular momentum associated with an 1 = 1 LG mode is h per photon (Simpson, Dholakia, Allen and Padgett [1997]). This experiment confirms that both spin and orbital angular momentum of light are transferred to a particle in a mechanically equivalent fashion.
8
9. Rotational Frequency Shift
As discussed in 6 5 , a n/2 mode converter introduces orbital angular momentum into the light beam. This is analogous to a quarter-wave plate introducing spin angular momentum to plane polarized light. Similarly a n mode converter is analogous to a half-wave plate and they reverse the handedness of the orbital and spin angular momentum, respectively. A rotating half-wave plate was shown, some 20 years ago, to shift the frequency of a circularly polarized light beam by twice the rotation frequency of the wave plate (Garetz and Arnold [ 19791). A simple polarization analysis shows
IV, Q 91
ROTATIONAL FREQUENCY S H F T
327
that this is due to a corresponding rotation of the electric field vectors at twice the rotation frequency of the wave plate. If the beam has a circularly symmetric intensity distribution, such a rotation of the electric field vector is equivalent to a rotation of the beam. Consequently, this frequency shift can be considered to arise from a rotation of a light beam which possesses angular momentum and is equal to the rotation frequency of the beam multiplied by the angular momentum per photon in units of fi. Recently, a similar behavior was predicted for a Laguerre-Gaussian beam (Nienhuis [ 19961). Although analyzed in terms of an energy exchange, a rotating n mode converter was shown to introduce a frequency shift of twice the rotation frequency. It has been shown (Courtial, Dholakia, Robertson, Allen and Padgett [ 19981) that both of these effects are examples of the recently highlighted rotational frequency shift (BialynickiABirula and Bialynicka-Birula [19971) or angular Doppler shift (Garetz and Arnold 119791). This shift should not be confused with the translational Doppler shift observed for rotating objects, which is due to the rotation having a linear velocity with respect to the observer. Unlike the translational Doppler shift which is maximal in the plane of rotation, this rotational effect is maximal in the direction of the angular velocity vector, where the linear Doppler shift is zero. The translational Doppler shift is equal to the linear velocity between source and observer multiplied by the linear momentum per photon. By contrast, the rotational Doppler shift is equal to the rotational velocity between source and observer multiplied by the angular momentum per photon. The rotation of a source or detector without the introduction of slight off-axis motion is difficult to achieve. But a rotating Dove prism and half-wave plate can be combined to simultaneously rotate the electric field vector and phase structure of the beam. This provides beam rotation while the source remains stationary. Both the spin and orbital angular momentum of the photon contribute in an additive and interchangeable way to the rotational Doppler shift. For circularly polarized Laguerre-Gaussian modes, the origin of this result is particularly simple to understand by examining the transverse form of the electric field; see fig. 11. In each case the field distribution is (I + a)-fold rotationally symmetric. For circularly polarized light, the electric field rotates at the optical frequency and a rotation of the beam at f2 introduces an additional (I + a ) phase cycles per revolution. The sense of this rotation relative to the circular polarization results in an up-shift or down-shift in frequency of (I + a)B. This shift has been measured directly using a highly accurate frequency counter and a “light” source in the mm-wave region of the spectrum (Courtial, Robertson, Dholakia,
328
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
a=-I
0=+1
I= 1
1=2
I= 3
Fig. 11. Vector plots of the transverse electric field for circularly polarised LG beams, showing the (I + a)-fold rotational symmetry.
Allen and Padgett [1998]). Any arbitrary field distribution can be expressed as a superposition of circularly polarized Laguerre-Gaussian modes and if rotated will therefore give rise to a frequency spectrum consisting of sidebands about the unshifted frequency. In a similar fashion to the optical spanner, we find that both the spin and orbital components of angular momentum act in an equivalent fashion. $j 10. Atoms and the Orbital Angular Momentum of Light
It is well known that the interaction of conventional laser light with a free
rv, §
101
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
329
atom can give rise to electromagnetic pressure forces which act on its center of mass. Such forces have been the subject of much investigation in both theory and experiment (Ashkin [1970a,b], Letokhov and Minogin [19871, Kazantsev, Surdutovitch and Yakovlev [1990], Arimondo, Phillips and Strumia [ 19921, Metcalf and van der Straten [1994], Adams and Riis [1997]). The basic features can be understood in terms of a simple model comprising a two-level atom subject to a plane electromagnetic wave which gives rise to two kinds of force. These are a dissipative force arising from the absorption of the light by the atom, followed by its spontaneous emission, and a reactive, or dipole, force arising from the non-uniformity of the field distribution. These forces underpin the manipulation of atoms by lasers in a variety of beam configurations. The dissipative force has been exploited in the Doppler cooling of the atomic motion (Wineland and Dehmelt [1975], Hansch and Schawlow [1975]) and the dipole force used for trapping (Chu, Bjorkholm, Ashkin and Cable [1986]). It appears likely that the interaction of atoms with beams possessing orbital angular momentum should lead to new effects. Theoretical studies of the interaction of LG beams with atoms have been conducted recently (Babiker, Power and Allen [ 19941, Allen, Bablker and Power [ 19941, Power, Allen, Babiker and Lembessis [ 19951, Allen, Lembessis and Babiker [ 19961, Allen, Babiker, Lai and Lembessis [1996], Lai, Babiker and Allen [1997], van Enk [1994], Masalov [1997], Kuga, Torii, Shiokawa and Hirano [1997], Wright, Jessen and Lapeyere [19961) to examine how the main features of Doppler cooling and trapping are modified when a plane wave or a fundamental Gaussian beam is replaced with LG light. It is also desirable to consider the role LG beams are likely to play in the emerging field of atom optics (Dowling and Gea-Banacloche [ 19961). To study the effects of the orbital angular momentum of light on atoms it is necessary to consider the theory of forces due to LG light and their effects on a two-level atom. This has been done by Allen, Babiker, Lai and Lembessis [ 19961, who also extended their investigations on the orbital angular momentum effects to more than one beam, in order to explore more fully the effects of the orbital angular momentum on atomic motion. The theory has been developed in terms of the optical Bloch equations (OBE) (Cook [ 19791, Letokhov and Minogin [ 19871, Dalibard and Cohen-Tannoudji [1985]) which allow the ab initio inclusion of relaxation effects and incorporate saturation phenomena naturally. The solution of the OBE in the adiabatic, or constant-velocity, approximation gave insight into the time evolution of angular momentum effects for an atom in an LG beam. It is useful to outline a derivation of the force acting on a two-level atom in the presence of monochromatic coherent light beam based on the density matrix formalism. The coherent light beam is assumed to have a complex amplitude and
330
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!§
10
a Laguerre-Gaussian (LG) spatial distribution. The Hamiltonian of the system is
where HA and HF are the zero-order Hamiltonians for the unperturbed atom and field, respectively, and are explicitly given by:
P2 HA = -+ ACIIOJC~JC, 2M HF
= hoata.
(10.2) (10.3)
Here A4 is the atomic mass, P is the momentum of the center-of-mass, and nt and n are the lowering and raising operators of the internal states of the atom with wo the transition frequendy. In eq. (10.3), a and at are the annihilation and creation operator$ of the light of frequency u.The coupling of the atom to the electromagnetic field is given in the electric dipole approximation by:
Hint = -d . E(R),
(10.4)
where E(R) is the electric field evaluated at the position R of the atom. d is the atomic electric dipole moment operator, which may be written as (10.5)
d=D,2(n+d),
with 0 1 2 the dipole matrix element. The electric field for a Laguerre-Gaussian mode propagating along the z-axis can be written as E(R) = i [uZ &kb(R)eioWp@) - h.c.1 ,
(1 0.6)
where 3 is the mode polarization vector and &klp(R)and Okb(R) are, respectively, the mode amplitude function and phase function which are explicitly given by Beijersbergen, Allen, van der Veen and Woerdman [1993] and, fiom eq. (2.19),
k 2 Z
@k[p(R)=
q.2
+
z;)
+ Z@ + (2p + Z + 1) arctan
(10.8)
d p m
Here EkOO is the amplitude for a plane wave of wavevector k;Cb = is a normalization factor; and w(z) is defined in terms of the Rayleigh range ZR
w 6 101
ATOMS AND THE ORBlTAL ANGULAR MOMENTUM OF LIGHT
33 1
by w2(z) = 2(z2 + z i ) / , ~ The . integers 1 and p are indices characterizing the LG mode, as described in 4 5. The time evolution of the system may be determined by transforming to the interaction picture governed by the unperturbed field Hamiltonian hat,. In this picture the field operator a(t) is time-dependent, with dependence given by
a(t) = exp(iwa+at)aexp(-iwatat)
= ae-lWf,
(10.9)
with a similar equation for at(t). In the classical limit in which the field forms a coherent beam, we may replace the field operators by c-numbers: a(t)4 cre-lw';
at(t) --f a*eIwt.
(10.10)
The corresponding interaction Hamiltonian is thus given by
Hint = -d .E(R) = -ifi [$+af(R)
- h.c.1 ,
(10.11)
where we have made use of the rotating-wave approximation and introduced 5 andf(R) by jt = ZelWt
(10.12)
(10.13) We can make use of the semiclassical approximation by replacing the position and momentum operators R and P by their expectation values Ro and PO, respectively, while maintaining a quantum treatment for the internal dynamics of the atom. The validity of the semiclassical approximation requires that the spatial extent of the atomic wavepacket be much smaller than the wavelength of the radiation field and that the uncertainty in the Doppler shift be much smaller than the upper-state linewidth of the atom. This is the case for most atoms (Letokhov and Minogin [19871) if the recoil energy of the atom is much smaller than the upper-state linewidth. Within the semi-classical approximation, the atomic density matrix can be written as
P = W R - Ro)
w - PO)P(t),
(10.14)
where now p(t) contains the internal dynamics of the atom and the evolution of p(t) is in accordance with the well-known relation (10.15)
332
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
"5
10
where the relaxation dynamics enters via the term Rp. Substituting for H and using eq. (I 0.1 I), we obtain the following equations for the atomic density matrix elements: (10.16)
dP21 - -(r - iAo)P21 + af(Rol6-722 - p1 I ), dt
--
(1 0.1 7)
where A0 = o - wo is the detuning of the field frequency from atomic resonance and P21 = ( 5 ) .Equations (10.16) and (10.17) are the optical Bloch equations for the two-level system interacting with the light. The average radiation force acting on the atom can be shown to be given by
Substitution of eq. (10.1 1) into eq. (10.18) and the use of eq. (1 0.13) shows that the force can be written as the sum of two terms, (F)= (Fdiss) + (Fdipole). Here ( F d i s s ) is the dissipative force, given by (Fdiss) = -h\J@(RO)
and
(pdipole)
{ P12(t) af(RO>+P21(t> a*f*(RO)},
(10.19)
is the dipole force, given by (10.20)
where we have introduced a positionally dependent Rabi frequency as (10.21) In the adiabatic approximation (Cohen-Tannoudji, Dupont-Roc and Grynberg [ 1975]),the velocity of the atom, defined as V = Po/M, is assumed to be constant during the time taken for the dipole moment to relax to its steady-state value. The position Ro of the atom at time t is then given by:
Ro
= ro
+ Vt,
(10.22)
where ro is the initial position of the atom when the beam was switched on. Thus,
N,§ 101
333
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
-f(rO) exp[iV@(ro) . Vt] ,
(10.24)
where we have assumed that the change in the field amplitude is negligible during the time taken for the dipole moment to relax to its steady-state value. Within the adiabatic approximation, the optical Bloch equations become dP22 dt
-- -- 2 m 2
dfi21 dt
~
-
0lf(ro)fil2-
= -[r-iA(ro,
V)lfi21 +af(ro)(p22 - P I I ) ,
where A is the total detuning, A(r0, V ) = AO - VO(r0) . V , and p21 exp[-itV . VO(ro)].The forces can now be written as
(10.25) (10.26) b21
=
(10.28) For given initial conditions, the solution of the optical Bloch equations (10.25) and (10.26) leads formally to the determination of the forces by direct substitution in eqs. (10.27) and (10.28). Torrey [ 19491 gave detailed solutions of the original Bloch equations. He also recognized that there were three special cases of interest which have relatively simple solutions. These were for strong collisions when the natural lifetime of the state may be replaced by the collision-shortened lifetime, for exact resonance, and for intense external fields. His approach has been applied (see Allen and Eberly [ 19751) to the optical Bloch equations. Consequently, the evolution of the forces from the instant the light beam is switched on can also be examined for a number of special cases. Such effects have been examined in detail for atoms excited by plane-wave light (Al-Hilfy and Loudon [ 19851) where the cases considered were (i) an atom with all relaxation constants equal to zero; (ii) a weak beam; (iii) exact resonance and (iv) steady state achieved by an intense field. This treatment may be readily generalized for Laguerre-Gaussian light. We shall settle simply for the consideration of the steady-state case because the general time dependence of the density matrix elements for arbitrary parameter values can be determined more readily by the numerical solution of the optical Bloch equations (10.25) and (10.26). This enables the evolution of the corresponding forces to be displayed. We display the results for a LaguerreGaussian mode with I = 1 andp = 0 such that Q(r0) = r;A0 = -r and w(0)= 3 5 4
334
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
0
2
4
6
0
10
6
8
10
rt
0
2
4
rt Fig. 12. Variation with time of (a) the average dissipative force, and (b) the average dipole force for a stationary atom in a single LG beam. The time variation of the corresponding torque would be the same as in (a). The time is in units of r-'.
where L=28Onm is the atomic transition wavelength. The results shown in figs. 12a and b depict the evolutions of the dissipative and dipole forces as given by eqs. (10.27) and (10.28), respectively. These figures show clearly that the dipole moment, and hence force components, relax to their steady-state values within a time of the order of r-'. The steady state corresponds to all time derivatives in the optical Bloch equations being set equal to zero, and coincides with the long time limit. It is straightforward to show that the steady-state solutions to the optical Bloch
n! §
101
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
335
equations (10.25) and, (10.26) give rise to the following expressions for the steady-state forces: ( F ) = (Fdiss) + (Fdipole)
Y
(10.29)
where (10.30)
where the notation is redefined, so that R now stands for the position of the atom and not ro. It is the proportionality to r that signifies the dissipative nature of the force in eq. (10.30) and its association with spontaneous emission. These results apply to an arbitrary field distribution, including the well-investigated plane wave case. Nevertheless, despite the generality of the results, it is possible to draw some conclusions about the characteristics of the forces. First, it is not difficult to see, by virtue of its proportionality to A and to VSZ, that the dipole force (10.31) would attract the atom to regions of intense field when the laser is tuned below resonance and repel the atom from these regions when tuned above resonance. It is this property of the reactive force that is frequently exploited in atom trapping experiments. The dissipative force in eq. (10.30), on the other hand, contains the factor AVO which corresponds to the momentum imparted by the light to the atom, which then reradiates spontaneously in a random direction. The probability of spontaneously emitting a photon in a given direction is the same as that for the opposite direction. The direction of absorption is well defined, so there is a net momentum change per absorbed photon of magnitude h IVO 1 when averaged over a large number of photons. As the maximum rate at which an atom may spontaneously emit photons is r, the maximum dissipative force on the atom is fir [DO!. It is instructive to consider first the familiar simple case of an atom interacting with a linearly polarized plane wave of wavevector k and polarization z. The expressions entering eqs. (10.1l), (10.13) and (10.21) appropriate for this case are: O(R) = k .R,
(10.32)
and (10.33)
336
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!5
10
where N is a plane-wave normalization factor. It is then readily shown that the phase factor in eq. (10.24) corresponds to the familiar Doppler shift, (10.34)
6=k.V.
The main influence of this effect is to change the detuning parameter from A O = O - W Oto A, where A = A 0 - 6 . The dipole force defined by eq. (10.31) for a plane wave is identically zero, which follows tivially from the fact that Vst=O. The dissipative force, eq. (10.30), on the other hand, can be written succinctly in the form (10.35) where I is a saturation parameter defined by I
=2
~ ~ / r ~ .
In the saturation limit corresponding to I --t the maximum dissipative force on the atom:
(F&)
= hkT.
(10.36) 00,
we obtain the usual result for
(10.37)
The dissipative force due to a plane wave produces zero torque on the atom about an axis parallel to the direction of propagation. This property stems from the uniformity of the plane wave which precludes the presence of non-axial forces on the atom. By contrast, as we pointed out at the outset and discuss in detail shortly, LG beams have non-trivial influence on the non-axial atomic motion. When explicit reference to a specific Laguerre-Gaussian mode is made, expressions (10.30) and (10.31) lead to the steady-state force on a moving atom due to a single Laguerre-Gaussian beam propagating along the positive z-axis in the form (10.38) where
rv, 8
337
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
101
and
where R(t) now denotes the current position vector of the atom and V = W d t . The total detuning dklp(R, V ) is both position- and velocity-dependent: dklp(R, V ) = do - I/. V@+(R, V ) .
(10.41)
The evaluation of v@k[p from eq. (10.8) can be carried out in cylindrical coordinates. We find
Substitution for VOkb yields straightforwardly the Doppler shift 6 = VOk, in cylindrical coordinates:
6=
-
(-)+ ( kr2+ krZ z;
22
-
.V
vr-T1v4 vr-7
[ 4 - z 2 ] + ( 2 p + l + 1)zR + k) vz, 2 2 +z; 2(z2 z;) 2 2 + 2;
(10.43)
where V,, V4 and V, are the radial, azimuthal and axial components of the atomic velocity, respectively. The Doppler shift divides naturally into four types of contribution: an axial contribution along the z-direction, a contribution due to the Gouy phase, a contribution due to the beam curvature, and finally an azimuthal contribution. We may write:
6 = &xial+
&ouy
+ &rve + dazimuth.
(10.44)
The axial component simply corresponds to a Doppler shift that would arise from a plane wave traveling along the beam axis, &axial = kVz.
(10.45)
This is normally the dominant shift, provided the atom has a substantial velocity component along the beam axis. The shift caused by the Gouy phase is: (10.46)
338
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!8
10
It is easily seen that, as typically ZR >> W O ,the Gouy shift is very small for practically all Laguerre-Gaussian beams. The shift arising from the beam curvature is given by ~CU,",
=
(-)
krZ
22
+z;
vr+
(
kr2
2(z2
+ z;)
[
Z 22
q
+ z;
)
v,.
(10.47)
This is a sum of contributions due to the spreading of the beam in the radial and axial directions. These contributions are well understood. They have the same origin as the corresponding shifts in conventional (0,O) mode Gaussian beams and arise from the curvature of the wavefront. They may, in certain circumstances, have observable consequences. Finally, the azimuthal Doppler shift is: (10.48) The important feature of this azimuthal shlft is that it is directly proportional to the orbital angular momentum quantum number 1 of the Laguerre-Gaussian mode; it occurs for motion which is azimuthal to the overall beam propagation. Further insight into the physical meaning of the azimuthal Doppler shift can be gained by equating ZA to the z-component of orbital angular momentum of the LG beam. We have,
lh = ( r x P ) =~ rpb,
(10.49)
where p is the beam linear momentum, formally given by eq. (2.20), and p$ is its azimuthal component, which can be written as
where k@(r)is the local azimuthal component of the wavevector. Equation (10.48) can now be written as (10.51) Thus the azimuthal Doppler shft has the same form as the usual translational Doppler shift, but the shift is now due to motion at right angles to the beam direction. It is the azimuthal component of the spiraling Poynting vector that produces this frequency-independent shift and is an example of the rotational ffequency shift (Bialynicki-Birula and Bialynicka-Birula [ 1997]),
n! §
101
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
339
discussed W h e r in Q 14. The azimuthal Doppler shift is a potentially observable characteristic of the internal motion of the atom arising from its interaction with the LG beam. We shall see that there is also a light-induced torque associated with the forces acting on the atom due to such beams. This, by contrast, is a measurable property of the gross motion of the atom. A different approach to the distinction between internal and gross motion in atoms due to fields possessing orbital angular momentum is also possible (van Enk and Nienhuis [ 19921). Substitution of V@lp in the expression for the dissipative force shows that, in general, there are non-zero force components in all three directions (?,$,?) of the cylindrical coordinates. In particular, a significant contribution arises in the form of an azimuthal component. This is responsible for a non-vanishing torque around the beam direction, given by (10.52) where FZissis the @-component of the dissipative force. This torque has a magnitude that can be written explicitly in the form (10.53) In the saturation limit I + 00, we obtain l(T)klp/
(10.54)
This result is as remarkably simple as the plane wave saturation force in eq. (10.37). We have seen that for the case of a single Laguerrffiaussian beam the forces are modified relative to the usual case of plane wave linearly polarized light. In particular, an azimuthal component of the dissipative, or radiation pressure, force exists which in the steady state leads to a non-zero torque acting on the atom around the beam axis. In order to elucidate fbrther the nature of the interaction between the LG beam and the atom, we consider analytically the low-velocity limit of the dissipative and dipole forces. In the computational evaluation of the dynamics of the atom, this approximation need not be made. The assumption involved in the lowvelocity limit is that the Doppler shift induced by the motion of the atom is smaller than the atomic width, V 00 << r. In this case, the denominators of eqs. (10.39) and (10.40) may be expanded, retaining terms up to those linear in
340
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[IV, 8 10
the velocity. We can thus write each force as the sum of static and dynamic, or velocity-dependent, components. The static components are given by:
(10.55) (10.56)
(10.57) The dynamic components are given by
(10.58)
(10.59)
(10.60) (10.61) We make the additional assumption that the atom moves in a region of the beam for which z << ZR and we can then ignore the z-dependence in Qklp(R)and set qklP(R)= 0 and &lp(R)= 0. We may also write to a good approximation:
In the low-velocity limit with z << zR, the static dissipative and dipole forces become (10.63)
n! 5
101
ATOMS AND THE ORBITAL ANGULAR MOMENTUM OF LIGHT
34 1
whilst the dynamic dissipative and dipole forces become
Equations (10.63) and (10.65) show that the dissipative force has static components in both the axial and azimuthal directions; the latter is equivalent to a torque about the beam axis. These forces combine with dynamic components in the axial and azimuthal directions. Note that within this approximation, eq. (10.65) shows that there is a reciprocal relationship between the axial and azimuthal motions. An atom moving initially in the z-direction will induce a force in the azimuthal direction and vice versa. It may be seen from eqs. (10.64) and (10.66) that the dipole force consists of static and dynamic components, both of which are in the radial direction. The static component of the dipole force, given by eq. (10.56), attracts the atom to the high-intensity regions of the field when the detuning is below resonance. This force can be derived from a potential (10.67)
such that (F&) = -V (U(R))kb.This potential exhibits minima in the highintensity regions of the beam for an atom tuned below resonance where A0 < 0. For A0 > 0, we have trapping in the low-intensity, or dark, regions of the field. As an illustration, we consider the LG mode for which 1= 1, p = 0. The potential is (10.68)
342
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
",
5 11
At the beam waist, z = 0, the miaimum occurs at r = ro, where WO
(10.69)
r o = 75'
For a beam propagating along the z-axis it is easy to verify that the locus of the potential minimum in the x-y plane is a circle given by (10.70) Expanding the potential in powers of (r - ro) we have the parabolic approximation (10.71) where UOis the potential depth given by (10.72) and
is an effective elastic constant given by (10.73)
The atom is considered trapped if its kinetic energy is less than U O and will exhtbit a quasi-harmonic vibrational motion about r = ro. The characteristic angular frequency is equal to where A4 is the atomic mass. These features are illustrated in Q 12 for a magnesium ion.
Jm,
$j 11. Atoms and Multiple Laguerre-Gaussian Beam Configurations
We have seen that an atom interacting with a Laguerre-Gaussian beam will experience a dissipative force that is predominantly in the direction of propagation and a dipole force in the radial direction. If a second beam propagating in the opposite direction is added, we have a 1D counter-propagating beam configuration. The beams are assumed to be independent of each other; their phases are not locked.
n! 5
111
ATOMS AND MULTIPLE LAGUERRE-GAUSSIAN BEAM CONFIGURATIONS
343
For independent counter-propagating LG beams we can write the mean force on the atom as a sum of forces due to individual beams,
(11.1)
(11.2)
where we have assumed that P I =p2 = p and either I1 =-Z2 = 1 or I , = 12 = 1. In the low-velocity regime, for an atom close to the beam waist, we may make use of eqs. (10.63)-(10.66). The total static dissipative and dipole forces are then given by
(11.3)
and the total dynamic dissipative and dipole forces are
(11.5)
344
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
tn!
§ 11
(11.6) From eq. (I 1.4) we see that the velocity-independent dipole force is simply double that of the single beam case. The dissipative force, however,' depends on the relative signs of ZI and 12. For ZI ' 1 2 = I , we have from eq. (11.3): (11.7)
(11.8)
Thus for ZI = Z2 = I , we have a torque about the beam axis and an axial cooling or heating force, depending on the sign of do. For the case 11 = -12 = -1, we have
(11.10) The static force in this case is zero whilst the velocity-dependent force contains extra terms which arise from the orbital angular momentum of the counterpropagating Laguerre-Gaussian beams. As with the one-beam case, we again see force components arising from the reciprocating interchange between the axial and azimuthal motions. The 2D configurations arise when a second pair of counter-propagating LG beams is arranged orthogonal to the first. The total force can again be written as a sum of forces from each of the beams. However, in addition to the reciprocity between the azimuthal and axial motions in each pair of beams, the azimuthal atomic motion associated with one beam is part of the axial motion in the other. In other words, there is an additional level of reciprocity between the components of the motion arising from the presence of two pairs of counter-propagating beams.
n! §
MOTION OF Mg' IN MULTIPLE BEAM CONFIGURATIONS
121
345
There are also two overlapping dipole potential distributions arising from the orthogonal beams. It is easy to see for beam pairs for which 11 =-I2 and p1 =p2 = 0, and where the axes are such that one pair is along the z-axis and the second along the x-axis, the potential minima are four times as deep as that of a single beam. The minima are situated at the space points defined by the two equations (11.11) (11.12) These equations apply the additional constraint x = f z . Atoms subject to such 2D counter-propagating beams will congregate at points lying on the curve defined by two intersecting circles, one on the plane x + z = 0 and the other on the plane x-z=o. When a third set of beams is arranged orthogonal to the other two pairs, we have 3D counter-propagating LG beams. The common potential minima in this case occur at eight distinct points defined by (11.13) and they are six times as deep as the potential due to one beam. However, in this case the detailed polarization gradients are such as to make further study of this configuration non-trivial. Another interesting case is that of three coplanar beams (Grynberg, Lunis, Verkerk, Courtis and Salomon [1993]) in a symmetric configuration in the x-y plane such that the angle between adjacent beams is 2x13. This leads to three overlapping circles which meet at two distinct points at x=o;
y=o;
WO
z=*z.
(11.14)
The potential well is then three times as deep as for a single beam.
9
12. Motion of Mg+ in Multiple Beam Configurations
In the theoretical analysis presented above we were able to infer that an atom in a configuration of several beams is subject to axial forces and various forms of static and dynamic rotational forces and that axial and rotational
346
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[n!5
12
motions influence each other in, a rather intricate way, Furthermore, a system of multiple Laguerre-Gaussian beams presents an atom with well-defined potential landscapes which depend on the angular momentum quantum numbers and beam configuration. For example, in the 1D case for a given set of parameters, a given atom should have well-defined quasi-harmonic vibrational states associated with the potential profiles. To illustrate these features we consider the case of Mg+ in LaguerreGaussian light. The Mg+ mass is M = 4.0 x 10-26kg; the transition wavelength is A.= 280.1 nm and its halfwidth is r = 2.7 x 10' s-' . To illustrate the theory, typical beam parameters are exemplified by the choices A o = - ~ ;&i?kOo= 1.648r and wo = 35h. The equation of motion of a Mg' ion in multiple LG beams is written as
(12.1) where Q is the ionic charge and we have included the last term on the righthand side to allow for the possibility of an applied external magnetic field. The summation indicates the vector addition over force contributions arising from individual beams. The forces from each beam are taken in their unapproximated forms given by eqs. (10.39) and (10.40). First consider 1D counter-propagating beams in the absence of a magnetic field. Figure 13a displays the dipole force as given by eq. (1 1.2) as a function of radial distance r at y = 0 and z = 0. The beam quantum numbers are such that 11 = -12 = 1 andpl = p2 = 0, and the parameters are A0 =-r,QkOO = 1.648r and w o= 35h. Figure 13b displays the corresponding radial potential distribution. The maximum intensity is located at points where r = W O / =~24.75A. As expected, we see that for A0 < 0 the dipole potential exhibits a minimum at points where the intensity is maximum. The vibrational states in the parabolic approximation have an elastic constant whch is twice that for the one-beam case as given by eq. (10.73). The vibrational frequency corresponding to the above parameters is (12.2) For the parameter values specified above, t h s yields Y = 2 . 0 1~0 4 r . Figure 14a displays the trajectory of the ion as a function of time and fig. 14b depicts its projection onto the x-y plane. The initial position is at Ro = 8 A 6 and the initial velocity components are V, = 5.0ms-', V, = 0 = V,. It is clear from
w § 121
347
MOTION OF Mg+ W MULTIPLE BEAM CONFIGURATIONS
10
I
0
,
1
25
50
75
100
r/A
L
-0.3
ci
1 b L
-0.6
-0.9 L 0
I
1
I
25
50
75
100
?-/A Fig. 13. (a) Radial distributionof the dipole force due to 1D counter-propagating LG beams at z=O. Here ZI = 4, = 1; pl = p 2 = 0 and the parameters are A0 = -r, sZko0 = 1.648r and wo = 35A. (b) The radial potential distribution corresponding to (a).
the figure that the atom, subject to an axial friction force, has been slowed axially. Once the atom is moving sufficiently slowly, it starts a vibrational motion about the radial coordinate r = wo/fi, accompanied by a slow rotational motion. The latter, according to eq. (10.69, is attributable to the azimuthal component of the dissipative force induced by the axial motion. Figure 15 displays the evolution of the velocity components. The axial velocity is seen to decay almost to zero. However, close inspection, as shown by the inset to fig. 15a, reveals that the axial motion exhibits periodic oscillations which are attributable to a reciprocating force arising from the periodic azimuthal motion, depicted in fig. 15b. The frequency associated with these figures is
348
i n k U K U I I A L ANUULAK M U M E N I U M
ur
LIUHI
50
25
<3 o -25
-50 -50
-25
0
25
50
x/h Fig. 14. (a) The trajectory of a Mg+ ion in 1D counter-propagatingLG beams with I , = -12 = 1 and = p2 = 0. The initial position is Ro = 812 and the initial velocity components are Vz = 5.0m s-l ; V, = 0 = Ve. (b) The projection in the x-y plane of the ion trajectory shown in (a). In this and subsequent figures, the initial position is indicated by a solid circle.
349
MOTION OF Mg' IN MULTIPLE BEAM CONFIGURATIONS
1.00
0
.-
2
4
6
8
1
0
-0.02 -0.04 -0.06
-0.08 -0.10
0
2
4
6
8
*104
rt 7
10
1.0
: 0.5
0.0 -0.5 -1.0
I
I
1
0
2
4
1
1
1
6
8
10
rt Fig. 15. Variations of the velocity components for the case displayed in fig. 11: (a) Evolution of V, indicating axial cooling; the inset to this figure shows small oscillations o f V, due to reciprocating effects. (b) Evolution of VQ. (c) Evolution of V,. Both (b) and (c) indicate the rapid onset of oscillatory motions of the same period.
350
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
tN 6
12
indeed about 2.0x104f as in the estimate based on eq. (12.2). An important feature displayed by the results depicted in figs. 14 and 15 is that changing the sign of the angular momentum quantum nuinber 1 from +1 to -1, which is readily achevable, causes the change in the rotational motion from clockwise to the opposite, anticlockwise, sense. Figure 16 represents the case 11 = 12 = +1 and pl =p2 = 0 in the presence of a magnetic field IBI = 1 T directed along the positive z-axis. From eqs. (1 1.2) and (1 1.3) we deduce that, besides the ion cyclotron motion due the magnetic field, the main effects are in the form of an axial friction force provided that A0 < 0 and a static torque about the beam axis which acts on the ion azimuthally. Figure 16a displays the evolution of velocity components for the case A0 = - f . The initial ion position is R(0) = - 8 A j and the initial velocity components are V, = 0, V, = -8.9ms-', V, = 5.0ms-'. We see that the torque due to the LG beams generates a braking effect on the cyclotron motion while the axial motion is gradually cooled by the axial friction force. All these features can be inferred from the trajectory shown in the inset to fig. 16a. If the sign of 1 in both beams were to be changed, but B kept in the same direction, we would have heating of the azimuthal motion while the axial motion would still be cooled. The results are depicted in fig. 16b. Clearly the former case amounts to a decrease in angular motion due to the LG beam, whle the latter is equivalent to the enhancement of the w l a r motion. These phenomena are attributable only to the angular momentum properties of the LG beams (Babiker, Lembessis, Lai and Allen [ 19961). Figure 17 is concerned with counter-propagating beams with the next hgherorder LG modes, 11 =-12 = 1 and p1 =p2 = 1, with no external magnetic field. Other parameter values are &OO = 1.648 and A0 = - f . Figures 17a and b display the radial distribution of the dipole force and corresponding potential, respectively. There are now two potential wells with minima at r=0.468wo = 16.38A = 52.86A. The ion oscillates about one of these points, dependmg and r = 1SWO on the initial conditions. Th~scan be seen in fig. 18 for an atom with V, = 0 = V, and Y, = 5.0 m s-' . The inner curve depicts the trajectory when the atom begins at Ro = lOAP and the outer curve when it begins at Ro = 40Aj. Figure 19a shows the trajectory in the 2D counter-propagating beam case with 11 = -12 = 1, p1 =p2 = 0 and fig. 19b shows the corresponding projection in the x-y plane. The initial position is at Ro = 1012 and the initial velocity components are (0.02,0.02,0.05) ms-' . Figure 20 shows the evolution of the velocity components. From figs. 19 and 20 it can be seen that the atom is subject to hction forces from all directions which result in it coming to rest at a point withm the potential profile. We have shown earlier that the locus of the dipole
Iv,
0
35 1
MOTION OF Mg+ IN MULTIPLE BEAM CONFIGURATlONS
121
10
5 h 4
I
-10 0
I
I
I
I
2
4
6
8
%/r
10
*lo3
10
-20
8 h 1
-15
I
-E"
6
h .-(
I
-10 a"
0 0
2
4
6
8
-5 10
Fig. 16. (a) Evolution ofthe velocity components of a Mg+ ion subject to LG ID counter-propagating beams with 11 = 12 = 1 and pl = p2 = 0 and IB\ = 1 T. Initially the ion possesses both azimuthal and axial velocity components V, = 5.0 m s-l, V, = -8.9 m s-'. The trajectory of the Mg' ion is shown in the inset to this figure. (b) As in (a), but with 11 = 12 = -1.
352
[Di: 0
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
12
10
5 0
-5 - 10
0
25
50
75
100
r/X
-0.9
'
0
I
I
I
25
50
75
100
r/h Fig. 17. (a) Radial distributionof the dipole force due to ID counter-propagating LG beams at z=O. Here I1 = -12 = 1; pi = p2 = 1, and the parameters are A0 = Q ~ O O = 1.648r and wo = 35h. @) The radial potential distribution corresponding to (a).
-c
potential minimum for the 2D case with (1,O) beams is in the form of two intersecting circles, satisfymg eqs. (1 1.11) and (1 1.12). This is shown in fig. 21 for the case w o= 35A. The trajectory endpoint for the case depicted in figs. 19 and 20 lies on the curve shown in fig. 21. Thus for given initial conditions this theory assigns predetermined end points. That this is the case can be seen from table 1, where the coordmates of the trajectory endpoint recorded for various starting points satisfy eqs. (11.11) and (11.12) for the two intersecting circles shown in fig. 2 1.
n! 5
353
MOTION OF Mg+ IN MULTIPLE BEAM CONFIGURATIONS
121
80
40
<;h o -40
-80 -80
-40
0
40
80
s/h Fig. 18. (a) Trajectories of a Mg' ion in 1D counter-propagating LG beams, for I I = -12 = 1 and p1 = p 2 = 1 and two different initial positions; one at Ro = 10Af and the other at Ro = 40AE.The initial velocity components in both cases are V, = 5.0ms-l; V, = 0 = Ve. (b) The projections in the x-y plane of the ion trajectories shown in (a).
354
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
50
25
-50 -50
-25
0
25
50
x/h Fig. 19. (a) Trajectory of a Mg' ion in 2D counter-propagating LG beams involving two orthogonal pairs; one pair has the z-axis as a common axis with 11 = -12 = 1 and pl = p2 = 0; the second pair has the y-axis as a common axis with 13 = -14 = 1 and p3 = p4 = 0. The initial position of the Mg' ion is at Ro = 20L2 and the initial velocity components are (0.02,0.02,0.05) m s-' . (b) Projection in the x-y plane of the trajectory in (a).
rv, 0
355
MOTION OF Mg+ IN MULTIPLE BEAM CONFIGURATIONS
121
7m E
0.10 0.05
0.00
bN
-0.05
-0.10
0
5
10
15
2(
*104
rt 0.5
4
0.3 0.1 1” -0.1
-0.3 -0.5
0
5
10
15
rt 7
i
20
+104
0.5 0.3 0.1
Lh
-0.1
-0.3 -0.5
0
5
10
rt
15
20
*104
Fig. 20. Evolution of the velocity components for an initial position Ro = 101f: (a) V,(t); (b) V e ( f ) ; (c) V,(t). Note that all components of velocity go to zero at sufficiently long times.
356
[IV, 8 13
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
Fig. 21. Locus of spatial points where the dipole potential profile due to two orthogonal sets of counter-propagating LG beams possesses the lowest minimum. Table 1 Coordinates xf,yf,qof the trajectory endpoints against initial coordinates xo,yo,zo for an Mg' ion for two orthogonal sets of counter-propagating LG beamsa
20
0
0
15.96
18.92
15.96
24.75
24.75
24.75
20
20
0
2.84
24.61
2.84
24.75
24.15
24.75
-20
20
10
-14.27
20.22
14.27
24.75
24.75
24.75
-20
20
-13.81
20.54
24.75
24.75
24.75
-10
-13.81
All distances are in units of the wavelength A. The last three columns demonstrate that the ion endpoints always lie on the two intersecting circles shown in fig. 21.
a
0
13. Atoms and Circularly Polarized Light
The one-dimensional case of two counter-propagating beams with the same linear polarization is one of the simplest and most investigated configurations. In this configuration, the radiation pressure forces from the two beams provide a frictional force to cool the atom, while the dipole forces provide a trapping force near the standing-wave maxima. This scheme, however, has been shown to lead to a prohibitive diffusion of the atomic momentum, which arises from the coherent re-distribution of photons between the two beams (Gordon and Ashkin [ 19801). To overcome the problem, Dalibard, Reynaud and Cohen-Tannoudji [ 19841 proposed a laser configuration consisting of two counter-propagating beams with opposite circular polarizations. In this scheme, the conservation of spin angular momentum forbids the coherent re-distribution of photons between the two beams for an atom with a J = O ground state and a J = 1 excited state. The approach was significantly developed subsequently (Dalibard and CohenTannoudji [1989], Ungar,Weiss, Riis and Chu [1989]) and applied to atoms with
n! 8
131
ATOMS AND CIRCULARLY POLARIZED LIGHT
357
a ground state value of J > 0. The restricted minimum temperature predicted by the Doppler-cooling limit had been found to be violated (Lett, Phillips, Rolston, Tanner, Watts and Westbrook [1989]), and this is now explicable in terms of the degeneracy of the levels involved. This development has led to o+-o_ beams becoming the dominant configuration for trapping and cooling. It was, for example, used with both traveling and standing waves in the first observation of Bose-Einstein cooling (Anderson, Ensher, Matthews, Wieman and Cornell [ 19951).We apply the scheme of Dalibard, Reynaud and Cohen-Tannoudji [ 19841 for a J = 0 to J = 1 transition to the case of two counter-propagating LaguerreGaussian beams with opposite circular polarizations. In this scheme, the interaction of the atom with laser light is treated classically, while the spontaneous decay of the atomic states is accounted for by the quantized vacuum modes of the electromagnetic field. By taking the trace over the quantized vacuum modes of the electromagnetic field, it may be shown that the evolution of the atom-laser system is governed by the following Liouville equation (Louise11 [ 19901): (13.1) where H is the atom-laser Hamiltonian and L is a relaxation operator. As previously, the atom-laser Hamiltonian consists of two parts, H
= HA+Hint,
(13.2)
where H A is the atomic Hamiltonian and Hintis the atom-laser coupling. For a J=O --t J = 1 atomic transition, the atomic Hamiltonian may be written as (13.3) where P is the atomic momentum. The states Ie,) = {le*), 10.) are the angular momentum states of the atom with transition frequency 00 and mass M . The coupling of the atom to the laser in the electric dipole approximation is Hint= -d .E(R, t), where now E(R, t ) is the total electric field evaluated at the position R of the atom. For counter-propagating Laguerre-Gaussian beams with O+ and u- polarizations, the total electric field is written as a sum of two parts corresponding to the u+ and u- LG fields:
358
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
tn! § 13
where
Ei(R,t)=--&klp(R) { ~ + e x p ( -[mt-@kllpl(R)]) +c.c.},
(13.4)
Ekhp~(R) E@, t) = -~ 21/2
(13.5)
2d5
{E-
exp(- [mt - @k12pz(R)])+ c.c.}.
In eqs. (13.4) and (13.3, c* are unit circular polarization vectors and &klp(R)and @k[p(R)are, respectively, the spatial field distribution and phase of the LaguerreGaussian mode with orbital angular momentum fh, d e h e d in eqs. (10.7) and (10.8). We restrict our attention to modes with indices 1211 = 1121 = 1 and PI =p2 =O. For such modes the field distribution and phase may be written as (13.6) (13.7)
d
m
where q ( z ) = max(l,.fl) is the distance of the peak intensity from the beam axis at a distance z from the beam waist and EO is a normalization constant. In eq. (13.7), we have neglected the small contribution due to the Gouy phase. Following Dalibard, Reynaud and Cohen-Tannoudji [19841, we introduce new unit polarization vectors defined by: (13.9a) (13.9b) The total electric field may be written in terms of these unit polarization vectors as
E(R, t)
=
ii&kr(R)[E’edwt- e’*eiW‘],
(13.10)
where we have used the condition that ) f , I = I l 2 J = 12). It should be noted here that the resulting field amplitude is fi larger than the individual beam amplitudes and that the spatial field distribution is the same as that of the individual beams.
n! 8
131
ATOMS AND CIRCULARLY POLARIZED LIGHT
359
This is in sharp contrast to the case of standing-wave linearly polarized modes, for which the spatial field distribution varies rapidly along the axis of the beams on the scale of a wavelength. A Laguerre-Gaussian mode defined by the orbital angular momentum index Z has a helical wave front making a full turn every ZA. This feature should be contrasted with another type of rotation associated with the polarization, or spin angular momentum, in the o + - c configuration. In this configuration the total wave polarization is everywhere linear and has a direction that rotates with a repeat distance A. We now introduce new basis states (Dalibard, Reynaud and Cohen-Tannoudji [ 19841) defined by
( 13.12)
In terms of these new basis states, the atom-laser coupling can be written in the rotating-wave approximation as Hint = i
f i ~ l ; [Qget ~ , ( ~-)Qe’g]
9
(13.13)
where QI(R)=D&(R)/~A, with D the dipole matrix element of the atomic transition, and Qge,= eiWt18) (e’1. We can now, as before, calculate the radiation forces within the semiclassical approximation using the definition
(F)= - (VHint) .
(13.14)
Substitution of eq. (13.13) into eq. (13.14) and use of eqs. (13.11) and (13.12), allows us to write the force as (F)= (Fdipole) + (Fdiss), where (Fdipole) is the dipole force given by
360
tIv,
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
8
13
r)
The evolution of the atomic,density matrix elements po = Q,i is governed by the optical Bloch equations, which are obtained with the help of the Liouville equation (13.1) and the atom-laser coupling (10.13). For 11=-12 =1, we find
where 6; = kE + (l/r)V$ is the Doppler shft of the atomic resonant frequency, V , and Vq are the axial and azimuthal components of the atomic velocity, r is half the atomic linewidth, and A0 = o - wo is the detuning of the laser frequency from resonance. For the other case, where 11 = 12 = I, we find
pgelf
where
=
$pget - [i (A0 - 14) + I']pgeff+ Q I ( R ) ~ ~ , ~ , , ,
4 = V,/r and 6: = kV, is the axial Doppler shift.
(13.25)
N, § 131
361
ATOMS AND CIRCULARLY POLARIZED LIGHT
For a stationary atom situated at the position Ro, the steady-state solutions to the optical Bloch equations are obtained readily. The atom is subject to a dipole force which can be written as
VS l+S’
1 2
(13.30)
(Fdipole) = - - ~ A o -
and a vanishing dissipative force for the case 1I = -12 = 1. In the case 11 = 12 = I, the atom experiences an azimuthal force, or equivalently a torque, similar to that for two linearly polarized beams: S l+S
(Fdiss) = firHere S = 2Q:(Ro)/ VS is given by
VS = 2 s
(!) 4.
(r2+ A;)
[ (! )’4 -
(13.3 1)
r
is defined as the local saturation parameter and
i.] -(I11
(2)
Z
+ 1) 2i. 2 +z;
(13.32)
For a negative detuning, (A0 < 0), the dipole force attracts the atom to the highintensity regions of the field distribution. T h s force can be derived from a potential of the form U(R)= ~ZZAOln(1 + S), where (Fdipole) = - ( V U ( R ) ) . For a moving atom, analytic solutions to the optical Bloch equations may be obtained to first order in the atomic velocity. The radiation forces for the case 11 = -12 = I are given by
Fdiss = fib:
(,i+ :$)
+
(13.33) (13.34)
2Ao/T
S ~
Ai/r2 + [ 1 + S(A;/r2 + 1)/412 ’
while for the case 11 = 12 =I, they are given by
-
kA; 2 (A; + r2)
vs
1 -s- --(I r2 +S)] A;
[
(L)
v,, (13.35)
362
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
240
tn! P
13
1
Equations (13.33) to (13.36) show clearly the additional components to the radiation force due to the orbital angular momentum properties of the two circularly polarized Laguerre-Gaussian beams. Furthermore, from eqs. (13.34) and (13.36) for the dissipitive component of the force, the existence of an axial friction force on the atom for negative detuning is seen clearly. We pointed out earlier that the static, velocity-independent, terms are identical to the corresponding static terms arising in the linear-linear LG case described earlier. It can be seen from eqs. (13.33H13.36) that the velocity-dependent terms are quite different from those arising in the linear-linear LG case. The dissipative force in eq. (I 3.36) agrees with the corresponding linear-linear forces in the limit of small concentration parameter S << 1, while the velocity-dependent dipole force in eq. (13.35) agrees with the corresponding terms in the linear polarization case in the limits S << 1 and A0 >> 1". To investigate the influence of the dissipative and dipole forces on an atom, we need to solve the optical Bloch equations together with the classical equation of motion for the atom, (13.37) where (Fdipole) and (Fdiss) are the unapproximated forces given by eqs. (13.15) and (13.16) or (13.17). In fig. 22a we plot the trajectory of a magnesium atom for the case 11 =-I2 = 1 moving under the influence of both the dissipative and dipole forces. The cooling of the axial motion is clearly evident in the z-direction, as is the subsequent trapping of the atom near the high-intensity regions of the Laguerrffiaussian beams. The dipole force here results in a radial oscillatory motion of the atom, while the azimuthal dissipative force causes the atom to rotate. This is shown in the x-y plane projection in fig. 22b. In figs. 23a and b we show the corresponding trajectory for 11 " 1 2 = 1. Unlike the previous case, the atom is now subject to a static torque which eventually
IV, 5 141
3 63
SPIN-ORBIT COUPLMG OF LIGHT I
I
I
30
20
10
*
c
......... ........... .,
...
.. .,,.......
-1t
-2(
-3( I
I
I
I
I
I
I
-30
-20
-10
0
10
20
30
X
Fig. 22. (a) The trajectory of a Mg atom moving under the influence of the dipole and dissipative forces in the Dalibard et al. u+-u- configuration with LG beams with I I = -12 = 1. The initial position of the atom is Ro = 2 0 d j and its initial velocity is 2.0im s-I . The parameter values are such that S = 1; q 1 = 251 and A,, = -r. Here I = 2 8 0 n m and r = 2 . 5 ~ 1 s-'. 0 ~ (b) The projection of (a) onto the x-y plane.
takes it out of the dipole potential well before the atom becomes significantly slowed.
8
14. Spin-Orbit Coupling of Light
In 5 2, we pointed out that spin and orbit contributions to the angular momentum in free space circularly polarized LG beams arise in the linear momentum
IN: § 14
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
364 I
I
I
I
100
50
L
O
-5c
-lo(
I
I
-100
-50
1 0
50
100
X
Fig. 23. (a) As in figure 22a, but with I , = 1, = 1, A0 = -lor, S = 8, and V, (b) The projection of (a) onto the n-y plane.
=
0.5ms-'
density. We now examine whether coupling between spin and orbit is also manifest in the interaction of such light with atoms. We have shown earlier (Allen, Lembessis and Babiker [1996]) that there is an azimuthal dissipative force due to the orbital angular momentum, given by: (14.1)
IV, Q 141
SPIN-ORBIT COUPLING OF LIGHT
365
If we now write u=ue", where u and 8 are real functions, it can be shown that the azimuthal component of the dissipative force is given by (14.2) where Qk00 is the equivalent plane-wave Rabi fiequency. We see that the product of spin and orbital angular momentum, a,l, appears. It manifests itself in the @-componentof force due to the dipole induced in the direction of propagation of the light and is due to the z-component of the electric field associated with the Laguerrdaussian mode. Clearly, just as reversing the direction of the orbital angular momentum changes the direction of the force and its accompanying torque (Babiker, Power and Allen [1994]), so changing the handedness of the Rolarization would be expected to change only the internal state of the atom but not, very remarkably as here, the gross motion. However, the detailed form and magnitude of the effect shows that such an interaction yields only a small correction to the motion. In general, the bracketted quantity in eq. (14.2) is a function of the Laguerre polynomials and their derivatives, but in the entirely experimentally valid case of p = 0, we find that: (14.3) Clearly, when r = w@ this term and the azimuthal force go to zero. This is as expected, because this is when the beam intensity is at a peak and d lu12/dr is zero. The magnitude of the effect is best considered by comparing the azimuthal force due to the spin-orbit coupling with that due to the transverse component of the electric field. To a good approximation for circularly polarized light when 0, = f1, we see, assuming that the atom is equally polarizable in all directions, (14.4) where 3, is the transition wavelength. l k s expression is not true when r = 0, but for a p = 0 mode the light intensity is in any case zero at that position. It may be seen that the term is of the order l/k2w2and is comparable to many other terms usually ignored in trapping calculations, but there is a physically meaningful size of beam waist w for which the term is of the order However, the existence of a spin-orbit term arising from light alone is intriguing.
366
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
[IY5 15
It may be the case that the presently chosen milieu is not the most appropriate to display it. Effects such as the geometric phase ascribed to the interaction of the spin of the photon and its orbital motion (Liberman and Zel’dovich [1992]) within optically inhomogenous media, such as an optical fiber, are outside the scope of this review.
8
15. Conclusions
A simple paraxial approximation approach to the propagation of a beam for a field with an azimuthal phase exp(iZ9) shows that a linearly polarized beam possesses an orbital angular momentum of lh per photon in the direction of propagation. The total angular momentum for a circularly polarized beam is (I + o)h per photon. This ready separation of spin and orbital contributions appears to be contrary to the requirements of gauge invariance. The solution of the full set of Maxwell equations for a similar field distribution shows that there is indeed a correction term, but it is small and depends only on spin. Thus for linearly polarized light, such a beam does have a uniquely defined component of orbital angular momentum along the direction of propagation even for a tightly focussed beam. Physically realizable beams with the appropriate azimuthal form can be made in the laboratory in the guise of Laguerre-Gaussian beams. We have shown how these may be created from the commonly occurring Hedte-Gaussian modes emitted by laboratory lasers. The transformation is possible in a number of ways, but the use of a mode converter consisting of cylindrical lenses shows that, in t h s context at least, asymmetric optical components are to orbital angular momentum what birefnngent components are to spin. Certain other beams without an azimuthal phase can also possess orbital angular momentum. In the case of elliptic beams, the magnitude is not quantized as it is in a Laguerre-Gaussian beam, but the angular momentum per photon can be tens of thousands of R. Unlike Laguerre-Gaussian beams, the orbital angular momentum density changes from point to point in the beam, but in a dfferent way to the variation of the local spin density. The interaction of Laguerre-Gaussian modes with nonlinear materials leads to second-harmonic generation and the conservation of orbital angular momentum in the light beam. This is because the output beam can acquire any value of 1; specifically, one related to that of the input field. This is in contrast to intrinsic spin, where the beam cannot have an angular momentum magnitude greater than
n! P
151
CONCLUSIONS
361
one. For modes with p = 0 the second harmonic beam is also a LaguerreGaussian. The behavior of higher-order modes is more complicated, but in all cases a mode transformation takes place. The mechanical equivalence of spin and orbital angular momentum has been demonstrated for I = 1 within an optical tweezer and this has led to the idea of an optical spanner for the manipulation of microparticles. The angular Doppler shift observed by Garetz [19811 for circularly polarized light, the aziumuthal Doppler shft, the shift in frequency predicted for light with orbital angular momentum passing through a rotating asymmetric lens system, and the frequency shift for an atom in a rotating potential, have been shown experimentally to be examples of the rotational frequency shift that arises when a beam of light is rotated. Spin and orbital contributions to the shift are additive and interchangeable. We have explored the nature of the radiation forces and their influence on atomic motion for Laguerre-Gaussian laser light, both in the form of a single beam and for multiple beams in a number of configurations. The orbital angular momentum aspects which characterize these modes give rise to new physical phenomena when such light interacts with atoms. A variety of forces has been shown to come into play when the Laguerre-Gaussian light is arranged in well-defined multiple beams, particularly linear, 2D-orthogonal and symmetric coplanar three-beam configurations. For simplicity, only coaxial multiple Laguerre-Gaussian beams of the same order, whose beam waists coincide and which have the same magnitude of 1, have been considered. Notwithstanding the simplification inherent in these symmetric configurations, the physics is intricate and has been shown to give rise to new effects associated with the orbital angular momentum. Laguerre-Gaussian light generates a potential arising from the dipole force, while the dissipative force provides a mechanism to cool the atom axially. A torque has been predicted that can be utilized to cool or heat the azimuthal motion. There are also reciprocating forces whch arise from the interplay between forces in orthogonal directions, which can generate oscillatory and precessional motion. It appears likely that the dynamically induced stabilization of the atomic motion, the so-called supermolasses configuration, that arises from a small dwplacement of the molasses field (Chu, Prentiss, Cable and Bjorkholm [ 19871) can be related to the azimuthal forces arising from orbital angular momentum, as probably can the macroscopic vortex force due to the offset beams in the spinpolarized spontaneous force-atom trap (Walker, Feng, Hoffinan and Williamson I11 [ 19921). However, the orbital angular momentum of the Laguerre-Gaussian
368
THE ORBITAL ANGULAR MOMENTUM OF LIGHT
rw 0
15
modes is explicit and their influence. on atomic motion more straightfonvard to interpret. Also, Laguendaussian modes offer the possibility of dark field selective excitation previously achieved by blocking the light (Anderson, Petrich, Ensher and Cornell [ 1994]), and there are proposed novel optical traps (Kuga, Torii, Shiokawa and Hirano [ 19971). Spin angular momentum plays a role in the internal dynamics of the atomic interactions and selects the appropriate transition. The orbital contribution of the beam is predicted to affect the gross motion of the atom and gives rise to an azimuthal Doppler shift proportional to 1 and to the frequency of the atom about the beam axis. The interaction of spin and orbit contributions plays only a small role in the force on the atoms. Nevertheless the spin-orbit interaction of the photon in a light beam, in a way analogous to that of the electron, is a new concept likely to acquire a measure of significance. So far only dipole interactions have been considered. A careful study of multipolar radiation is likely to lead to unusual and interesting results. Spin will still select the nature of the magnetic sub-levels involved, while orbital angular momentum might now be expected to be involved in the internal motion when, for example, a pure quadrupole transition of 1=2 is involved. The transformation of Gaussian light beams has been cited (van Enk [ 19931) as a way to yield yet another form of geometric phase. The first predicted geometric phase (Pawharatnam [1956]), originates from cyclic changes of the polarization vector of a plane wave propagating in a fixed direction and is explicable in terms of the transfer of spin angular momentum (Tiwari [1992]). However, van Enk [ 19931 has predicted that although a similar analogy can be made for the transfer of orbital angular momentum for HermitAaussian modes where (n + m)= 1, this appears not to be true when (n + m) > 1. A geometric phase should still be found to occur as the modes are transformed, but it appears that once again spin and orbital contributions may not behave identically. In the literature of beams with discontinuities in their most general form, the effect of orbital angular momentum has been discussed essentially only by implication. It appears that much of the language of the literature of vortices in light beams should be susceptible to being re-written explicitly in terms of orbital angular momentum. This has not yet been done and this review is not the place to attempt it. It would appear that the overlap is likely to be increasingly significant; the conservation of orbital angular momentum in soliton motion in non-linear media (Firth and Skryabin [ 19971) has been demonstrated theoretically. We have seen that spin and orbital angular momentum are sometimes interchangeable and sometimes not. It might appear that there is an important difference between spin and orbital angular momentum because o is independent
NI
REFERENCES
369
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Acknowledgements It is a pleasure to thank Professor Stephen M. Barnett for reading and commenting on a preliminary version of this review.
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Harris, M., C.A. Hill and J.M. Vau&an, 1994, Opt. Commun. 106, 161. Haus, H.A., 1984, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, NJ). He, H., M.E.J. Friese, N.R. Heckenberg and H.Rubinsztein-Dunlop, 1995, Phys. Rev. Lett. 75,826. He, H., N.R. Heckenberg and H. Rubinsztein-Dunlop, 1995, J. Mod. Opt. 42, 217. Heckenberg, N.R., R. McDuff, C.P. Smith, H. Rubinsztein-Dunlop and M.J. Wegener, 1992, Opt. Quantum Electron. 24, S951. Jackson, J.D., 1962, Classical Electrodynamics (Wiley, New York). Jaucb, J.M., and F. Rohrlich, 1976, The Theory of Photons and Electrons, 2nd Ed. (Springer, New York). Kazantsev, AX, (3.1. Surdutovitch and VP. Yakovlev, 1990, Mechanical Action of Light on Atoms (World Scientific, Singapore). Kuga, T., Y. Torii, N. Shiokawa and T. Hirano, 1997, Phys. Rev. Lett. 78, 4713. Lai, W., M. Babiker and L. Allen, 1997, Opt. Commun. 133, 487. Lax, M., W.H. Louisell and W.B. McKnight, 1975, Phys. Rev. A 11, 1365. Letokhov, VS., and VG. Minogin, 1987, Laser Light Pressure on Atoms (Gordon and Breach, New York). Lett, PD., W.D. Phillips, S.L. Rolston, C.E. Tanner, R.N. Watts and C.I. Westbrook, 1989, J. Opt. SOC.Am. B 6, 2084. Liberman, VS., and B.Ya. Zel’dovich, 1992, Phys. Rev. A 46, 519. Louisell, W.H., 1990, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York). Mandel, L., and E. Wolf, 1995, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge). Marcuse, D., 1972, Light Transmission Optics (Van Nostrand, New York). Masalov, A.V, 1997, Laser Phys. 7, 751. McGloin, D., N.B. Simpson and M.J. Padgett, 1998, Appl. Opt. 37, 469. Metcalf, H., and P. van der Straten, 1994, Phys. Rep. 244, 203 Nienhuis, G.,1996, Opt. Commun. 132, 8. Nienhuis, G., and L. Allen, 1993, Phys. Rev. A 48, 656. Padgett, M., J. Ark, N. Simpson and L. Allen, 1996, Am. J. Phys. 64, 77. Padgett, M.J., and L. Allen, 1995a, Opt. Commun. 121, 36. Padgett, M.J., and L. Allen, 1995b, quoted by B. Amos and P. Gill, 1995, Meas. Sci. Technol. 6, 248. Pancharatnam, S., 1956, Proc. Ind. Acad. Sci. 44, 247. Phillips, W.D., 1992, Fundamental systems in quantum optics, in: Proc. Int. School of Physics, Les Houches, Session LIlII, 1990, eds J. Dalibard, J-M. Raimond and J. Zinn-Justin (Elsevier, Amsterdam). Power, W.L., L. Allen, M. Babiker and YE. Lembessis, 1995, Phys. Rev. A 52, 479. Poynting, J.H., 1909, Proc. R. SOC.London, Ser. A 82, 560. Rigrod, W.W., 1963, Appl. Phys. Len. 2, 51. Rose, M.E., 1955, Multipole Fields (Wiley, New York). Siegman, A.E., 1986, Lasers (University Science Books, Mill Valley, CA). Simmonds, J.W., and M.J. Guttmann,1970, States, Waves and Photons (Addison-Wesley, Reading, MA). Simpson, N.B., K. Dholakia, L. Allen and M.J. Padgett, 1997, Opt. Len. 22, 52. Stoler, D., 1981, J. Opt. SOC.Am. B 71, 334. Tamm, C., 1988, Phys. Rev. A 38, 5960. Tamm, C., and C.O. Weiss, 1990%J. Opt. SOCAm, B 7, 1034. Tamm, C., and C.O. Weiss, 1990b, Opt. Commun. 78, 253. Tiwari, S.C., 1992, J. Mod. Opt. 39, 1097.
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E. WOLF, PROGRESS IN OPTICS XXXIX @ 1999 ELSEVIER SCIENCE B.V ALL RIGHTS RESERVED
V
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS BY
h R E A S SIZMANN AND
GERDLEUCHS
Lehrstuhl fur Optik, Physikalisches Institut. Universitat Erlangen-Niirnbee, Staudtstr. 7/B2, 91508 Erlangen, Germany email: [email protected]
373
CONTENTS
PAGE
Q 1 . INTRODUCTION
. . . . . . . . . . . . . . . . . . .
375
. . . . . . . . . . . . . .
377
. . . . . . . . . . . . .
380
Q 2 . HISTORICAL PERSPECTIVE
Q 3. THE OPTICAL KERR EFFECT
Q 4. QUANTUM OPTICS IN FIBERS PRACTICAL CONSIDERATIONS . .
. . . . . . . . . .
388
Q 5. QUADRATURE SQUEEZING . . . . . . . . . . . . . .
397
Q 6. QUANTUM NONDEMOLITION MEASUREMENTS . . . . .
418
. . . . . . . . . . . .
435
Q 7. PHOTON-NUMBER SQUEEZING Q 8. FUTURE PROSPECTS
. . . . . . . . . . . . . . . . . .
458
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . .
460
REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
460
374
0
1. Introduction
If one wished to study nonlinear optical effects, one would not choose silica glass as the nonlinear medium at first sight. Second order effects are not observable because the bulk material is symmetric with respect to inversion. Of all materials, silica glass has about the smallest nonlinear coefficient in third order. Nevertheless, this review deals with the nonlinear optical Kerr effect in precisely this material. It does so for several reasons. Firstly, the nonlinearity of silica optical fibers has increasing importance for optical communication applications. Secondly, the fiber allows for the study of soliton dynamics which is important in many nonlinear wave propagation models. Thirdly, on the fundamental side, it allows for the study of novel quantum optical effects such as intrapulse quantum correlations. Furthermore, the squeezing achieved in fibers, i.e., the reduction of the quantum uncertainty of light, has improved substantially in the last decade. The long term goal to achieve an order of magnitude noise reduction (10 dB) now seems within reach. Furthermore, the devices generating squeezing are becoming more and more practical. This is potentially important for all applications which are limited by the shot noise of a laser beam. The experimental methods which have evolved allow for quantum measurements such as back-action evading or quantum nondemolition detection. The in-depth study of the rich nonlinear dynamics of quantum solitons in fibers and the measurement of the quantum characteristics of optical pulses has only just started. One practical application that bears high economic potential is in optical communication. In the following, this example will be used to present some aspects of quantum optics in fibers. In an optical data transfer fiber channel a minimum number of photons is necessary per bit of information, depending on the maximum allowed bit error rate. With increasing data transmission rates the peak intensity of a pulse carrying a bit is increasing consequently. The small value of the nonlinear coefficient is compensated for by the light power confinement in both the tranverse spatial and the temporal dimensions. The latter occurs if the nonlinear interaction leads to the formation of soliton pulses in the optical fiber. The point has been reached where the fiber nonlinearities can no longer be neglected in the systems which are being presently field tested. On the other hand, the concept of optical solitons makes use of this nonlinearity 375
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THE OPTICAL KERR EFFECT A N D QUANTUM OPTICS IN FIBERS
[v, §
1
to counter dispersion. Solitons have already been tested in a commercial long distance data transmission line (Robinson, Davis, Fee, Grasso, Franko, Zuccala, Cavaciuti, Macchi, Schiffini, Bonato and Corsioni [ 19981). In addition, solitons in fibers are an interesting and rewarding test-bed for studying the dynamical evolution of the quantum properties of optical pulses in a nonlinear environment. Data transmission is an area where optical technology has taken over from electrical technology because of the higher modulation band width and the much reduced crosstalk between nearby channels. One limitation of optical data transmission today is based on the analog nature of the optical systems. In order to guarantee error-free transmission over long distances, it is eventually necessary to regenerate the signal in a repeater station where the signal must be converted to the electronic level and then back into the optical domain. The finite bandwidth of the electronic circuit is the bottleneck limiting the transmission rate. Working with solitons, passive spectral filters, and erbiumdoped fiber amplifiers, one can overcome this limitation (Essiambre and Agrawal [ 19971). Based on their particle-like stability, solitons build a digital more than an analog system. For networks already installed it is difficult to follow this route because they are not designed to support solitons. An all-optical signal regenerator working much like an amplifying discriminator would boost the achievable data rates enormously. In order to explore this possibilty, existing and new functional devices must be studied with respect to their noise properties. The performance of up-to-date optical transmission lines is limited by the quantum noise introduced by the light generation process in the source, by the attenuation in the fiber, and by optical amplifiers. Therefore, it is especially important to study the mechanisms for the reduction of noise, quantum and classical as well. In particular, this refers to squeezing and to the generation of spectral and intermodal quantum correlations. Along with amplitude noise reduction, the functional behavior of a transmission device may show optical limiting, which is a necessary ingredient towards building an optical discriminator. This article reviews experiments which have been setting the stage for quantum optics in fibers by studying the effect of the nonlinear optical Kerrinteraction and of stimulated Raman scattering on the quantum properties of light in silica fibers. To begin, the historical development is summarized in tj 2. The following two chapters prepare for the main part of the review, starting in 9 3 with an outline of the basic characteristics of the optical Kerr effect. Section 4 reviews the essential properties of optical solitons and thermal noise sources in fibers. The central part of the review begins in tj 5 with a discussion of quadrature squeezing in fibers using self-phase modulation (SPM). It has been the first and so far most intensively researched avenue of fiber squeezing. Recent progress
v, § 21
HISTORICAL PERSPECTIVE
377
includes the use of cross-phase modulation (XPM) for quadrature squeezing. Section 6 is devoted to quantum nondemolition (QND) measurements in fibers, where XPM is used to achieve quantum entanglement of two modes or two pulses. Here the concepts of XPM and SPM are used to diswss progress and perspectives for QND. New squeezing mechanisms are discussed in $ 7 . Only three years ago there emerged the transition from quadrature squeezing and coherent signal extraction using a local oscillator, to directly detectable squeezing. A number of recent experiments which produced photon-number squeezing will be discussed in Q 7. Again, the nonlinear Kerr effect is the basic underlying squeezing mechanism. In Q 8 the perspectives emerging from these experiments, including the potential impact on all-optical signal processing, are discussed.
0
2. Historical Perspective
Nonclassical states of the electromagnetic field, i.e., states which have no classical analog, have been intensively studied experimentally over the past 20 years I . Early experiments showed subpoissonian photon statistics in resonance fluorescence (Teich and Saleh [ 19881). Parametric, phase-sensitive deamplification of noise in a beam of light was first studied in four-wave mixing and was observed in the classical domain (Levenson, Shelby and Perlmutter [19851, Maeda, Kumar and Shapiro [1985], Levenson and Shelby [1985], Slusher, Hollberg, Yurke and Mertz [ 19851).Then, the first true squeezing in the quantum domain was achieved in degenerate four-wave mixing in a Na atomic beam (Slusher, Hollberg, Yurke, Mertz and Valley [ 19851). Next, squeezing through four-wave mixing in glass fibers was observed (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]) shortly before the generation of squeezing in a x ( ~ ) nonlinear parametric oscillator (Wu, Kimble, Hall and Wu [1986]). The main experimental difficulty in all systems is that the weak nonlinear optical processes must be efficient enough to compete with attenuation, diffraction and material-dependent noise sources. Squeezed state generation
I Squeezed states of light are discussed in a number of textbooks, e.g. Walls and Milbum [1994], Bachor [1998], Scully and Zubairy [1997], and Meystre and Sargent 111 [1990]. In addition, the subject was covered in review articles (Leuchs [1986], Loudon and Knight [1987a], Teich and Saleh [1989], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990], Reynaud, Heidmann, Giacobino and Fabre [1992], and Tanas, Miranowicz and Gantsog [1996]). Several special journal issues were dedicated to squeezed light (Kimble and Walls [1987], Loudon and Knight [1987b], and Giacobino and Fabre [1992]).
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THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
lX§2
based on the third-order nonlinearity. of single-mode fibers was proposed as a promising system (Levenson, Shelby, Aspect, Reid and Walls [1985]). In contrast to enhanced near-resonant nonlinearities in atomic systems, where noise from the creation of an excited-state population is related to the squeezing interaction itself, the extreme off-resonant nonlinearity in fiber must compete with thermal noise not related to the squeezing interaction. The hope was that the thermal noise could be eliminated whle the power confinement and long interaction length in the highly transparent medium would generate detectable squeezing. After all, traveling-wave geometries offer a larger bandwidth for squeezing and have a natural interface to fiber-optic communication systems, thus making optical fibers attractive for classical and quantum noise reduction. In the early experiments, classical thermal noise unique to fibers was found to be a severe limitation. Elimination of this noise was regarded as a technological (Levenson, Shelby, Aspect, Reid and Walls [1985]) rather than a fundamental physical problem. The development of squeezing experiments in fibers shows that of the many techniques for thermal noise reduction, some were successful, and even fibers with particularly low intrinsic noise were found (Bergman, Haus and Shirasaki [19921). Also, new squeezing mechanisms in fibers which seem to be immune to intrinsic phase noise were discovered recently (Friberg, Machida, Werner, Levanon and Mukai [ 19961). The hstory of fiber squeezing is illustrated in fig. 1. After the first continuous wave (CW) results (Shelby, Levenson, Perlmutter, DeVoe and Walls [ 1986]), the second generation of fiber squeezing experiments used high-peak power pulses and coherent detection (Rosenbluh and Shelby [ 19911, Bergman and Haus [1991]), and a variety of more-or-less successful methods of intrinsic noise reduction were reported. The squeezing results achieved over the years, however, did not show a continuous development towards more noise reduction, as instrinsic noise properties of fibers varied to a great extent. T h s review draws a distinction between photon-number and quadrature squeezing. The former refers to reduced amplitude noise and may be measured by direct detection. The latter refers to measuring a specific field quadrature using a local oscillator. Basically, two classes of pulsed quadrature squeezing experiments were performed in fibers: those using sub-picosecond optical solitons (Rosenbluh and Shelby [1991]) and those using 10 to loops pulses near zero dispersion. The latter have matured experimentallyand have reached the (inferred) 7 dB detection limit for chirped pulses (Bergman and Haus [1991]). In the negative dispersion regime, more than lOdB of noise reduction is predicted for solitons; however, this remains to be demonstrated. With solitons, the need for inducing a large nonlinear phase shift over a short
v, 5 21
HISTORICAL PERSPECTIVE
0
379
cw quadrature squeezing and four-mode squeezing
v$/
pulsed quadrature squeezing pulsed photon-number squeezing (spectral filter)
v m pulsed photon-numbersqueezing (asymmetric lwp) Fig. 1. Progress in squeezing experiments with fibers, quantified by the squeezing ratio R of observed photocurrent noise reduction relative to the shot-noise limit (R=1). Different symbols and shaded areas represent the four generations of squeezing experiments, from the first generation of continuous-wave experiments in cryogenic fibers to the present generation of pulsed squeezing experiments at room temperature. The individual experimental results are reviewed in 5 5 (quadrature squeezing) and 5 7 (photon-number squeezing).
length of fiber had led to the use of sub-picosecond pulses already by 199 1. The limited amount of squeezing observed raised the question of whether stimulated Raman scattering masked the noise reduction. This was not the case, as more complete quantum soliton models showed. Therefore, the goal of more than 10 dB of squeezing from fibers seems feasible. The latest quadrature squeezing experiments use extremely short fibers and short pulses in order to bring out the quantum properties of solitons. There has also been recent progress in novel approaches for quadrature squeezing using the cross-Kerr effect, thus avoiding mode matching problems that have been thought to limit earlier soliton squeezing experiments. The third generation of fiber squeezing experiments produced directly detectable photon-number squeezing via spectral broadening, and will be discussed in 9 7. In these experiments, strong squeezing was observed with spectral filtering (Friberg, Machida, Werner, Levanon and Mukai [19961, Spalter, Burk, StroBner, Sizmann and Leuchs [ 19981). However, according to numerical calculations the spectral filtering technique is not expected to yield more than 8 dE3 of squeezing.
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THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, 5 3
The fourth generation of experiments makes use of two-beam interference where at least one of the interfering pulses has experienced nonlinear interaction (Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [1998]). In this case, numerical calculations predict more than 10 dB of photon-number squeezing. The stable, directly detectable squeezing combined with the promising predictions will lead to continued experimental effort in this field. Today, the main experimental effort in developing sources of squeezed light is directed towards efficient, stable, and compact devices that are versatile and which can be tailored to specific measurement problems. In this context, the fiber is an attractive system, because it is easy to use and because of its promise for efficient squeezing. Moreover, recent progress in the study of optical quantum solitons and of quantum limits in fiber-optic communication fuels the continued interest in quantum optics in fibers2. One goal is to generate quantum solitons with more than an order-of-magnitude quantum noise reduction (>lo dB). With such a highly nonclassical state, basic paradigms of quantum measurement and nonlinear quantum field theory can be tested. Another perspective is to establish the quantum noise performance in fiber-optic components, such as filters, loop mirrors or switches, in comparison with which actual devices may be evaluated. Before presenting the state of the art of non-classical effects in fibers, we begin by reviewing the basic properties of the nonlinear interaction in fibers.
0 3.
The Optical Kerr Effect
Small perturbations in physical systems close to equilibrium tend to lead to harmonic oscillations around the point of equilibrium. If the deviation from equilibrium is large enough, most systems show anharmonic behavior which leads to a change in the oscillation period and to the appearance of new frequencies and overtones. It is well known that this also happens for atomic valence electrons when excited by an intense light beam. This opens the rich field of nonlinear optics. The response of the medium is described by the nonlinear polarization (Bloembergen [ 19651, Hanna, Yuratich and Cotter [ 19791, Loudon [1983], Ducuing [1977]). With regard to silica fibers, the following discussion is
For a review on quantum optics in fibers see e.g. Milbum, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], Drummond, Shelby, Friberg and Yamamoto [1993], Haus [1995], Friberg [1996], Spalter, van Loock, Sizmann and Leuchs [1997], Sizmann [1997], and Sizmann, Spalter, Burk, SlroBner, Bohm and Leuchs [1998].
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THE OPTICAL KERR EFFECT
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restricted to a medium showing inversion symmetry so that the polarization of the medium can be written as an expansion in odd powers of the electric field3:
where E represents the total electric field, which may consist of components having different frequencies and may also be the sum of the electric fields of two beams intersecting spatially. The linear ) term just transfers these frequencies to the polarization of the medium. The nonlinear x ( ~term ) mixes different frequencies, generating new spectral components. Most processes induced by ~ ( ~ such 1 , as third-harmonic generation, require a special effort for matchmg of the phase velocities of all optical fields involved. The only one for which this does not hold is The optical Ken effect which is, therefore, the dominant nonlinear interaction in fibers. When the effect was first discovered by Kerr [ 18751, two of the electric fields were constant and one was an optical field. The effect was used for fast optical switches. It is because of its short time constant that the optical version of the effect generates squeezing in an extremely large frequency band4. If only one optical electromagnetic field is applied, then there is a component of the induced nonlinear polarization, P , which effectively changes the refractive index for the propagating light beam. This can be shown by rewriting eq. (3.1), and is known as the optical Kerr effect (Agrawal [1995]):
x('
with I = iceonoE2 being the intensity of the light beam, c the vacuum speed of light, and no the refractive index given below. The Kerr effect may be described by an effective susceptibility which leads to the following intensity-dependent
A single silicon dioxide molecule does not have this symmetry. The inversion symmetry of silica glass is a property of the bulk material where the molecules are statistically oriented. It is, however, possible to pole a silica fiber so that the second-order term appears in the nonlinear polarization and a second-harmonic wave can be generated (Kazansky,Dong and Russell [1994], Kazansky, Russell and Takebe [1997]). Phase matching may be obtained by using spatially periodic poling. s for a If the nonlinear response is dominated by electrons the time constant is less than nonresonant light field (Owyoung, Hellwarth and George [1972]).
382
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y 0 3
x(')
refractive index5, assuming that the imaginary parts of and x(3)are negligible and that the intensity-dependent term is much smaller than 1+x('),
or
n = no + nzI.
(3.3)
As a result of the optical Ken- effect, a light field will experience an intensitydependent phase shift. For a light pulse this leads to a phase varying with time, producing a frequency chup across the pulse. If a light beam contains different intensity components, each will be phase shifted differently. This explains the initial evolution of a coherent state of th8 light field in a Kerr medium as will be discussed in more detail below. The intensity-dependent phase shift leads to a wide variety of phenomena and applications such as pulse compression, solitons, optical switclung (Agrawal [ 19951) and quantum measurement (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [19901). The full richness of the optical Ken- effect is revealed when taking into account field quantization (Loudon [ 19831). The operator for the electric field of one mode is a superposition of the photon annilulation and photon creation operator6 ,
(3.4) The energy density of the field is given by the product of the electric field and the dielectric displacement vector (Bloembergen [19651, Drummond and Walls [1980]):
The contribution of the nonlinear interaction to the Hamilton operator is, therefore, a linear combination of products of four field operators. The order of Here the refractive index is given by n2 EOE= EOE+P V describes the quantization volume. In the case of a fiber, V is given by the cross-section of the fiber core A e f f times the length of the fiber AL over which the photons are distributed. For pulses of width At, V = A , e c A t (Imoto [1989], Imoto, Jeffers and Loudon [1992]).
v, 5 31
383
THE OPTICAL KERR EFFECT
1
1
2
3
4
5
6
Fig. 2. Schematic energy level diagram for glass. The low- and the high-lying excited states (e) shall correspond to infrared and ultraviolet absorption bands (g = ground state). The arrows pointing up and down represent absorbed and emitted photons. Here all photons have the same frequency. The various groups of arrows describe the four photon processes contributing to the optical Kerr effect and to the nonlinear refractive index.
the field operators in the product depends on the level structure and the electric dipole transition matrix elements of the medium (Bloembergen [ 19651, Hanna, Yuratich and Cotter [1979], Loudon [1983], Ducuing [1977]). Figure 2 shows diagrams of elastic four-photon interactions; i.e., interactions where the state of the medium is not changed. These diagrams contribute to the optical Kerr effect. Some of the diagrams yield non-vanishing contributions to the interaction energy, which leads to divergence when assuming instantaneous response of the medium. This may be solved by renormalization (Abram and Cohen [ 19941). The interaction energy may be calculated in fourth order perturbation theory and is determined by a sum of products of four electric dipole matrix elements and three energy denominators. The processes described by diagrams 1 and 4 may be enhanced by a near-resonant two-photon transition. In the case of silica glass and a wavelength in the near infrared, diagrams 1 and 4 can be neglected with respect to diagrams 2, 3, 5 and 6 which are all enhanced because the initial state is also-one of the intermediate states. Using the commutator relation [a, a t ]= 1, each of these four operator products can be written as a linear combination of fa)^, a two-operator product (ata), and a constant term. The two latter terms
384
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
(4
[Y 8 3
(b)
Fig. 3. Phase diagram for the state of the light field. The shaded area indicates the region in phase space which the state occupies. The circular region in (a) describes a coherent state and the distorted region in (b) represents a state squeezed by the nonlinear Ken interaction.
produce an overall phase factor which does not change the character of the light field (Imoto, Haus and Yamamoto [ 19853). The quartic term leads to characteristic changes of the quantum uncertainty; i.e., of the distribution function in phase space describing the state (Ritze and Bandilla [1979], Kitagawa and Yamamoto [1986], Gerry and Knight [1997]). This shall be illustrated with the example of a light field which is in an initially coherent state. To this purpose, the electric field may also be written as E(t) = E(0)cos W t +
.
~
0
sin W t .
The phase diagram in fig. 3a shows the area in phase space occupied by a coherent state the size of which is determined by the uncertainty relation. For a coherent state the contour line is a circle. The two-dimensional phase space of one mode of the light field is spanned by the amplitude of the cosine wave, a,=&(O), and the amplitude of the sine wave, a,=d,E(O)lo (Leuchs [1988], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). The mean values of the two conjugate variables a, and a, are the expectation values of (a-at)/2i and (a+at)/2, respectively. In other words, the figure shows the contour of the Wigner hnction characterizing the light field (Teich and Saleh [1990], Walls and Milburn [1994]). The length and angle of the arrow in the phase diagram represent the amplitude and the phase of the field, respectively. As a result of the nonlinear interaction the circular distribution is initially squeezed to an ellipse, as shown in fig. 3b. This is the basis for the quantum measurements reviewed in 5 5 and 5 6. The beginning of this evolution is readily understood based on the intensity-dependent refractive index causing an amplitude-dependent phase shift. The 111 evolution of the
Y 4 31
THE OPTICAL KERR EFFECT
385
coherent state of the 1ight.field can be calculated by applying the time-evolution operator, u(t)
= exp {-iHff/h},
to the coherent state:
and
Obviously, the time evolution operator does not change a pure Fock state, apart from an overall phase factor. A superposition of Fock states such as a coherent state changes its character, however, because the phase factors introduced by the interaction are different for different Fock states. The discrete nature of the Fock-state basis, In), gives rise to unique quantum features in the absence of dissipation, which shows an interesting formal analogy to the quantum revival in a one-atom-maser (Eberly, Narozhny and Sanchez-Mondragon [ 19801, Knight and Radmore [1982], Yo0 and Eberly [1985], Rempe, Walther and Klein [1987]). The evolution of the coherent state turns out to be periodic (Ritze and Bandilla [1979], Walls and Milburn [1994], Buzek and Knight [1995]). After the time fza given by [ 3 ~ ( ~ ) h u ~ /V)]fzn ( 4 ~ o= 2n, the time evolution takes the light field back to the coherent state. After one quarter of that time, td2 = t2J4, the field is in a superposition of two coherent states lao) and /-ao).Figure 4 shows this evolution. This quantum superposition state is highly sensitive to dissipation (Buzek and Knight [1995], Yao [1997]). The diagrams in fig. 3 differ from those in fig. 4 in the following way. The former &splay contour lines of the Wigner function and the latter show contour lines of the @function,
plotted versus the real and imaginary part of a. The a’s are the eigenvalues of the coherent states onto which the light field is projected. The Q-function is the Wigner function convoluted with the Q-function for coherent states, which is a Gaussian distribution (Walls and Milburn [ 19941). For currently available silica
386
THE OPTICAL KERR EFFECT AND OUANTUM OPTICS IN FIBERS
IV, § 3
T 1
-.
Rea
VIII
IX
Fig. 4.Evo!ytion of an initially coherent light field due to the nonlinear Kerr interaction. Figure 3b shows only the very beginning of this evolution. The diagrams show contour lines of the Q-function as a function of the real and imaginary parts of a. The origin is at the center of each diagram. Starting from a coherent state a. = 2 in the diagram at the top left, the light field evolves as indicated in the following diagrams going from left to right and top to bottom. The interaction time elapsed between two consecutive diagrams is At = t2,/32. In the last diagram at the bottom right the light field has evolved into a superposition of two coherent states. After twice the evolution time (16At) the field evolves back to a coherent state -ao, and after tzn:=32At, back to a0 (Courtesy of N. Korolkova [1998]).
fibers the nonlinearity seems to be too low and the dissipation too large to allow for the observation of this unique quantum evolution7. The dissipative attenuation of about 0 . 2 d B h in a low-loss optical fiber near 1.5 pm is already close to the fundamental limit largely determined by Rayleigh scattering from density fluctuations*. Although the generation of
' For other optical materials see Hilico, Courty, Fabre, Giacobino, Abram and Oudar [1992]. The dissipation in optical fibers due to absorption by and scattering off impurities and host density fluctuations and possibly inhomogeneously oriented silica molecules has been reduced dramatically during the last three decades (Drexhage [1991]). While in 1973 the ultimate loss limit was believed to be 1.2dBkm for fused silica (Pinnow, Rich, Ostetmayer Jr and Di Domenico Jr [ 1973]), 0.2 dskm
Y
o 31
THE OPTICAL KERR EFFECT
387
quantum superposition states might find applications in quantum communication, this will not be discussed further in this review because such studies are currently lacking experimental viability in the case of optical fibers. When working with sub-picosecond pulses and long fibers, one must also consider stimulated Raman scattering (Yariv [ 1987]), which shifts the spectrum towards longer wavelengths (Stolen and Ippen [19731, Weiner, Heritage and Stolen [1988]) and introduces fluctuations in the blue and red wings of the spectrum which are negatively correlated. Stimulated Raman scattering is an inelastic process that excites optical phonons in the fiber. The interaction Hamiltonian is btala1 + bataz, describing e.g. the process where a photon of frequency w~ is absorbed and a photon of lower frequency w2 and a phonon ( B ) are generated. The frequency of the emitted phonon is equal to the difference of the two optical frequencies, wl - w2 in this case. The experiments reviewed in the following sections deal with the initial elliptical deformation of the area occupied in phase space (fig. 4). This picture uses a single-mode Hamiltonian where the four field operators refer to the same mode (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901). Alternatively, one may use a multimode picture (Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], Tanis, Miranowicz and Gantsog [19961). As already mentioned, the intensity of a laser pulse which varies across the pulse leads to a phase change varying with time. This in turn corresponds to a frequency shift varying across the pulse. The diagrams of fig. 2 must be modified slightly (fig. 5 ) ; e.g., diagram 2 changes to represent absorption of a photon of frequency w1, emission at w2, absorption again at 01 and emission at 013. The corresponding four-field operator product is aialaial. The requirement on the optical Kerr interaction to be elastic simply means that energy is conserved; i.e., 201 - 02 - w3 = 0. In general, the two pump photons may have different frequencies. This model not only shows that the optical Kerr effect generates new frequency components but also suggests that the fluctuations in the red and blue detuned sidebands are positively correlated because photons are created in pairs. The special case where the interaction is described by the four-field operator product aiazafal is referred to as cross-phase modulation (XPM) because the intensity of one mode changes the phase of the other.
was reached six years later (Miya, Terunuma,Hosaka and Miyashita [1979]). Currently,the minimum loss is approaching 0 . 0 6 d B h (Sakaguchi, Todoroki and Sbibata [1996], Sakaguchi and Todoroki [1997]). With reference to footnote 3 one may wonder whether it is possible to improve upon this fundamental dissipation limit by reducing the orientation inhomogeneity by electric field poling of the silica molecules preferentially during solidification in the fiber pulling process.
388
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
E
[v, § 4
e
1
1
2
3
4
5
6
Fig. 5 . Schematic energy level hagram for glass as in fig. 2. Here the emitted photons have frequencies different from the absorbed pump photons.
0 4. Quantum Optics in Fibers - Practical Considerations 4.1. KERR-NONLINEARITY AND POWER CONFINEMENT
A nonlinear, i.e., intensity-dependent phase shift produced by the optical Kerr effect was among the first optical nonlinearities proposed for squeezing and quantum nondemolition measurement. Even though the Kerr nonlinearity of silica glass is small compared to that of other nonlinear media, its effect is greatly enhanced in fibers by strong power confinement, chirp-free high-peakpower soliton propagation, and low attenuation at the wavelength of 1.55 pm. The magnitude of nonlinear effects in waveguides can be compared with bulk nonlinear systems using a figure of merit defined as F =I.Ler (Agrawal [1995]). Here Z=P/A,ff- is the intensity and L,ff is the effective interaction length of the nonlinearly coupled modes having an effective cross section of A e r = nwg. In a nonlinear waveguide the effective length L,R is limited by losses only. The lowest attenuation in silica glass fibers is about 0.2dBh-1 around a wavelength of 1.55 pm, with a spectral window approximately 25 THz (200 nm) wide. If attenuation is required to be less than 1% (-0.044dB) for quantum measurements, the corresponding fiber may be up to 220m long (fig. 6). For comparison, a 9pm-diameter focus of a Gaussian beam near the wavelength of 1.5 pm is diffraction-limited to a confocal length of Lee= nnw;/& = 60 pm.
Y 5 41
QUANTUM OPTICS IN FIBERS - PRACTICAL CONSIDERATIONS
389
220 rn
Fig. 6. The dimensions of a 150-fs soliton in a single-mode fiber with a 8-pm core and a 9 y m mode field diameter. The optical high-intensity pulse remains self-confined without experiencing spreading and travels more than 200m with negligible (
In single-mode silica glass fibers the tiny cross-sectional area Aeff of the mode and low attenuation enhance the figure of merit by F f i b e r / F b u l k =4x lo6 when comparing with a diffraction-limitedbeam. However, the highly transparent glass fiber comes with a small nonlinearity. The refractive index of a germania-doped silica glass fiber, n(Z) = no + n2I, has a nonlinear index coefficient of approximately n2 = 3 x cm2W which is smaller by two or more orders of magnitude when compared with most other nonlinear media (Agrawal [ 19951). However, the nonlinear response of silica glass is very fast because most of it (81%) is due to a far off-resonant electronic response of the medium on a time scale of less than 5 fs. Only a small fraction (19%) of the nonlinear response originates from nuclear vibrations on a time scale of 100 fs and gives rise to stimulated Raman scattering (Blow and Wood [1989]). Furthermore, the Kerr effect does not require phase matching. Therefore, ultrashort pulses from a mode-locked laser with high peak intensities and a Terahertz bandwidth can be used. A nonlinear peak phase shift of n is obtained in less than 2 m of fiber for an ultrashort pulse of 1 kW peak power. However, the peak power may decay quickly when ultrashort pulses experience dispersion. Taking advantage of the soliton phenomenon, the pulse energy remains confined in a small volume and the high peak power is maintained over a long propagation distance (fig. 6). 4.2. OPTICAL SOLITONS IN FIBERS
Among the many linear and nonlinear pulse-shaping effects in glass fibers, temporal pulse spreading due to group-velocity dispersion (GVD) and spectral
390
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS R‘I FIBERS
!x§4
broadening due to self-phase modulation (SPM) are the most important ones for picosecond pulses near 1.5 pm center wavelength. In the absence of nonlinear effects, i.e., at low power, the rate of pulse broadening depends on the groupvelocity dispersion (GVD)parameter k” = d2kldw2and on the spectral width of the pulse. A useful length scale over which pulse broadening becomes important is given by the dispersion length L ~ = T $ l k “ l , where To describes the width of the pulse9. In the absence of GVD,i.e., at the zero-dispersion wavelength AD = 1.3 pm where the GVD of silica glass fiber changes sign, the rate of spectral broadening is given by the nonlinear peak phase shift per unit length, &LIZ = n;?koIo= yPo, with PO being the peak power of the pulse. The corresponding nonlinear length scale is LNL=lI( yP0) (Agrawal [ 1995]), which is the propagation distance for a 1-radian nonlinear peak phase shift. In the presence of both GVD and SPM, the initially dominant pulse-shaping effect is determined by the shortest length scale for a given input pulse (T0,Po) and fiber (k”,y). Pulse propagation in optical fibers is modeled by the nonlinear Schrodinger equation (NSE) (Agrawal [1995]), which is solved numerically for a given input pulse. The parameters To, LD and fi are convenient for scaling time T, propagation distance z and pulse amplitude A( T ,z), respectively, to dimensionless parameters. The lmensionless NSE simultaneously solves the pulse propagation problem for a variety of physical situations in which TITO,l j = z l L ~ , N =d m and initial pulse amplitude U(T/To, 0) = P;”2A(T/To, 0) are identical. One class of solutions stands out. It represents stationary or periodically recurring pulse envelopes. These are the “soliton” solutions, the only solutions for which the NSE can be solved analytically (Zakharov and Shabat [ 19721). Solitons are a phenomenon in many areas of nonlinear physics lo. Because of their particle-like stability, optical solitons in fibers bear great potential for tomorrow’s broadband digital-optical telecommunication systems ’. ~ = T F W H M1.763To = istheFWHMvalueofasech2(T/To)pulse withspectral widthAv=0.31/t,
PO is the peak power, y = n 2 w o / ( c A e f f )is the nonlinear coefficient of the fiber, and A , ~ = n w $is the effective cross-sectional area of the mode. lo The term “soliton” was invented by Zabusky and Kruskal [1965]. In a numerical solution to the Korteweg-de Vries equation, Zabusky and Kruskal found particle-like stability in solitary waves which preserved their identity even through numerous interactions with one another. ‘I After soliton propagation of ultrashort laser pulses in optical fib& was predicted (Hasegawa and Tappert [1973]), the self-trapping of the pulse energy in an optical fiber soliton was observed experimentally (Mollenauer, Stolen and Gordon [ 19801). Fiber solitons, their properties and applications are discussed in several books (Taylor [1992], Hasegawa and Kodama [1995], Hasegawa [1998]), book chapters (Agrawal [1992, 199.51, Mollenauer, Gordon and Mamyshev [1997]), and review articles (Kodama and Hasegawa [1992], Haus [1993], Kodama, Maruta and Hasegawa [1994], Haus and Wong [1996], +nd Essiambre and Agrawal [1997]).
Y P 41
QUANTUM OPTICS IN FIBERS - PRACTICAL CONSIDERATIONS
391
Several soliton properties are intimately connected with the nonlinear and dispersion length scales. First, a fundamental ( N = 1) soliton is created when LD =LNL in the anomalous GVD regime (k” < 0 for A > 1.3 ym). By adjusting the peak power to the pulse width, the opposite chirps created by GVD and SPM exactly cancel each other. A fundamental soliton may emerge from a variety of envelopes and pulse energies; however N > 0.5 is required for asymptotic soliton formation from a sech input pulse (Zakharov and Shabat [ 19721). Second, the peak power of a fiber soliton scales as Po,sT; = const. as a consequence of LD=LNL.Shorter solitons require more energy, and the nonlinear effects are enhanced dramatically. Next, higher-order solitons are created for integer values of N = d m ,where N is the order of the soliton. The fundamental ( N = 1) soliton propagates without change in pulse shape, but higher-order ( N = 2,3,. . . ) solitons undergo periodic changes in pulse width with a period of zp = ~ J G L D the, so-called soliton period. Finally, the distance a soliton of width To must travel in order to acquire a in nonlinear phase shift is also given by the soliton period, zp, which is proportional to Ti/I k” I and independent of n2, the fiber nonlinearity. The latter determines the energy content of the soliton, which is proportional to I k” I /( y To). The soliton period, which is also the length required for solitonic self-stabilization, depends on the (linear) GVD parameter of the fiber. Therefore, short pulses and strong dispersion are required for fast soliton dynamics in short pieces of fiber. The propagating soliton may experience damping and Raman scattering. Fortunately, the minimum loss window at 1.55 ym lies in the anomalous GVD regime (k” < 0) where solitons can form. As an example, the soliton period of a 1-ps pulse in a standard fiber is zp=25m. This distance is short enough to neglect attenuation (zp << L 3 & = 15 km).As long as zp is small compared with a length scale over which significant damping occurs, the soliton stays intact. Using a 100-fs soliton, the peak power is two orders of magnitude higher compared with the 1-ps soliton and the same soliton dynamics are obtained in only 25 cm of fiber. However, other nonlinear effects become important, predominantly stimulated Raman scattering (SRS) (Agrawal [19951). Nevertheless, the subpicosecond soliton remains a stable particle-like entity while it experiences a continuous downshift of the center frequency 1 2 . Therefore, with picosecond and sub-picosecond solitons around the wavelength of 1.55 ym, the three
’*
Shortly after the discovery of the soliton self-frequency shift (Mitschke and Mollenauer [1986]), the Raman effect was modeled in the nonlinear Schrodinger equation with great precision by including a delayed nonlinear response (Gordon [ 19861).
392
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, § 4
requirements for quadrature quantum measurements in glass fibers, namely high peak intensity at low average power, chirp-free and loss-less propagation, are all fulfilled elegantly. In contrast to classical solitons, coherent quantum solitons l 3 are not stationary under propagation. Only the mean values corresponding to the classical amplitude, phase, position and momentum are stationary. Phase and position evolve as a function of amplitude and momentum, respectively. Consequently, phase and position uncertainties continue to spread with time, whereas the initial amplitude and momentum uncertainties remain unchanged. In this evolution process the uncertainty in a particular field quadrature may be reduced below the vacuum level, while the uncertainty in its conjugate variable is increased according to the minimum uncertainty principle (see Q 3). Such a quantum state of light is called a quadrature-squeezed state. Its “nonclassical” nature can be observed \using a balanced homodyne receiver (Yuen and Chan [ 19831). Quadrature squeezing is discussed in Q 5. Furthermore, colliding solitons become quantum-mechanically entangled. The solitons acquire a permanent phase and position shift proportional to the energy of the collision partner (Zakharov and Shabat [ 1972]), whch allows for back-action evading measurements of the photon-number of a soliton (Haus, Watanabe and Yamamoto [19891; see Q 6). The experimental methods used for the observation of quadrature-,squeezing and for the back-action evading detection of the photon number were similar to the methods used for coherent CW quantum measurements, with the added advantage of the h g h peak power of solitons. In contrast to these types of measurements, the method of spectral filtering (described in Q 7) is particularly suitable for observing pulsed squeezing. When a spectral filter was applied to solitons, duectly detectable photon-number squeezing (Friberg, Machida, Werner, Levanon and Mukai [ 19961) and spectral quantum correlations were observed (Spalter, Burk, Konig, Sizmann and Leuchs [1998]). The filter method did not require chirp-free pulse propagation and showed remarkable nonclassical features even for pulses below and above the fundamental soliton energy and for Raman-shifted solitons. Quantum solitons and their observation were covered in several reviews (Drummond, Shelby,
l 3 In this review, the term “quantum soliton” refers to optical solitons which possess or acquire nonclassical properties, such as a quadrature squeezed quantum uncertainty which can be detected in a homodyne receiver (Drummond, Shelby, Friberg and Yamamoto [1993]). Another definition of optical “quantum soliton” refers exclusively to photonic number states for the optical soliton (Lai and Haus [1989]), a state which is preserved by nonlinear propagation (see 8 3). In the following, the broader defimtion of optical “quantum soliton” is used.
v, P 41
QUANTUM OPTICS IN FIBERS - PRACTICAL CONSIDERATIONS
393
Friberg and Yamamoto [1993], Friberg [1996], Spalter, van Loock, Sizmann and Leuchs [1997], Sizmann [1997], Sizmann, Spalter, Burk, StroBner, Bohm and Leuchs [1998]). 4.3. GUIDED ACOUSTIC-WAVE BRnLOUIN SCATTERING (GAWBS)
Before the first experiments had been carried out, glass fibers seemed to be a promising system for quantum optical studies based on the optical Kerr effect. The only drawback had been the small nonlinear coefficient of silica, which was readily compensated for by the spatial and temporal confinement in a singlemode fiber as discussed above. Shelby, Drummond and Carter [1990] wrote: “Because the nonlinearity is nonresonant and electronic in origin, nearly ideal behavior was expected, with usefbl amounts of squeezing at easily attainable pump powers”. As one often finds, the idea was too nice to be true. Previously little understood or even unknown thermal noise sources were found to mask the quantum phase noise in the fiber. M.D. Levenson, R.M. Shelby, and coworkers identified these sources and eventually found ways to fight them. The first type of noise is caused by thermally activated relaxation modes of the silica glass structure (Perlmutter, Levenson and Shelby [ 19881) whch have been modeled by “two-level modes” (Hunklinger and v. Schickfus [1981]). It was the main obstacle in the first attempts to observe squeezing with a continuous-wave (CW) pump laser. This noise has an unstructured spectrum which dominates at low frequencies and tends to fall off as llf, but this spectrum is not well understood (Perlmutter, Levenson and Shelby [ 19901). The second type of noise is forward Brillouin scattering, which came as a surprise (Shelby, Levenson and Bayer [1985a]). When light scatters off acoustical phonons, energy and momentum conservation can be fulfilled in forward scattering only for zero-frequency phonons. As a result, no phase noise is expected to be introduced to the pump beam. In general, when scattering a photon of frequency w off an acoustical phonon of frequency Q, the inelastically scattered photon travels under an angle q with respect to the pump photon. S2 and q are related according to (Kittel [1971]):
- 2cacoustwnsin(q/2), . a= Copt
(4.1)
where cacoust and cOptlnare the phase velocities of the phonon and of the photon. For small Q, the phonon travels perpendicular to the photons. In bulk glass material Brillouin-scattering is observed only in the backward direction. In order to avoid pump attenuation by the stimulated version of this process, the CW
394
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y § 4
pump power must be kept below a certain limit. Forward scattering is only possible in the case of a Raman-process, where pump laser photons are scattered off optical phonons having the same phase velocity as the light. However, these arguments are true only in bulk material. In a single-mode fiber the lateral beam confinement leads to a discrete transverse mode structure and the first mode has an angular uncertainty of (Young [ 19931): sin(Aq)
1.2
Copt nmnrcore ’
where rcoreis the core radius. Consequently, acoustical phonons of frequency up to 52,
2
1.2-Cacoust rcore
(4.3)
will contribute to forward scattering and thus to phase noise. This process was named “guided acoustic-wave Brillouin scattering” (GAWBS); Shelby, Levenson and Bayer [ 1985b1). For a typical communication fiber, Y,,,, = 5 pm, yielding 52, M 1 GHz. In these fibers the phonons are confined in the transverse direction by the glass cladding which has a diameter of 125 pm. The resulting discrete transverse phonon modes lead to spectrally structured phase noise in the transmitted pump beam, the lowest mode being around 52 = 20 MHz. The radial phonon modes are rotationally symmetric with all silica molecules vibrating in the radial direction only. Photons scattered off these modes have the same polarization as the pump photon. These polarization-preserving radial modes are referred to as polarized GAWBS-modes. Figure 7 shows a typical polarized GAWBS spectrum in a long jacketed optical fiber. The spectrum was obtained by heterodyne detection of the light emerging from the fiber with a local oscillator derived directly from the laser source (Poustie [1993]). There are other modes where the molecules are sheared in a torsional way and which do not preserve the polarization of the light. They are called depolarized GAWBS-modes. Owing to their different nature the two types of modes can be distinguished experimentally. The experimental set-up for the measurement of depolarized GAWBS is simpler. At the output end of the fiber a polarizer is used to turn polarization noise into directly detectable amplitude noise. There are residual depolarized GAWBS resonances visible in the polarized GAWBS spectrum (fig. 7), because optical heterodyning was sensitive to both types of GAWBS modes. One attempt to reduce the thermal phase noise was to cool the fiber down to helium temperature (Perlmutter, Levenson and Shelby [1990]). This reduced the noise, but not to an insignificant level and not more
Y 5 41
QUANTUM OPTICS IN FIBERS - PRACTICAL CONSIDERATIONS
395
Frequency (MHz) Fig. 7. The spectrum of GAWBS in 900 m of optical fiber showing the broad bandwidth, measured with a CW laser pump having a power of 3.25 mW (Poustie [1993]).
than 1 dB of squeezing could be observed (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]). Another idea to reduce GAWBS noise in the radio-frequency measurement window is to increase the quality factor of the phonon modes. This can be done by removing the plastic jacket from the silica fiber (Shelby, Levenson and Bayer [1985b], Poustie [1993], Nishizawa, Kume, Mori and Goto [1994], StroBner, Konig, Spalter, Sizmann and Leuchs [1997]). As a result the GAWBS peaks in the noise spectra will substantially reduce in width. This increases the peak amplitude, but also reduces the noise floor between peaks. In this way frequency windows may be identified where GAWBS noise is reduced (Bergman, Haus and Shirasaki [1992]). When using pulse trains from a modelocked laser, the repetition rate is an important issue (Poustie [ 19921, Hardman, Townsend, Poustie and Blow [1996]). The spectrum of the pulse train consists of modes separated by the repetition rate, which is in most cases smaller than the upper frequency limit of GAWBS. Consequently a higher frequency noise side band will contribute to the noise at a lower frequency by beating with another longitudinal mode (Poustie [ 19921). As a result it will be more difficult to find a noise reduced frequency window. Figure 8 shows that broad GAWBS noise windows can be found even at low repetition rates if the repetition rate of
396
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
120
130
140
150 160 Laser repetition rate (MHZ)
K § 4
170
180
Fig. 8. The apparent frequencies of GAWBS modes as a function of laser repetition rate as calculated From GAWBS measurements at various laser repetition rates. Horizontal lines represent the actual frequency of the corresponding GAWBS mode. The inclined lines correspond to higher-frequency modes which are transferred to lower frequencies by beating with neighboring longitudinal modes of the laser. The thickness of the lines indicates the strength of the GAWBS peaks. At a laser repetition rate of 145 MHz there is a particularly wide GAWBS free frequency window From 40 to 55 MHz (Strofiner, Konig, Spalter, Sizmann and Leuchs [1997]).
the mode-locked laser is adjusted properly (StroRner, Konig, Spalter, Sizmann and Leuchs [ 19971). Yet another idea to reduce GAWBS further was to use more intense pulses (Carter, Drummond, Reid and Shelby [1987], Shelby, Drummond and Carter [ 19901). With a higher peak intensity the same nonlinear phase shift is acquired in a shorter fiber. GAWBS, being a linear process, introduces less noise in a shorter fiber since it grows in proportion to the fiber length and the relative noise is independent of intensity. In the case of solitons, a higher peak power requires a shorter pulse duration. As a result, the advantage of high peak power of solitons for GAWBS noise reduction is limited, because stimulated Raman scattering becomes more prominent for shorter pulses. The fourth idea is to subtract out GAWBS noise. To this purpose two pulses separated by a short distance are sent through the fiber. They will pick up very similar thermal noise, provided their temporal separation is substantially less than the highest GAWBS noise oscillation period. In a differential measurement the thermal phase noise will largely cancel out, and a pulse separation of loops yields a 12dB reduction of polarized GAWBS noise (Townsend and Poustie [1995]). This was used in quadrature squeezing (Bergman, Doerr, Haus and
v, P
51
QUADRATURE SQUEEZING
391
Shirasaki [ 19931) and,quantum nondemolition (Friberg, Machida and Yamamoto [1992]) experiments and will be discussed in 9 5 and 9 6, respectively.
0
5. Quadrature Squeezing
The success of various fiber quadrature squeezing experiments is displayed in table 1. After the first demonstration of continuous wave (CW) squeezing, pulsed squeezing experiments showed very promising results. The brief review of CW experiments (9 5.2) gives insight into physical as well as technological problems that soon led to pulsed squeezing (9 5.3). Before reviewing the experimental development, some properties of Kerr squeezing are discussed (9 5. l), such as the choice of local oscillator phase, the squeezing characteristic as a function of the nonlinear phase shift, and the enhancement of the nonlinear phase shift with pulses from a mode-locked laser. 5.1. PROPERTIES OF KERR QUADRATURE SQUEEZED STATES
5.1.1. Single-mode interaction Hamiltonian Quadrature squeezing is defined as the squeezing of the uncertainty ((Aa,)2)of a field quadrature, a,, below its vacuum or coherent state level, ((Aac,vac)2) = 114; i.e., ( ( A a C l 2<) 1/4. In a minimum uncertainty state the conjugate field quadrature, a,, becomes anti-squeezed according to the minimum uncertainty principle, ( ( A a c ) 2 ) ( ( A a s )32 )1/16. The standard way to describe the generation of squeezed light is to use a quadratic interaction Hamiltonian, H I = ( ~ ? 2 / 2 ) k ( a+t )x~* a 2 ] .A coherent or vacuum state evolves into a squeezed state through the unitary transformation, or U=exp[(ilh)(z/c)Hr]. The nonlinear interaction described by H I is a x(3)photon-pair interaction; e.g., degenerate parametric down-conversion and second-harmonic generation, (x cx x(2)E p ) , or degenerate four-wave mixing, 01 c( x(3)Ei). The unitary transformation for the Kerr effect involves a quartic interaction Hamiltonian (Drummond and Walls [ 19801, Kitagawa and Yamamoto [ 1986]), H I = hX(ata)’ = hXn2, with 0:x(3) 0:n2 (see 9 3), which describes phenomena different from the quadratic squeezing Hamiltonian. In general, the quartic Hamiltonian produces crescent-shaped squeezed states which resemble slightly tilted photon-number squeezed states (see initial evolution in fig. 4, 0 3). Consequently, such squeezing is less restricted in photon-number noise reduction ((An’) >, (n)1’3)than quadrature squeezed states ( ( A n 2 )2 (n)”/’)(Kitagawa and
x(’)
x
398
[v, 5 5
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS M FIBERS
Table 1 Experimental results for quadrature squeezing and four-mode squeezing in various fiber experimentsa Ref.
Rb
R b (dB) Pump light
fiber
config.
~
h
d
(radian) [l] [2]
[0.34] 0.87
[-4.71 -0.58
continuous wave (CW), 6471x11 lOOm 114m, 2 K phase modulated CW, 647 nm
linear
0.5
linear
0.62
[3]
0.80d
-0.97d
phase modulated CW, 647nm
linear
0.4
phase modulated CW, 6761x11
114m, 2 K
linear
0.1
[4]
0.68
-1.7
0.2 ps soliton, 1.55pm
5m, 77K
ring
4.1
114m, 2 K
0.78
-1.1
0.2ps soliton, 1.55 pm
5m
ring
4.1
[5]
0.32
-5.0
100ps pulse, k” = 0, 1.3 pm
50 m
ring
1.5
[6]
0.50
-3
lOOpsgulse, k”=O, 1.32pm
50 m
ring
2.7
[7]
0.65
-1.9
0.32 ps pulse, k”
1 m, DS
ring
n.d.
0.69
-1.6
0 . 3 2 ~ pulse, s k N = O , 1.48pm
40cm, DS
ring
n.d.
0.74
-1.3
0.32 ps pulse, k” =0, 1.48 pm
40cm, DS
linear
n.d.
[8]
0.63
-2.0
100ps pulse, k” > 0, 1.06 pm
50 m
[9]
0.31
-5.1
17ps pulse, k” = 0, 1.3 1 pm
90 m
ring ring
4.2
0.40
-4
17ps pulse, k“ = 0, 1.31 pm
90 m
ring
4.2
0.50
-3
0 . 1 5 4 . 2 ~ ssoliton, 1.55pm
8 cm
ring
n.d.
[-21 -3f
phase modulated CW, 1.56pn
7.7km, DC
linear
0 . 1 5 ~ ssoliton, 1.55pm
20 cm
linear
[lo]
[111 [0.61 [12] 0.50f
= 0,
a R is the ratio of observed photocurrent noise power to the shot-noise level. The pulse widths are FWHM values ( Z F ~ = H 1.763To for a secb pulse). The fibers were at room temperature unless otherwise indicated. Noise reduction values in square brackets refer to classical noise reduction, where R is the ratio of observed photocurrent noise power to the classical input noise level.
References [l] Levenson, Shelby and Perlmutter [1985] [2] Shelby, Levenson, Perlmutter, DeVoe and Walls [1986] [3] Schumaker, Perlmutter, Shelby and Levenson [ 19871 [4] Rosenbluh and Shelby [1991] [5] Bergman and Haus [1991] [6] Bergman, Doerr, Haus and Shirasaki [I9931
1.48 pm
2.7
0.3 -1
DS, dispersion-shifted fiber; DC, dispersion-compensating fiber. Four-mode squeezing. n.d., no data. Squeezing based on cross-phase modulation.
[7] Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [1993] [8] Nishizawa, Kume, Mori, Goto and Miyauchi [1994al [9] Bergman, Haus, Ippen and Shirasaki [1994] [lo] Bergman [I9961 [I 11 Lorattanasane and Kikuchi [1997a] [12] Margalit, Yu,Ippen and Haus [1998]
v, § 51
QUADRATURE SQUEEZING
399
Yamamoto [1986]; see aiso table 1 therein). This limit, however, cannot be reached with fiber squeezing experiments 1 4 . In the practical situation of fibers, where the small nonlinearity requires a large photon number, both descriptions produce quadrature squeezed states; i.e., the quartic and the quadratic interaction Hamiltonian yield equivalent results if the nonlinear self-phase shift of the pump is taken into account in the parametric model” (Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987], note 6 ; Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990], p. 116). Some unique properties of Kerr squeezing can already be understood from the properties of the quartic interaction Hamiltonian. Firstly, Kerr squeezing is a self-squeezing effect of the pump mode. One consequence is that it is not squeezing the vacuum state. The vacuum state is the lowest energy eigenstate of the field and therefore it cannot act as a pump for the nonlinear process. In comparison with a Kerr-squeezed state, a squeezed vacuum is easier to characterize experimentally, because the condition that the local oscillator amplitude should be substantially higher than the signal amplitude can be fulfilled readily without saturating the detectors. A special effort is required to generate a squeezed vacuum via the Kerr effect; e.g., by interference of two bright squeezed states in a balanced fiber loop (Shuasaki and Haus [1990]), as will be discussed in $5.3.2. Another consequence of self-squeezing of the pump is that the four photons involved in the elementary process have nearly the same frequency so that phase matching conditions are greatly relaxed. The large bandwidth of the far off-resonant nonlinear susceptibility in glass fibers
l 4 The creation of a detectable crescent shape of the quasiprobability distribution (QPD) would require a Kerr nonlinearity so strong that a few pump photons would suffice to induce squeezing. In practical situations of fiber squeezing, however, -(n) = lo8 photons are required for detectable squeezing. In order to achleve a detectable curvature or to reach the quadrawe squeezing limit with fibers, more than 27dB= 110 l ~ g [ ( n ) - ” ~ ]of[ photon-number noise reduction below the standard quantum limit is required. It will be difficult to observe more than lOdB of squeezing due to detection efficiency and other limiting factors. To date, the observed squeezing from fibers is still near 5 d8. l 5 In order to correctly predict the observed orientation of the squeezed ellipse, the nonlinear phase shift of the pump, E,, must he taken into account (Levenson, Shelby and Perlmutter [1985]). The
a k a ~ aA , is degenerate four-wave mixing Hamiltonian is then modified to H I c c ~ ( ~ ) [ E ~ I ’ where the amplitude quadrature operator as defined by Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987, p. 14771, in the rotating frame of the pump wave. As can be seen from this modified Hamiltonian, only amplitude fluctuations of the total field contribute to the nonlinear interaction, and the nonlinear interaction cannot alter amplitude fluctuations because of [ a A , H [ ]= 0 (Milhum, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987]), as is expected for Kerr squeezing.
400
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, 0 5
thus allows for ultrashort pulses with high peak powers to be used for attaining a strong nonlinear effect in short fibers. Secondly, [a,HI]= 0 implies that the photon number is a constant of motion, and therefore the Poisson distribution of a coherent input state will remain unchanged. This is obvious physically if the glass fiber is seen as a highly transparent medium whch produces only an intensity-dependent phase shift of the beam. Photon-number squeezing from a coherent input beam will not occur in a Kerr interaction unless additional measures are taken to rotate the ellipse. In the CW case, this rotation was demonstrated with strong dispersion; e.g., by reflection from a cavity (Levenson, Shelby and Perlmutter [ 19851) and by highly dispersive fibers (Lorattanasane and Kikuchi [ 1997a1). Interference with a weak coherent beam has also been proposed for this purpose (Kitagawa and Yamamoto [ 19861). In the case of high-power pulses, self-phase modulation (SPM) leads to spectral broadening allowing for other' mechanisms, e.g., spectral filtering, to be used for obtaining photon-number squeezing (see 8 7). Thirdly, the quartic Hamiltonian introduces higher moments into the quasiprobability distribution (QPD) of the field. Numerical calculations performed with small photon number reveal the unique crescent-shaped form of the QPD of Kerr squeezing (fig. 4), which is different from quadrature-squeezed QPDs. This unique feature of Kerr-squeezed states cannot be detected with present technology. The large photon number and small Kerr-induced phase diffusion allow for linearization of the quantum fluctuations. The following linearized semiclassical approach is sufficient to model the detectable squeezing characteristics as a function of the average nonlinear phase shift. 5.1.2. Single-mode linearized approach
The intensity-dependent phase shift provides a physical picture for the squeezing mechanism, which will be presented here in more detail because it will also be useful for the discussion of QND experiments in 0 6 . In the absence of losses, a coherent beam of constant average power PO=IoA,ff acquires a nonlinear phase shift of (see eq. 3.3):
where A,R is the effective cross-sectional area of the mode, y = ( n t / A , ~ko, ) and ko = 27~112.In the following, we use A = a A for the amplitude quadrature and B = a$ for the phase quadrature to avoid complexity in the formal presentation. As is shown in fig. 9a, two different amplitudes A1
Y P 51
QUADRATURE SQUEEZING
401
/I Fig. 9. (a) In a linearized picture, the uncertainty region of coherent light becomes squeezed to an ellipse because a lower amplitude ( A l ) produces a phase delay and a higher amplitude ( A 2 ) produces a phase advance relative to the average nonlinear phase. As a result, the initially (z = 0) uncorrelated amplitude and phase quadratures become progressively correlated. @) The correlation of amplitude and phase quadratures deamplifies field fluctuations below the vacuum level at phase angle 8.(The figure is not drawn to scale. The uncertainty region should be lo4 of the amplitude.)
nonlinear phase shifts q4 ,NL < $ ~ , N L . Consequently, the amplitude uncertainty of the coherent state will lead to spreading of the phase uncertainty and will transform the coherent state into a quadrature-squeezed state. The amplitude uncertainty, however, will remain unchanged. The optimum phase angle 8 for sub-shot-noise measurements can be derived from linearizing the nonlinear phase shift #NL of eq. (5.1) for a small amplitude uncertainty A A << ( A1,
402
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS h'FIBERS
where we have used IAI'=Po/(h@) and
[v, 0 5
7= hooy. The rms-amplitude uncer-
tainty, ([AA(0)]2)1'2 = $, represents the amplitude uncertainty of a coherent state, which is the size of vacuum field uncertainty. The conjugate, i.e., phase quadrature uncertainty, AE, can be expressed in terms of the phase uncertainty consisting of both the initial phase uncertainty, A@(O),and the nonlinear phase diffusion, A@NL. Using AB = A@(z)IA I = [A@(O)+A@NL(z)] [ AI, we obtain: M(z) = U ( 0 ) + 2 7 IAI2 ~ AA(0) = hB(0) + 2
4 AA(0). ~ ~
(5.3)
The second term on the right-hand side shows that phase difision increases with nonlinear phase shift. This self-phase-modulation term progressively correlates amplitude and phase, which are initially uncorrelated ((AA(0) AB(0))= 0). Quadrature squeezing is observed when the local oscillator projects out the correct combination of phase and amplitude noise. This occurs under a phase angle 8 where the amplitude and phase fluctuations nearly cancel each other; i.e., where AA(z) cos(8) M -AB(z) sin(@)as graphed in fig. 9b. The optimum squeezing angle can be derived easily from minimizing the uncertainty in a rotated frame (X,Y). The fluctuations in the &rotated frame (X,Y) are related to fluctuations in frame (A& by AX(,-) = +A&) cos(8) +All(,-)sin(8) =
{AA(O)[cot(B)+ 2@NL]+ h B ( 0 ) ) sin(@,
AY(z) = -A&) =
sin(8) + M ( z ) cos(8)
{AA(O)[tan(B)- 2@NL]+ AB(0)) cos(8),
(5.4a) (5.4b)
where AA(0) and AB(0) are the input fluctuations of the coherent field ((AA(0)2)(AB(0)')= 1/16). By minimizing (AX(z)') with respect to the phase angle 8, one finds the optimum squeezing angle: 8(z) = -iarccot(@NL) = -$arccot(yPoz>.
(5.5)
Figure 10 shows the evolution of squeezing with propagation in the fiber. The squeezing ellipse rotates from O = - $ X at z=O towards 8=0 for z -+ 00 as it becomes progressively squeezed, while the amplitude variance (AA2) remains unchanged. Expressing 8 in terms of @NL,the squeezed and antisqueezed variances are simply (5.6a) (5.6b) and the minimum uncertainty of the squeezed state is preserved in the (X,Y)frame; i.e., ((AX~I)*)((AYO)') = 1/16.
v, o 51
403
QUADRATURE SQUEEZING
--________-
4 O([
O
,-
.I
'.
_ c -
-
/
-
-
--I
I
W
I.
0
.
. 0.5
.
1
1
1.5
2
2.5
3
nonlinear phase shift QNL (rad)
Fig. 10. Evolution of squeezing and antisqueezing with nonlinear phase shift (or propagation distance z in units of LNL= l/(yPo)) in the fiber, derived from a linearized single-mode model. A and B represent the amplitude and phase quadratures, as shown in the previous figure. The squeezing ellipse rotates from 8=-45" near #NL = O (z=O) towards 0" for increasing h~as it becomes progressively squeezed, as shown on top. The amplitude variance (AA2) remains unchanged. The squeezing and antisqueezing ratios, R,Y(&)= (AX,(&,)2)/(AX(0)2) and Ry(@"L) (see eqs. 5.6), are shown below. Their product corresponds to a minimum uncertainty state as indicated by the dashed line.
For example, 10 dB of squeezing requires a nonlinear phase shift of 1.42 radian (fig. 10). With a standard telecommunication fiber of L=200m (<1% loss) at a wavelength of A = 1.55 pm and a nonlinearity coefficient of y = 3 W-' km-', a power of more than 2.4W is required to achieve this squeezing value. An enhancement of the nonlinear phase shift can be expected if mode-field diameters smaller than the assumed 7-9 pm are used. To this end, and because
404
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
tv, 8 5
of 7 0: ( w ; / A e ~shorter ), wavelengths may be of advantage for demonstration of squeezing as long as overall losses remain small. As another example, the effect of a nonlinear phase shift in a long-distance transmission system is estimated. Losses must be included in the derivation of the nonlinear phase shift for long fibers. The path-averaged power leads to an effective propagation distance, Zeff = exp[-az] dz = (l/a)( 1 - exp[-az]), such that POis the input power eq. (5.1) and z is replaced by z , ~ .Assuming a 50 km amplifier spacing and a = 0.2 d B h , the effective length of each segment is well approximated by z,ff M l l a = 22 km. A launched power of 1 mW into a 2500 km transmission system produces a total phase shift of approximately ; r ~over the 50 segments. Squeezing of the input quantum uncertainty will be eliminated by attenuation and noise from amplifiers. However, the nonlinear phase shift will convert classical input amplitude noise and amplifier noise into phase noise, which limits the performance of phase-shift keying coherent communication systems (Gordon and Mollenauer [19901). The squeezing mechanism discussed in this chapter will also “squeeze” the classical input and amplifier noise in a certain quadrature below the level of a linear system (Lorattanasane and Kikuchi [ 1997a1). In the presence of GVD, the squeezing ellipse rotates such that the combined effect of SPM and GVD leads to directly detectable enhancement or reduction of amplitude noise, which is important for incoherent systems (Kikuchi [1993], Lorattanasane and Kikuchi [ 1997b1). We have neglected fiber properties such as dispersion, losses, stimulated Brillouin scattering (SBS), guided acoustic-wave Brillouin scattering (GAWBS) and stimulated Raman scattering (SRS) in t h s simple squeezing model. A detailed feasibility study for squeezing with fibers was performed (Levenson, Shelby, Aspect, Reid and Walls [1985]) even before the first fiber experiment. In the first squeezing experiments, the CW power level was limited by stimulated Brillouin back-scattering. A solution to this problem was to use a phasemodulated pump (Shelby, Levenson, Perlmutter, DeVoe and Walls [ 19861) or to use ultrashort pulses with a correspondingly high peak power and broad spectrum in later experiments. In both cases, the spectrum is broadened to keep the optical power per mode below the Brillouin threshold. With the high peak power of ultrashort pulses, an enhanced nonlinear phase shift is acquired in shorter pieces of fiber which add less GAWBS noise to the transmitted field. Therefore, pulsed squeezing combines several advantages over CW squeezing in fibers.
Ji
5.1.3. Power enhancement with ultrashort pulses
The enhancement of nonlinear phase shift by using ultrashort pulses is illustrated
Y P 51
QUADRATURE SQUEEZING
405
briefly. A mode-locked laser emits pulses of duration z (FWHM) at a repetition rate f rep. The pulse peak power, PO,is then related to the average power by: (5.7)
assuming a sech2 pulse shape. Pulses of 1 ps duration at frep = 100 M H z yield a power enhancement of PolPavg = 0.88 x lo4. Now, only 21 mW of average power in 2 m of fiber seem sufficient to achieve a nonlinear phase shift of 1.42radian (10 dB squeezing). However, this is the nonlinear phase shift of the peak of the pulse only. When pulses are used in the zero-dispersion regime, SPM introduces a chirp across the pulse. Therefore, the local oscillator phase t) is not optimized for the pulse as a whole (Shirasaki and Haus [1990]). This limits the amount of observable squeezing to a maximum of approximately 7 dB assuming a Gaussian pulse shape and 100% detection efficiency (Bergman, Haus, Ippen and Shirasaki [19941), as is shown in fig. 15 in Q 5.3.5. When fundamental solitons are used, the pulses remain free of chrp and a phase-shifted local oscillator soliton can efficiently project out the squeezed quadrature (Drummond, Carter and Shelby [1989], Haus and Lai [1990], Lai [1993]). The fact that fundamental solitons propagate without chrp makes soliton squeezing conceptually and experimentally similar to CW squeezing. In order to exploit this analogy for a squeezing estimate, the nonlinear phase shift, $NL,S,of a fundamental soliton must be used. It is related to the soliton peak power, PO,^, by (Agrawal [1995]): (5.8) i.e., the soliton phase shift is half the value of the phase shift of a square pulse with same peak power. The enhancement of nonlinear phase shift of 1-ps solitons from a 100-MHz mode-locked laser compared with continuous wave light of the same average power is then ~NL,S(Z)/$NL,CW(Z)=4.41 x lo3, and 10 dB of squeezing is predicted in the simple model for 2.4 mW average power in 45 m of fiber. In general, pulsed fiber squeezing is governed by the quantum nonlinear Schrodinger equation (QNSE), which takes into account both SPM and GVD effects. Quadrature squeezing of QNSE solitons is well described in the literature (Haus, Watanabe and Yamamoto [ 19891, Drummond, Shelby, Friberg and Yamamoto [1993], Lai [1993]). An important question is: What is the
406
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y 0
5
optimum pulse width of a soliton for quadrature squeezing? Shorter pulses with correspondingly higher peak power produce the same nonlinear phase shift in shorter fibers than picosecond solitons. Losses may then be neglected in short pieces of fiber. When intrinsic fiber noise, such as GAWBS noise is taken into account, a clear advantage of shorter solitons is predicted over longer pulses (Shelby, Drummond and Carter [ 19901; see also $ 4.3). However, for ultrashort pulses of z < 1ps duration, stimulated Raman scattering becomes important. Simulations of sub-picosecond soliton quadrature squeezing still show a potential for squeezing exceeding lOdB even for pulses as short as lOOfs (Drummond, Shelby, Friberg and Yamamoto [1993], Yu and Lai [1995]) and 18Ofs (Drummond and Hardman [1993], IGirtner, Dougherty, Haus and Ippen [ 19941, Werner [ 1996133).The experimental results with sub-picosecond solitons, so far limited to 3 dB of inferred squeezing, will be discussed in $ 5.3.4. 5.2. EXPERIMENTS WITH CONTINUOUS-WAVELASER LIGHT
Continuous-wave squeezing in fibers (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]) was achieved shortly after the first observation of squeezing of quantum noise (Slusher, Hollberg, Yurke, Mertz and Valley [1985]). The problems and prospects of squeezing in fibers were studied in great detail (Levenson [lp84], Levenson, Shelby, Aspect, Reid and Walls [1985]) and the squeezing process in fibers was demonstrated with classical input noise (Levenson, Shelby and Perlmutter [1985], Levenson and Shelby [ 19851) before quantum noise squeezing was observed for the first time. Squeezed state generation in single-mode optical fibers requires a nonlinear phase shift that develops fast enough as compared with damping and intrinsic fiber noise. The intrinsic fiber noise sources are due to two acousto-optic scattering processes: stimulated Brillouin scattering (SBS) and weak spontaneous guided acoustic-wave Brillouin scattering (GAWBS), which was discovered in connection with the squeezing experiments as a new classical noise source physically connected with fiber media (Shelby, Levenson and Bayer [ 1985a], Shelby, Levenson and Bayer [1985b]; see also $4.3). Both must be suppressed in order to observe squeezing. In the first fiber squeezing experiment (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]), a 114-m fiber had to be immersed in liquid helium in order to reduce GAWBS noise. Yet, scattering noise could not be reduced far enough and remained the dominant noise source that limited the photocurrent noise reduction, even though the measurement frequency range was chosen to detect squeezing between GAWBS resonance frequencies. A further problem
Y 9 51
QUADRATURE SQUEEZING
401
was the reduced SBS threshold at liquid helium temperature, lowered to approximately 1OmW. In order to send 100-200mW through the fiber for a typical squeezing measurement, SBS had to be suppressed. This was acheved with phase-modulating the pump beam, thus distributing its power over approximately 25 frequency components. The detection of Kerr quadrature squeezing required a conversion into a state with reduced amplitude fluctuations. To this end, the quadrature squeezed light was reflected off a phase-shifting cavity (Levenson, Shelby and Perlmutter [1985], Galatola, Lugiato, Porreca, Tombesi and Leuchs [1991]). The effect of the cavity can be understood by looking at the squeezed state as three modes: a strong classical carrier mode at the pump frequency and two sideband modes above and below the carrier frequency, the offset being the frequency of the measured noise band. The strong cavity dispersion introduces a phase shft of the sideband frequencies relative to the carrier mode. The sidebands generate the photodetector noise current by a heterodyne beat with the pump frequency, which acts as a local oscillator when incident on a detector. This scheme required the phase modulator to be driven at exactly the mode-spacing frequency of the phaseshfting cavity so that all 25 modes became resonant simultaneously (Shelby, Levenson, Perlmutter, DeVoe and Walls [ 19861). Up to 12.5% (0.58 dB)of photocurrent noise reduction below shot noise was observed. Due to various imperfections in the detection channel, e.g., linear attenuation and imperfect spatial mode overlap, only approximately 50% of the noise suppression was actually detected. A squeezing of 25% (1.2 dB) can therefore be inferred for the transmitted light. There are three outstanding points in this first fiber squeezing experiment. Firstly, and fundamentally important, optical fibers are indeed capable of deamplifying quantum noise below the vacuum level. Secondly, GAWBS, a material-dependent light scattering process that is not physically related to the squeezing process, severely limited the amount of squeezing. A special effort must be made to eliminate this classical noise source in order to acheve much improved quantum noise reduction. Finally, a third and technical result was the demonstration of the phase-shifting cavity as a hghly dispersive element for turning Kerr quadrature squeezing into below-shot-noise amplitude fluctuations. Another configuration, the fiber ring resonator (Shelby, Levenson, Walls and Aspect [ 198611, was investigated for squeezing with fibers, but bistability and strong phase noise made it impractical for observation of squeezing. This configuration was later used with more success for a quantum nondemolition measurement (Bachor, Levenson, Walls, Perlmutter and Shelby [ 19881; see Q 6).
408
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
rYP5
The early theoretical and experimental results of fiber squeezing are presented in a comprehensive review (Milburn, Levenson, Shelby, Perlmutter, DeVoe and Walls [1987]). Yet another experimental squeezing configuration involved the coupling of two laser beams in an optical fiber and detection of a quantum correlation (termed four-mode squeezing) with a dual-.frequency detector (Schumaker, Perlmutter, Shelby and Levenson [ 19871, Levenson and Shelby [ 19871). The experimental method was very similar to the first fiber QND apparatus (Levenson, Shelby, Reid and Walls [1986]), which was a modification of the first fiber squeezing experiment (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]). Up to 20% (0.97 dB) of squeezing was achieved (Schumaker, Perlmutter, Shelby and Levenson [1987]). Four-mode squeezing has the advantage of increased immunity to GAWBS noise and achieved additional noise suppression by using more than one pump frequency (Schumaker [1985, 19861). 5.3. EXPERIMENTS WITH ULTRASHORT PULSES
5.3.I . Ultrashort pulses for GAWBS noise suppression Ultrashort pulses generate much less SBS and GAWBS, thus removing the limitations of the CW experiment to a great extent. Linear scattering noise such as GAWBS scales with the product of fiber length and average power. Squeezing, however, scales with the nonlinear phase shift which is proportional to the peak power-length product, therefore giving an advantage to short pulses (Shelby, Drummond and Carter [ 19901). Due to the broad spectrum of ultrashort pulses, the SBS threshold is not reached. The detection of quadrature squeezing is a coherent measurement; i.e., it involves a local oscillator to extract the signal from a combination of pulse amplitude and phase (fig. 9). Therefore, chirp-free propagation becomes an important issue. Temporal or spectral broadening and chrping of the pulse is expected due to group-velocity dispersion (GVD) and self-phase modulation (SPM) in the fiber. There are two classes of fiber squeezing experiments with ultrashort pulses. One class of experiments (Rosenbluh and Shelby [ 199I], Bergman [ 19961, Margalit, Yu, Ippen and Haus [1998]) employed optical solitons, which propagate free of c h q and free of distortion by balancing GVD with SPM, even in the subpicosecond domain. These experiments were the first to demonstrate squeezing with ultrashort pulses and will be described in 9 5.3.4.
v, 5 51
QUADRATURE SQUEEZING
409
The second class of experiments, to be discussed in $5.3.5, did not take advantage of soliton effects. Instead, these experiments employed pulses centered near 1 pm (Nishizawa, Kume, Mori, Goto and Miyauchi [1994a]), near the zero-dispersion wavelength (1.3 pm) of a standard fiber (Bergman and Haus [1991], Bergman, Doerr, Haus and Shirasaki [1993], Bergman, Haus, Ippen and Shirasaki [1994]), or near zero dispersion (A= 1.48 pm) of a dispersionshifted (DS) fiber (Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [ 19931). Compared with solitons at 1.5 pm wavelength, there were two advantages to shorter wavelengths. Firstly, the smaller mode-field diameter enhanced the nonlinear phase shift (see $ 5.1.2), and secondly, the quantum efficiency of the detectors at 1.3 pm was considerably higher than near 1.5 pm l 6 (Shirasaki and Haus [1990]). When the pulse duration is long enough to allow for a narrow spectrum, the pulse will maintain its temporal shape. The disadvantage of pulse propagation at zero dispersion is the variation of the squeezing phase across the pulse, O=O(T). The local oscillator phase cannot be optimized for the pulse as a whole, which fundamentally limited squeezing to 7 dB. Up to 5.1 dB was observed with 85% total detection efficieny, showing the maturity of these experiments (Bergman, Haus, Ippen and Shirasaki [ 19941). Another consideration for pulsed fiber squeezing is the spectral properties of the residual GAWBS noise. When pulses are used, the large number of high-frequency GAWBS resonances are down-shifted into the radio-frequency measurement window owing to beating of components between the GAWBS frequencies and laser modes, as discussed in $4.3 (fig. 8). This problem can be solved by broadband suppression of GAWBS noise with a dual-pulse method (Townsend and Poustie [1995], Bergman, Doerr, Haus and Shirasaki [1993]), by proper adjustment of the laser frequency (Stroher, Konig, Spalter, Sizmann and Leuchs [ 19971) or by a high repetition rate of the laser (Bergman, Haus, Ippen and Shirasaki [ 19941). 5.3.2. Generation and detection of pulsed quadrature squeezing using a balanced Sagnac loop The detection of pulsed squeezing using a phase-shifting cavity, as demonstrated with CW experiments, would be feasible. However, it will be more difficult to implement than in the 25-mode case of the phase-modulated CW pump.
l6 Today, the quantum efficiency of squeezing detectors at 1.5 wm exceeds 90%, thanks to progress in lightwave communication in this wavelength region.
410
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, 8 5
Fig. 11. Balanced Sagnac interferometersand homodyne detector for generation and detection of a squeezed vacuum. A pump pulse is split into two counterpropagating pulses by a 50/50 beam splitter (BSl). A squeezed vacuum (SV) is generated from the two bright squeezed states at the transmission output port of the interferometer whde the pump light re-emerges from the reflection port. A fraction of the light that is reflected from the fiber loop is used as a local oscillator (LO) for detecting squeezing at phase angle 8 in the balanced homodyne receiver (50/50beam splitter BS2) (Shirasaki and Haus [1990]).
Approximately lo4 modes are excited in sub-picosecond pulse trains with a 2-THz spectral width and a 200-MHz repetition frequency. The cavity mode spacing would have to be exactly tuned to the pulse repetition frequency, so that all modes of the pulse spectrum resonate simultaneously. A &fferent squeezing and detection scheme, used for almost all pulsed quadrature squeezing experiments, is shown in fig. 11. The balanced loop was initially proposed by Shirasaki and Haus [1990]. It was implemented for the first time by Rosenbluh and Shelby [1991] in the soliton squeezing experiment, and by Bergman and Haus [1991] for pulses centered at the zero-dispersion wavelength. It marked the beginning of a series of pulsed quadrature squeezing measurements. The balanced Sagnac loop generates a squeezed vacuum and a local oscillator by interference of two bright quadrature-squeezed states (fig. 11). Two pulses of equal intensity propagate in each direction around the loop and become squeezed. Their linear and nonlinear phase evolution are identical because the pulses propagate through the same fiber length with the same average power. This configuration is self-stabilized against any index fluctuations on a time scale longer than the transversal time. Upon recombination at the beam splitter, a squeezed vacuum is generated by destructive interference at the dark output
v, § 51
QUADRATURE SQUEEZING
41 1
port. Similarly, the constructive interference at the other output port generates a bright pulse. The full power of the reflected pump pulse or a fraction of it is used as a local oscillator. The squeezed vacuum and the local oscillator are then overlapped and detected in a balanced homodyne receiver. By varying the phase 8 of the local oscillator with respect to the squeezed vacuum state, the squeezing signal is optimized. Low-frequency phase noise from thermal or acoustic refractive index variations in the fiber are greatly reduced in this differential method, because the local oscillator is derived from the pump light that re-emerges from the loop. The bandwidth of acoustic and thermal noise suppression is proportional to the inverse of the loop length, thus giving another advantage to high peak power pulses and short loops. 5.3.3. Generation and detection of pulsed quadrature squeezing using a linear conjgwation An alternative, linear configuration also generates a squeezed vacuum from self-
stabilized interference of two bright squeezed states (Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [1993]) (fig. 12). The advantage of the linear setup is
TPBS
Fig. 12. Experimental outline for the generation and detection of a squeezed vacuum with a selfstabilized, balanced linear interferometer. A pump pulse is split into two orthogonally polarized pulses of equal energy, which propagate at different speeds on the slow and fast axis of a polarizationmaintaining fiber. Upon reflection, the pulses exchange polarization (QWPis a quarter-wave plate). The two pulses are automatically temporally and spatially mode-matched when they exit at the input end. The special polarizing beam splitter (SPBS) reflects 100% of the squeezed vacuum and 10% of the pump which is used as a local oscillator (LO). The adjustment of the LO phase 0, the overlap with the squeezed vacuum (SV) and the balance of the homodyne detector are acheved with the quarter-wave plate (QWP),half-wave plate ( H W P ) and polarizing beam splitter (PBS) (Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [1993]).
412
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
tv, § 5
HWP Laser
I I
Fig. 13. Experimental outline for quadrattue-squeezed vacuum generation via cross-phase modulation: A linearly polarized pump pulse is launched into a low-birefringent fiber. The orthogonally polarized field is in the vacuum state. The cross-phase modulation squeezes the vacuum fluctuations of the mode orthogonally polarized to the pump. A polarizing beam splitter (PBS) separates pump and squeezed vacuum (SV). A fraction of the pump is used as the local oscillator (LO) for detecting squeezing at phase angle 0 in a balanced homodyne detection scheme (HWP is a half-wave plate, P is a polarizer, and BS is a 50/50 beam splitter) (Boivin and Haus [1996], Margalit, Yu, Ippen and Haus [1998]).
the improved mode overlap of the local oscillator with the squeezed radiation. Instead of using a loop, self-stabilized interference is achieved in a straight piece of polarization-maintaining (PM) fiber which propagates two orthogonally polarized pulses at different group velocity due to the h g h birehgence. The two pulses are launched simultaneously, originating from one linearly polarized pulse oriented at 45” to the fast axis of the PM fiber. A quarter-wave plate and a mirror at the fiber end exchange slow and fast axes for the two pulses upon reflection. The pulses emerging from the fiber input are then automatically mode matched both spatially and temporally. Another linear scheme generates a squeezed vacuum in transmission instead of reflection and employs cross-phase modulation (XPM) in a low-birefringent fiber (Boivin and Haus [1996]) instead of SPM in a PM fiber (fig. 13). This method of XPM squeezing was pioneered with semiconductors (Udo, Zhang and Seng-Tiong [ 19941, Fox, Dabbicco, von Plessen and Ryan [ 19951).Experimental results of both linear configurations (Doerr, Lyubomirsky, Lenz, Paye, Haus and Shirasaki [1993], Margalit, Yu, Ippen and Haus [ 19981) will be discussed below.
5.3.4. Experiments with solitons
In the first soliton squeezing experiment, 200-fs solitons centered at a wavelength
v, B 51
QUADRATURE SQUEEZING
413
of 1.55 pm were propagated through 5 m of optical fiber (Rosenbluh and Shelby [1991]). This length corresponds to approximately five soliton periods. The short pulses required only 36mW average power (in each direction) in the fiber at a repetition rate of 168MHz in order to form solitons with a peak power of 1kW. Through this power enhancement, a 4.1-radian nonlinear soliton phase shift was achieved in the short loop. In the measurement range of 3 to 30 MHz, 1.7 dB or 32% of photocurrent squeezing below shot noise was observed at a fiber temperature of 77 K. A squeezing of the vacuum of 2.4 dB (43%) can be inferred, if known losses and the detector quantum efficiency (total efficiency of 75%) are taken into account. The significance of these experimental results is three-fold: Firstly, progress was achieved in obtaining stronger noise reduction in fibers and in demonstrating for the first time the nonstationary quantum properties of optical solitons. Secondly, the suppression of thermal and acoustic phase noise (GAWBS) was demonstrated successfully with enhanced peak power and a short fiber loop. Compared with the first CW squeezing experiment, the phase shift enhancement of ultrashort solitons allowed for a significant reduction of the fiber length (from 114m to 5 m) and of the average power (36 mW instead of 100-200 mW were necessary for squeezing; however, the interferometric generation of a squeezed vacuum was less energy-efficient as it required 2 x 36mW pump power). As a result, two to three times stronger squeezing was observed with the loop at 77 K than at 2 K in the fist CW experiment. Thirdly, a new noise source was identified from comparison of observed squeezing with predicted squeezing. A 4-radian nonlinear phase shift should produce more than lOdB of squeezing even in presence of GAWBS noise. Raman scattering, not included in the prediction, was seen as a possible and likely explanation for the limited amount of observed squeezing because the 200-fs solitons showed a self-frequency shift. Theoretical studies of squeezing in Raman-active waveguides were initiated (Carter and Drummond [199 l]), showing that even in the presence of Raman scattering more than lOdB of squeezing should be achieved with the balanced loop (Drummond and Hardman [ 19931). A comparison of theory and experiment of quadrature squeezing with sub-picosecond solitons is shown in fig. 14. Shorter pulses produced much less squeezing than predicted, even with the Raman effect included. A slight imbalance of the Sagnac loop was identified as a technical problem that limited squeezing. The limitation to soliton squeezing due to Raman scattering was studied in more detail (Kiirtner, Dougherty, Haus and Ippen [1994], Werner [1996b]) and an analysis of pulsed squeezing in a variety of propagation
414
[v, 5 5
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
100
$';
0)
-5-m
Zg
*-
1
- - - - - - _ _ _ _ _ _ _ 0.35 -----__
PS
0.1
0
2
4
6
8
10
Fiber Length (m)
Fig. 14. Comparison of experimental soliton-squeezing data (corrected for detection efficiency) with theoretical predictions (solid lines). The experimental data are plotted for pulses of 0.35 ps (squares) and 0.2 ps (filled circles). Both the noise d n i m a and maxima are shown for the experimental data and for the simulations of 0.1 ps, 0.2ps and 0 . 3 5 ~ spulses. The simulations include stimulated Raman scattering and GAWBS in a fiber at 77 K. The dashed line indicates a simulation for 0.35 ps pulses in the absence of thermal noise. For shorter pulses, better noise suppression is predicted because thermal effects are weaker for short propagation distances (Drummond, Shelby, Friberg and Yamamoto [1993]).
regimes, including third-order dispersion was presented (Yu and Lai [ 1995]), also predicting a high potential for soliton quadrature squeezing. A new experiment with only 8 cm of fiber in the balanced Sagnac loop (Bergman [ 19961) is under way to overcome present limitations. Preliminary measurements show up to 3 dE3 of noise reduction below the shot-noise level. The generation of a squeezed vacuum without a Sagnac ring was demonstrated recently with a 150-fs pump pulse in a fiber of only 20 cm length (approximately one soliton period) (fig. 13). The vacuum mode was orthogonally polarized to the pump and was squeezed by cross-phase modulation (XPM). The novel linear configuration produced 3 dB of photocurrent noise reduction below the shotnoise limit (Margalit, Yu, Ippen and Haus [1998]). The XPM squeezing mechanism was fist demonstrated in semiconductor materials (Udo, Zhang and Seng-Tiong [1994], Fox, Dabbicco, von Plessen and Ryan [1995]) and was later also proposed for optical fibers. The mechanism can be understood as periodic deamplification of a quadrature due to nonlinear polarization rotation in a low-birefringent (LB) fiber (Boivin and Haus [1996]). The advantage of this experimental setup over the loop configuration is that the output modes of the squeezed vacuum and local oscillator are matched automatically. A disadvantage is the reduced nonlinear effect due to the smaller cross-Kerr coefficient (XPM coefficient is of SPM coefficient). For a given
4
v, 8 51
QUADRATURE SQUEEZlNG
415
optical power, the linear scheme is less efficient only by a factor of $, because the loop divides the optical pump power among two pulses for squeezing (50150 beam splitter) (fig. 1 l), whereas the full pump power is used for squeezing in the linear configuration (fig. 13). The main limitation to the achievable squeezing in the XPM scheme is due to residual birefiingence, which caused a phase mismatch between pump and squeezed vacuum. The observed noise reduction of 3 dB corresponds to 5 dB of inferred squeezing when the detection efficiency is taken into account (Margalit, Yu, Ippen and Haus [1998]). The expected noise reduction was 6dB. This experiment demonstrated for the first time squeezing in fibers via XPM, using a simple and compact setup, which also for the first time employed a fiber laser (Haus, Tamura, Nelson and Ippen [1995]) as a source for pulses. 5.3.5. Experiments with non-solitonic ultrashort pulses (k" M 0)
So far, the best quadrature squeezing in fibers was obtained with pulses which propagated without GVD at 1.3 pm. Some common characteristics of experiments with nonsolitonicpulses were relatively long fibers (50-90 m Sagnac loop), 17 to loops Gaussian input pulses, and low measurement frequencies (Bergman and Haus [1991], Bergman, Doerr, Haus and Shirasaki [1993], Nishizawa, Kume, Mori, Goto and Miyauchi [1994a], Bergman, Haus, Ippen and Shirasaki [1994]). The squeezed quantum noise was measured between 40 kHz and 200 kHz. Quantum measurements at low frequencies are possible with a balanced homodyne detector which cancels classical noise from the laser, unlike directly detected amplitude squeezing 1 7 . The low-frequencyrange is particularly interesting for high-precision interferometric measurements, such as gravitational-wave detectors or laser gyroscopes, which employ low-frequency modulation, e.g., of a mirror position. If fiber gyros are used with pulsed squeezed input, however, the effects of the nonlinearity in the gyro in cascade with a Sagnac squeezer ring puts additional constraints on the system design (Haus, Bergman and Lai [1991]). In a first experiment, squeezing of up to 5dB was observed with a nonlinear phase shift of approximately 1.1 rad (Bergman and Haus [1991]). The measurement seemed unaffected by G A W S noise which was attributed to
l 7 When the excess noise of the pump source is large compared with the shot-noise level, the Sagnac ring reflector must be sufficiently well balanced. If the coupling ratio is slightly different from 1:1, the squeezed vacuum is contaminated with noisy pump light. Some part of the pump noise appears in the detector output (Nishizawa, Kume, Mori, Goto and Miyauchi [1994b]).
416
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y § 5
detection at low frequencies and to a low-noise fiber sample (Bergman, Doerr, Haus and Shrasalu [1993], Bergman, Haus and Shirasaki [1992]). The application of a squeezed vacuum to improve the sensitivity of a MachZehnder interferometer and the active cancellation of phase noise from GAWBS in the fiber ring was proposed (Shirasaki and Haus [1992]) and implemented (Bergman, Doerr, Haus and Shu-asaki[ 1993]), using the previously demonstrated 50 m fiber ring squeezer (Bergman and Haus [199 11). Classical noise in the squeezed vacuum was eliminated in a dlfferential measurement with two consecutive pulses. The two pulses pick up the same classical phase noise and are then combined to cancel any such noise. When injected into the Mach-Zehnder interferometer, the sensitivity was improved by 3 dB beyond its shot-noise limit (Bergman, Doerr, Haus and Shuasaki [1993]). While two pulses separated by 500 ps were used in this experiment for measurements at low frequencies, shorter delays of less than loops are required for bmadband cancellation of GAWBS noise by more than 12dB (Townsend and Poustie [1995]). A difficulty in this scheme is the fast phase modulation for switching the phase of one pulse by n in order to cancel GAWBS noise upon detection. Another approach for eliminating GAWBS noise in the measurement window used the high repetition rate of a mode-locked laser (Bergman, Haus, Ippen and Shirasaki [1994]). If the repetition frequency is hlgher than the bandwidth of G A W S , there will be no aliasing of GAWBS peaks into the low-frequency measurement range, according to the sampling theorem. The observed GAWBS spectrum is then similar to the spectrum obtained with a CW pump (Poustie [1992]). Squeezing can then be measured between GAWBS resonances, combined with the advantage of short pulses; i.e., the large nonlinear phase shift and suppression of SBS. With 17-ps Gaussian pulses centered at 1.3 pm from a 1-GHz mode-locked laser, 4.2radian nonlinear phase shft were obtained in a 90-m Sagnac ring, resulting in 5.1 dB (69%) of observed squeezing in the low-frequency range (80-100 kHz)(Bergman, Haus, Ippen and Shirasaki [ 19941). 7 dB of squeezing was inferred for perfect detection from the 85% detection efficiency. Slightly less squeezing (4dB, 60%) was observed in the 10-30MHz range. Figure 15 shows the saturation of detected squeezing. The variation of the optimum squeezing phase 13' across the temporal pulse profile prevented the local oscillator phase from being optimized for the pulse as a whole (Shrasaki and Haus [1990]). These experiments were reviewed by Haus [ 19951. The strategy of achieving an enhanced power confinement in a smaller core area was followed up with 100-ps pulses at 1.064pm in the positive dispersion regime. 2.0 dB of squeezing was observed at low pump power (Nishizawa, Kume,
v, 5 51
417
QUADRATURE SQUEEZING
h
E m
15-
-0
shot-noise level
..._.-...._ .._ ......._
4 W
a:
-20 I
0
I
I
I
I
.._ ...._.
I
PbLSE GONLIdEAR F"HASE irad)
I
6
Fig. 15. Comparison of experimentally observed squeezing data (filled circles) with analytical predictions (solid lines) based on the assumptions of Gaussian pulse shape and 85% detection efficiency (Bergman, Haus, Ippen and Shirasaki [1994]). The experimental data represent photocurrent noise power relative to shot-noise, measured between 80 and 1OOkHz. 5.1 dB of noise reduction was aciueved, with a maximum of 210mW (4.2radian nonlinear phase shift) propagating in each direction in the 90m fiber ring. The dashed curve represents expected optimum squeezing as shown in fig. 10 for a pulse of constant intensity with perfect detection.
Mori, Goto and Miyauchi [1994a]). Another strategy was to use fiber6 shorter than 1m. A dispersion shifted fiber in a balanced linear (fig. 12) and in a balanced ring interferometer was used (Doerr, Lyubomirsky, Lenz, Paye, Haus and Slurasaki [1993]). Here, the linear configuration was expected to yield a perfect mode match of squeezed vacuum and local oscillator. To this end, the two modes were imaged colinearly onto the detectors. However, squeezing in the linear fiber configuration was limited primarily by losses at the fiber end where the polarizations were exchanged. The Sagnac loop with only 1 m of fiber produced up to 1.9dB (35%) squeezing. The experiments show that a variety of GAWBS suppression techniques can be combined with a balanced interferometer squeezing apparatus. Theoretical models predict more than lOdB of squeezing if solitons are used instead of Gaussian pulses. Practical problems, such as a slight imbalance in the loop or in the linear configuration and a residual mode mismatch may have limited soliton quadrature squeezing so far. Only much later it was discovered that soliton squeezing works well in an inbalanced, i.e., asymmetric interferometer. This approach will be discussed in 0 7.4.
418
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
0 6.
[\!
86
Quantum Nondemolition Measurements
The Kerr effect gives rise to an intensity-dependent phase shift, due not only to self-phase modulation (SPM) but also to the intensities of other modes copropagating in the fiber. T h s coupling of modes, referred to as cross-phase modulation (XPM), produces a variety of interesting phenomena in classical optics (Agrawal [ 19951). In quantum measurements, XPM has been used recently for squeezed vacuum preparation (Margalit, Yu, Ippen and Haus [1998]) as discussed in 0 5.3.3. More importantly, in quantum optics in fibers XPM has been investigated intensively for so-called “back-action evading” (BAE) or “quantum nondemolition” (QND) detection of the photon number, the subject of this chapter. 6.1. CONCEPT
AND REALIZATION OF A QND MEASUREMENT OF THE PHOTON
NUhIBER
6.I . I . Cross-phase modulation as a QND interaction
In general, a QND measurement aims at detecting an observable X , to better than the standard quantum limit while leaving the observed quantity unperturbed. Repeated m,easurements on X , should then yield identical results. Consequently, two back-action evading measurements are necessary to demonstrate QND detection, one for preparing a quantum state and the second to measure this state. A QND measurement involves two coupled quantum systems described by the conjugate pairs X,,Y , (“signal system”) and X,,Y , (“probe system”) and by the total Hamiltonian, H =Ho + H I .The observable X , is referred to as “signal” and is, e.g., a field quadrature in an optical beam (“signal beam”). The interaction H I and the QND variable X , must satisfy certain con&tions to allow for a QND measurement. A QND variable X,is a constant of motion, [Ho,X,] = 0, of the unperturbed system described by Ho. The QND measurement is accomplished through QND coupling to Y,; i.e., HI,^,] #O, where Y , is an observable of another system referred to as “probe” and H I is the interaction Hamiltonian, depending on X,. If the commutator HI,^,] vanishes’*, the signal remains unperturbed and is
I * The original definition (Unrub [1979], Braginsky, Vorontsov and Thome [1980]) allowed for a deterministic perturbation of X, by the measurement. However, the condition HI,^,] = 0 is generally used.
Y 0 61
QUANTUM NONDEMOLITION MEASUREMENTS
419
at the same time copied onto Y,. The input-output relations for the signal and probe observables of interest are:
where G is the QND gain. The probe quadrature Y, contains a copy of the signal and becomes a noiseless or “macroscopic” copy in a perfect QND scheme (G + m). The inevitable quantum mechanical back-action noise due to the measurement is channelled into the quadrature Y,, the conjugate variable to X,. After the first proposals of QND measurements (Braginskii and Vorontsov [ 19741, Unruh [19781, Unruh [19791, Braginsky, Vorontsov and Thorne [ 1980]), various QND schemes for optical systems were discussed (Milburn and Walls [1983], Imoto, Haus and \Yamamoto [1985], Yurke [1985]). Photon-number or amplitude QND measurements through the Kerr effect were among these first proposals. The first QND experiment was realized, with some success, with the optical Kerr effect in fibers (Levenson, Shelby, Reid and Walls [ 19861). In fibers, the signal-to-noise ratio of mformation modulated onto the optical power of a mode is conserved in the presence of SPM and XPM when losses and linear dispersion are neglected. Yet, XPM provides a readout for the power, P,,of the signal mode, s, if the nonlinear phase shift, (PNL,~, of a probe mode, p, is measured, according to:
where yp = (n2/&) ko,,, with Aeff the effective mode field cross-section which is assumed to be approximately the same for both modes, and k~,,the vacuum wavevector of the probe (Agrawal [1995]). The factor of 2 in the XPM term applies to the case where signal and probe are two parallel polarized continuouswave modes at different frequencies in a linearly birefiingent lossless fiber. The factor of 2 reduces to 2/3 if orthogonally polarized beams are used instead, and any XPM coefficient between these two extremes can be realized with elliptically birefringent fibers (Agrawal [ 19951). The basic idea for a QND measurement in fibers is that the Z(PM effect known from classical fiber optics could also be used for detection of a quantum signal encoded in the photon number without degrading the S N R even at the quantum level. The corresponding fiber QND scheme is shown in fig. 16. This idea must be tested within the framework of quantum mechanics. If proven right, a semiclassical approach will be taken to derive some basic properties of fiber QND.
420
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
r--T----
I I I I
Measurement
--I Back action I
signal Fig. 16. The QND scheme using cross-phase modulation in optical fibers. The intensity-dependent refractive index change due to a signal is imprinted on the phase of a probe beam and vice versa. Thus a QND measurement of the signal photon number below shot-noise can be performed by detecting the probe phase, leaving the signal photon number unperturbed. The quantum-mechanical back action due to the measurement adds noise to the the signal phase.
In the following, we use n as the photon number detected in a measurement time interval At,
where L is the corresponding length of a beam segment in free space. In the case of n being the photon number per pulse, At is related to the pulse width. A 1-ps soliton typically contains los photons.’9 The quantum nondemolition measurement properties of a fiber, i.e., the signal nondemolition, the measurement and the probe back action, can be examined using the SPM and XPM Hamiltonians (Imoto, Haus and Yamamoto [1985], Kitagawa, Imoto and Yamamoto [1987])20.
l9 For solitons, A t = 1.137, with t = 1.763To being the full width at half maximum. Because of the scaling properties of soliton parameters, Pg,s Ti= (k”(/y=const., the photon number of a soliton is n c c I / t (see 54.2). A typical 1-ps soliton contains n=1.1x108 photons for A=1.55pm, k” = -20 p s 2 h and y = 3 (W km-l in a standard single-mode fiber. 2o As in the case of squeezing, the fiber Ken nonlinearity for the QND interaction can be described by a four-wave mixing Hamiltonian (Levenson, Shelby, Reid and Walls [1986], Levenson and Shelby [1987]) or by the Hamiltonian (Imoto, Haus and Yamamoto [1985]) used in this section.
v, 8 61
QUANTUM NONDEMOLITION MEASUREMENTS
42 1
Firstly, the quantum nondemolition property expressed in eq. (6.la) is confirmed for the signal photon number n,. Using the Hamilton operator for the beams in the fiber2', H
=H ,
+H , + H I=
+ hXpni + hX,nsnp,
it can be seen immediately that there is no back action onto the signal due to the measurement because of [Hr,n,] = 0. Also, there is no secondary back action from the phase of the signal beam in the time evolution to follow because of [Hs,n,l = 0. Secondly, XPM copies the photon number of the signal onto the phase $p of the probe because H I is a function of n, and [HI,S,] # 0. Here S, is the sine operator (Carruthers and Nieto [1965]) with [n, S] = iC, where C is the cosine operator. For (n,) >> 1, a "phase operator", @,, can be used such that S , =sin(&) (Vogel and Welsch [ 19941).For a small phase shift the sine component represents the probe phase shift. A problem arises from SPM: because H , contains the probe photon number np, it follows from [n,S]=iC that [Hp,Sp]+.O, and consequently the probe phase is perturbed by np. The effect of SPM in the QND experiments with fibers will be discussed in 0 6.1.3. Thirdly, the Kerr QND interaction provides minimum back-action phase noise; i.e., a sub-shot-noise detection of the signal preserves a minimum uncertainty state with ((AnF')2)((A&'"t)2)= 1/4 in the absence of SPM, as was shown by Imoto, Haus and Yamamoto [1985] and Kitagawa, Imoto and Yamamoto [1987]. 6.I .2. Semiclassical approach When the fluctuations are small compared with the mean field, a linear treatment of photon-number and phase fluctuations can be used (Reynaud, Heidmann, Giacobino and Fabre [1992], Courty and Reynaud [1992]). The input-output relations for the variables of interest are (Courty, Spalter, Konig, Sizmann and Leuchs [19981):
An':
= An:,
A#':
= A$
+ o A n r + KAnF,
(6.4a,b)
where K represents the QND gain (G in eq. 6.lb) for the signal, An,, in the probe phase output, A@.':
The ordering of the creation and annihilation operators in H depends on the medium (see Q 3). If the ordering is changed, e.g. from H,=hXs(aba,)2 = h ~ , n $to H , = ~ ~ ( a ~ ) *=Xsns(n, ( a ~ ) * - I), the difference is the appearance of additional terms which only provide a constant phase shift which is unimportant (Imoto, Haus and Yamamoto [1985]). 2'
422
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
rv, 5
6
Signal-to-noise ratios ( S N R s ) are, practical quantities for determining the actual signal degradation and the measurement error, because they are based on readily available quantities in a single back-action evading experiment. The transfer of the S N V of the signal input to the S N V t of the signal output and to the S N V ' of the probe output allows one to compare a QND experiment with the performance of a perfect beam splitter (SNF$"' + S w t = S N V ) . The beam splitter serves as a reference at the borderline between classical and quantum regimes, and
is required for a quantum measurement. An independent proposal (Holland, Collett and Walls [1990]) for characterizing a nonideal QND measurement required knowledge of the input fluchations which cannot be obtained in a single back-action evading experiment. The criterion (6.5) was fist introduced by Imoto' and Saito [1989; eq. (16) therein] and was later adopted as one of two criteria for benchmarking of QND experiments (Grangier, Courty and Reynaud [1992], Poizat, Roch and Grangier [ 19941). The input signal in eq. (6.4a) may be taken as a photon-number variation, A n y d , and the input noise as quantum noise, An: ,such that An: =An: + A n y d . If we assume the fiber to be perfectly transparent, the input signal-to-noiseratio, S w = ((AnF"d)2)/((An:)2), is not degraded; i.e., S w ' = S w . The phase fluctuations in eq. (6.4b) are obtained from eqs. (6.2) and (6.3) or explicitly from
The coefficients used in eq. (6.4b) are therefore:
These expressions suggest a physical interpretation of the QND gain, namely K being the cross-phase modulation of the probe per photon in the signal. Analogously, (J represents the self-phase modulation per photon of the probe.
v: 5 61
QUANTUM NONDEMOLITION MEASUREMENTS
423
SPM is a noise source that degrades the output signal-to-noise ratio S N V ' of the probe, ((Anf"d>2) S T " '
= K2
((A@:)2) + a2 (
+ K~ ((An:)2))
The experimental challenge is to maximize the QND gain K and to suppress SPM noise a2((AnF)2)and other background phase noise sources such as GAWBS. In the limit of infinite gain, we find that S N V ' = S N V and in eq. (6.5), S N V ' + SNRF' = 2 S N V (perfect QND detection). Various fiber experiments aiming at back-action evading measurements of the signal intensity are listed in table 2. Two types of experiments were performed. One type of experiments used an on-off modulation as the signal (classical modulation transfer). The measurements in the classical regime were performed to test various setups and parameters for QND experiments. The other type of experiments were measurements in the quantum regime and used shot-noise limited power fluctuations as a quantum signal (quantum modulation transfer). These quantum nondemolition experiments were limited by background phase noise that made it difficult to achieve a probe phase readout that is dominated by the signal photon number. The general state of the art of QND detection is a probe readout that was used for a noise reduction of the signal of 3.5dE3 (55%) below the shot-noise limit with cold atoms in a trap (Roch, Vigneron, Grelu, Sinatra, Poizat and Grangier [ 19971). This raises the questions: What are the ultimate bounds, and what must be done to perform a perfect QND measurement with fibers? These questions will be discussed in 0 6.1.3 and 9 6.1.4.
6.1.3. Self-phase modulation noise in the QND measurement Before reviewing the experimental results, three strategies for optimizing a QND measurement based on cross-phase modulation are recalled brief3y: Firstly, the SPM-free case ( a=0) is presented, then SPM is included (0# 0) for the case of a fiber QND measurements, and finally a strategy to eliminate SPM noise in a fiber QND experiment is presented.
P h) P
Table 2 Experiments aiming at fiber QND detectiona Fiber length
L,R
Experimental configuration
3
Classical modulation transfer results 1 . 3 8 ~ 1 0 -continuous ~ wave, 12.6mW@ 1.31 pm/2OpW@ 1.52pm 111 n.d. continuous wave, 1mW @ 1.06 pm I -mW @ 1.3 1 pm [1,21 ~2~31 n.d. continuous wave, 1mW @ 1.06 pm / -mW @ 1.31 pm
500m 10km 10km
500m 10km 10 km
Mach-Zehnder interferometer Sagnac interferometer modified Sagnac interferometer
Quantum modulation transfer results phase modulated CW, 130mW @ 676 nm / 60 mW @ 647 nm [4,5] 0.37 n.d. [6] 0.26e n.d. continuous wave, 15 mW @ 676 nm / 16 mW @ 647 nm [7] 0.39 1.22 solitons, 2.6ps, 15pJ@1460,7nm / 3.6ps, [email protected] [8] n.d.c 1.12 solitons, 2.6ps, IOpJ@ 1460.5nm / 5ps, 6pJ@ 1456.3nm
I 14m, 2 K 13m 400m 1500m
114m lOOm llOm 150m
linear travelling wave resonant ring linear travelling wave linear travelling wave
~~~~
3 rn
sF
~
Classical modulation transfer results were obtained with 0-ff modulation of the signal beam. The quantum noise transfer results were obtained with a shot-noise limited signal pump beam. The correlation coefficient C is the fraction of probe readout photocurrent rms noise due to sigml shot-noise. n.d., no data. SignaYprobe center wavelength and power or pulse energy. Results for temporarily optimum polarization conditions. a
References [l] Imoto, Watkins and Sasaki [1987] [2] Itnoto [1990] [3] Saito and Imoto [1988] [4] Levenson, Shelby, Reid and Walls [1986]
Effective interaction length, different from fiber length due to resonator finesse (Bachor, Levenson, Walls, Perlmutter and Shelby [1988]) or pulse wak-off (Friberg, Machida and Yamamoto [1992], Friberg, Machida, Imoto, Watanabe and Mukai [1996]). The soliton collision in the experiment of Friberg, Machtda and Yamamoto [1992] occurs in an effective length of L e =~3.53LWwhere the pulse separation is less than the average pulse width (ZFWHM) = 1.763(To),and L,= (To)/(11ug2- l/ugl), with l/u, = dk/kw (Agrawal [1995]).
3 m rn
P, [5] Levenson and Shelby [I9871 [6] Bachor, Levenson, Walls, Perlmutter and Shelby [1988] [7] Friberg, Machida and Yamamoto [1992] [8] Friberg, Machda, Imoto, Watanabe and Mukai [1996]
-s w
m
v, 5 61
QUANTUM NONDEMOLITION MEASUREMENTS
425
If a coherent probe dpes not experience SPM-induced phase spreading in the measurement process ( 5= 0), e.g., in a resonant Kerr nonlinearity (Imoto, Haus and Yamamoto [1985]), the transfer of SNRs of eq. (6.8) becomes 1
S N V ‘ = SNR:
1+
((A@>’) . ((An:)’)
K2
In this case, the best measurement strategy is to use an arbitrarily strong probe beam, thus allowing its phase uncertainty to disappear (for a coherent probe, ((A@)2)=1/(4(np)) + 0 for (np) + 00). Then eq. (6.4b) becomes (a=O, (.p)
--$
00):
A$rt
=
KAnr,
(6.10)
and K , the cross-phase modulation (XPM) per photon of such a strong probe, represents a gain that generates a macroscopically observable copy of the quantum signal; i.e., S N Y ‘ = S N F . The strong probe produces a back action onto the phase of the signal beam (Kartner and Haus [ 19931) without degrading the transmitted s i b a l (SNRT’= S N V ) . In optical fibers, however, u f 0 and SPM phase noise increases with probe power (((A~SPM,~)’) = d ( n p ) for a coherent probe with ((An,)’) = (np)). Photonnumber fluctuations of the probe input coherent state feed continuously into phase fluctuations via SPM (fig. 9). The increasing phase uncertainty reduces the XPM sensitivity in a phase measurement (see fig. 17). Now the best strategy is to use a small probe power in order to minimize the nonlinear SPM phase spreading, but still enough power to ensure a small probe phase uncertainty [((A@)’) = l/(4(np))]; i.e., to maximize the signal-to-noise ratio transfer expressed in eq. (6.8) with respect to (np). The optimum probe photon number is found to be (n$‘) = 1420) for a coherent probe, and the transfer becomes S N ~ OoptU=~S~N R ~ 1 1 = SNR; U , 1 1+ 1+ K 2 ((An!)’) W w , / q J ((An!)’) (6.1 1) using ” K / 5= 2 ( 0 , / 0 , ) . Although the choice (o,/op) > 1 is of some advantage, If pulses at different velocities are used for the QND measurement, the effective propagation distance z for SPM and XPM is not the same in eqs. (6.6) and (6.7). The ratio KIU is diminished by the relatively small length L X ~ of M pulse overlap compared to the total length L s p ~over which the probe is self-phase modulated, K / U = 2(o,/op)(LXpM/LSpM). 22
426
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
Signal
Probe
I
-I I I I I I
Fig. 17. Self-phasemodulation (SPM) noise in a QND measurement: (a) The probe beam experiences both self-phase and cross-phase modulation. Two different signal amplitudes As.1 AS,^ can be resolved in the probe phase when its uncertainty is small enough. Phase difksion due to SPM of the probe becomes a limitation to the QND measurement. (b) A measurement of the probe phase quadrature B is contaminated with phase diffusion noise. A better signal-to-noise ratio of the QND readout is obtained in a measurement of the X(+) quadrature. The combined amplitudephase measurement of the probe cancels SPM noise with correlated amplitude noise. (The figure is not drawn to scale. The uncertainty region should be lo4 of the amplitude.)
a further improvement to the signal transfer to the probe beam requires a correspondingly larger QND gain K to dominate over SPM noise of the probe. The key to a noiseless QND experiment in fibers is the measurement of a linear combination of phase and amplitude quadratures (see fig. 17) (Levenson, Shelby, Reid and Walls [1986], Levenson and Shelby [1987], Imoto [1990], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 1990]), as was already implemented in the first continuous-wave fiber QND experiments (Levenson, Shelby, Reid and Walls [1986], Levenson and Shelby [1987], Bachor, Levenson, Walls, Perlmutter and Shelby [ 19881). A similar improvement towards
v, § 61
427
QUANTUM NONDEMOLITION MEASUREMENTS
noiseless quantum measurement was also predicted (Drummond, Breslin and Shelby [1994], Courty, Spalter, Konig, Sizmann and Leuchs [1998]) for soliton collision QND experiments (Friberg, Machida and Yamamoto [ 19921, Friberg, Machida, Imoto, Watanabe and Mukai [1996]). Since the optimized detection scheme requires a quadrature measurement, the input-output relations (6.4a,b) are transformed into an amplitude (AGUA)and phase ( B = q ) quadrature representation as used in 9 6.1.2,
AAF'
= AAF,
(6.12a)
AB;~= AB$ + 2 a (n,) AA;
+2 4 n , )
(6.12b)
(n,)AA:,
where AA=U(2&$)An and A B = m A @was used. The SPM and XPM coefficients are related to the nonlinear phase shifts (6.6) and (6.7) by 2a(n,) = 2@SPM,pand 2 K d m = 2 @ ~ p ~ , , respectively. Now the probe output fluctuations can be measured in the quadrature X that is rotated by #I with respect to the probe amplitude A; i.e., = AA;'COS(W) + hB;%n(v)
d m ,
xPt(v)
=
{AA;
[cot(W) + 2 a (n,)] +
B; + 2K
(n,) (n,)AA:
.I
sin(v).
(6.13) The self-phase modulation noise term, 2a(n,)AAr, can be eliminated by choosing [cot(?+!~)+2o(la,)]=O in eq. (6.13); i.e., 1 ~ ,= -arccot(2a (n,))
= -arccot(2@spM,,>.
(6.14)
The probe output quadrature, X$'( W), then contains only the probe input phase quadrature noise23 and the QND copy of the signal, similar to the SPM-free (a= 0) interaction discussed above. Therefore, it is not surprising that the transfer of the signal-to-noise ratio from the signal beam to the probe beam is (6.15)
which is identical to eq. (6.9) if quadratures are converted to phase and photonnumber representation. This semiclassical analysis shows that SPM-noise-free QND is indeed possible with the combined amplitude-phase measurement. The local oscillator phase I)is different from the squeezing phase 0 of the probe (see eq. 5.5 in §5.1.2), but the difference vanishes in the limit of strong squeezing. Therefore, in schematic graphcal representations (Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [ 19901, Sizmann [1997], Courty, Spalter, Konig, Sizmann and Leuchs [1998], and fig. 17), I)is shown as the squeezing angle. 23
428
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
lY§6
6.2. EXPERIMENTS WITH CONTINUOUS-WAVE LASER LIGHT
The first quantum-nondemolition detection of photon-number was performed by the IBM group (Levenson, Shelby, Reid and Walls [1986]) using a modification of the first fiber squeezing experiment (Shelby, Levenson, Perlmutter, DeVoe and Walls [1986]), described in 6 5.2. Prisms were used for combining and separating the probe (60mW, 676nm) and signal (130mW, 647nm) beams copropagating in the cryogenic fiber (1 14m at 2 K). Additional detectors and electronics were used to analyze the individual and combined photocurrents of signal and probe beams. The probe beam was reflected off a phase-shifting cavity, as in the squeezing experiment, in order to detect a low-noise (SPM-suppressed) superposition of amplitude and phase quadratures (Levenson and Shelby [ 19871). The signal intensity was measured and the sum and difference of signal and probe photocurrents were compared. Approximately 37% of the rms fluctuations of the probe were caused by shot noise (quantum noise) of the signal. A correlation dip of approximately 1 dB below the average combined noise was observed, similar to the “four-mode squeezing” experiment with almost identical setup and parameters (Schumaker, Perlmutter, Shelby and Levenson [19871). GAWBS and polarization-decorrelation in the low-birefringence fiber limited the amount of detected quantum correlation. In terms of QND parameters introduced later by Grangier and coworkers (Grangier, Courty and Reynaud [1992], Poizat, Roch and Grangier [1994]), a quantum correlation was measured for the first time in the history of optical QND experiments, although the transfer of signal-to-noise ratios as measured in the photocurrents was too small to put the experiment into the QND quadrant. The detected quantum correlation led to a conditional variance, V , 1 = 0.95, or a 5% reduction of noise below the standard quantum level of the signal (Levenson and Shelby [1987]) when the ratio of signal and meter gains was optimized. Simultaneously, the NTT group (Imoto, Watkins and Sasaki [ 19871, Saito and Imoto [1988], Imoto [1990]) reported on a different test apparatus preparing for a QND measurement. A classical modulation transfer from an on-off modulated signal beam to a probe beam was detected in three different setups with long fibers. In a first setup (Imoto, Watkins and Sasaki [1987]), a nonlinear Mach-Zehnder interferometer with 500 m polarization-maintaining fibers in both arms was used. The signal was co-propagating with the probe in one arm. The modulation transfer from signal power to probe phase was used to measure the optical Kerr coefficient for the nonlinear coupling of both waves. For detection of quantum fluctuations of the signal, the measurement
V,
8 61
QUANTUM NONDEMOLITION MEASUREMENTS
429
accuracy was too low by a factor of lo3. Greater stability was achieved with a Sagnac ring interferometer made of 10 km of fiber (Imoto, Watkins and Sasaki [1987], Imoto [1990]). The probe and reference beams were split off the same input beam and counterpropagated through the ring where their average phase difference was self-stabilized. By duect power detection of the probe at the transmission output port of the interferometer, the modulation transfer from the signal to the co-propagating probe beam was clearly observed because the counter-propagating reference beam only sensed the average signal photon number. In the third experiment, the 10-km fiber was cut in half, rotated by 90 degrees and spliced in order to cancel polarization group dispersion (Saito and Imoto [1988]). Yet, the observed signal-to-noise ratio was not sufficient for detecting a quantum noise transfer to the probe. Following the experiments, the criteria and regimes of operation of a QND measurement in a lossy Kerr medium were investigated theoretically for an optimized experimental QND setup. T h s analysis resulted in the first formulation of practical criteria for characterizing a nonideal QND experiment (Imoto and Saito [1989]), as discussed in $ 6.1.2. These first measurements of the transfer of quantum noise and of classical modulation suggested that the long interaction length and high power density of optical fibers enables strong coupling between signal and probe. GAWBS noise was observed in experiments of both groups. The IBM group worked in a GAWBS-noise measurement window with a cryogenic fiber (Levenson, Shelby, Reid and Walls [1986]). The NTT group used a Sagnac fiber ring. It was a fiber configuration that was later proposed (Shxasaki and Haus [1990]) and successfully implemented for pulsed squeezing (see $5.3). It was also proposed for QND measurements with solitons (Yu and Lai [ 1996]), combining the advantage of pulses (no stimulated Brillouin scattering) with the advantage of detecting the optimized SPM-free probe quadrature. A key to the success of the IBM group was the detection of a superposition of the phase and amplitude quadrature of the probe beam, which suppresses SPM noise and optimizes the QND readout (Levenson and Shelby [1987], Yamamoto, Machida, Saito, Imoto, Yanagawa, Kitagawa and Bjork [1990]). A third, theoretically more promising approach in terms of QND gain employed a fiber ring cavity (Bachor, Levenson, Walls, Perlmutter and Shelby [ 19881). A theoretical treatment and preliminary measurements (Shelby, Levenson and Perlmutter [ 19881, Shelby, Levenson, Walls and Aspect [ 19861) showed that QND measurements using a fiber ring cavity to enhance the nonlinear coupling were advantageous. Compared with the traveling-wave scheme (Levenson, Shelby, Reid and Walls [1986]), an enhancement fac-
430
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
1% § 6
tor of 400 for the QND gain was expected with a resonator finesse of 10 for both modes. The added difficulty in the experimental realization (Bachor, Levenson, Walls, Perlmutter and Shelby [1988]) of the fiber ring resonator, compared with a traveling-wave scheme, is the boundary condition of the resonator. It causes phase shifts which can mix amplitude and phase quadratures. If signal phase noise from SPM and from classical phase noise sources is mixed with amplitude noise, the back-action evasion property is lost and added amplitude noise may appear in the signal. The signal frequency therefore needed to be locked exactly to the center of the resonator mode. Furthermore, low-frequency depolarized light scattering in conjunction with polarization-selective optical elements in the resonator generated amplitude fluctuations, and therefore classical probesignal correlations through the QND interaction, which masked a quantum effect. The polarization conditions varied over a time scale of minutes, so that QND measurements were possible only under temporarily optimum QND conditions. The best quantum correlation coefficient achieved was C = 0.26, exceeding the classical noise correlation limit of C, = 0.14 due to excess noise. The experimental results were obtained with two krypton laser beams at 647 nm (probe) and 676 nm (signal), both resonantly enhanced simultaneously in a 13-m fiber resonator of low birefringence, with a finesse of 7 and 12, respectively. This resulted in an effective interaction length of approximately 100m, and in a circulating power of 112 and 240 mW for probe and signal. In order to optimize the QND readout (Levenson and Shelby [1987]), the probe was phase-shifted by reflection from an external cavity before detection, as in the traveling-wave squeezing (Shelby, Levenson, Perlmutter, DeVoe and Walls [19861, Schumaker, Perlmutter, Shelby and Levenson [ 19871) and QND schemes (Levenson, Shelby, Reid and Walls [1986]). Intrinsic excess noise, locking error and depolarization fluctuations limited the performance of the system. Excess noise in the fiber ring resonator was generated through thermal light scattering processes. GAWBS reduction by cooling to liquid helium temperature and by removing the polymer jacket was not feasible because of the complexity of the setup. The experimental problems of QND detection and of squeezing are related. Therefore, it is not surprising that elimination of GAWBS noise and a major improvement in signal-to-noise ratios was expected from using short soliton pulses (Haus, Watanabe and Yamamoto [ 19891, Sakai, Hawkins and Friberg [1990]) instead of CW laser light. The following fiber QND detection experiments employed picosecond solitons and a differential dual-pulse method for elimination of GAWBS noise at room temperature.
v, 0 61
43 1
QUANTUM NONDEMOLITION MEASUREMENTS
6.3. EXPERLMENTS WITH SOLITONS
Interacting solitons experience a permanent phase and position shift proportional to the intensity of the collision partner (Zakharov and Shabat [1972]). The quantum theory of solitons shows that the soliton collision can be used to perform quantum nondemolition measurements (Haus, Watanabe and Yamamoto [1989]). A quantum nondemolition (QND) measurement of the signal photon number detects the phase shft of the probe soliton, using a reference soliton that does not interact with the signal soliton, as is shown in fig. 18. Three major experimental challenges must be taken up for such a QND measurement: (1) the preparation of packets of three solitons (signal, probe and reference) in the fiber, (2) the elimination of classical phase noise, and (3) the readout of the QND signal with suppressed quantum phase diffusion noise of the probe and reference solitons. After a first proposal for an experimental apparatus based on a nonlinear interferometer (Haus, Watanabe and Yamamoto [ 1989]), a detailed feasibility Negative Dispersion Probe
Reference
From Two-Color Soliton Source
k
/ Signal
w
Grating
Iy I \
A
I\
Soliton Overlap
i j\ r..........
...A
Sianal Soliton Shot-Noise
!
RF Spectrum Analyzer
Delay Line, 30 m
Fig. 18. Outline of a soliton quantum-nondemolition measurement of the photon number. After the probe-signal interaction in the fiber, probe and reference are overlapped in a Mach-Zehnder interferometer with a ~ / relative 2 phase delay in addition to the group delay. The phase difference between probe and reference is a readout of the signal soliton photon number (Friberg, Machida and Yamamoto [19921).
432
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
!x§6
study of a different soliton-collision interferometer was performed (Sakai, Hawkins and Friberg [19901). With this experimental scheme, Friberg, Machda and Yamamoto [1992] realized the first, and so far only, quantum nondemolition measurements with optical solitons in fibers. Both the experimental demonstration of a single back-action evading (BAE) measurement (Friberg, Machida and Yamamoto [ 19921) and fist steps towards repeated BAE detection were reported (Friberg, Machida, Imoto, Watanabe and Mukai [19941, Friberg, Machida, Imoto, Watanabe and Mukai [ 19961). The experimental apparatus for the single (fig. 18) and repeated BAE measurements were very similar. A different pulse sequence and a longer fiber were used to realize two collisions. The following discussion of the first soliton QND experiment shows how the problems of packet preparation and GAWBS noise elimination were solved ($0 6.3.1 and 6.3.2). The problem of quantum phase diffusion noise (6 6.3.3) was addressed in recent proposals to overcome thls remaining limitation ($ 6.3.4).
6.3.1. Pulse preparation All three pulses involved in the measurement - signal (2.6 ps), probe (3.6 ps), and reference (3.6 ps), with proper spacing and relative velocity - were prepared from a single-SPM-broadenedhigh-power pulse using a novel spectral filter and delay-line technique (Friberg, Machida and Yamamoto [19921). The spectrally separated signal, probe and reference pulses were obtained by using a two-color bandpass filter and by cutting two slices (approximately 0.8 nm wide) out of the 10-nm spectrum of the input pulse. Using delay lines behind the spectral filter, the timing of signal, probe and reference was adjusted to their relative velocity (spectral offset of signal and probe) so that a complete collision between signal and probe occurred in the 400m long QND fiber. For shorter solitons in a shorter QND fiber, a more energy-efficient pulse preparation scheme is needed. Shorter solitons require more bandwidth and energy. A promising source may be a two-color mode-locked chromiumYAG laser. For a titanium-sapphire laser this mode of operation was already demonstrated (de Barros and Becker [1993], Evans, Spenche, Burns and Sibbett [ 19931, Leitenstorfer, Furst and Lauberau [19951). However, when probe and signal are derived from one laser source either by spectral filtering of spectrally broadened pulses or by two-color mode-locking in one laser cavity, unwanted correlations between the pulses and excess noise may appear. This was already observed experimentally. Spectral filtering of a shot-noise-limited SPMbroadened pulse revealed spectral correlations (Spalter, Burk, Konig, Sizmann
Y 9 61
QUANTUM NONDEMOLITION MEASUREMENTS
433
and Leuchs [1998]) and added up to 15 dB of excess noise into the transmitted pulses when the nonlinear filter losses were large (Spalter, Burk, StroBner, Sizmann and Leuchs [1998]). 6.3.2. Elimination of GAWBS noise in the QND detection
The second problem, the elimination of GAWBS noise in the probe phase measurement, was solved with the probe soliton closely following the reference soliton so that both pulses pick up essentially the same GAWBS phase fluctuations. In the QND experiment, Friberg, Machida and Yamamoto [1992] placed the reference only 30ps ahead of the probe pulse (fig. 18). This eliminated room-temperature GAWBS noise in the measurement. As was shown by Townsend and Poustie [1995], excellent reduction of GAWBS noise by more than 12 dB is achieved when the two pulses are separated by less than 100ps and detected differentially in the phase measurement. On the detection side, the signal was separated from probe and reference pulses by a grating and the QND signal was extracted using a Mach-Zehnder interferometer (fig. 18). The experiment showed a 0.25 dB noise reduction below the combined noise level due to the correlation between signal intensity and probe phase, where the signal intensity was shot-noise limited (Friberg, Machida and Yamamoto [1992]). The achieved signal-to-noise transfer (see 0 6.1.2) was S N V ' = 0.98SNR: for the signal after the collision with 0.1 dB fiber losses (98% transmission). When output coupling losses and linear losses at the grating are included, 8 1% of S N V were transferred to the signal output. The transfer of S N V to the probe detector was S y t = O . 1 5 S N V , derived from the 0.7 dB (15%) increase in probe phase noise when the shot-noise-limited signal beam was turned on (Friberg, Machida and Yamamoto [1992]). The total transfer, S N V ' + S N Y ' , is larger (quantum domain) or just below S N V (classical domain), depending on whether output coupling losses and grating losses for the signal are included. An advantage of soliton QND compared to other QND detection methods is the straightforward and efficient extension to repeated QND. In the doublecollision experiment (Friberg, Machida, Imoto, Watanabe and Mukai [1996]), the signal soliton stayed in the fiber for both interactions. There is no additional mode matching or coupling loss; only propagation loss (0.3dB) degrades the signal before the second collision. In the experiment it was shown that probe and reference pulses picked up the same quantum cross-phase modulation to within 13 dB. An extension to repeated BAE detection requires individual readouts of the meter pulses of the first and second collision, which can be
434
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, 0 6
implemented easily in this scheme,with an additional soliton that does not collide with the signal soliton. 6.3.3. Quantum noise of the probe The soliton QND experiments were limited by detection efficiency and by phase diffusion quantum noise of the probe beam. The two limitations were inherent in the Mach-Zehnder and pulse delay detection scheme. Firstly, uncorrelated vacuum fluctuations entered the interferometer via beam splitters, because in this pulse delay technique only 50% of each pulse, probe and reference, overlapped. In the context of all-optical soliton switching, a more efficient QND scheme using orthogonally polarized probe and reference pulses was proposed (Friberg [ 19931). However, GAWBS noise will at best be reduced by 7dB; i.e., not as perfect aswith parallel polarized pulses (Townsend and Poustie [19951). A solution to this probkm will be presented below in connection with quantum phase noise suppression of probe and reference, the third major challenge that must be taken up. Secondly, taking the quantum nature of the probe soliton into account, the optimum probe power is found to be approximately 1/10 of the signal power (Drummond, Breslin and Shelby [ 19941). The best signal-to-noise ratio that can he achieved with a probe phase measurement yields a correlation dip of 20% below the shot-noise limit of the signal. This is the absolute lower bound on the conditional variance with a fiber length of approximately five times the soliton interaction length to allow the solitons to separate after the collision (Drummond, Breslin and Shelby [1994]). Therefore, any improvement of the QND measurement requires a different approach. The key to coming closer to an ideal soliton QND experiment is to detect the probe with an optimum superposition of phase and amplitude quadratures (see fig. 17) as in the continuous-wave fiber QND experiments (Levenson and Shelby [1987]). Drummond, Breslin and Shelby [1994] showed that perfect QND detection with optical solitons requires equal amplitudes of signal and probe and low relative velocities in addition to the combined amplitude-phase measurement. 6.3.4. Recent proposals Practical schemes for SPM noise elimination in a soliton QND measurement have been discussed recently (Yu and Lai [1996], Spalter, van Loock, Sizmann and Leuchs [ 19971). A new detection scheme (Courty, Spalter, Konig, Sizmann
v, 5 71
PHOTON-NUMBER SQUEEZING
435
and Leuchs [ 19981) prqmises a simultaneous reduction of both classical GAWBS noise and quantum SPM noise. It uses the combined phase-amplitude measurement together with a delayed-pulse method. A frequency-selective phaseshifting cavity (Levenson, Shelby and Perlmutter [ 19851, Galatola, Lugiato, Porreca, Tombesi and Leuchs [1991]) allows for a shift of the phase of the field fluctuations relative to the phase of the carrier field. The carrier acts as a local oscillator, turning phase fluctuations into amplitude fluctuations such that the SPM noise cancels out (Levenson and Shelby [1987]). A subsequent direct intensity measurement and electronic subtraction of the two noise signals (probe and reference) eliminates the classical GAWBS noise in the QND measurement. With the detection problems identified and promising solutions at hand, new and more efficient QND detection using solitons in optical fibers can be taken on.
0
7. Photon-Number Squeezing
7.1. SPECTRAL FETEIUNG
In 1995, an unexpected intensity noise reduction mechanism was discovered by NTT researchers Friberg, Machida and Levanon [1995] and was discussed in further detail by Friberg, Machida, Werner, Levanon and Mukai [1996]. The noise reduction mechanism is based on the spectral filtering of pulses after propagation through a fiber over a length of several soliton periods. Depending on pulse width, energy and fiber length, the pulse spectrum will broaden or contract. A bandpass filter centered around the output pulse spectrum introduces intensity-dependent losses and creates a nonlinear input-output transfer function for the pulse energy (fig. 19). From the slope of the energy transfer function, the classical intensity noise transfer behavior can be derived (Leuchs [ 19861). In the quantum domain, spectral filtering may deamplify photon-number fluctuations below the shot-noise level at those input energies where classical noise reduction is also found. In contrast to the quadrature squeezing discussed in 6 5, the combination of spectral broadening and spectral filtering does not conserve the photon number and produces directly detectable squeezing of the photon-number fluctuations. In direct detection, phase noise and frequency chirp are not limitations. Indeed, the new squeezing mechanism seems to be immune to GAWBS noise and is applicable to fundamental solitons as well as to chirped pulses, thus extending the range of pulse propagation regimes for sub-shot-noise measurements. However,
436
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, § 7
Input energy (n,J
Fig. 19. Schematic diagram of (a) the energy transfer and (b) the noise transfer characteristic for spectrally filtered solitons. Noise reduction is expected to occur at certain input energies where the energy transfer function shows a reduced slope. The transfer of photon-number uncertainty from a coherent input to-the output described by the squeezing ratio R = (An2)/(.), must be derived from a quantum model.
the largest noise reduction predicted so far falls short of the lOdB goal. Nevertheless, it is one of the most significant recent developments in quantum fiber optics as it allows for incoherent sub-shot-noise measurements in a wide range of nonlinear pulse evolution, it has led to the best squeezing with subpicosecond pulses (67.3.2) and it has made spectral quantum correlations of solitons accessible to observation (6 7.3.3). Furthermore, spectral filtering is a key element in high-performance fiber-optic communication, e.g., for reduction of Gordon-Haus timing jitter, thus bringing quantum fiber optics closer to present-day incoherent optical communication (see § 8). 7.1.1. Ampllfication and deamplijcation of quantum noise
The nonlinear transfer function and the squeezing mechanism can be understood in terms of nonlinear pulse evolution in the fiber with subsequent filtering; e.g., at the soliton energy. The classical pulse dynamics in the fiber tend to stabilize the soliton. If the input pulse energy or shape differs from that of the fundamental
v, o 71
PHOTON-NUMBER SQUEEZING
Time
437
Frequency
Fig. 20. Photon-number squeezing from spectral filtering of solitons. (a) The photon-number uncertainty An of the coherent input soliton becomes correlated with an uncertainty in spectral width (b) during propagation through the fiber. When the out-lying sidebands are removed through spectral filtering (c), the photon-number uncertainty is reduced (R< 1) below the shot-noise limit in the transmitted pulse (d).
soliton, the self-stabilizationdynamics (Hasegawa and Tappert [ 19731) will cause oscillations in the temporal and spectral width as the soliton propagates. In the long run, a certain fraction of the input energy ends up in a fundamental or higher-order soliton with increased or reduced spectral width24. If a spectral bandpass filter is located at a propagation distance where excess energy leads to spectral broadening, the filter will remove the excess energy (fig. 20) and will allow for increased transmission in the case of lower pulse energies due to spectral narrowing. As a result, the filter acts as an optical limiter, transmitting a constant power for a small range of input powers, thus reducing fluctuations in the transmitted pulse (fig. 19). The classical output noise properties can be derived from the nonlinear transfer function. The maximum noise reduction occurs at the points of zero slope of the input-utput function. In contrast to a reduced slope, an enhanced slope in the nonlinear input-output curve is expected to amplify the input fluctuations (Leuchs [ 19861). A l l l y quantum-mechanical treatment of the input-output noise transfer shows that even fluctuations at the quantum level may be deamplified at certain input energies in close analogy to classical noise reduction. However, there are
24 A soliton amplitude of N > 0.5, i t . at least a quarter of the fundamental soliton energy is required
for asymptotically creating a soliton out of a sech2 input pulse (Zakharov and Shabat [1972]).
438
[Y 8 7
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
Tab!e 3 Predictions for reduction in photon-number noise of spectrally filtered solitons with and without stimulated Raman scattering (SRQa Reference
Rb
R~ ( d ~ ) pump pulse’
Nd
ce
Without SRS
Werner [1996al
0.22
-6.5
soliton
1.o
3.0
Werner [ 1996bl
0.30
-5.2
soliton
1.o
2.6
Werner [1996b]
0.2 1
-6.8
soliton
1.1
2.0
Friberg, Machida, Werner, Levanon and Mukai [ 19961
0.26
-5.9
C = -0.5f,
1.2
4.5
soliton
Mecozzi and Kumar [1997]
0.22
-6.5
soliton
1.o
3.0
Werner and Friberg [1997a]
0.19
-7.1
soliton
1.2
4.0
Werner [1997a]
0.15
-8.1
soliton
1.3
6.7
Spalter [1998]
0.23
-6.3
soliton
1.o
2.9
Spalter [1998]
0.359
-4.59
soliton
1.o
Mecozzi and Kumar [ 19981
0.41
-3.9
soliton
1.0
Werner [1996b]
0.37
-4.3
1.8 ps soliton
1.o
Werner [1996b]
0.34
-4.7
1.8 ps soliton
1.1
2.1
Werner and Friberg [1997a]
0.61
-2.1
1.8ps sech’, k” > 0
1.2
14.0
Werner and Friberg [1997a]
0.33
-4.8
1.8ps soliton
1.o
3.0
3.3 99
SRS (r = 300K) included 2.6
The filter function is a bandpass filter unless otherwise indicated. R is the squeezing ratio. The pulse widths are FWHM values ( ~ F W H M= 1.763To). N is the amplitude in soliton units (soliton order for k” < 0). is the propagation distance in units of soliton periods. C is the chirp parameter as defined by Agrawal [1995]. g The filter function is a high-pass or a low-pass filter. The same noise reduction was found for both filter functions. a
also operating regimes where quantum noise enhancement is found even though a classical noise reduction is expected, demonstrating the different qualities of quantum and classical noise models. Therefore, a quantum analysis of this system is required for a prediction of quantum noise reduction. Measurements of quantum noise reduction and enhancement cannot be emulated by classical noise transfer models. Predictions of quantum noise reduction by the novel filtering technique are summarized in table 3. For fundamental (N = 1) solitons of the nonlinear
439
PHOTON-NUMBER SQUEEZING
-g
0
-1
Y
& -2 BQ -3 0
.-$
-4
-ca,
-5
0
-6
0
2
4
6
8 1 0 1 2 1 4
Propagation distance
<
Fig. 2 1. Photon-number variance of spectrally filtered fundamental solitons vs. propagation distance in units of soliton periods (Mecozzi and Kumar [1997]). An optimum photon-number squeezing of 6.5 dB (R=0.22) was predicted for 18% losses introduced by a bandpass filter' at a fiber length of 3 soliton periods (Werner [1996a], Mecozzi and Kumar [1997]).
Schrodinger equation without Raman noise, a maximum noise reduction of 6.5 dB below the shot-noise limit was predicted using the generalized positive P representation (Werner [ 1996a1) and a linearized quantum model (Mecozzi and Kumar [19971). Both models predicted this amount of squeezing to occur with a bandpass filter positioned at a distance of 3 soliton periods with an optimum filter loss of 18%. The analytical results of the linearized model reveal that the interference of the soliton with the quantum-noise continuum limits the amount of squeezing and produces periodic variations of photon-number noise reduction along the fiber (fig. 21). In a classical analysis, similar oscillations with decaying amplitude were predicted for the interference of the soliton with a dispersive wave (Gordon [1992]). Another limit to noise reduction is due to stimulated Raman scattering. When Raman noise at room temperature is included, numerical simulations for 1.8-ps (FWHM) fundamental solitons predict up to 4.8dB noise reduction below the shot-noise limit (Werner and Friberg [1997a]); i.e., 1.7 dB less squeezing than without Raman scattering. An important question is whether quantum noise reduction by spectral filtering is related to the soliton phenomenon or whether similar results are achievable with pulses which do not form solitons. Non-solitonic pulses propagate either in the zero (A x 1.3 pm) or positive (A < 1.3 pm) dispersion regime, or are pulses in the anomalous dispersion regime whch are too weak to form a soliton. Asymptotic soliton formation requires input amplitudes of N > 0.5 (sech shape assumed). The theoretical work of Werner and Friberg [1997a1showed much less squeezing for pulses in the positive dispersion regime, as expected from the fact that the decaying peak power allows only for a very limited distance of nonlinear propagation (see $7.3.4 for experiments). Moreover, the simulations showed
440
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y 5 7
a quantum phase transition, i.e., an abrupt transition in intrapulse quantum correlation, related to the classical phase transition between dispersive pulse decay and soliton formation. The quantum effects observed by spectral filtering are therefore particularly pronounced when solitons are used. More squeezing was asserted for more energetic solitons corresponding to N = 1.2. Without Raman scattering, up to 7.1 dB (Werner and Friberg [ 1997a1) of noise reduction was predicted. Experiments (Spalter, Burk, Stroher, Sizmann and Leuchs [ 19981) showed the best noise reduction for higher-energy solitons (N = 1.3) and long fibers of approximately 100 soliton periods. In numerical simulations, up to 8.1 dJ3 of squeezing was found for N = 1.3 near 7 soliton periods of propagation (Werner [1997a], Friberg and Werner [1998]). Because of numerical difficulties, the long-fiber results of the experiments could not be simulated. Recently, Mecozzi and Kumar applied the analytical model (Mecozzi and Kumar [ 19971) to the long fiber limit and found about 3.9 dB of squeezing for 25% filter loss at a propagation distance of 99 soliton periods (Mecozzi and Kumar [ 19981). 7.1.2. Experimental apparatus and results: an overview
The experimental method for observing photon-number squeezing by spectral filtering is outbned in fig. 22. Table 4 summarizes the experimental results. Up
Fig. 22. Experimental apparatus for photon-number noise reduction by spectral filtering. An optical pulse experiences spectral broadening by traveling in an optical fiber. A grating and two variable knife edges are used for band-, high- or low-pass filtering. A slight increase in input energy excites out-lying sidebands that are removed by the fdter. The transmitted pulse is energy-stabilizedand may show photon-number squeezing in the sum current whereas the difference current of the balanced two-port receiver measures the shot-noise reference level.
v, Q
71
44 1
PHOTON-NUMBER SQUEEZING
Table 4 Experimental results for reduction in photon-number noise of spectrally filtered solitons and nonsoliton pulses
Cd
Ref.
Ra
R a (dB) Pump pulseb
NC
Fiber
[I]
0.59
-2.3
2.7 ps soliton, 1.46 pm
1.2
1.5h
4.5
BP
0.79
-1.0
2.7 ps soliton, 1.46 pm
1.1
1.5km
4.5
BP
[2]
0.60
-2.2
133fs soliton, 1S O pm
0.92
3.3 m
3.8
BP
[3]
0.48
-3.2
160 fs soliton, 1.51 pm
0.90
10 m
10
BP
0.50
-3.0
160 fs soliton, 1.51 pm
0.90
10 m
10
HP
0.58
-2.4
160fs soliton, 1.51 pm
0.90
10 m
10
LP
Filter typee
[41
[0.011
[-231
1.5ps soliton, 1.54 pm
1.16
1.5 km
27
BP
[5]
0.42
-3.8
135fs soliton, 1.51 pm
1.3
90 m
100
BP
0.48
-3.2
130 fs soliton, 1.5 1 pm
1.o
90 m
110
HP
0.59
-2.3
130fs soliton, 1.51 pm
1.o
2.9 m
3.6
HP
[6]
0.64
-1.9
130fs soliton, 1.51 pm
1.o
2.9 m
3.6
N
[7]
0.82f
-0.84f
130fs soliton, 1.51 pm
1.o
2.7 m
3.3
HP&LP
[8]
0.76
-1.2
29 fs sech2, k” z 0, 809 tun
0.9
2cmg
2
BP
~
~
~~
R is the ratio of observed photocurrent noise power to the shot-noise level. The noise reduction values in square brackets refer to classical llf-noise reduction, where R is the ratio of observed photocurrent noise power to the IF-noise level. The pulse widths are FWHM values (TFWMH = 1.763To). N is the amplitude in soliton units (soliton order for k” < 0). f=z/zp is the propagation distance z scaled to the soliton period zp = n / 2 (T;Ilk”I). BP, HP and LP are band-, high- and low-pass filters, respectively. N is a notch filter. Sub-shot-noise signal of the sum photocurrent of two spectrally filtered pulse halves g Effective length of nonlinear propagation, limited by peak power decay in the normal dispersion regime. The total fiber length was 2 m. a
References [I] Friberg, Machida, Werner, Levanon and Mukai [I9961 [2] Spatter, Sizmann, StroDner, Burk and Leuchs [I9971 [3] Spatter, Burk, StroRner, Bobm, Sizmann and Leuchs [I9971 [4] Friberg and Machida [1997,1998], Friberg [1998a,b] [5] Spatter, Burk, StroRner, Sizmann and Leuchs [I9981 [6] Spatter, Burk, Konig, Sizmann and Leuchs [I9981 [7] Spatter, Korolkova, Konig, Sizmann and Leuchs [1998] [8] Konig, Spalter, Shumay, Sizmann, Fauster and Leuchs [I9981
442
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
rv, 0
7
to 3.83~0.2dB of noise reduction below the shot-noise limit was measured with sub-picosecond solitons (Spalter, Burk, Stroher, Sizmann and Leuchs [ 19981). Extrapolating based on an overall detection efficiency of 76%, loss-free detection would yield a quantum noise reduction of 6.4f0.8 dB (inferred) in this system. An important difference between experiments with picosecond (4 7.2) and subpicosecond (§ 7.3) pulses is the influence of the Raman effect. Sub-picosecond pulses show a Raman self-frequency shift (Mitschke and Mollenauer [1986], Gordon [ 19861) already in short pieces of fiber.
7.2. SPECTRAL F L E R I N G OF PICOSECOND PULSES
In the pioneering experiment (Friberg, Machida, Werner, Levanon and Mukai [1996]), bandpass filtering of 2.7-ps solitbns produced 2.3 dB of noise reduction below the photocurrent shot-noise limit in direct detection. The discovery of the new squeezing mechanism constitutes a major breakthrough and a clear contrast to “traditional” quadrature squeezing experiments, because of (i) the noise reduction being observed in direct detection, (ii) the simplicity of the apparatus, and (iii) potential applications. Pulses from a mode-locked NaCl color center laser propagated down a 1.5-kmpolarization-maintainingfiber with energies above the fundamental soliton. After spectral filtering through a grating and a slit, the transmitted light was detected by a balanced two-port detector (fig. 22). A 30-m electric delay line was used for frequency-dependent phase shifting of one detector current before combining it with the other photocurrent. The sum of the photocurrent fluctuations, recorded between 5 and 30MHz, measures the photon-number fluctuations whereas their difference represents the shot-noise reference level. The observed 2.3 dB of squeezing correspond to an inferred photon-number noise reduction of 3.7 dB for vanishing detection losses. Alternating amplification and dearnplification of vacuum fluctuations were clearly observed as the input energy was varied from slightly below to far above the fundamental soliton energy (fig. 23). Another transition from noise reduction to enhancement was observed when the filter bandwidth was narrowed down at fixed input energy. From 2.7 dB of inferred noise reduction below the shot-noise limit to up to 10.9 dB of inferred excess noise above the shot-noise limit were reported at filter bandwidths of 1.4nm and 0.4nm, respectively. In both cases, the input pulses were shot-noise limited. Ths shows that strong filtering in the presence of nonlinear pulse evolution can generate strong quantum excess noise. All these phenomena were predicted by the quantum soliton model (Friberg, Machida, Werner, Levanon and Mukai [1996], Werner [l996b]) and can be
v, Q 71
443
PHOTON-NUMBER SQUEEZING
5
6
7
8 9 10 11 12 13 14 Energy per pulse (pJ)
5
Fig. 23. Variation of squeezing as a function of the soliton energy at a fixed filter bandwidth and propagation distance. The fundamental soliton energy is 8pJ. At an input energy of 12pJ, 1.3 to 1.4dB of squeezing were observed. In another data run the best noise reduction of 2.3 dB was achieved at this input energy (Friberg, Machida, Werner, Levanon and Mukai [1996]).
understood qualitatively by the noise transfer characteristics of the nonlinear energy input-output function. In another experiment, more than 23dB of classical noise reduction at a frequency of 1 MHz was observed with picosecond pulses from a fiber laser (Friberg and Machida [1997, 19981, Friberg [1998a, b], demonstrating the practical potential of this method. Spectral filtering proved to be a simple and effective method for removing low-frequency classical noise of a fiber laser, but with a broader range of applications to precision measurement and optical communications. Because of the speed of the nonlinearity one may expect the noise reduction mechanism to be effective up to approximately 100 THz. 7.3. SPECTRAL. FILTERING OF SUB-PICOSECOND PULSES
The best noise reduction through spectral filtering was observed with subpicosecond pulses (Spalter, Burk, Stroher, Sizmann and Leuchs [ 1998]), even though stimulated Raman scattering becomes a dominant effect. The setup was essentially the same as shown in fig. 22. 130-16Ofs solitons from a saturable-absorber mode-locked, all-solid-state chromium-YAG laser (Spalter, Bohm, Burk, Mikulla, Fluck, Jung, Zhang, Keller, Sizmann and Leuchs [ 19971) were launched into fiber lengths of up to 90 m, and different filter types such as high-pass, low-pass, band- and notch filters were applied. In order to optimize squeezing, the power, fiber length and filter losses were varied. The signature of the Raman self-frequency shift of the soliton was clearly observed in the spectral dependence of squeezing and of intrapulse correlations.
444
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, § 7
7.3.1. Noise reduction and enhancement as a function ofjilter type and cut-of wavelength
In a first experiment with sub-picosecond pulses (Spalter, Sizmann, StroDner, Burk and Leuchs [1997]), squeezing of solitons just below the fundamental soliton energy ( N < 1) was investigated with a variable bandpass filter, in contrast to the N > 1 squeezing regime of the NTT experiment. Up to 2.2 dB of squeezing was observed with 3.25 m of fiber and a bandpass filter removing 15% of the pulse energy. In a second experiment, low-pass and high-pass filters were applied for the first time to look for an asymmetry in the spectral intensity noise (Spalter, Burk, StroDner, Bohm, Sizmann and Leuchs [1997]). With 160-fs pulses in 10m of fiber, corresponding to twice thg propagation distance of the NTT experiment in terms of soliton periods, up to 3.2 dB of noise reduction in the photocurrent
-4
-6
i
t 0
10
20
30
40
50
Input power (mW) Fig. 24. (a) Nonlinear energy transfer characteristic and (b) variation of squeezing, as a function of the soliton energy at a fixed cut-off frequency for low-pass filtering. The fundamental 160-fs soliton energy corresponds to 25 mW average power. Repeated squeezing was observed where the slope of the energy-transfer function is reduced (arrows). At 20mW input power, 2.4dB of squeezing were observed. In another data run,the best noise reduction of 3.2 dB was acheved with a bandpass filter at 20mW input power (Spalter, Burk, StroBner, B o b , Sizmann and Leuchs [1997]).
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PHOTON-NUMBER SQUEEZING 6 4.5
3 1.5
0 -1.5
-C Frequency high-pass
-3 16
12 8 4 0
-4 1460
1480
1500
1520
1540
1560
1580
Wavelength (nm) Fig. 25. Photon-number squeezing vs. cut-off frequency of a high-pass and a low-pass filter. The noise levels show the expected asymmetry due to stimulated Raman scattering. The soliton selffrequency shift per photon increases with fiber length, leading to an increased asymmetry of the quantum noise levels. The input energy was fixed near the energy of a fundamental soliton, the fiber length was 2.9m in (a) and 90m in (b). Input and output pulse spectra are shown for comparison (Spalter, Burk, StroBner, Sizmann and Leuchs 119981).
fluctuations was obtained, again for pulse energies below the fundamental soliton energy. When a low-pass filter was applied to the short-wavelength side, a recurring noise reduction as a function of the launch power was observed whch is clearly related to the slope of the nonlinear energy-transfer function (fig. 24). The Raman self-fiequency shift led to an asymmetry of squeezing in low- and high-pass filtering, which was investigated in more detail in following experiments. The impact of the Raman effect on the nonlinear filter squeezing mechanism was investigated with a variable edge filter at fixed power in 10m of fiber (Spalter, Burk, Stroflner, Bohm, Sizmann and Leuchs [1997]) and in a third experiment with up to 90m of fiber (Spalter, Burk, Stroflner, Sizmann and Leuchs [1998]). Figure 25 shows the noise levels relative to shot noise as a function of the filter cut-off wavelength, at a fixed pump for fundamental solitons. The noise reduction is clearly asymmetric in magnitude and asymmetric
446
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y 8 7
with respect to the center wavelength of the output pulse spectrum. This may be interpreted qualitatively in the nonlinear input-output picture: a quantumnoise induced self-frequency shift of a fundamental soliton supports an optical limiting effect when the edge filter is placed on the long-wavelength side as compared with the short-wavelength side of the spectrum. However, the selffrequency shift may not be able to enhance squeezing because it is physically connected with noise from stimulated Raman scattering. Simulations by the NTT group confirmed the asymmetric squeezing behavior for high- and low-pass filtering in the presence of Raman scattering for these experimental parameters (Werner [1997a]) and resulted in less squeezing as compared to simulations which omit the Raman effect. Therefore, the net effect of stimulated Raman scattering is a deterioration of squeezing at these short propagation distances. An interesting experimental result is the dramatic nonstationary and asymmetric evolution of the internal spectral correlsltions of the fundamental soliton, as is shown in fig. 25, while the soliton remains stable in shape despite of the self-frequency shft (Spalter, Burk, StroBner, Sizmann and Leuchs [1998]). The observation of squeezing as a function of the fiber length is expected to improve the understanding of the squeezing mechanism. 7.3.2. Noise reduction as a function o f j b e r length
Theoretically, an optimum fiber length of approximately 3 soliton periods was predicted for maximum noise reduction of fundamental solitons (Werner [ 1996a,b], Werner and Friberg [ 1997a1, Mecozzi and Kumar [ 19971). With the sub-meter soliton periods of 130-fs pulses, much longer loss-free soliton propagation distances, can be realized than with picosecond pulses, however, at the risk of degrading the squeezing by Raman scattering noise. The third experiment (Spalter, Burk, StroBner, Sizmann and Leuchs [1998]) of the Erlangen group investigated noise reduction as a function of the propagation distance for fundamental solitons to up to 110 soliton periods (90 m), and surprisingly, strongest squeezing has been found for the longest propagation distance. In this series of measurements, a fiber was gradually shortened from 100 soliton periods down to almost zero. As shown in fig. 25, a frequency high-pass and a low-pass filter were applied, at either side of the spectrum. The photocurrent fluctuations were recorded for different cut-off wavelengths, and only the value for the noise minimum for each type of filter is shown in fig. 26 as a function of the fiber length, together with the results for optimum bandpass filtering. As a result of the propagation-dependent squeezing measurement, a local maximum amount of squeezing was observed for high- and bandpass frequency
c,
Y5
447
PHOTON-NUMBER SQUEEZING
71 0 -0.5
g z
-1 -1.5
a
.I -2
z -
g
-2.5 -3 -3.5 0
2
4
6
a
10
12
90
Fiber length (rn)
Fig. 26. Photon-number squeezing vs. propagation distance, optimized for three different filter functions. Best squeezing of 3.2 dB (4.5 dB inferred, corrected for linear losses) is observed for high-pass filtering after 90 m of fiber, a propagation distance that corresponds to approximately 110 soliton periods. The fist noise minima near 3 and 3.6 soliton periods occur with low-, respectively high- or bandpass filtering (Spalter, Burk, StroBner, Simann and Leuchs [1998]).
filtering near 3.6f0.4 soliton periods. The noise minima might correspond to the predicted maximum squeezing for the 3-soliton-period propagation distance (Werner [1996a,b], Mecozzi and Kumar [1997]), shown in fig. 21. The experimental results for the long propagation distance (90m fiber), however, show substantially stronger noise reduction (3.2 dB,corresponding to 4.5dB if corrected for linear losses) than for short fibers. If the predictions for Raman-free propagation of fundamental solitons ( N = l), reaching up to 15 soliton periods in fig. 2 1 are extrapolated to longer fiber lengths, up to 4 dB may be expected. Indeed, recent calculations show approximately 3.9 dB of squeezing at 99 soliton periods of propagation (Mecozzi and Kumar [1998]). Therefore, the observed noise reduction at 110 soliton periods represents an unexpectedly large value if compared with the prediction and to the observed squeezing near 3 soliton periods. The best squeezing with sub-picosecond pulses was observed for higherenergy solitons ( N = 1.3) in a long fiber of 100 soliton periods (Spalter, Burk, StroBner, Sizmann and Leuchs [ 19981). The photocurrent fluctuations at 20MHz were reduced to 3.8f0.2dB below shot-noise. Taking an overall detection efficiency of 76% into account, the inferred quantum noise reduction is 6.4f0.8 dE3. Simulations of higher-order solitons ( N = 1.3) propagating over longer distances (S; M 7) predicted 8.1f0.3 dB of squeezing; i.e., the strongest noise reduction predicted for spectral bandpass filtering to date (Werner [ 1997a1).
448
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS 1N FIBERS
[v, 0 7
A perspective for approaching optimum noise reduction with the filtering technique is to examine the long-fiber limit, which is difficult to model with present numerical methods and capabilities. It is an experimentally promising regime for pulses whlch are long enough to suppress stimulated Raman scattering and yet short enough to propagate over many soliton periods without appreciable loss. 7.3.3. Intrapulse spectral correlations
Another way to optimize squeezing by the filtering technique is to optimize the filter function. The knowledge of the internal spectral correlations of a pulse should allow one to find an optimum filter hnction that may be more sophlsticated than presently used filters. In a fourth experiment, the Erlangen group found evidence for intraspectral quantum noise correlations extending over about 10 nm (Spalter, Burk, Konig, Sizmann and Leuchs [1998]) in 130-fs fundamental solitons after traveling over only 2.9m of fiber. A further experiment (Spalter, Korolkova, Konig, Sizmann and Leuchs [1998]) allowed for refinement of the method in order to establish a high-resolution map of spectral quantum correlations for fundamental 130-fs solitons. These experimental data provide a better and more complete characterization of the quantum properties of a soliton as compared with what is possible using only the spectral distribution of photon-number variances. The data can be used to apply arbitrary filter functions and to optimize the expected squeezing result for these particular experimental parameters. Furthermore, a complete characterization of the quantum soliton has been proposed that includes quantum correlations of spectral components for arbitrary quadrature phase components (Spalter, Korolkova, Konig, Sizmann and Leuchs [1998]). An experimental realization with the use of a frequency-selective phaseshifting cavity (Levenson, Shelby and Perlmutter [ 19851, C o w , Spalter, Konig, Sizmann and Leuchs [19981) seems feasible. Photon-number noise reduction by interference of two solitonic fields (or one solitonic field with a dispersive field) as produced by the asymmetric Sagnac interferometer (see 0 7.4) results in spectral filtering of certain quadrature phase projections, and is predicted to produce stronger photon-number noise reduction than the spectral intensity filtering method discussed so far. Recently, the problem of optimum noise filtering of quantum solitons has been solved analytically for any propagation distance, using the nonlinear Schrodinger equation and a linearized approximation of quantum fluctuations (Levandovsky, Vasilyev and Kumar [1998]).
v, o 71
PHOTON-NUMBER SQUEEZING
449
7.3.4. Spectral jTltering of pulses in the normal group-velocity dispersion regime
Significant quantum noise reduction has also been observed in the normal dispersion regime with sub-picosecond (Konig, Spalter, Shumay, Sizmann, Fauster and Leuchs [1998]) and picosecond (Friberg [ 1998~1)pulses. Predictions showed more than 2 dB of photon-number squeezing for spectrally filtered 1.8-ps, N = 1.2 pulses in the presence of the Raman effect (Werner and Friberg [ 1997a1). Extrapolating from the calculations, more squeezing is expected for higher-energy pulses. The process, however, seems much less effective in the normal dispersion regime than for solitons in the anomalous dispersion regime. A limiting factor is the decay of the peak intensity due to temporal broadening of the pulse in the normal dispersion regime that terminates all nonlinear effects after a certain propagation distance. In a first experiment, slightly chirped 29-fs pulses from a titanium-sapphire laser, centered at 808nm, were propagated down a 2-m fiber (Konig, Spalter, Shumay, Sizmann, Fauster and Leuchs [1998]). The input pulse energy is such that the length scales for dispersive and nonlinear effects are comparable (N M 1). The dispersive effects are estimated to terminate the nonlinear interaction after the initial 2cm of fiber. The pulse spectra showed symmetric broadening and a bandpass filter was applied and optimized for squeezing. The best squeezing was observed for relatively large filter losses of 50%. Up to 1.2 dB of squeezing (2.6dB inferred) were observed. The pulse energy was only limited by the saturation threshold of the detectors. With pulses in the picosecond regime, higher N values can be realized and more squeezing can be expected at lower pulse energies, thus avoiding the detector saturation limit. Although this squeezing process is not as effective as squeezing of solitons, it is important to examine this propagation regime as it complements the picture of quantum noise in optical pulse propagation in fibers. 7.4. ASYMMETRIC FIBER SAGNAC INTERFEROMETER
In this section, the latest developments in quantum soliton preparation will be discussed. Experiments on photon-number squeezing of solitons from an asymmetric Sagnac fiber loop were initiated by one of the authors (A.S.). Simultaneously and independently, the theory of soliton photon-number squeezing from a highly asymmetric Sagnac fiber loop was presented for the first time by Werner [1997b] of NTT. As a result of the experimental and theoretical preparations, photon-number squeezing was first demonstrated experimentally by
450
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
Input energy (n,")
Fig. 27. Schematic diagram of (a) the energy transfer and (b) the noise transfer characteristic of an asymmetric fiber Sagnac interferometer. Noise reduction is expected to occur at certain input energies where the energy transfer function shows a zero slope. The transfer of photon-number uncertainty An from a coherent input to the output, described by the squeezing ratio R = (An2)l(n), must be derived from a quantum model.
Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [19981 of the Erlangen group. The asymmetric loop is a practical and promising method for preparation of quantum solitons. The slope of a nonlinear energy transfer function seems to be closely related to amplification and deamplification of photon-number noise, as the previous results with spectrally filtered solitons suggest. Also, early theoretical work (Ritze and Bandilla [1979], Kitagawa and Yamamoto [1986]) described an amplitude-stabilizing Kerr-nonlinear interferometer as a source for photonnumber squeezed light. Therefore, a variety of systems with a nonlinear inputoutput characteristic, such as all-optical switches or pulse shapers, may be efficient photon-number squeezers. Because of the different qualities of quantum and classical noise, a complete quantum model must be applied to the system for accurate squeezing predictions. An asymmetric fiber interferometer is well known for its optical switching capability. The intensity-dependent refractive index leads to self-induced phase shift in the fiber. Subsequent interference of the two modes produces a highly nonlinear energy transmission characteristic (fig. 27). A Kerr-nonlinear asymmetric Sagnac interferometer was first proposed for realizing several
v I 71
PHOTON-NUMBER SQUEEZING
45 1
optical functions (Otsuka [1983]). Switching of optical pulses in a nonlinear optical loop mirror (NOLM), the fiber-integrated form of an asymmetric Sagnac interferometer, was investigated by Doran and Wood [ 19881. Similarly, Kerr-nonlinear polarization rotation (Maker, Terhune and Savage [1964]), a mechanism useful for mode-locking of fiber lasers (Hofer, Fermann, Haberl, Ober and S c h d t [1991], Haus, Fujimoto and Ippen [1992]), produces a nonlinear transmission function behind a polarizer at the output end of a fiber (Winful [ 19851) and might generate photon-number squeezing. In the following, the early theoretical quantum analyses of an asymmetric nonlinear interferometer with two interfering modes will be reviewed (9 7.4.1) and the extension to ultrashort pulses will be discussed, where spectral filtering plays a role (Q 7.4.2). Then, recent theoretical preditions and experimental observations will be reviewed (Q 7.4.3). 7.4.1. Single-mode analysis of a Kerr-nonlinear interferometer An interferometer with an intensity-dependent phase difference between two
modes may operate in an optical limiting regime of the power-dependent transmission (fig. 27). In a small range of input powers, an increase in intensity will shift the differentialphase of the two modes towards destructive interference, thus keeping the output power constant. Therefore, output energy fluctuations will be reduced. On the quantum level, photon-number fluctuations may be suppressed in the output port. This squeezing scheme was first proposed by Ritze and Bandilla [1979] for both a Mach-Zehnder and a polarization interferometer. The photon antibunching of a sufficiently weak output field was suggested to be detected with the Hanbury-Brown-Twiss method. Later, Kitagawa and Yamamoto [ 19861 analyzed the quasiprobability density of the output field for this scheme. In their Mach-Zehnder interferometer, the highly asymmetric beam splitter combines the nonlinearly propagated beam with a small fraction of the coherent reference beam. Upon interference, the squeezed field is rotated in the phase diagram such that its minimum uncertainty is perpendicular to the in-phase quadrature of the combined field, leading to photon-number squeezing (fig. 28). Kitagawa and Yamamoto [ 19861 showed that, in principle, the photon-number uncertainty of a coherent input can be reduced far below the limit for quadrature squeezed states because of the crescent-shaped quasiprobability distribution of the output field. In a practical implementation with fibers, however, a small nonlinearity and large photon numbers prevent the crescent shape from being observed with reasonable degrees of squeezing. Nevertheless, the nonlinear asymmetric Mach-
452
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
Fig. 28. Photon-number squeezing from a nonlinear interferometer.(a) One arm of the interferometer creates a Kerr-squeezed state through SPM. (b) A small fraction of the original coherent field is added upon interference at the interferometer output &d the output energy is stabilized below the shot-noise limit (The figure is not drawn to scale. The uncertainty region is approximately 10'' of the amplitude.)
Zehnder interferometer seems to be a promising method for generating strong and directly detectable squeezing. 7.4.2. Considerationsfor pulsed squeezing A realization of this scheme with fibers would employ a self-stabilized Sagnac loop or polarization interferometer and ultrashort pulses with correspondingly high peak powers. While the CW or quasi-CW case discussed above produces the best possible interference contrast, pulse shaping and frequency chirping due to group-velocity dispersion (GVD) and self-phase modulation (SPM) tend to deteriorate the interference contrast. Solitons show an effective and cascadable switching behavior similar to square pulses (quasi-CW case), because the interplay of GVD and SPM maintains a chirp-free pulse. The switching energy of solitons can therefore be determined to good approximation by the analytic soliton phase model (Blow and Doran [1992]) that defines an average energy-dependent phase for a particular pulse energy. In contrast to solitons, switching of non-soliton pulses at zero GVD is inefficient because strong frequency chirping deteriorates the visibility of interference (Doran and Wood [ 19881). When pulses are used in an asymmetric fiber interferometer, both a differential phase shift and differential spectral broadening contribute to the nonlinear inputoutput characteristic. Therefore, spectral filtering is involved in the squeezing mechanism, which is not contained in the CW model.
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453
PHOTON-NUMBER SQUEEZING
Table 5 Experimental and theoretical results for reduction in photon-number noise in the transmission port of an asymmetric Sagnac loop interferometer Ref.
Ra
R a (dB) Pump pulseb
N C Fiber
Cd
Splitting ratio
100
88/12
Experiment
[l]
0.60
-2.2
130 fs soliton, 1S O pm
0.7
80m
[2]
0.48
-3.2
120fs soliton, 1S O pm
1.0
6.40m
9
90.519.5
[3]
0.41
-3.9
126fs soliton, 1.51 pm
1.0
6.40m
8
90/10
soliton
1.5
1.9
90/ 10 90/10
Theorye
[4]
0.08
-1 1
[5]
0.13‘
-8.8‘
176fs soliton
1.5
1.9
[6]
0.13
-8.7
soliton
1.85
1
90110
[6]
0.29f
-5.4‘
200 fs sech2, k” > 0
3.0
1.8
90/10
[3]
0.08
soliton
1.1
8
92.517.5
a
-1 1
R is the observed photocurrent squeezing ratio, or the theoretically predicted value. The pulse widths are FWHM values (ZFWMH = 1.763To). N is the soliton order. <=z/z, is the propagation distance z in units of the soliton period zp =n12 (T$lk”l). Theoretical models do not include stimulated Raman scattering unless otherwise indicated. Includes stimulated Raman scattering at T = 300 K and fiber loss of 0.1 d B h .
References [l] Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [1998] [2] Schmitt, Ficker, Konig, Merk, Wolff, S h a n n and Leuchs [1998] [3] Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [1998] [4] Werner [1997b] [5] Werner and Friberg [1998] [6] Werner [1998]
7.4.3. Pulsed photon-number squeezing9orn an asymmetric Sagnac loop
Table 5 shows the first theoretical and experimental results for the noise property of optical pulses transmitted through a highly asymmetric fiber loop. In a slightly asymmetric Sagnac loop (NOLM), large excess noise was predicted theoretically (Werner and Friberg [1997b]) and found experimentally by F. Konig of the Erlangen group with an 80m loop. Later, squeezing was predicted by Werner [1997b] of NTT and was demonstrated experimentally for the first time by Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [ 19981 of the Erlangen group in the transmission output port of the highly asymmetric Sagnac loop.
454
THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[v, 5 1
A photon-number noise reduction of 11dB (Werner [ 1997b1) and recently up to 12 dB (Werner [1998]) was predicted €or short fibers and N = 1.7 solitons. For longer fibers, 11 dB was predicted to occur even just below the fundamental soliton energy (Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [19981). Theoretical models for soliton squeezing were based on the positive P representation and on linearized quantum fluctuations. The linearized approach (Drummond and Carter [1987], Haus and Lai [1990], Doerr, Shirasaki and Khatri [ 19941) was used to model the experimental apparatus of Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [1998]. The positive P representation (Carter and Drummond [1991]) was used for a complete quantum analysis, including stimulated Raman scattering and losses for pulse lengths of 180 fs to 1.8 ps (FWHM) (Werner [1997b, 19981, Werner and Friberg [1998]). The quantum analysis of the asymmetric Sagnac loop is interesting in several ways. Firstly, it was pointed out by the NTT group (Werner and Friberg [1997b]) that in some cases the classical picture of noise reduction fails to predict the qualitative behavior of quantum noise transfer. In contrast to the classical picture for a 60140 NOLM, quantum theory predicts 15dB of excess noise generated from a coherent input at a zero-slope point of the transfer function. Secondly, the Sagnac loop is a model for a variety of nonlinear quantum systems due to the universal nature of the nonlinear Schrodinger equation. Werner [1998] states that “any scheme which involves the interference of bosonic fields (different in either direction or polarization) and where at least one has undergone evolution according to an effective (1+1)D nonlinear Schrodinger equation (NSE) could exhibit similar behavior”, and points out that this includes a variety of atomic and optical systems. The photon-number squeezing experiment is outlined in fig. 29. Subpicosecond pulses from a chromium-YAG laser, centered near 1.5 vm, were injected into a Sagnac loop interferometer made of polarization preserving fiber. The light exiting through the transmission output port was analyzed in a balanced two-port detection scheme. In a fist experiment, 80m of fiber and a splitting ratio of 88/12 were used (Schmitt, Konig, Mikulla, Spalter, Simann and Leuchs [ 19981). The long fiber allowed for observation of periodic noise reduction and enhancement as the pump power was increased. Up to 2.2dB of squeezing were obtained with pump pulse energies of about half the soliton energy in the fiber. In order to observe squeezing near the fundamental soliton energy in one arm of the interferometer, a fiber of approximately 8 soliton periods was used with a 90110 beam splitter in the second and third experiment (Schmitt, Ficker, Konig, Merk, Wolff, Sizmann and Leuchs [1998], Schmitt, Ficker, Wolff, Konig,
Y
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Fig. 29. Experimental apparatus for photon-number squeezed solitons from an asymmetric Sagnac loop. The asymmetric beam splitter BS1 (splitting ratio a/(l -a)) prepares two pulses of different energy from the pump pulse. The two pulses acquire different intensity-dependent phase shifts and spectral sidebands in the loop before interfering at BS 1. The transmitted pulse is energy-stabilized at certain pump powers and may show directly detectable squeezing in the sum current. BS2 is a 50/50 beam splitter. The difference current of the balanced two-port receiver measures the shot-noise reference level (Schmitt, Ficker, WOW, Konig, Sinnann and Leuchs [1998]).
S i n a n n and Leuchs [1998]). Up to 3 . 9 B of noise reduction were achleved, corresponding to 6.0 dF3 of inferred squeezing with linear detection losses of 21% taken into account (not including the unknown mode mismatch at BS1). The experimental result is shown in fig. 30. As expected from a heuristic classical picture, noise reduction occurs at pump energies where the slope of the inputoutput transfer function is zero. At higher energies, the interference contrast and noise level variation disappear. This is expected when stimulated Raman scattering induces intensity-dependent spectral and temporal shifts such that interference is avoided. A variation of the splitting ratio (90/10,88/12 and 80/20), showed best sqeezing for the 90/10 case (Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [19981). By modeling the experiments with linearized quantum fluctuations without stimulated Raman scattering, a slightly more asymmetric splitting ratio was found to produce more squeezing (fig. 31). The theoretical results for the 90/10 splitting ratio were in good agreement with the inferred data from the experimental observation. Up to 11 dB of noise reduction was predicted for a splitting ratio of 92.5/7.5 and a loop length of 8 soliton periods, where a particularly broad input energy range is obtained for optical limiting in the energy transfer function (Schmitt, Konig, Mikulla, Spalter, Sizmann and Leuchs [1998], Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [1998]). The Raman effect
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THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
120 = 90/10 = 6.40m = 126 fs = 20 MHz
100
80 60
40 20 0 6 4 2
0
-2 -4
0
50 100 150 Total launched pulse energy (pJ)
Fig. 30. Nonlinear energy transfer and squeezing as a function of the total launched pulse energy, at a fixed fiber length and splitting ratio. The fundamental soliton energy is (56*4)pJ. (a) The energy transfer functions shows an optical limiting effect at input energies near 53 pJ and 83 pJ. (b) The photocurrent noise is reduced up to 3.9 dB below the shot-noise level near a pump energy where optical limiting occurs (Schmitt, Ficker, Wolff, Konig, Sizmann and Leuchs [1998]).
and a residual mode mismatch may account for the discrepancy between theory and experiment. Independent work of other groups is in progress. Recent results of remarkable squeezing were reported with an asymmetric Sagnac loop (Krylov and Bergman [1998]). A noise reduction of 2dB from a linear (i.e., topologically linear, as
c: 8 71
451
PHOTON-NUMBER SQUEEZING
15
Relative noise power (dB) 10 5 0 -5
-10 70
10
-
75
zi%
80
m ' D 5
x
85
2 8' Q
90 95 -1 0 100
Input pulse energy (soliton units)
Fig. 3 1. Photon-number squeezing vs. input energy for coherent 126-fs solitons and for a fixed fiber length of approximately 8 soliton periods. The numerical predictions are based on a linearization of quantum fluctuations. (a) Theoretical model compared with data inferred from observation. The experimental data are corrected for linear losses of 21%. (b) Contour plot of the predicted noise power in dB relative to shot noise, as a function of input energy N 2 and splitting ratio a/(l -a). The bold contour represents the shot noise level, the dashed line indicates the cross section at a/(l - a)=90/10 shown in (a) (Schmitt, Ficker, Wolff, Konig, SiZmann and Leuchs [1998]).
opposed to a loop) interferometer was reported recently (Margalit, Ippen and Haus [ 19981). These first promising theoretical as well as experimental results open new avenues towards stable and efficient generation of quantum solitons and nonclassical light. In contrast to the balanced Sagnac loop, the highly asymmetric fiber loop does not require a local oscillator for extraction of the squeezed signal. Instead, squeezing can be detected directly. In adhtion, the highly unbalanced loop is insensitive to small changes in the splitting ratio. These are practical advantages for the unbalanced loop. The mode-matching problem at the beamsplitter can be solved by using a fiber coupler or an asymmetric linear interferometer similar to those used for quadrature squeezing (Doerr, Lyubomirsky, Lenz, Paye, Haus and Shlrasalu [1993], Margalit, Yu, Ippen and Haus [1998]) or photon-number squeezing (Margalit, Ippen and Haus
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THE OPTICAL KERR EFFECT AND QUANTUM OPTICS IN FIBERS
[Y § 8
[1998]), which may eventually lead to a reliable source of highly squeezed quantum solitons.
8
8. Future Prospects
Quantum optics research in optical fibers is of both fbndamental and practical relevance. The above discussion of the recent experimental and theoretical advances of the field demonstrates how rapid the progress has been and that there is still ample room for hrther improvement and development. On the fundamental side, one may study and improve upon the quantum measurement process, including back-action evading measurements of a variable such as the number of photons in a pulse. This leads to quantum nondemolition measurements which - provided they are sensitive enough - would ultimately project the state of the light pulse onto a Fock state. A less ambitious goal would be to project an initially coherent pulse onto a state showing a photon number uncertainty reduced substantially below the shot noise level. On the applied side, this will allow for the repeated measurement of a signal without any attenuation of the signal. A more speculative but highly interesting topic would be the study of quantum superposition states of the light pulses in a fiber or of the quantum properties of solitons as quasi-particles. However, the ratio between the effective nonlinearity and the losses in a fiber is much too small. But in view of the tremendous progress already made, the losses may be decreased further, the nonlinearity may be increased and even these exotic quantum effects might contribute to quantum communication in the more distant future. A fiber-optic communication system is limited by the effects of loss, dispersion, nonlinearity and by noise from amplifiers. This demands development of the optical functional elements to improve the bit-error-rate. Research on the quantum noise limitation and a thorough understanding of it will be particularly helpful for developing fiber-optic components which reach the ultimate device performance. One way to overcome the noise accumulated in optical communication transmission lines is to use soliton pulses. The nonlinear dynamics of the pulseforming process in a fiber transmission line with a series of amplifiers leads to solitons having a stable amplitude in spite of fiber attenuation and amplifier noise. The remaining problem of timing jitter, giving rise to the GordonHaus limit (Gordon and Haus [1986]), was solved in test systems by imposing boundary conditions on the spectrum of the solitons using a series of filters
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or by forced synchronization using modulation techniques worlung in the time domain (Nakazawa, Yamada, Kubota and Suzuki [1991], Haus [1993], Jacob, Carter and Menyuk [1998]). Ideally, one would use specially designed fibers when installing new soliton transmission lines. It is, however, also possible to achieve soliton transmission in existing long-distance fiber links (Robinson, Davis, Fee, Grasso, Franko, Zuccala, Cavaciuti, Macchi, Scluffini, Bonato and Corsioni [ 19981). The theoretical investigation of the physical limits of soliton-based Terabaud communication systems has started only recently (Corney, Drummond and Liebman [ 19971). It was predicted that the timing jitter due to quantum position diffusion becomes increasingly important when shortening the pulse duration into the sub-picosecond regime. The Raman-enhanced position diffusion due to the initial quantum uncertainty of solitons was shown to exceed the GordonHaus noise over very short distances. Although sub-picosecond pulses may not be used for communication in the near future, the relevance of quantum effects for the timing jitter becomes obvious as higher bit rates are realized. A worthwhde goal for experimental quantum optics with solitons in fibers is to investigate quantum position diffusion limits and to find ways to reduce the timing jitter below the standard quantum limit. This direction of research will be highly relevant for the development of fiber-optic communication systems if the data transfer rates are to be increased substantially. Along those lines, one of the objectives is the development and realization of noiseless components for improved optical information processing and transmission. Ten years ago, it was believed that the future of optical communication would be coherent optical transmission lines, because amplifiers and detectors show better performance than for intensity modulation and direct detection (IWDD). Some of the proposed applications of squeezed light were consequently aiming in this direction. Other ideas were to use twin beams or even one-photon Fock states (Giacobino, Fabre and Leuchs [1989]). The latter is now receiving increasing attention in connection with quantum cryptography (Tittel, Ribordy and Gisin [1998]). Meanwhile, the optical communication community has decided that high bit-rate communication systems will use incoherent transmission because it is technologically simpler, leaving only small niches for coherent systems; e.g., in communication between satellites. With respect to incoherent communication, the recently discovered mechanisms for quantum noise reduction in fibers discussed above are promising, and their potential future impact can only be guessed. These very first noise reduction devices, such as the nonlinear fiber optical loop mirror (NOLM), can be fully integrated using optical fiber
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[v
components and can reduce intensity noise below the shot-noise level under reliable, easy-to-control operating conditions.
Acknowledgements It is our great pleasure to acknowledgemany helpfid discussionswith K. Bergman, J.M. Courty, C . Fabre, S.R. Friberg, M.W. Hamilton, N. Imoto, N. Korolkova, P. Kumar, J.-P. Poizat, S . Spalter, D.-G. Welsch and M.J. Werner, as well as the technical assistance by M. Eberler, J. Ficker and G. Gardavsky.
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AUTHOR INDEX FOR VOLUME XXXIX
A Abarbanel, H.D.I. 215,218, 224, 227 Abbas, G.L. 73 Abram, I. 383, 386 Abramochkin, M. 3 11 Adam, G. 153-157, 185 Adam, P. 91 Adams, C.S. 175, 329 Aden, A.L. 230 Agarwal, G.S. 87, 89, 193, 194 Agrawal, G.P. 381, 382, 388-391, 405, 418, 419,424,438 Agrawal, I? 376, 390 Aharonov, Y. 87, 148 Akulin, VM. 99 Al-Hilfy, H. 333 Albert, D.Z. 87 Allen, L. 295-297, 300-303, 306, 308, 309, 311, 313-316, 318-323, 325-327,329, 330, 333, 350, 364, 365 Alonso, M.A. 16, 19, 21, 27, 50 Alter, 0. 148 Alvarez-Estrada, R.F. 219, 247, 249, 254 Anandan, J. 148 Anderson, M.E. 74, 79, 111, 112 Anderson, M.H. 173, 357, 368 Andrews, M.R. 173, 174 Andrianov, VA. 42 Antesberger, G. 98, 126, 151 Anyutin, A.P. 23 Apresyan, L.A. 17 Aragon, S.R. 264 Aravind, P.K. 309, 311 Arimondo, E. 329 Ark, J. 313, 318 Amaud, J. 4, 44, 48 Arnaud, J.A. 4, 24, 44, 48, 319 Amold, S. 326, 327
Amold, VI. 26 Amott, W.P 281 Arsaev, I?E. 40 Arthurs, E. 85, 87 Asano, S. 230 Asatryan, A.A. 49, 50, 52 Ashburn, J.R. 185 Ashkin, A. 325,329,356 Aspect, A. 378, 404, 406, 407, 429 Assion, A. 158 Au, C.K. 87 Avdeev, VG. 27, 52 Averbukh, I.Sh. 162, 180
B Babich, VM. 3, 49 Babiker, M. 329, 350, 364, 365 Bachor, H.A. 377,407, 424,426,429, 430 Bacry, H. 191 Bagnato, VS. 86 Bajer, J. 194 Baker, A. 215, 228 Balykin, V 183 Ban, M. 139 Banaszek, K. 76, 90, 122, 123, 150 Band, N? 100, 115, 135 Band, Y.B. 161 Bandilla, A. 94, 384, 385, 450, 451 Banerjee, H. 216, 224, 247 Bardroff, P.J. 97, 125, 151, 170, 171, 175 Barnett, S.M. 73,86, 133, 139, 140,303,306 Baseia, B. 86 Basistiy, 1.V 322 Bassichis, WH. 249 Baumberg, J.J. 73 Baumert, T. 158 Bayer, RN? 393-395, 406 Bayvel, L.P. 234 471
472
AUTHOR INDEX FOR VOLUME XXXIX
Bazhenov, VY. 322 Beck, M. 65, 70, 74, 76, 79, 102, '103, 105, 106, 140, 175 Becker, F!D. 432 Beijersbergen, M.W. 296, 297, 300, 309, 311, 314316,319, 320, 324, 330 Belanger, P.A. 42, 44 Bennet, J.A. 39 Berg, H.C. 325 Bergman, K. 378, 395, 396, 398, 405, 408410, 414417, 456 Bergou, J. 140 Bergquist, J.C. 165 Berlad, G. 241 Bernstein, H.J. 77 Berriman, B.J. 227 Berry, M.V. 8, 369 Berstein, I.B. 23 Bertani, P. 77 Bertoni, H. 48 Bertom, H.L. 50 Bertrand, J. 101 Bertrand, P. 101 Beth, R.A. 294 Bhandari, R. 264 Bialymcka-Birula, Z. 133, 134, 140,327,338 Bialynicki-Birula, I. 133, 134, 140, 327, 338 Biedenharn, L.C. 302, 304 BijediC, N. 87 Bjork, G. 377, 382, 384, 387, 399, 426, 427, 429 Bjorkholm, J. 367 Bjorkholm, J.E. 325, 329 Blair, D. 325 Blankenbecler, R. 227 Blatt, R. 164, 165, 167 Block, S.M. 325 Blockley, C.A. 164 Bloembergen, N. 380, 382, 383 Blow, K.J. 73, 389, 395, 452 Bodendorf, C.T. 98, 126, 151 Boggavarapu, D. 74, 111, 112 Bohm, M. 380, 393, 441, 443445 Bohmer, B. 143 B o b , J. 115, 183 Bohren, C.F. 234, 257 Boivin, L. 412, 414 Bolda, E.L. 173, 174 Bollinger, J.J. 164 Bonato, L. 376, 459
Bonse, U. 186 Booker, H.G. 3 Born, M. 5 , 6, 19, 24, 25,216 Boron, S. 269 Borovoi, A.G. 215, 273, 281 Bouche, D. 20, 52, 53 Bourrely, C. 237, 241, 247, 251-253, 2 7 4 276 Bouwkamp, C.J. 294 Box, M.A. 273 Bradley, C.C. 173 Braginskii, VB. 419 Braginsky, V.B. 98,418, 419 Brand, G.F. 312 Braunstein, S.L. 74, 89 Bravo-Ortega, A. 42 Breitenbach, G. 102, 110, 111, 140 Bremmer, H. 3 Breslin, J. 427, 434 Brevik, I. 294 Brinkmeyer, E. 278 Brown, K. 42 Brown, L.S. 182 Brown, R.H. 137 Brune, M. 96-98, 125, 151 Brunner, W. 70 Brunner, WH. 70 Bruns, H. 216 Bryant, ED. 281 Bucksbaum, PH. 181 Budden, K.G. 3, 4, 16, 25,42 Burk, M. 379, 380, 392, 393, 432, 433, 440-448 Burns, D. 432 Burrows, C.R. 10 Busch, P. 115 Buiek, V 87, 118, 125, 140, 153-157, 185, 194, 385 Byron, F.W. 215, 218, 225-229 C
Cable, A. 329, 367 Cahill, K.E. 191, 193 Cai, B. 48 Calvo, M.L. 219, 247, 249, 254, 278-280 Campos, R.A. 70 Car, R. 230 Carmichael, H.J. 70, 74, 76 Carnal, 0. 127, 175, 183 Carri, J. 75
AUTHOR INDEX FOR VOLUME X M I X
Carroll, J.E. 84, 86 Carruthers, P. 421 Carter, G.M. 459 Carter, S.J. 393, 396, 405,406,408,413,454 Casimir, H.B.G. 294 Cas!aiio,V.M. 216 Castillejo, L. 227 Cavaciuti, A. 376, 459 Caves, C.M. 89, 91, 93, 95 Censor, D. 23, 39 Cerveny, V 49 Champion, J.V 281 Chan, VW.S. 73, 93, 392 Chang, T.N. 215 Chaturvedi, S. 87, 89 Chen, T.W. 229, 239, 240, 243-245, 247, 251,264268,270,272 Chen, X. 162, 180 Cherkashin, Yu.N. 17 Chiao, R.Y. 91 Chiappetta, P. 237, 241, 245, 247, 251-253, 273-276, 283 Chlzhov, A.V 82, 87 Choudhary, S. 4, 13, 38 Chu, S. 325, 329, 356, 367 Chun, K.-Y. 41, 43 Ch+lek, P. 281 Cicuta, G.M. 230 Cirac, J.I. 99, 127, 164, 167 Cline, R. 151, 185 Cline, R.A. 185 Coene, W. 216 Coerwinkel, R.P.C. 314 Cohen,E. 383 Cohen-Tannoudji, C. 183, 305, 329, 332, 3563.59 Collett, M.J. 73, 422 Collins, M.D. 42 Connor, K.A. 23,24, 53 Conover, C.W.S. 181 Cook, R.J. 183, 329 Cooper, J. 70, 74, 76, 102, 106, 140 Corbett, J.V 115 Comell, E.A. 173, 357, 368 Corney, J.F. 459 Corsioni, R. 376, 459 Cotter, D. 380, 383 Cotton, R.W. 281 Coudreau, T. 108 Courtial, J. 295, 318, 320, 321, 323, 327
473
Courtis, J.-Y. 345 Courty, J.M. 386, 421, 422, 427, 428, 434, 448 Couture, M. 42, 44 Cowley, J.M. 216 cummings, EW. 97 D Dabbicco, M. 412, 414 Dakna, M. 140-142, 150 Dalibard, J. 329,356359 Danakis, S. 309, 311 D’Arimo, G.M. 86, 104, 107-109, 113, 138, 143, 144, 146148, 150, 195 Dasgupta, B. 277 Davidovii., D.M. 87 Davidovich, L. 96, 98, 127, 169, 172 Davies, E.B. 87 Davies, K.B. 173 Davis, G. 376, 459 Davis, L.W. 296, 304 de B m s , M.R.X. 432 de Boor, C. 107 de Matos Filho, R.L. 168, 171, 172 Debi, S. 215,241,273 Dehmelt, H. 163, 165, 329 Demin, A.V 27, 52 Demkov, Yu.N. 216 Derka, R. 153-157, 185 Deschamps, G.A. 4, 13, 18, 38, 39, 44,4749, 51 d’Espagnat, B. 115 DeVoe, R.G. 377, 378, 380, 387, 395, 398, 399,404,40&408,428, 430 D’Helon, C. 165, 166 Dhiani, A-A. 179 Dholakia, K. 295, 313, 318-323, 325-327 Di Domenico Jr, M. 386 DiManio,F. 258 Dodonov, VV 125, 145, 184 DMIT, C.R. 396, 398, 409, 411, 412, 415417,454, 457 Domokos, P. 97, 125, 151 Dong, L. 381 Doran, N.J. 451, 452 Doron, E. 43 Dougherty, D.J. 406, 413 Dowling, J.P. 194, 329 Draine, B.T. 281 Draine, B.T.D. 282
474
AUTHOR INDEX FOR VOLUME XXXIX
Drexhage, M. 386 Dreyer, I. 125 Drobn?, G. 153, 154, 185, 194 Drummond, P.D. 73, 89, 93, 380, 382, 392, 393, 396, 397, 405, 406, 408, 413,414, 427, 434,454,459 Ducuing, J. 380, 383 D ~ n nT.J. , 158-160 Dupont-Roc, J. 305, 332 Durfee, D.S. 173, 174 Dutra, S.M. 127 D~tta-Roy,B. 224,247 Dykaar, D.R. 181 Dziedzic, J.M. 325
E Eberbard, PH. 91 Eberly, J.H. 333, 385 Einziger, PD. 24, 42, 44, 53 Ekstrom, C.R. 175 El-Hewie, M.F. 42 Elwenspoek, M. 264 Englert, B.-G. 140 Englert, F. 2 15 Ensher, J.R. 173, 357, 368 Epstein, PS. 3 Epstein, S.T. 192 Essiambre, R.-J. 376, 390 Evans, B.T.N. 282 Evans, J.M. 432
F Fabre, C. 377, 386, 421,459 Farn, LeKien 99 Fano, U. 100, 183 Faridani, A. 65, 102, 103, 105, 106 Fauster, T. 441, 449 Feam,H. 70 Federici, J.F. 181 Fedoriuk, M.V. 21, 26 Fee, J. 376,459 Feenberg, E. 115 Felsen, L.B. 4, 13, 20, 23-25, 38, 42, 44, 48, 50, 53 Felsteiner, J. 186 Feng, G. 48 Feng, F! 367 Fermann, M.E. 451 Feynman, R.P 183 Ficker, J. 453457
Fiddy, M.A. 247 Fields, B.D. 115 Finer, J.T. 325 Firth, W.J. 368 Fischer, D. 178 Fischer, E. 163 Flannery, B.P. 197 Flatau, P.J. 281, 282 Fluck, R. 443 Flugge, S. 248 Fontana, T. 49 Forbes, G.W. 16, 17, 19, 21, 27, 49, 50 Fougkres, A. 84, 143 Fournier, G.R. 282 Fox, A.M. 412, 414 Franco, V 215 Franko, F! 376, 459 Freyberger, M. 84, 86,99, 128, 130, 171, 178 Friberg, S.R. 378-380, 392, 393, 397, 405, 406,414,424,427,430435,43843,446, 449,453, 454 Friedman, C.N. 115 Friese, M.E.J. 325 Frishat, S.D. 43 Fritsch, W. 185 Frolov, VV 53 Fujimoto, J.G. 451 Fukumitsu, 0. 49, 53 Fiirst, C. 432 Fymat, A.L. 273, 282 G Gabrielse, G. 182 Galatola, P. 407, 435 Gale, W. 115, 184 Galetti, D. 194 Gantsog, T. 377, 387 Gao, X.J. 42 Gardiner, C.W. 73, 89 Garetz, B.A. 326, 327, 367 Garibaldi, U. 216 Garrison, J.C. 17 Gate, L.F. 281 Gauss, C.F. 151 Gea-Banacloche, J. 329 Geisler, M. 158 Gelles, J. 325 George, N. 381 Gerjuoy, E. 218 Gerlach, W. 183
AUTHOR INDEX FOR VOLUME XXXLX
Gerry, C.C. 384 Gersten, J.I. 215 Ghione, G. 42 Ghosh, G. 247, 248, 284, 285 Giacobino, E. 108, 377, 386, 421, 459 Gibson, N.D. 185 Gien, T.T. 218 Giese, R.H. 273, 274 Gilmore, R. 26 Gisin,N. 459 Glasser, A.H. 42 Glauber, R.J. 95, 188, 189, 191, 193, 195, 215, 218, 219, 222-224, 226 Glogower, J. 140 Goldman, 1.1. 215 Golovchenko, J.A. 186 Golub, G.H. 197 Gomez, A. 216 Gordon, J.E. 282 Gordon, J.P. 356, 390, 391, 404, 439, 442, 458 Goto, K. 42 Goto, T. 395, 398, 409, 415, 416 Gottfiied, K. 294 Gowan,E. 49 Goy, F! 96 Grabow, J. 74, 75 Gracia-Bondia, J.M. 194 Gradshteyn, I.S. 260 Grangier, P. 73, 422,423, 428 Granovslai, Ya.1. 284 Grasso, G. 376, 459 Greenberg, J.M. 215 Greene, B.I. 181 Grelu, P. 423 Grimshaw, R. 3, 37 Grossmann, A. 191 Gruner, T. 189 Grynberg, G. 305,332,345 Gusein-Zade, S.M. 26 Guth, E. 115, 184 Guttmann, M.J. 294, 300
H Haberl, F. 451 Hagley, E. 125 Hahn, Y. 227 Hall, J.L. 377 Hallett, F.R. 273 Hallett, J. 281
475
Hamilton, W.R. 216 Hammond, T.D. 179 Hanna, D.C. 380, 383 Hansch, T. 329 Haramaty,Y. 42 Hardman, A.D. 406, 413 Hardman, P.J. 395 Hamad, J.P. 216 Haroche, S. 9 6 9 8 , 125, 151 Harris, M. 311, 318 Hart, R.W. 282 Hasegawa, A. 390,437 Hashimoto, M. 44, 53 Haus, H.A. 84, 87, 88, 297, 298, 304, 378, 380, 384, 390, 392, 395, 396, 398, 399, 405,406,408-421,425,429431,451,454,
457459 Havener, C.C. 185 Hawkins, R.J. 430, 432 He, H. 313, 325 Heading, J. 4, 31 Heam, D.J. 43 Heckenberg, N.R. 312, 313, 325 Heethaar, M.R. 281 Heffels, C. 281 Heidmann, A. 377, 421 Heinzen, D.J. 164 Heitzmann, D. 281 Helbing, J. 158 Heller, W. 272 Hellwarth, R.W. 381 Helstrom, C.W. 87, 155 Heritage, J.P. 387 Herkommer, A.M. 99, 100, 128, 130 Herzog, U. 135, 146 Hessel, A. 50 Heurtley, J.C. 49 Heyman, E. 24, 42,44 Hilico, L. 386 Hill, C.A. 311, 318 Hillery, M. 140 Hirano, T. 329, 368 Hirleman, E.D. 281 Ho, S.T. 91 Hodgkinson, R.J. 269 Hoekstra, A.G. 281 Hofer, M. 451 Hoffman, D. 367 Hohenstatt, M. 163 Holevo, AS. 87, 155
476
AUTHOR INDEX FOR VOLUME XXXIX
Holford, R.L. 36 Holland, M.J. 422 Hollberg, I,. 377 Hollberg, L.W. 377, 406 Holt, A.R. 267, 281 Hong, C.K. 70, 92 Hoock, D.W 251 Hosaka, T. 387 Hovenac, E.A. 42 Hradil, 2. 84, 87, 139, 143, 157 Huffman, D.R. 234,257 Hulet, R.G. 165, 173 Hunklinger, S . 393 Hunziker, W. 215 Hurst, C.A. 115 Husimi, K. 87, 192, 193 Huttner, B. 73 I Ikuno, H. 42 Imamoglu, A. 148 Imoto, N. 123, 377, 382, 384, 387, 399, 419422, 424429, 432, 433 Ippen, E.P. 387, 398, 405, 406, 408, 409, 412418, 451, 457 Ishihara, T. 42 Itano, W.M. 116, 122, 151, 163-165, 167170 Ito, M. 44 Itzykson, C. 215, 224, 227 IvanoviC, I.D. 115, 185 Izmest'ev, A.A. 44 J Jackson, J.D. 294, 305 Jacob, J.M. 459 Jain, A. 185 Jakob, J. 48 James, G.L. 52 Janicke, U. 175 Janszky, 3. 91, 167 Jauch, J.M. 294, 304 Javanainen, J. 173 Japes, E.T. 97, 153 Jeffers, J.R. 382 Jennings, B.R. 264 Jessen, P.S. 329 Jex, I. 77, 91, 92, 101, 132 Joachain, C.J. 218, 225-229 Jonas, P.R. 281
Jones, A.R. 234 Jones, D.S. 39, 42 Jones, KR.W 144, 155, 157, 185 Jones, R.M. 4, 16, 39 Jones, R.R. 180, 181 Jull, G.W. 4 Juncos del Egido, F! 219, 254, 278-280 Jung, I.D. 443 Jung, Y.D. 216
K Kaloshin, V.A. 36 Kano, Y. 87 Kaplan, D.R. 186 Karal, F.C. 3, 37 Kirtner, EX. 406, 413, 425 Katriel, J. 282 Kazansky, P. 381 Kazansky, P.G. 381 Kazantsev, A.P. 183, 294, 329 Keitel, C.H. 87 Keith, D.W. 175 Keller, J.B. 3, 4, 27, 37, 44,45, 52 Keller, U. 443 Kelley, P.L. 195 Kelly Jr, J.L. 85, 87 Kemble, E.C. 115 Kerker, M. 230, 234 Kerr, J. 381 Ketterle, W. 173, 174 Khalili, F.Y. 98 Khatri, F.I. 454 Kienle, S.H. 178 Kikuch, K. 398, 400, 404 Kim,C. 78 Kim, M.S. 78, 88, 95, 98, 123, 126, 145, 151 Kimble, H.J. 75, 77, 127, 377 Kinber, B.E. 40 Kincaid, B. 186 Kmg, B.E. 116, 122, 151, 163, 164, 167-170, 172 Kiss, T. 91, 92, 109, 110, 114, 132, 146, 148, 150 Kitagawa, M. 148, 377, 382, 384, 387, 397, 399,400, 420, 421,426,427,429,450,451 Kittel, C. 393 Klauder, J.R. 70, 95, 191 Klein, N. 96, 385 Kleiner, W.H. 195 Klett, J.D. 273,281, 282
AUTHOR INDEX FOR VOLUME XXXIX
Knight, P.L. 87, 97, 122, 125, 127, 153-157, 185, 377, 384, 385 Knoll, L. 140, 189 Kochahski, P. 86, 88 Kodama, Y. 390 Kogelnik, H. 47,48, 319 Kogelnik, H.W. 47, 48 Kokhanovsky, A.A. 231, 281 Kokorowski, D.A. 178, 179 Konig, F. 380, 392, 395, 396, 409, 421, 427, 432,434,441,448450,453457 Kopilevich, Yu.1. 53 Korrnilitsyn, B.T. 32 Korolkova, N. 386, 441, 448 Kouyoumjian, R.G. 52 Kowalczyk, P. 158, 159 Krahmer, D.S. 113, 122 Kravtsov, Yu.A. 4 6 , 8, 17, 18, 22-27, 30, 31, 33, 39, 40, 44, 45, 5G52 Krebes, E.S. 43 Kreinovitch, VJa. 115 Kristensen, M. 314 Krotkov, R.V 215 Kruskal, M.D. 390 Krutikov, VA. 215,273 Krylov, D. 456 Kubota, H. 459 Kudou, T. 48, 53 Kuga, T. 329, 368 Kiihn, H. 77, 79, 107 Kujawaski, E. 227 Kujawski, A. 48 Kumar, P. 78, 3 7 7 , 4 3 8 4 0 , 4 4 6 4 8 Kume, S. 395, 398,409,415,416 Kum, D.M. 173, 174 Kurtsiefer, Ch. 175-177 Kumetsov, VV 281 Kwiat, P.G. 91 L La Porta, A. 91 Lagendijk, A. 180 Lahti, P.J. 115 Lai, W. 329 Lai, W.K. 329, 350 Lai, Y. 84, 87, 88, 392, 405, 406, 414, 415, 429,434,454 LaloviC, D. 87 Lamb, W.E. 115 Lambrecht, A. 108
477
Lamy, P. 251, 274 Landau, L.D. 215 Lan&ck,R. 325 Lane, AS. 91 Lapeyere, G.J. 329 Latimer, l? 281 Lauberau, A. 432 Lax, M. 296 Lee, C.T. 135 Lee, S.W. 49 Lefevre, V 98 Legendre, A.M. 151 Lehner, J. 121 Leibfried, D. 116, 122, 151, 168-170 Leichtle, C. 162, 170, 171, 180 Leighton, R.B. 183 Leine, L. 97, 124 Leitenstorfer, A. 432 Lemaire, T. 237, 241, 247, 251-253, 214, 275 Lembessis, VE. 329, 350, 364 Lenz, G. 398, 409,411, 412, 417, 457 Lebn, 1. 216 Leonhardt, U. 70, 74, 76, 78, 80, 84, 8G88, 95, 100, 101, 109, 110, 113, 114, 121-123, 130, 143, 145, 146, 148, 150, 162, 175, 185, 194, 195 Letokhov, VS. 294, 329, 331 Lett, P.D. 357 Leuchs, G. 377, 379, 380, 384, 392, 393, 395,396,407,409,421,427,432435,437, 44W50, 453457,459 Levandovsky, D. 448 Levanon, A. 378, 379, 392, 435, 438, 4 4 443 Levenson, M.D. 377, 378, 380, 387, 393395,398400,404,406408,419,420,424, 426, 42&430,434,435,448 Levesque, R. 186 Levi, A.C. 216 Levy, B.R. 52 Levy, M. 215,227 Lewenstein, M. 189 Lewis, J.T. 87 Lewis, R.M. 23 Li, T. 48 Liberman, VS. 366 Liebman, A. 459 Lifshitz, E.M. 215 Lin, C.D. 185
478
AUTHOR INDEX FOR VOLUME XXXIX
Lin, F.C. 247 Lindell, 1.V 44 Liu, C. 281 Liu, Y. 281 Lock, J.A. 42 Lohmann, A.W. 176 London, F. 139 Lopatin, VN. 281 Lorattanasane, C. 398, 400, 404 Louck, J.D. 302, 304 Loudon, R. 70, 73, 333, 377, 380, 382, 383 Louisell, W.H. 296, 357 Lu, B. 48 Lu, L.T. 48 Ludwig, D. 26, 52 Lugiato, L.A. 407, 435 Luis, A. 70, 84, 88 Luki, A. 139 Lundborg, B. 42, 53 Lunis, B. 345 Lutterbach, L.G. 98, 127, 169 Lynch,R. 139 Lyubomirsky, I. 398,409,411,412,417,457
M Maali, A. 125 Macchi, M. 376, 459 Macchiavello, C. 86, 108, 147, 148, 150 Maccone, L. 143 Machado Mata, J.A. 69 Machida, S. 377-379, 382, 384, 387, 392, 397, 399,424,426,427,429,431433,435, 438,441443 Maciel, J. 42 Maeda, M.W. 377 Mahood, R.W. 281 Maker, P.D. 451 Makin, LA. 125 Mallik, S. 216 Mamyshev, PV 390 Mwcini, S. 104, 107, 109, 116, 173, 174, 182 Mandel, L. 70,73, 84,92, 137, 143, 195,294 Man’ko, 0.V. 104 Man’ko, VI. 104, 107, 109, 116, 125, 184 Marchiolli, M.A. 145, 194 Marcuse, D. 306 Marcuvitz, N. 20 Margalit, M. 398, 408, 412, 414, 415, 418, 457
Marks, J.R. 177 Marte, €! 127 Martell, E.C. 185 Martienssen, W. 70 Maruta, A. 390 Masalov, A.V 329 Maslov, VP. 3, 21, 22, 26, 29, 43, 52 Maslowska, A. 281 Mast, P.E. 47, 48 Matijevih, E. 230 Matthews, M.R. 173, 357 Mattle, K. 77 Mayr, E. 97, 125, 151 McAlister, D.F. 79, 80, 121, 136, 137, 150, 175 McCall, S.L. 70, 95 McDuff, R. 312 McGloin, D. 316 McKeller, B.H.J. 273 McKnight, W.B. 296 McLaughlin, D.W. 17, 35 Mease, K.D. 282 Mecozzi, A. 438-440, 446, 447 Medeiros, J.A. 215 Meekhof, D.M. 116, 122, 151, 163, 164, 167-170, I72 Meeten, G.H. 281 Meixner, A. 186 Menyuk, C.R. 459 Merk, R. 453, 454 Mertz, J.C. 377,406 Meschede, D. 96 Metcalf, H. 329 Mewes, M.-0. 173, 174 Meystre, P. 98, 126, 127, 377 Michler, M. 77 Mie, G. 230 Miesner, H.-J. 173, 174 Migdal, A.B. 2 15 Mikulla, B. 380, 443, 450, 453455 Milburn, G.J. 89, 165, 166, 377, 380, 384, 385, 387, 399, 408, 419 Miller, W.H. 43 Minogin, VG. 294, 329, 331 Miranowicz, A. 377, 387 Mishchenko, A S . 22 Mitschke, EM. 391, 442 Mittelbrunn, J.R. 216 Mittleman, M.H. 215 Mittra, R. 20, 52, 53
AUTHOR INDEX FOR VOLUME XXXIX
Miya, T. 387 Miyashita, T. 387 Miyauchi, A. 398,409,415,416 Mizrahi, S.S. 145 Mynek, J. 102, 110, 111, 175, 183 Mogilevtsev, D. 157 Moliere, G. 215 Molinet, F. 20, 52, 53 Mollenauer, L.F. 390, 391, 404, 442 Mollow, B.R. 95 Monroe, C. 116, 122, 151, 163, 164, 167170, 172 Montroll, E.W. 215, 282 Montrosset, I. 42 Moodie, A.F. 216 Moon, B.R. 281 Moore, EL. 164 Moore, R.J. 226 Mori, M. 395, 398,409,415,416 Moroz, B.Z. 115 Moms, V.J. 264 Mostowski, J. 158, 159 Mount, K.E. 8 Mourikis, S. 186 Moussa, M.H.Y. 86 Moya-Cessa, H. 122, 127 Mukai, T. 378, 379, 392, 424, 427, 432, 433, 435, 438, 441-443 Mukamel, S. 158-160 Miiller, G. 96 Muller, T. 102, 110, 111 Mund, E.H. 227, 228 Munroe, M. 74, 109-112, 114, 148, 150
N Nagourney, W. 165 Nakagaki, M. 272 Nakazawa, M. 459 Napper, D.H. 281 Naraschewski, M. I73 Narozhny, N.B. 385 Nasalski, W. 53 Nath, N.S.N. 215, 279 Natterer, F. 101 Nayfeh 229 Nelson, L.E. 415 Neuhauser, W. 163, 165 Newton, R.G. 99, 127, 184, 234,236 Nicoletopoulos, B.R. 215
479
Nienhuis, G. 302, 305, 307, 308, 319, 327, 339 Nieto, M.M. 421 Nijhof, E.-J. 281 Nikoskinen, K.I. 44 Nishimoto, M. 42 Nishizawa, N. 395,398,409,415,416 Noel, M.W. 180 Noh, J.W. 84, 143 Noordam, L.D. 180, 181 Norris, A.N. 48, 49 Nussenzveig, H.M. 43, 282
0 Ober, M.H. 451 O’Connell, R.F. 87 Ohmori, T. 42 Ono, T. 273, 274 Opatrny, T. 80, 81, 114, 116, 117, 120, 136, 141, 142, 150, 151, 194 Orenstein, M. 282 Orlov, Yu.1. 5, 6, 18, 22-27, 30, 31, 33, 36, 39, 40, 51, 52 Orlowski, A. 115 Orszag, M. 172 Orta, R. 42 Osberghaus, 0. 163 Osterberg, H. 49 Ostermayer Jr, F.W. 386 Ostrovskii, L.A. 23 Ostrovskii, VN. 216 Otsuka, K. 451 Ottewill, R.H. 281 Ou, Z.Y. 70, 77, 92 Oudar, J.L. 386 Owyoung, A. 381 Ozaki, H.J. 277
P Padgett, M. 3 18 Padgett, M.J. 295, 301, 302, 313, 314, 316, 318-323, 325-327 Pancharatnam, S. 368 Paris, M.G.A. 82, 86, 87, 90, 108, 113, 119, 138, 148 Park, J.L. 100, 115, 135 Parkins, A.S. 127, 164 Pathak, PG. 52
480
AUTHOR INDEX FOR VOLUME XXXIX
Paul, H. 70, 76, 84, 86, 87, 91, 92, 94, 95, 97, 109, 113, 115, 121, 132, 143, 145, 146, 195 Paul, W. 163 Pauli, M. 247 Pauli, W. 66, 100, I15 PaviEiC, M. 115 Pavlova, L.N. 281 Paye, J. 398,409, 411,412,417, 457 Pedersen, M.A. 26 Pegg, D.T. 86, 133, 139, 140 Penndorf, R. 282 Penning, EM. 182 Pereka, L.C.P. 13, 49 Pereira, S.F. 102, 110, 111 Perelman, A.Y. 284, 285 Perelomov, A. 191 Peiina, J. 84, 88, 137, 157, 193 Peiinovi, V 139 Perlmutter, S.H. 377,378,380,387,393-395, 398400, 404,40&-408,424,426, 428430, 435,448 P d , J.M. 241,245,251, 274 Petrich, W. 368 Petroff, M.D. 91 Pfau, T. 175-177, 183 Phillips, L.S. 140 Phillips, W.D. 294, 329, 357 Phoenix, S.J.D. 73 Pinnow, D.A. 386 Platunann, PM. 277 Plebanski, J. 192 Poe, R.T. 215 Poizat, J.-P. 422, 423, 428 Poizat, J.-Ph. 102 Polzik, E.S. 75 Popov, M.M. 49 Popper, K.R. 87 Porreca, M.G. 407, 435 Poston, T. 26 Potasek, M.J. 73 Poustie, A.J. 394-396, 409, 416, 433, 434 Power, W.L. 329, 365 Powers, S.R. 258, 259, 269 Poyatos, J.F. 167 Poynting, J.H. 294 Prasad, S. 70 Prentiss, M.G. 367 Press, W.H. 197 Pritchard, D.E. 175, 178, 179
PrugoveEki, E. 115 Psencik, I. 49 Punina, VA. 282
Q Quigg, C. 218 Quirbs, M. 216
R Ra, J.W. 48 Radmore, P.M. 385 Radon, J. 101 Radzewicz, C. 158, 159 Raimond, J.M. 96-98, 125, 151 Rajagopal, A.K. 87 Raman, c. 181 Raman, C,V 215,279 Ramsey, N.F. 179 Ray,P. 215 Raymer, M.G. 65, 70, 74,76, 79, 80, 87, 102, 103, 105, 106, 109-114, 121, 136, 137, 140, 148, 150, 162, 175, 177 Raz, S. 24, 44 Reading, J.E 249 Reck, M. 77 Reichenbach, H. 115 Reid, M. 378, 404, 406, 408, 419, 420, 424, 426, 428430 Reid, M.D. 396 Rempe, G. 96, 385 Renwick, S.P. 185 Reynaud, S. 183, 356-359, 377, 421, 422, 428 Ribordy, G. 459 Rich, T.C. 386 Richter, G. 70 Richter, Th. 109, 110, 113-115, 121, 122, 134, 136, 148, 150, 162 Rigrod, W.W. 3 11 Riis, E. 329, 356 Risken, H. 65, 77, 101, 164, 194 Risley, J.S. 151, 185 Ritze, H.-H. 384, 385, 450, 451 Eva, F. 230 Robbins, M.P. 186 Roberts, A.D. 179 Robertson, D.A. 295, 314, 318, 327 Robinson, D.J.S. 197 Robinson, N. 376, 459 Roch, J.-F. 422, 423, 428
AUTHOR INDEX FOR VOLUME XXXlX
Rohrl, A. 173 Rohrlich, F. 294, 304 Rohwedder, B. 179 Rolston, S.L. 357 Rose, M.E. 294 Rosenbluh, M. 378, 398,408,410, 413 Rouze, N. 185 Roy, A. 283, 284 Roy, A.K. 257 Roy, T.K. 245,247, 248, 258, 260-262 Royer, A. 115, 122, 139, 148 Rum, Y.Z. 48 Rubenstein, R.A. 179 Rubinow, S.I. 3 Rubinsztein-Dunlop, H. 312, 313, 325 Runge, I. 216 Russell, P.S.J. 381 Ryan, J.F. 73, 412,414 Rytov, S.M. 6, 25, 229 Ryzhik, LM. 260
S Sackett, C.A. 173 Saghafi, S. 44 SaitB, N. 87 Saito, S. 377, 382, 384, 387, 399, 422, 424, 426-429 Sakaguchi, S. 387 Sakai, Y. 430, 432 Saleh, B.E.A. 70, 377, 384 Salomon, C. 345 Sanchez-Mondragon, 5.3. 385 Sbchez-Soto, L.L. 70 Sandberg, J. 165 Sanders, B.C. 43, 78, 88, 145 Sands, M. 183 Sargent 111, M. 377 Sarkar, S. 228 Sasaki, Y. 424,428, 429 Saunders, C.P.R. 281 Sauter, Th. 165 Savage, C.M. 451 Saxon, D.S. 217, 227, 249 Sayasov, Yu. 3, 25 Scarlett, 9. 281 Schatzberg, A. 24,44 Schawlow, A. 329 Schenzle, A. 173 Schiff, L.I. 43, 215, 227, 249 S c h i f i , A. 376,459
48 1
Schiller, S. 102, 110, 111, 140 Schleich, W. 140 Scbleich, W.P. 84, 86, 97, 99, 100, 125, 130, 151, 162, 170, 171, 178, 180, 194 Schmidt, A.J. 451 Schmidt-Kaler, F. 125 Schmitt, S. 380, 450,453-4,, Schneider, S. 100, 113, 130, 162 Schrade, G. 170, 171 Sclrodinger, E. 191 Schubert, M. 190 Schiilke, W. 186 Schulp, W.A. 216 Schumaker, B. 408 Schumaker, B.L. 73, 95, 398,408, 428,430 Scully, M.O. 70, 377 Seckler, B.D. 3 Seifert, N. 185 Sekistov, VN 42 Self, S.A. 48 Seng-Tiong, H. 412,414 Senior, M. 281 Senit&, I.R. 192 Seyfned, V: 158 Shabat, A.B. 390-392, 431, 437 Shapiro, J.H. 69, 73, 88, 93, 143, 377 Shapiro, M. 161, 162, 180 Sharma, S. 238,269 Sharma, S.K. 215, 216, 218, 224, 237, 238, 241,243-245,247,248, 250, 251,257-263, 265, 269, 273, 277, 283-285 Shatalov, B.E. 22 Shatalov, YE. 22 Shelagin, A.V: 216 Shelby, R.M. 377, 378, 380, 387, 392-396, 398400,404408,410,413,414,419,420, 424,426430,434,435,448 Shepherd, J.W. 267, 281 Shepherd, T.J. 73 Sheppard, C.J.R. 44 Shibata, S. 387 Shifiin, K.S. 235, 282 Shmizu,K. 282 Shin, S.Y. 25, 44, 48 Shiokawa, N. 329, 368 Shirasaki, M. 378, 395, 396, 398, 399, 405, 409412,415417,429, 454,457 Shore, B.W. 97 Shumay, I. 441, 449 Sibbett, W. 432
482
AUTHOR lNDEX FOR VOLUME XXXlX
Siegman, A.E. 296 Sigel, M. 175 Simmonds, J.W. 294, 300 Simmons, R.M. 325 Simpson, N. 318 Simpson, N.B. 316, 319, 322, 325, 326 Sinah, A. 423 Singer, M. 115 Sitenko, A.G. 215 Simann, A. 379, 380, 392, 393, 395, 396, 409, 421, 427, 432434, 440450, 453457 Skryabia, D.Y 368 Sleator, T. 183 Slusher, R.E. 73, 91, 377, 406 Smith, B.J. 43 Smith, C.B. 273 Smith, C.P. 312 Smith, E.T. 179 Smith, G.M. 314 Smith, L.W. 49 Smith, M.S. 42 Smith, W.S. 239, 245 Smithey, D.T. 65, 70, 74, 76, 79, 102, 103, 105, 106, 140 Somerford, D.J. 218,238,243,245,250,251, 257-262, 265, 269, 284, 285 Sommerfeld, A. 216 Song, S. 91 Soskin, M.S. 322 Spadacini, R. 216 Spalter, S. 379, 380, 392, 393, 395, 396,409, 421,427,432434, 438,440450,453-455 Speiser, S. 282 Spenche, D.E. 432 Spreeuw, R.J.C. 296, 297, 300, 309 Spudich, J.A. 325 Srinivasan,V 87 Steinberg, A.M. 91, 108 Stenholm, S. 77, 85, 87 Stepanov, A.Y 216 Stepanov, N.S. 23 Stephens, G.L. 256, 263, 281 Stem, 0. 183 Stemin, B.Yu. 22 Sterpi, N. 147, 148, 150 Steuemagel, 0. 82, 87, 131 Stewart, N. 26 Stolen, R.H. 387, 390 Stoler, D. 73, 76, 308 Ston, M. 284
Stone, C.D. 185 Streekstra, G.J. 281 Streifer, W. 4, 44, 45 StroOner, U. 379, 380, 393, 395, 396, 409, 433,440-447 Stroud Jr, C.R. 180 Strumia, F. 329 Stulpe, W. 115 Sucher, J. 215, 227 Sudarshan, E.C.G. 191, 193 Sugar, R.L. 227 Sukenik, C.I. 181 Sukhy, K. 23,25, 39 Surdutovitch, (3.1. 294, 329 Susskind, L. 140 Sutherland, R.A. 282 Suzuki, K. 459 Swift, A.R. 224, 226-228 Szajman, J. 258
T Takahashi, K. 87 Takebe, H. 381 Takenaka, T. 49 Tamm, C. 311, 315 Tamura, K. 415 Tan, S.M. 114, 151, 173, 174 Tanas, R. 377, 387 Tang, C.C.H. 230 Tanguy, C. 183 Tanner, C.E. 357 Tappert, F. 390,437 Tapster, P.R. 3 11, 3 18 Tasche, M. 107 Taylor, J.R. 390 Tegmark, M. 183 Teich, M.C. 70, 377, 384 ten Wolde, A. 180 Tenney, A. 215 Terhune, R.W. 451 Terry, ED. 16, 25,42 Terunuma,Y. 387 Teukolsky, S.A. 197 Tew, R.H. 43 Thomas, B.K. 218 Thorne, K.S. 418, 419 Tinin, M.V. 17, 27, 52 Tittel, W. 459 Titulaer, U.M. 188 Tiwari, S.C. 368
AUTHOR INDEX FOR VOLUME XXXlX
Tobocman, W. 247 Todomki, S. 387 Todoroki, S.4. 387 Tollett, J.J. 173 Tombesi, P. 104, 107, 109, 116, 173, 174, 182,407,435 Tommei, G.E. 216 Toni, Y. 329, 368 Torma, P. 77, 91, 92, 132 Torrksani, B. 276, 283 Torrey, H.C. 333 Toschek, RE. 163, 165 Townsend, C.G. 173, 174 Townsend, P.D. 395, 396,409, 416, 433, 434 Trammell, G.T. 115, 184 Tribe, L. 43 Tnppenbach, M. 161 Truffin, C. 215 Turchette, Q.A. 175 Turnbull, G.A. 314 Turski, L.A. 143 Twiss, R.Q. 137
U Udo, M.K. 412,414 Ueda,M. 148 Uhtsev, P.Ya. 52 Ungar, J. 356 Unger, H.G. 48 Unruh, W.G. 418,419 V v. Schickfus, M. 393 Vaccaro, J.A. 131, 133, 143 Vaidman, L. 148 Valley, J.F. 377, 406 Van de Hulst, H.C. 217, 234, 235, 261, 283 van der Burgt, P.J.M. 151, 185 van der Straten, P. 329 van der Veen, H.E.L.O. 309, 311, 315, 316, 319, 320, 330 van Druten, N.J. 173, 174 Van Dyck, D. 216 van Enk, S.J. 302, 305, 307, 308, 319, 329, 339, 368 van Linden van den Heuvell, H.B. 180 van Loan, C.F. 197 van Loock, P. 380,393,434 Varchenko, A.N. 26 Virilly, J.C. 194
483
Varsimashvili, K.V 216 Vasilyev, M.V 448 Vasnestsov, M.V 322 Vassallo, C. 53 Vaughan, J.M. 311,318 Verkerk, P. 345 Vetterling, WT. 197 Vigneron, K. 423 Vogel, K. 65, 77, 84, 86, 101, 194 Vogel, W. 70, 74, 75, 77-83, 85, 86, 90, 91, 93, 97, 107, 114, 116, 117, 120, 122-124, 136, 140, 143, 145, 150, 151, 161-164, 166, 168, 171, 172, 189, 190, 196,421 Vogt, A. 115 Volostnikov, V 311 von Neumann, J. 65, 153 von Plessen, G. 412, 414 Vorontsov, Y.I. 98, 418, 419
W Wagner, S.S. 93, 143 Wait, J.R. 230 Waldie, B. 269 Walker, N.G. 77, 78, 82, 84, 86-88 Walker, T. 367 Wallace, S.J. 224, 225, 227, 228 Wallach, M.L. 272 Wallentowitz, S. 90, 91, 114, 116, 117. 122, 123, 150, 151, 163, 164, 166,.171, 172 Wallis, H. 173 Walls, D.F. 164, 173,174,377,378,380,382, 384, 385, 387, 395, 397-399,404,406-408, 419, 420, 422, 424,426,42&430 Walmsley, LA. 158-162 Walser, R. 99, 127, 167, 173, 174 Walther, H. 96, 98, 126, 151, 385 Wang, J. 273 Wang, M.D. 325 Wang, W.D. 13, 51 Wang, Z.S. 42 Watanabe, K. 392,405,424,427,430-433 Watkins, S. 424, 428, 429 Watts, R.N. 357 Waxer, L.J. 161, 162 Weaver, W.D. 185 Wegener, M.J. 312 Weigert, S. 115, 185 Weiner, A.M. 387 Weinfurter, H. 77 Weiss, C.O. 311, 315
484
AUTHOR INDEX FOR VOLUME XXXIX
Weiss, D.S. 356 Weiss, K. 273, 274 Weiss, U. 224, 226-228 Weiss-Wrana, K. 276 Welsch, D.-G. 70, 74, 77-83, 85, 86, 97, 107, 114, 116, 117, 120, 124, 136, 140-143, 145, 150, 151, 166, 189, 190, 194, 196, 421
Werner,M. 440 Werner, M.J. 378, 379, 392, 406, 413, 435, 438443,446,447, 449, 453, 454
Westbrook C.I. 357 Westerveld, W.B. 151, 185 Westwood, E.K. 42 Weyl, H. 193 White, D.W. 26 Wiedemann, H. 154 Wieman, C.E. 173, 357 Wiesbrock, H.-W. 115 Wigner, E.P. 106, 184, 193 Wilkens, M. 98, 126, 127, 175 Williams, A.C. 224 Williamson 111, R.S. 367 Wineland, D. 329 Wineland, D.J. 116, 122, 151, 163-165, 167170, 172
Winful, H.G. 451 Wiscombe, W.J. 282 Wodkiewicz, K. 76,86-88,90, 122, 123, 150 Woerdman, J.P. 296, 297, 300, 309, 311,
Yamamoto, Y.
148, 377, 380, 382, 384, 387, 392,397,399,400,405,406,414,419421, 424427, 429433, 450, 451 Yanagawa, T. 377, 382, 384, 387, 399, 426, 427, 429 Yang, L.M. 266, 267 Yao, D. 385 Yariv, A. 387 Yarygin, A.P. 27, 52 Yashin, Yu.Ya. 24 Yates, A.C. 215 Yeazell, J.A. 162, 180 Yee, T.K. 73 Yin, H. 325 Yokota, M. 48, 53 YOO,H.-I. 385 You, D. 181 Young, B. 99, 127, 184 Young, M. 394 Yourgrau, W. 100, 115, 135 Yu, C.X. 398, 408,412, 414, 415,418,457 Yu, S.S. 429, 434 Yu, S.-S. 406, 414 Yuen, H.P. 69, 73, 88, 93, 148, 392 Yukutake, K. 42 Yuratich, M.A. 380, 383 Yurke, B. 70, 73, 76, 91, 95, 377, 406,419 Yushin, Y. 91 Yushin, Y.Y. 167
314-316, 319,320,330
Wolf, E. 5, 6, 19, 24, 25, 137, 193, 216, 294 Wolf, M. 252, 274 Wolf, M. 4 5 3 4 5 7 Wong, W.S. 390 Wood, D. 389, 451,452 Wootters, WK. 87, 115, 194 Wright, E.M. 17, 329 Wu,H. 377 WU, L.-A. 377 Wu, R.S. 44 Wiinsche, A. 87, 106, 109, 113, 118, 134, 135, 148, 162
Y Yakovlev, VP. 178, 294, 329 Yakushkin, I.G. 22, 36, 38 Yamada, E. 459 Yamamoto, G. 230
Z
Zabusky, N.J. 390 Zagury, N. 98, 172 Zak, J. 191 Zakharov, YE. 390-392, 431, 437 Zanon, D. 230 Zaugg, T. 127 Zege, E.P. 231, 281 Zeilinger, A. 77 Zel’dovich, B.Ya. 366 Zemov, N.N. 42, 53 Zerull, R.H. 273, 274 Zhang, G. 443 Zhang, W. 43 Zhang, X. 412, 414 Zhang, Z.L. 48 Zhou, W. 48 Zhu, T. 41, 43 Zoller, P. 99, 127, 164, 167 Zolotov, I.G. 235
AUTHOR INDEX FOR VOLUME XXXIX
Zubairy, M.S. 377 Zuccala, A. 316, 459 Zucchetti, A. 82, 83, 107
Zukowski, M. 77 Zumer, S. 281 Zurek, W.H. 87
485
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SUBJECT INDEX FOR VOLUME XXXIX
- - _ optical scattering 230-268
A Airy function 26, 30 anomalous diffraction approximation (ADA) 217, 237,238
-equation 5,6,39,216,217 - -, complex 17 - - in anisotropic medium 24 eikonal-Born series 228,229 evanescent wave 4
B back-action evading measurement 418, 432 balanced homodyning 73, 91, 92, 143, 151 Bernoulli process 196 - transformation 146 Bessel function 29, 223, 256, 260 Born approximation 221, 225,228 -series 225 Bose-Einstein condensate 68, 173 Brillouin scattering, guided acoustic wave 393-397, 404, 406
F Fermat’s principle 16, 17 fiber-optic communication, quantum limits in 380 Fock state 189 four-wave mixing 377 Fraunhofer diffraction 234,260, 268
G Gaussian beam 4, 11, 17,43-50 geometrical optics 3, 9, 10, 20, 314 - -, basic equations of 5-14 - -, complex 4, 5, 11, 19, 26, 42, 50-53 - -, ray equation of 5 - -, transfer equation of 5 - - approximation 2 17 Glauber amplitude 224 -formula 224 Gouy phase shift 315, 338 Green’s function 8, 19,221, 279
C Clebsch-Gordan coefficients 127, 174, 184 coherent state 191, 192 complex rays 3,4, 12 - -, properties of 1544 - -, selection rules for 17, 18 - -, space-time 23, 24 - - in physical problems 27-43 - - - weakly absorbing media 39-42
D displaced-photon-number statistics 90 Doppler cooling 295, 329 - shift 295,327,331,337-339,360,367
H Hamilton-Jacobi equation 17 Haakel function 260 Heisenberg picture 70 helicity 294 Helmholtz equation 6, 7, 303 Hermite-Gaussian beam 48 - - mode 309,366 Hilbert space 65
E eikonal approximation 215, 219-222, 227229 - -, applications of 268-282 - - in non-relativistic potential scattering 218-230 487
488
SUBJECT INDEX FOR VOLUME XXXIX
hologram, computer-generated 3 12, 3 13 -,volume 279 homodyne, balanced 392 - detection 63-100 - detector, eight-port 85, 89 - -, fow-port 70, 80, 85, 90 - -, multiport 89, 90 - -, six-port 84, 89 I ionosphere 3
J Jaynes-Cummings dynamics 167-1 7 1 - - model 97, 124, 164
K Kerr effect, nonlinear 377 - -, optical 375,376, 380-388 Kirchboff integral 46 - solution 5 Kramers-Kronig dispersion relation 249
L Laguerre polynomial 119, 121,323 LaguerreGitlrssian beam 295,296,300,301, 322, 327,338, 339, 342-352, 362-366 - - mode 300,301, 306,309, 322,323, 327, 330, 336, 338,359, 366, 368 - - -, generation of 309-319 LambDicke parameter 168, 170 - - regime 166 Legendre polynomial 236 Liouville equation 357 Liouvillian 143, 144 Lippman-Schwinger equation 2 18, 220, 222 Lorentz gauge 297
M Mach-Zehnder interferometer 318, 323, 416, 428 magnetic tomography 127 maser,one-atom 385 Maslov’s asymptotic theory 26 -method 21,27 Maximum-entropy principle 153, 154 M e coefficients 252, 253 -theory 252
0 optical Bloch equations 329, 333, 334, 362 - homodyne tomography 101, 102, 160 - homodyning 69-100
P P-function, Glauber-Sudarshan 193 - -, positive 88-90 parametric amplification 94 - oscillator, nonlinear 377 paraxial approximation 43, 44, 296302 - wave equation 296, 307 partial wave expansion 224 Penning trap 182 phase-sensitive measurement of light 69-100 photodetection 195-1 97 photon statistics, subpoissonian 377 propagator approximation 221 propensity 87, 91, 118
Q Q-function 66, 81-87, 89,95, 108, 193, 194, 385 -, reconstruction of 123 quadrature-component state 190 - squeezing 392, 397417,457 quantum nondemolition measurement 377, 418435 - - _ of the photon number 418-427 - state endoscopy 124 - - measurement,tomographic 65 --of matter ~ y S t e m s 157-187 - - reconstruction, tomographic 66 - - representation 189-195
R Rabi frequency, vacuum 97 radiation pressure 294 radiative transfer, theory of 17 Radon transform 101, 105, 160, 176, 185 Raman scattering, stimulated 376, 379, 387, 389,391,404,446 Raman-Nath approximation 279 Rayleigh range 44,300, 330 -scattering 386 RayleighGans approximation 234, 256 resonance fluorescence 377 Riccati-Bessel function 236 Riccati-Neumann function 236 rotating-wave approximation 331
SUBJECT MDEX FOR VOLUME XXXIX
Rutherford formula 226 , Rydberg wave packet 180 Rytov approximation 229 S s-parametrized functions 87, 192, 193 saddle-point method 5, 19, 2 1 Sagnac interferometer, asymmetric 4 4 8 4 5 1 Schrodinger equation 3, 113, 218, 219, 229 - -, nonlinear 390,405,439 second-harmonic generation 322 Sitenkmlauber approximation 2 15 soliton, experiments with 431435 -, quantum 380 - in optical fibers 375, 389 spiwrbit coupling of light 363-366 squeezed light 110 - vacuum 399, 410414 squeezing 377 -, amplitude 415 -, Kerr 399 -, photon-number 379, 380, 392, 435-458 -, pulsed 429, 452 -, quadrature 392,397417,457 stationary phase, method of 19
489
Stern-Gerlach apparatus 184 - - measurements 99 Struve function 256 superoperator 143 Susskind-Glogower operator 140 symplectic tomography 104
V vibrational wave packet 158 vibrations, anharmonic 160 von Neumann entropy 153
W Wigner function 67, 79, 87, 101, 108, 122, 131, 151, 169, 170, 172, 176, 177, 185, 193, 194, 385 Wittaker-Shannon sampling theorem 177 WKB approximation 221 -method 22 -phase 217 Y Yukawa potential 228,229
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CONTENTS OF PREVIOUS VOLUMES
VOLUME I (1961)
I I1
The Modem Development of Hamiltonian Optics, R.J. PEGIS Wave Optics and Geometrical Optics in Optical Design, K. MIYAMOTO 111 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 VI Interference Color, H. KUBOTA VII Dynamic Characteristics of Visual Processes, A. FIORENTIM VIII Modem Alignment Devices, A.C.S. VANHEEL
1- 29 31- 66 67-108 109- 1 53
1 55-2 10 211-251 253-288 289-329
VOLUME I1 (1963)
I
Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy, G.W. STROKE I1 The Metrological Applications of Diffraction Gratings, J.M. BURCH I11 Diffusion Through Non-Uniform Media, R.G. GIOVANELLI IV Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering, J. TSUJRTCHI V Fluctuations of Light Beams, L. MANDEL VI Methods for Determining Optical Parameters of Thin Films, F. ABELBS
1- 12 73-108 109-129 13t-180 181-248 249-288
VOLUME rn (1 964) I The Elements of Radiative Transfer, F. KOTTLER B. ROIZEN-DOSSIER I1 Apodisation, P. JACQUINOT, III Matrix Treatment of Partial Coherence, H. GAMO
1- 28 29-186 187-332
VOLUME IV (1965) I Higher Order Aberration Theory, J. FOCKE I1 Applications of Shearing Interferometry, 0. BRYNGDAHL 111 Surface Deterioration of Optical Glasses, K. KINOSITA IV Optical Constants of Thin Films, P. ROUARD, P. BOUSQUET V The Miyamot+Wolf Diffraction Wave, A. RUBINOWICZ VI Aberration Theory of Gratings and Grating Mountings, W.T. WELFORD VII Diffraction at a Black Screen, Part I: Kirchhoffs Theory, F. KOTTLER 49 1
1- 36 37- 83 85-143 145-197 199-240 241-280 281-314
492
CONTENTS OF PREVIOUS VOLUMES
VOLUME V (1966) I I1
Optical Pumping, C. CoHEN-TANNOUDn, A. KASTLER 1- 81 Non-Linear Optics, P.S. PERSHAN 83-144 111 Two-Beam Interferometry, W.H. STEEL 145-197 199-245 IV Instruments for the Measuring of Optical Transfer Functions, K. MURATA V Light Reflection from Films of ContinuouslyVarying Refractive Index, R. JACOBSSON247-286 VI X-Ray Crystal-Structure Determination as a Branch of Physical Optics, H. LIPSON, C.A. TAYLOR 287-350 351-370 VII The Wave of a Moving Classical Electron, J. PXCHT VOLUME VI (1967)
1- 52 Recent Advances in Holography, E.N. LEITH,J. UPATNIEKS 53- 69 II Scattering of Light by Rough Surfaces, P.BECKMANN III Measurement of the Second Order Degree of Coherence, M. FRANCON, S. MALLICK 71-104 105-170 IV Design of Zoom Lenses, K. YW 17 1-209 V Some Applications of Lasers to Interferometry, D.R. HERRIOT VI Experimental Studies of Intensity Fluctuations in Lasers, J.A. ARMSTRONG, 21 1-257 A.W. S m 259-330 VII Fourier Spectroscopy, G.A. VANASSE, H. SAKAI 331-377 VIII Diffraction at a Black Screen, Part II: Electromagnetic Theory, F. KOTTLER
I
VOLUME VII (1969) Multiple-Beam Interference and Natural Modes in Open Resonators, G. KOPPELMAN 1- 66 67-137 R.J. PEGIS II Methods of Synthesis for Dielectric Multilayer Filters, E. DELANO, 111 Echoes at Optical Frequencies, LD. ABELLA 139-168 N Image Formation with Partially Coherent Light, B.J. THOMPSON 169-230 231-297 V Quasi-Classical Theory of Laser Radiation, A.L. M I K A E L ~ M.L. , BR-MKAELIAN VI The Photographic Image, S. Oom 299-358 W Interaction of Very Intense Light with Free Electrons, J.H. EBERLY 359415 I
VOLUME VIII (1970) 1- 50 Synthetic-Aperture Optics, J.W. GOODMAN 51-131 The Optical Performance of the Human Eye, G.A. FRY 133-200 III Light Beating Spectroscopy, H.Z. Cuhmms, H.L. SWINNEY 201-237 IV Multilayer Antireflection Coatings, A. MUSSET,A. THELEN 239-294 V Statistical Properties of Laser Light, H. h s m VI Coherence Theory of Source-Size Compensation in Interference Microscopy, 295-341 T. YAMAMOTO 343-372 Vn Vision in Communication, L. LEVI 37340 VIII Theory of Photoelectron Counting, C.L. MEHTA
I I1
VOLUME IX (1971) I Gas Lasers and their Application to Precise Length Measurements, A.L. BLOOM 1 1 Picosecond Laser Pulses, A.J. D E W 111 Optical Propagation Through the Turbulent Atmosphere, J.W. STROHBEHN IV Synthesis of Optical Birefringent Networks, E.O. AMMA"
1- 30 31- 71 73-122 123-177
CONTENTS OF PREVIOUS VOLUMES
V Mode Locking in Gas ,Lasers, L. ALLEN,D.G.C. JONES VL. GINZBURG Vl Crystal Optics with Spatial Dispersion, VM. AGRANOVICH, VII Applications of Optical Methods in the Difiaction Theory of Elastic Waves, K. GNIADEK,J. PETYKIEWICZ VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based 6n Use of the Prolate Functions, B.R FRIEDEN
493 179-234 235-280 281-310 3 11407
VOLUME X (1972) I I1
III IV V
VI VII
Bandwidth Compression of Optical Images, T.S. HUANG The Use of Image Tubes as Shutters, R.W. SMITH Tools of Theoretical Quantum Optics, M.O. SCULLY, K.G. WHITNEY Field Correctors for Astronomical Telescopes, C.G. WYNNE Optical Absorption Strength of Defects in Insulators, D.Y. SMITH,D.L. DFXTBR Elastooptic Light Modulation and Deflection, E.K. SITI-IG Quantum Detection Theory, C.W. HELSTROM
1- 44 45- 87 89-135 137-164 165-228 229-288 28S369
VOLUME XI (1973) I
II 111
IV V
VI VII
Master Equation Methods in Quantum Optics, G.S. AGARWAL Recent Developments in Far Infrared Spectroscopic Techniques, H. YOSHINAGA Interaction of Light and Acoustic Surface Waves, E.G. LEAN Evanescent Waves in Optical Imaging, 0. BRYNGDAHL Production of Electron Probes Using a Field Emission Source, A.V CREWE Hamiltonian Theory of Beam Mode Propagation, J.A. A R N A ~ Gradient Index Lenses, E.W. MARCHAND
1- 76 77-122 123-166 167-22 1 223-246 247-304 305-3 37
VOLUME XI1 (1974) Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams, 0. SVELTO 1- 51 Self-Induced Transparency, R.E. SLUSHER 53-100 111 Modulation Techniques in Spectrometry, M. -WIT, J.A. DECKER JR 101-162 IV Interaction of Light with Monomolecular Dye Layers, K.H. DREXHAGE 163-232 V The Phase Transition Concept and Coherence in Atomic Emission, R. GRAHAM 233-286 VI Beam-Foil Spectroscopy,S. BASHKIN 287-344
I
II
VOLUME XIII (1976) I
On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium
Environment, H.P. BALTES The Case For and Against Semiclassical W a t i o n Theory, L. -EL III Objective and Subjective Spherical Aberration Measurements of the Human Eye, W.M. ROSENBLUM, J.L. CHRISTENSEN IV Interferometric Testing of Smooth Surfaces, G. SCHULZ, J. SCHWLDER V Self-Focusing of Laser Beams in Plasmas and Semiconductors, M.S. SODHA, A.K. GHATAK, V.K. TRIPATHI VI Aplanatism and Isoplanatism, W.T. WELFORD
II
1- 25 27- 68 69- 91 93-167 169-265 267-292
494
CONTENTS OF PREVIOUS VOLUMES
VOLUME X N (1976) I I1
The Statistics of Speckle Patterns, J.C. DAINTY High-Resolution Techniques in Optical Astronomy, A. LABEYRIE m Relaxation Phenomena in Rare-Earth Luminescence, L.A. RISEBERG, M.J. WEBER N The Ultrafast Optical Ken Shutter, M.A. DUGUAY V Holographic Diffkaction Gratings, G. SCHMAHL,D. RUDOLPH VI Photoemission, P.J. VERNIER VII Optical Fibre Waveguides - A Review, P.J.B. CLARRICOATS
1- 46 41- 87 89-159 161-193 195-244 245-325 327402
VOLUME XV (1977) I I1 111 N V
Theory of Optical Parametric Amplification and Oscillation, W. BRUNNER, H. PAUL A. MEESSEN Optical Properties of Thin Metal Films, P. ROUARD, Projection-Type Holography, T. OKOSHI Quasi-Optical Techniques of Radio Astronomy, T.W. COLE Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media, 3. VAN KRANENDONK, J.E. SIPE
1- 75 77-137 139-185 187-244 245-350
VOLUME XVI (1978) I
II III
Iv V VI VII
1- 69 Laser Selective Photophysics and Photochemistry, VS. LETOKHOV Recent Advances in Phase Profiles Generation, J.J. CLAJR, C.I. ABITBOL 71-117 119-232 Computer-Generated Holograms: Techniques and Applications, W.-H. LEE Speckle Interferometry, A E . ENNOS 233-288 Deformation Invariant, Space-Variant Optical Pattern Recognition, D. CASASENT, 289-356 D. PSALTIS 3 5 7 41 Light Emission From High-Current Surface-Spark Discharges, R.E. BEVERLY I11 Semiclassical Radiation Theory Within a Quantum-Mechanical Framework, 413-448 I.R. SEMTZKY
VOLUME XVII (1980) I Heterodyne Holographic Interferometry, R. DANDLIKER 1- 84 I1 Doppler-Free Multiphoton Spectroscopy, E. GIACOBINO, B. CAGNAC 85-161 111 The Mutual Dependence Between Coherence Properties of Light and Nonlinear 163-238 B. WILHELMI Optical Processes, M. SCHLJBERT, 239-277 IV Michelson Stellar Interferometry, WJ. TANGO,R.Q. 'Mss 279-345 V Self-Focusing Media with Variable Index of Refraction, A.L. MIKAELIAN
VOLUME XVIII (1980) I I1
Graded Index Optical Waveguides: A Review, A. GHATAK, K. THYAGARAJAN 1-126 Photocount Statistics of Radiation Propagating Through Random and Nonlinear 127-203 Media, J. PERINA 111 Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium, 20&256 VI. TATARSKII, VU. ZAVOROTNYI IV Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns, 257-346 M.V BERRY,C. UPSTILL
CONTENTS OF PREVIOUS VOLUMES
495
VOLUME xu( (1981) I
Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence, B.R. MOLLOW 1- 43 II Surface and Size Effects on the Light Scattering Spectra of Solids, D.L. Mrtts, K.R. SUBBASWAMY 45-137 111 Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids, S. USHIODA 13%210 IV Principles of Optical Data-Processing,H.J. BUTERWECK 21 1-280 281-376 V The Effects of Atmospheric Turbulence in Optical Astronomy, F. RODDIER
VOLUME XX (1983) I
Some New Optical Designs for Ultra-Violet Bidimensional Detection of AstronomF! ~CRUVELLIER, M. DETAILLE, M. WSSE 1- 61 ical Objects, G. C O U R ~ , II Shaping and Analysis of Picosecond Light Pulses, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE 63-153 III Multi-Photon Scattering Molecular Spectroscopy, S. KIELICH 155-26 1 IV Colour Holography, P. HARIHARAN 263-324 V Generation of Tunable Coherent Vacuum-Ultraviolet Radiation, W. JAMROZ, B.P. STOICHEFF 325-380
VOLUME XXI (1984) 1- 67 Rigorous Vector Theories of Diffraction Gratings, D. MAYSTRE Theory of Optical Bistability, L.A. LUGIATO 69-216 111 The Radon Transform and its Applications, H.H. BARRETT 217-286 IV Zone Plate Coded Imaging: Theory and Applications, N.M. CEGLIO, D.W. SWEENEY.287-354 V Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity, J.C. ENGLUND, R.R. SNAPP,W.C. SCHIEVE 355428 I I1
VOLUME XXII (1985) Optical and Electronic Processing of Medical Images, D. MALACARA 1- 76 II Quantum Fluctuations in Vision, M.A. BOL~MAN, W.A. VANDE GRIND,F! ZUIDEMA 77-144 III Spectral and Temporal Fluctuations of Broad-Band Laser Radiation, A.V MASALOV 145-196 IV Holographic Methods of Plasma Diagnostics, G.V OSTROVSKAYA, Yu.1. OSTROVSKY197-270 V Fringe Formations in Deformation and Vibration Measurements using Laser Light, I. YAh4AGUCHI 271-340 VI Wave Propagation in Random Media: A Systems Approach, R.L. FANTE 341-398 I
VOLUME XXIII (1986) I
Analytical Techniques for Multiple Scattering from Rough Surfaces, J.A. DESANTO, G.S. BROWN 1- 62 I1 Paraxial Theory in Optical Design in Terms of Gaussian Brackets, K. TANAKA 63-1 11 III Optical Films Produced by Ion-Based Techniques, PJ. MARTIN,R.P NETTER~~ELD113-182 IV Electron Holography, A. TONOMLJRA 183-220 V Principles of Optical Processing with Partially Coherent Light, F.T.S. Yu 221-275
496
CONTENTS OF PREVIOUS VOLUMES
VOLUME XMV (1987) Micro Fresnel Lenses, H. NISHRIARA, T. SUHARA Dephasing-Induced Coherent Phenomena, L. R~THBERF 111 Interferometry with Lasers, F! HAR~ARAN N Unstable Resonator Modes, K.E. ~UFHSTUN V Information Processing with Spatially Incoherent Light, I. GLASER I
I1
1- 37 ,39-101 103-164 165-387 389-509
VOLUME XXV (1988)
P. MANDEL, Dynamical Instabilities and Pulsations in Lasers, N.B. ABRAHAM, L.M. NARDUCCI 1-190 I1 Coherence in Semiconductor Lasers, M. OHTSU,T. TAKO 191-278 III Principles and Design of Optical Arrays, WANGSHAOMIN, 279-348 L. RONCHI N Aspheric Surfaces, G. SCHULZ 349-415 I
VOLUME XXVI (1988)
I I1 111 IV V
Photon Bunching and Antibunching, M.C. BICH, B.E.A. SALEH Nonlinear Optics of Liquid Crystals, I.C. KHOO Single-Longitudinal-Mode Semiconductor Lasers, G.P. AFRAWAL Rays and Caustics as Physical Objects, YuA. KRAVTSOV Phase-Measurement Interferometry Techniques, K. CREATH
1-104 105-161 163-225 227-348 349-393
VOLUME XXVII (1989) The Self-Imaging Phenomenon and Its Applications, K. PATORSKI 1-108 Axicons and Meso-Optical Imaging Devices, L.M. SOROKO 109-160 111 Nonimaging Optics for Flux Concentration, I.M. BASSETT,W.T. WELFORD, R. WINSTON 161-226 N Nonlinear Wave Propagation in Planar Structures, D. MIHALACHE, M. BERTOLOTTI, c. SIBMA 227-3 13 V Generalized Holography with Application to Inverse Scattering and Inverse Source Problems, R.P. PORTER 3 15-397
I I1
VOLUME XXVnI (1990) Digital Holography - Computer-GeneratedHolograms, 0. BRYNGDAHL, F. WYROWSKI 1- 86 Quantum Mechanical Limit in Optical Precision Measurement and Communication, S. WCHIDA, S. SAITO,N. IMOTO,T. YWAGAWA, M. KITAGAWA, Y. YAMAMOTO, G. BJORK 87-179 I11 The Quantum Coherence Properties of Stimulated Raman Scattering, M.G. RAYMER, LA. WALMSLEY 181-270 IV Advanced Evaluation Techniques in Interferometry, J. SCHWIDER 271-359 V Quantum Jumps, R.J. COOK 361-416
I I1
CONTENTS OF PREVIOUS VOLUMES
497
VOLUME XXIX (1991) I
Optical Waveguide Diffraction Gratings: Coupling between Guided Modes, 1- 63 D.G. HALL I1 Enhanced Backscattering in Optics, Yu.N. BARABANENKOV, Yu.A. KRAVTSOV, VD. O z m , A.I. SAICHEV 65-197 199-291 111 Generation and Propagation of Ultrashort Optical Pulses, I.P. CHRISTOV N Triple-Correlation Imaging in Optical Astronomy, G. WEIGELT 293-319 V Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics, C. FLYTZANIS, F. HACHE,M.C. KLEIN,D. RICARD, PH. ROUSSIGNOL 32141 1
VOLUME XXX (1 992) I
Quantum Fluctuations in Optical Systems, S . REYNAUD, A. HEIDMANN, E. GIACOBINO, C. FABRE I- 85 I1 Correlation Holographc and Speckle Interferometry,Yu.1. OSTROVSKY, VP. SHCHEPrNov 87-135 111 Localization of Waves in Media with One-Dimensional Disorder, VD. FREKIKHER, S.A. GREDESKUL 137-203 IV Theoretical Foundation of Optical-Soliton Concept in Fibers, Y. KODAMA, A. HASEGAWA 205-259 26 1-355 V Cavity Quantum Optics and the Quantum Measurement Process., P. MEYSTRE
VOLUME XXXI (1993) Atoms in Strong Fields: Photoionization and Chaos, P.W. MILONNI, B. SUNDARAM 1-137 II Light Diffraction by Relief Gratings: A Macroscopic and Microscopic View, E. POPOV 139-1 87 III Optical Amplifiers, N.K. DLITTA, J.R. SIMPSON 189-226 IV Adaptive Multilayer Optical Networks, D. PSALTIS,Y. QIAO 227-261 V Optical Atoms, R.J.C. SPREEW,J.P. WOERDMAN 263-3 I9 VI Theory of Compton Free Electron Lasers, G. DATTOLI, L. GIANNESSI, A. RENIERI, A. TORRE 321412
I
VOLUME XXXII (1993) Guided-Wave Optics on Silicon: Physics, Technology and Status, B.P. PAL 1- 59 61-144 II Optical Neural Networks: Architecture, Design and Models, F.T.S. Yu III The Theory of Optimal Methods for Localization of Objects in Pictures, L.l? YAROSLAVSKY 145-201 IV Wave Propagation Theories in Random Media Based on the Path-Integral Approach, VI. TATARSKII, VU. ZAVOROTNY 203-266 M.I. CHARNOTSKII, J. GOZANI, V Radiation by Uniformly Moving Sources. VavilwXherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, VL. GINZBURG 267-3 12 VI Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas, G. MANFRAY, C. MANUS 313-361 I
498
CONTENTS OF PREVIOUS VOLUMES
VOLUME XXXIII (1994) I
The Imbedding Method in Statistical Boundary-Value Wave Problems, W.KLY-
ATSKRi 1-127 129-202 I1 Quantum Statistics of Dissipative Nonlinear Oscillators, V. PEKLNOVA, A. LIJK~ Ill Gap Solitons, C.M. DE STERKE, J.E. SJPE 203-260 IV Direct Spatial Reconstruction of Optical Phase from Phase-Modulated Images, VI. VLAD, D. W A C A R A 261-317 J. Oz-VOGT 3 19-388 V Imaging through Turbulence in the Atmosphere, M.J. BERAN, VI Digital Halfioning: Synthesis of Binary Images, 0. BRYNGDAHL, T. SCHEERMESSER, F. WYROWSIU 389-463
VOLUME XXXIV (1995) I
II
Quantum Interference, Superposition States of Light, and Nonclassical Effects, V: B u ~ KPL. , KNIGHT Wave Propagation in Inhomogeneous Media:; Phase-Shift Approach, L.F! PRESNYAKOV
III The Statistics of Dynarmc Speckles, T. OWOTO, T. ASAKURA N V
1-158 159-1 8 1 183-248
Scattering of Light from Multilayer Systems with Rough Boundaries, I. OHLfDAL, K. N A ~ T I L M., O H L ~ A L 249-33 1 Random Walk and Diffusion-Like Models of Photon Migration in m b i d Media, A.H. GAND~AKHCHE, G.H. WEISS 333402 VOLUME. XXXV (1996)
1- 60 Transverse Paaerns in Wide-Aperture Nonlinear Optical Systems, N.N. ROSANOV Optical Spectroscopy of Single Molecules in Solids, M. ORRIT,J. BERNARD, R. BROWN,B. Lows 61-144 111 Interferometric Multispectral Imaging, K. ITOH 145-196 N Interferometric Methods for Artwork Diagnostics, D. PAOLETTI,G. SCHIRRIPA SPAGNOLO 197-255 V Coherent Population Trapping in Laser Spectroscopy, E. ARMONDO 257-354 VI Quantum Phase Properties of Nonlinear Optical Phenomena, R. TANAS,A. MIRANOWICZ, Ts. GANTSOG 355-446 1
II
VOLUME XXXVI (1996)
I
Nonlinear Propagation of Strong Laser Pulses in Chalcogenide Glass Films, V CHUMASH, I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLOTTI 1- 47 B.C. SANDERS 49-128 11 Quantum Phenomena in Optical Interferometry, F'. HARWARAN, C. DE MOL 129-178 III Super-Resolution by Data Inversion, M. BERTERO, 179-244 N Radiative Transfer: New Aspects of the Old TheoIy, Yu.A. KRAVTSOV,L.A. AFXESYAN V Photon Wave Function, I. BIALYNICKI-BIRULA 245-294 VOLUME XXXVII (1997) The Wigner Distribution Function in Optics and Optoelectronics, D. DRAGOMAN Dispersion Relations and Phase Retrieval in Optical Spectroscopy, K.-E. ~ I P O N E N , E.M. VARTIAI", T. ASAKURA III Spectra of Molecular Scattering of Light, I.L. FABELINSKU
I
1- 56
II
57- 94 95-1 84
CONTENTS OF PREVIOUS VOLUMES
G.P. AGRAWAL N Soliton Communication Systems, R.-J. ESSIAMBRE, V Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems, 0. KELLER VI Tunneling Times and Superlmninality,R.Y. CHIAO,A.M. STEINBERG
499 185-256 257-343 345-405
VOLUME XXXVIII (1998) 1- 84 Nonlinear Optics of Stratified Media, S. DLITTAGWTA 85-164 Optical Aspects of Interferometric Gravitational-Wave Detectors, €? HELLO 111 Thermal Properties of Vertical-Cavity Surface-Emitting Semiconductor Lasers, W. NAKWASKI, M. OsrErsia 165-262 D. MENDLOVIC, Z. ZALEVSKY263-342 IV Fractional Transformations in Optics, A.W. LOHMANN, V Pattern Recognition with Nonlinear Techniques in the Fourier Domain, B. JAVIDI, J.L. HORNER 343418 4 19-5 13 VI Free-space Optical Digital Computing and Interconnection, J. JAHNS
I I1
This Page Intentionally Left Blank
CUMULATIVE INDEX - VOLUMES I-XXXIX
F., Methods for Determining Optical Parameters of Thin Films ABELLA,I.D., Echoes at Optical Frequencies ABITBOL,C.I., see Clair, J.J. ABRAHAM, N.B., P. W m , L.M. NARDUCCI, Dynamical Instabilities and Pulsations in Lasers AGARWAL, G.S., Master Equation Methods in Quantum Optics AGRANOVICH, VM., VL. GINZBURG, Crystal Optics with Spatial Dispersion AGRAWAL, G.P., Single-Longitudinal-Mode Semiconductor Lasers G.P.,see Essiambre, R.-J. AGRAWAL, ALLEN, L., D.G.C. JONES,Mode Locking in Gas Lasers ALLEN,L., M.J. PADGETT, M. BABIKER, The Orbital Angular Momentum of Light AMMA", E.O., Synthesis of Optical Birefringent Networks A~RESYAN, LA., see Kravtsov, Yu.A. ARIMONDO, E., Coherent Population Trapping in Laser Spectroscopy Experimental Studies of Intensity Fluctuations in ARMSTRONG, J.A., A.W. SMITH, Lasers ARNAUD,J.A., Hamiltonian Theory of Beam Mode Propagation ASAKURA, T., see Okamoto, T. ASAKURA, T., see Peiponen, K.-E. ASATRYAN, A.A., see Kravtsov, Yu.A. &EL!%,
BAF~IKER, M., see Allen, L. BALTES,H.P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment BMANENKOV, Yu.N., W.A. Ikwrsov, W.O m ,A.I. SAICHEV, Enhanced Backscattering in Optics BARAKAT, R.,The Intensity Distribution and Total Illuminationof Aberration-Free Diffraction Images BARREIT,H.H., The Radon Transform and its Applications S., Beam-Foil Spectroscopy BASHKKN, BASSETT,I.M., W.T. WLFORD, R. WINSTON,Nonimaging Optics for Flux Concentration BECKMA",P., Scattering of Light by Rough Surfaces Imaging through Turbulence in the Atmosphere BERAN, M.J., J. OZ-VOGT, BERNARD, J., see Orrit, M. Catastrophe Optics: Morphologies of Caustics and their BERRY,M.V., C. UPSTILL, Diffraction Patterns BERTERO, M., C. DE MOL,Super-Resolution by Data Inversion BERTOLOTTI, M., see Mihalache, D. 501
11, 249 VII, 139 XVI, 71
xxv,
1
XI, 1 M, 235 XXVI, 163 XXXVII, 185 IX, 179 XXXM, 291 IX, 123 XXXVI, 179 XXXV, 257 VI, 211 XI, 247 XXXW, 183 XXXVII, 57
XXxM,
1
xxw(,
291
xm,
1
XXIX, 65
I, 67 XXI, 217 XII, 287 XXW, VI, XXXIII, XXXV,
161 53 319 61
XVIII, 257 XXXVI, 129 XXW, 227
502
CUMULATIVE INDEX - VOLUMES I-XXXIX
BERTOLOTTI, M., see Chumash, V BEVERLY 111, R.E., Light Emission From High-Current Surface-Spark Discharges BIALYNICKI-BIRULA, I., Photon Wave Function BJORK,G., see Yamamoto, Y. BLOOM, A.L., Gas Lasers and their Application to Precise Length Measurements BOW, M.A., W.A. VANDE GRIND,P. ZUIDEMA, Quantum Fluctuations in Vision F!, see Rouard, P. BOUSQUET, BROWN, G.S., see DeSanto, J.A. R., see Omt, M. BROWN, BRUNNER,W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation O., Applications of Shearing Interferometry BRYNGDAHL, BRYNGDAHL, O., Evanescent Waves in Optical Imaging O., F. WYROWSKI, Digital Holography - Computer-Generated BRYNGDAHL, Holograms BRYNGDAHL, O., T. SCHEERMESSER, F. WYROWSKI, Digital Halftoning: Synthesis of Binary Images BURCH, J.M., The Metrological Applicattbns of Diffraction Gratings BUTIERWECK, H.J., Principles of Optical Data-Processing BUZEK,V, P.L. KNIGHT, Quantum Interference, Superposition States of Light, and Nonclassical Effects CAGNAC, B., see Giacobino, E. CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical pattern Recognition CEGLIO,N.M., D.W. SWEENEY,Zone Plate Coded Imaging: Theory and Applications VI. TATARSKII, VU. ZAVOROTNY, Wave Propagation CHARNOTSKII, M.I., J. GOZANI, Theories in Random Media Based on the Path-Integral Approach CHIAO,R.Y., A.M. STEMBERG, Tunneling Times and Superluminality CHRISTENSEN, J.L., see Rosenblum, W.M. CHRISTOV, I.P., Generation and Propagation of Ultrashort Optical Pulses V., I. COJOCARU, E. FAZIO,F. MICHELOTTI, M. BERTOLO~I, Nonlinear CHUMASH, Propagation of Strong Laser Pulses in Chalcogenide Glass Films CLAIR, J.J., C.I. ABITEOL, Recent Advances in Phase Profiles Generation CLARRICOATS, P.J.B., Optical Fibre Waveguides - A Review Optical Pumping COWN-TANNOUDII, C., A. KASTLER, COIOCARU, I., see Chumash, V. COLE,T.W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see Froehly, C. COOK,R.J., Quantum Jumps C O U R ~G., S , P. CRWELLIER, M. DETAILLE, M. SAYSSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects CREATH, K., Phase-Measurement Interferometry Techniques CREWE,A.V, Production of Electron Probes Using a Field Emission Source CRWELLIER, P., see Court&s,G. CUMMINS, H.Z., H.L. SWINNEY, Light Beating Spectroscopy DAINTY, J.C., The Statistics of Speckle Patterns DANDLIKEK R., Heterodyne Holographic Interferometry
=,
1
XVI, MNI, XXVIII, IX, XXII,
357 245 87 1 77 IV, 145 XXIII, 1 XXXV, 61
xv, Iv,
1 37 XI, 167
XXVIII,
1
XXXIII, 389 11, 73 XIX, 211
XVII, 85 XVI, 289 XXI, 287 XXXII, 203 XXXW, 345 XIII, 69 XXIX, 199 XXXVI, 1 XVI, 71 X l V , 327
v,
1
MNI, 1 XV, 187 XX, 63 XXVIII, 361 XX,
1
XXVI,349 XI, 223 XX, 1 VnI, 133
m, XVII,
1 1
503
CUMULATIVE INDEX - VOLUMES I-XXXIX
DATTOLI,G.,L. GIANNESSI,A. RENIERI, A. TORRE,Theory of Compton Free Electron Lasers DE MOL,C., see Bertero, M. DE STEW, C.M., J.E. SPE, Gap Solitons DECKER JR, LA., see Harwit, M. DELANO, E., R.J. PEGIS,Methods of Synthesis for Dielectric Mdtilayer Filters DEMARIA,A.J., Picosecond Laser Pulses Analytical Techniques for Multiple Scattering from DESANTO,J.A., G.S.BROWN, Rough Surfaces DETAILLE, M., see Courtes, G. DEXTER, D.L., see Smith, D.Y. DRAGOMAN, D., The Wigner Distribution Function in Optics and Optoelectronics DREXHAGE, K.H., Interaction of Light with Monomolecular Dye Layers DUGUAY, M.A., The Ultrafast Optical Kerr Shutter DUTTA, N.K., J.R. SIMPSON,Optical Amplifiers DUTTA GUPTA,S., Nonlinear Optics of Stratified Media EBERLY,J.H., Interaction of Very Intense Light with Free Electrons ENGLUND,J.C., R.R. SNAPP,W.C. SCHIEVE, Fluctuations, Instabilities and Chaos in the Laser-Driven Nonlinear Ring Cavity ENNOS,A.E., Speckle Interferometry Soliton Communication Systems ESSIAMBRE, R.-J., G.P. AGRAWAL,
XXXI, XXXVI, XXXIII, XII, W,
321 129 203 101 67 M, 31
XXIII,
1 1 165 1 163 161 189 1
=, X,
xxxw, XII, XIV,
XXXI, XXXVIII,
w, 359 X X I , 355 XVI, 233 XXXW, 185
FABELINSKLI, I.L., Spectra of Molecular Scattering of Light FABRE,C., see Reynaud, S. F m , R.L., Wave Propagation in Random Media: A Systems Approach FAZIO,E., see Chumash, V FIORENTINI, A,, Dynamic Characteristics of Visual Processes FLYTZANIS, C., F. HACHE,M.C. KLEIN,D. hem, PH. ROUSSIGNOL, Nonlinear Optics in Composite Materials. 1. Semiconductor and Metal Crystallites in Dielectrics FOCKE,J., Higher Order Aberration Theory FORBES,G.W., see Kravtsov, Yu.A. FRANCON, M., S. WLICK, Measurement of the Second Order Degree of Coherence FREILWIER,VD., S.A. GREDESKUL, Localization of Waves in Media with OneDimensional Disorder FRIEDEN,B.R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of Picosecond Light Pufses FRY, G.A., The Optical Performance of the Human Eye
XXXw, 95
GABOR, D., Light and Information GAMO, H., Matrix Treatment of Partial Coherence GmmAKHcHE, A.H., G.H. WISS, Random Walk and Diffusion-Like Models of Photon Migration in Turbid Media G~SOG Ts.,,see TanaS, R. Graded Index Optical Waveguides: A Review GHATAK, A., K. THYAGARAJAN, GHATAK, A,&, see Sodha, M.S. GIACOBQJO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy
I, 109 111. 187
=, 1 XXU, 341 XXXVI, I I. 253
VI, 71 XXX, 137
IX,311 XX, 63 VIII, 51
XXXIV, 333 XXXV, 355 XVIII, 1 XIII, 169 XW, 85
504
CUMULATIVE INDEX
~
VOLUMES I-XXXIX
GrAcoBrNo, E., see Reynaud, S. GIANNESSI, L., see Dattoli, G. GINZBURG, V.L., see Agranovich, V.M. GINZBURG, VL., Radiation by Uniformly Moving Sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena GIOVANELLI, R.G., Diffusion Through Non-Uniform Media GLASER, I., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWCZ, Applications of Optical Methods in the Diffraction Theory of Elastic Waves GOODMAN, J.W., Synthetic-Aperture Optics GOZANI, J., see Charnotskii, M.I. GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission S.A., see Freilikher, VD. GREDESKUL, F., see Flytzanis, C. HACHE, HALL,D.G., Optical Waveguide Diffraction Gratings: Coupling between Guided Modes P., Colour Holography HARIHARAN, HARIHARAN, P., Interferometry with Lasers HARIHARAN, P., B.C. SANDERS, Quantum Phenomena in Optical Interferometry HARWIT, M., J.A. DECKER JR, Modulation Techniques in Spectrometry HASEGAWA, A,, see Kodama, Y. HEIDMANN, A., see Reynaud, S. HELLO, P., Optical Aspects of Interferometric Gravitational-Wave Detectors HELSTROM, C,W., Quantum Detection Theory HERRIOT, D.R., Some Applications of Lasers to Interferometry HORNER,J.L., see Javidi, B. HUANG, T.S., Bandwidth Compression of Optical Images IMOTO, N., see Yamamoto, Y. ITOH,K., Interferometric Multispectral Imaging JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive Index P., B. Ro?m-Dossrm, Apodisation JACQUINOT, JAHNS, J., Free-space Optical Digital Computing and Interconnection JAMROZ, W., B.P. STOICHEFF, Generation of Tunable Coherent Vacuum-Ultraviolet Radiation JAVIDI,B., J.L. HORNER, Pattern Recognition with Nonlinear Techniques in the Fourier Domain JONES,D.G.C., see Allen, L.
KASTLER,A,, see Cohen-Tannoudji, C. KELLER,O., Local Fields in Linear and Nonlinear Optics of Mesoscopic Systems MOO, LC., Nonlinear Optics of Liquid Crystals KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy KINOSITA, K., Surface Deterioration of Optical Glasses KITAGAWA, M., see Yamamoto, Y. KLEIN,M.C., see Flytzanis, C.
=, 1 XXXI, 321 IX,235 XXXII, 261 11, 109 XXIV. 389 IX, WI, XXXII, XI, XXX,
281 1 203 233 137
XXIX, 321 XXIX, 1 XX, 263 XXIV, 103 m 1 , 49 XII, 101 XXX, 205 XXX, 1 XXXVIII, 85 X, 289 VI, 171 XXXVIII, 343 x, 1 XXVIII, 87 XXXV, 145
V, 247 111, 29
XXXVIII, 419 XX, 325
xxxvm,343 IX,119 v, 1 XXXW, 257 XXVI, 105 XX, 155 IV, 85 XXVIII, 87 XXE, 321
505
CUMULATIVE INDEX - VOLUMES I-XXXIX
KLYATSKIN, VI., The Imbedding Method in Statistical Boundary-Value Wave Problems KNIGHT, RL., see Buiek, V KODAMA,Y., A. HASEGAWA, Theoretical Foundation of Optical-Soliton Concept in Fibers KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators K o m ~F., , The Elements of Radiative Transfer KO= F., Diffraction at a Black Screen, Part I: Kuchhoffs Theory KO=& F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory KRAVTSOV,Yu.A., Rays and Caustics as Physical Objects KRAVTSOV, Yu.A., see Barabanenkov, Yu.N. Radiative Transfer: New Aspects of the Old KRAVTSOV, Yu.A., L.A. APRESYAN, Theory KRAVTSOV, Yu.A., G.W. FORBES,A.A. ASATRYAN, Theory and Applications of Complex Rays H., Interference Color KUBOTA, LABEYRE, A,, High-Resolution Techniques in Optical Astronomy LEAN,E.G., Interaction of Light and Acoustic Surface Waves LEE,W.-H., Computer-Generated Holograms: Techniques and Applications S, Advances in Holography LEITH,E.N., J. U P A ~ K Recent VS., Laser Selective Photophysics and Photochemistry LETOKHOV, LEUCHS, G., see Sizmann, A. LEVI,L., Vision in Communication X-Ray Crystal-Structure Determination as a Branch of LIPSON,H., C.A. TAYLOR, Physical Optics LOHMA", A.W., D. M E ~ L O V IZ. C , ZALWSKY,Fractional Transformations in Optics LOWS, B., see Orrit, M. LUGIATO, L.A., Theory of Optical Bistability L u K ~A., , see Peiinova, V MACHIDA,S., see Yamamoto, Y. MAWFRAY,G., C. MANUS, Nonlinear Processes in Atoms and in Weakly Relativistic Plasmas MALACARA, D., Optical and Electronic Processing of Medical Images MALACARA, D., see Vlad, VI. MALLICK,S.. see FranGon, M. MANDEL, L., Fluctuations of Light Beams -EL, L., The Case For and Against Semiclassical Radiation Theory MANDFL,P., see Abraham, N.B. MANUS,C., see Mainfray, G. MARE.W., Gradient Index Lenses MARTIN, P.J., R.P. NETTER~~LD, Optical Films Produced by Ion-Based Techniques MASALOV, A.V, Spectral and Temporal Fluctuations of Broad-Band Laser Radiation MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings IMEESSEN, A., see Rouard, P. MEHTA,C.L., Theory of Photoelectron Counting MENDLOVJC, D., see Lohmann, A.W.
XXX, 205 VII, 111,
1 1
IV, 281 VI, 331 XXVI, 227 XXM, 65
XXXVI, 179
=,
1 I, 211
XTV, XI, XVI, VI, XVI, XXXIX, VIII,
47 123 119 1 1 373 343
V, 287 XXXVIII, 263 XXXV, 61 XXI, 69 XXXIII, 129 XXVIII, 87 XXXII, 313 XXII, 1 XXXIII, 261 VI, 71 11, 181 XIII, 27 XXV, 1 XXXII, 313 XI, 305 XXIII, 113 XXII, 145 XXI, 1 XV, 71 VIII, 373 XXXVIII, 263
506
CUMULATIVE INDEX - VOLUMES I-XXXIX
MEYSTRE, I?, Cavity Quantum Optics and the Quantum Measurement Process. MICHELOTTI, F., see Chumash, V MIHALACHE, D., M. BERTOLOTTI, C. SIBILIA, Nonlinear Wave Propagation in Planar Structures MIKAELIAN, A.L., M.L. RR-MWLIAN, Quasi-Classical Theory of Laser Radiation MIKAELIAN, A.L., Self-Focusing Media with Variable Index of Refraction MILLS,D.L., K.R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids MEOW, P.W., B. SUNDARAM, Atoms in Strong Fields: Photoionization and Chaos MIRANOWICZ, A., see Tanai, R. MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design MOLLOW, B.R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence MURATA, K., Instruments for the Measuring of Optical Transfer Functions MUSSET, A., A. THELEN, Multilayer Antireflection Coatings
I, Properties of Vertical-Cavity SurfaceNAKWASKI, W., M. O S ~ ~ S KThermal Emitting SemiconductorLasers NARDUCCI, L.M., see Abraham, N.B. NA~TIL K.,, see Ohlidal, I. NETTE&LD,R.P., see Martin, P.J. NISHMARA, H., T. SUHARA, Micro Fresnel Lenses I., K. N A ~ T I LM., OHL~DAL, Scattering of Light from Multilayer Systems with Rough Boundaries OHL~AL M., , see Ohlidal, I. OHTSU,M.,-T. TAKO,Coherence in Semiconductor Lasers OILWOTO, T., T. ASAKURA, The Statistics of Dynamic Speckles OKOSHI, T., Projection-Type Holography OOUE,S., The Photographic Image O P A T R T., ~ , see Welsch, D.-G. ORRIT,M., J. BERNARD,R. BROWN, B. L o w s , Optical Spectroscopy of Single Molecules in Solids O S ~ S KM., I , see Nakwaski, W. OSTROVSKAYA, G.V, Yu.1. OSTROVSKY, Holographic Methods of Plasma Diagnostics Yu.I., see Ostrovskaya, G.V OSTROVSKY, Correlation Holographic and Speckle OsTRovsKY, Yu.I., V€! SHCHEPINOV, Interferometry OUGHSTUN, K.E., Unstable Resonator Modes OZ-VOGT,J., see Beran, M.J. O z m , VD., see Barabanenkov, Yu.N.
=,
XXX, 261 1
XXW, 227 VII, 231 XVII, 279
m,
45
=, 1
m,355
I, 31
m,
1 V, 199 VIII, 201
XXXVIII, 165 1 XXXIV, 249 Xxm, 113 =, 1
OHLfDAL,
PADGETT,M.J., see Allen, L. PAL,B.P., Guided-Wave Optics on Silicon: Physics, Technology and Status PAOLETIT, D., G. S C ~ P SPAGNOLO, A Interferometric Methods for Artwork Diagnostics PATORSKI, K., The Self-Imaging Phenomenon and Its Applications PAUL,H., see Brunner, W. PEGIS,R.J.,The Modern Development of Hamiltonian Optics PEGIS,R.J., see Delano, E.
XXXIV, 249 XXXIV, 249 XXV, 191 XXXIV, 183 XV, 139 VII, 299 XXXIX, 63 XXXV, 61 XXXVIII, 165
XXII, 197 XXII, 197 XXX, XXIV, XXXIII, XXIX,
87 165 319 65
XXXIX, 291 XXXU, 1
CUhtULATIVE INDEX
~
VOLUMES I-XXXIX
PEPONEN,K.-E., E.M. VARTIAMEN, T. ASAKURA, Dispersion Relations and Phase Retrieval in Optical Spectroscopy PERINA, J., Photocount Statistics of Radiation Propagating Through Random and Nonlinear Media PEhovA, V,A. L d , Quantum Statistics of Dissipative Nonlinear Oscillators PERSHAN, P.S., Non-Linear optics PETYKIEWICZ, J., see Gniadek, K. PICHT,J., The Wave of a Moving Classical Electron 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 PRESNYAKOV, L.P., Wave Propagation in Inhomogeneous Media: Phase-Shift Approach PSALTIS,D., see Casasent, D. PSALTIS,D., Y. QIAO,Adaptive Multilayer Optical Networks QIAO,Y., see Psaltis, D. RAYMEII,M.G., I.A. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering RENIERI,A,, see Dattoli, G. C. FABRE,Quantum Fluctuations in REYNAUD, S., A. HEIDMA”,E. GIACOBMO, Optical Systems &CARD, D., see Flytzanis, C. RISEBERG,L.A., M.J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN,H., Statistical Properties of Laser Light F., The Effects of Atmospheric Turbulence in Optical Astronomy RODDIE& ROIZEN-DOSSIER, B., see Jacquinot, P. RONCHI,L., see Wang Shaomin ROSANOV, N.N., Transverse Patterns in Wide-Aperture Nonlinear Optical Systems Objective and Subjective Spherical ROSENBLUM,W.M., J.L. CHRISTENSEN, Aberration Measurements of the Human Eye ROTHBERG, L., Dephasing-Induced Coherent Phenomena ROUARD, P., P. BOUSQLET,Optical Constants of Thin Films ROUARD,P., A. MEESSEN, Optical Properties of Thin Metal Films ROUSSIGNOL, PH., see Flytzanis, C. RUBINOWICZ, A,, The MiyamotwWolf Diffraction Wave RUDOLPH, D., see Schmahl, G. SAICHEV, A.I., see Barabanenkov, Yu.N. SA~SSE, M., see Courtes, G. SAITO,S., see Yamamoto, Y. SAKAI,H., see Vanasse, G.A. SALEH, B.E.A., see Teich, M.C. SANDERS,B.C., see Hariharan, P. SCHEERMESSER, T., see Bryngdahl, 0. SCHEVE,W.C., see Englund, J.C. SCHIRRPASPAGNOLO, G., see Paoletti, D. SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings
507
XXXVII, 51 XVIII, 121 XXXIII, 129 V, 83 IX,281 V, 351
XXXI, 139 XXVII, 315 XXXIV, 159 XVI, 289 XXXI, 221
XXXI. 221
XXVIII, 181 XXXI, 321
=,
1
XXH, 321 XIV, 89
VIII, 239 XIX, 281 111, 29 XXV, 219 xxxv, 1
XIII, 69 XXIV, 39 IV, 145 xv, 71 XXIX, 321 IV, 199 XIV, 195 XXIX, 65 XX, 1 XXVIII, 81 VI, 259 XXVI, 1 XXXVI, 49 XXXIII, 389 XXI, 355 XXXV, 191 XIV, 195
508
CUMULATIVE INDEX - VOLUMES I-XXXIX
SCHUBERT, M., B. WILHELMI, The Mutual Dependence Between Coherence Properties of Light and Nonlinear Optical Processes S c m z , G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces SCHULZ,G., Asphenc Surfaces SCHWIDER, J., see Schulz, G. SCHWIDER, J., Advanced Evaluation Techniques in Interferometry SCULLY, M.O., K.G. W ~ YTools , of Theoretical Quantum Optics SENITZKY,I.R., Semiclassical Radiation Theory Within a Quantum-Mechanical Framework SHARMA, S.K., D.J. SOMERFORD, Scattering of Light in the Eikonal Approximation SHCHEPINOV, VP., see Ostrovsky, Yu.1. SIBILIA,C., see Mihalache, D. SIMPSON, J.R., see Dutta, N.K. SIPE, J.E., see Van Kranendonk, J. SIPE,J.E., see De Sterke, C.M. SITTIG, E.K., Elastooptic Light Modulation and Deflection SIZMANN, A., G. LEUCHS, The Optical Kerr Effect and Quantum Optics in Fibers SLUSHER, R.E., Self-Induced Transparency SMITH, A.W., see Armstrong, J.A. 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 Englund, J.C. SODHA, M.S., A.K. GHATAK, VK. TRIPATHI, Self-Focusing of Laser Beams in Plasmas and Semiconductors SOMERFORD, D.J., see Sharma, S.K. SOROKO, L.M., Axicons and Meso-Optical Imaging Devices Optical Atoms SPREEUW, R.J.C., J.P. WOERDMAN, STEEL,W.H., Two-Beam Interferometry STEINBERG, A.M., see Chiao, R.Y. STOICHEFF, B.P., see JattUOZ, w. STROHBEHN, J.W, Optical Propagation Through the Turbulent Atmosphere STROKE,G.W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY, K.R., see Mills, D.L. SUHARA,T., see Nishihara, H. SUNDARAM, B., see Milonni, P.W. SVELTO, O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams SWEENEY, D.W., see Ceglio, N.M. S ~ YH.L., , see Cummins, H.Z. TAKO,T., see Ohtsu, M. TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets Quantum Phase Properties of Nonlinear TANAS,R., A. MIRANOWICZ, Ts. GANTSOG, Optical Phenomena TANGO,W.J., R.Q. W ~ s sMichelson , Stellar Interferometry TATARSKII, VI., VU. ZAVOROTNYI, Strong Fluctuations in Light Propagation in a Randomly Inhomogeneous Medium TATARSKII, VI., see Charnotskii, M.I. TAYLOR, C.A., see Lipson, H. TEICH,M.C., B.E.A. SALEH,Photon Bunching and Antibunching
XVII, XIII, XXV, XIII, XXVIII, X,
163 93 349 93 271 89
413 213 87 227 189 XV, 245 XXXIII, 203 X, 229 XXXIX, 373 XII, 53 VI, 211 X, 165 x, 45 XXI, 355
XVI, XXXIX, XXX, XXVII, XXXI,
XIII, 169 XXXIX, 213 XXVII, 109 XXXI, 263 V, 145 m 1 , 345 XX, 325 1x. 73 11, 1 XIX, 45 xm, 1 XXXI. 1
xn, 1 XXI, 287 VIII, 133 XXV, 191 XXIII, 63
xxxv, 355 XVII, 239
XWI, 204 XXXII, 203 V, 287
m,
1
509
CUMULATIVE INDEX - VOLUMES I-XXXIX
%R-MIKAELIAN, M.L., see Mikaelian, A.L. THELEN, A., see Musset, A. THOMPSON, B.J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see Ghatak, A. TONOMURA, A,, Electron Holography TORRE,A., see Dattoli, G. VK., see Sodha, M.S. TRIPATHI, Tsunuc~r,J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering Wss, R.Q., see Tango, W.J.
VII, 231 VIII, 201 Vn, 169 mn, 1 XXIII, 183 XXXI, 321 XIII, 169
J., see Leith, E.N. UPATMEKS, UPSTILL,C., see Berry, M.V S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in USHIODA, Solids
VI, 1 XVIII, 257
VAMPOLJILLE, M., see Froehly, C. VANDE GRIND,W.A., see Bouman, M.A. VANHEEL,A.C.S., Modem Alignment Devices VANKRANENDONK,J., J.E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media Fourier Spectroscopy VANASSE, G.A., H. SAKAI, VARTIAINEN, E.M., see Peiponen, K.-E. VERNIER, P.J., Photoemission Direct Spatial Reconstruction of Optical Phase from VLAD,VI., D. MALACARA, Phase-Modulated Images VOGEL,W., see Welsch, D.-G.
XX, 63 XXII, 77 I, 289
WALMSLEY, LA., see Raymer, M.G. WANGSHAOMIN, L. RONCHI, Principles and Design of Optical Arrays WEBER,M.J., see Riseberg, L.A. WEIGELT, G., Triple-Correlation Imaging in Optical Astronomy WEISS,G.H., see Gandjbakhche, A.H. WELFORD, W.T., Aberration Theory of Gratings and Grating Mountings WELFORD, W.T., Aplanatism and Isoplanatism WELFORD, W.T., see Bassett, I.M. WELSCH, D.-G., W. VOGEL,T. O P A T R Homodyne ~, Detection and Quantum-State Reconstruction WHITNEY, K.G., see Scully, M.O. WILHELMI, B., see Schubert, M. WINSTON, R., see Bassett, I.M. WOERDMAN, J.P., see Spreeuw, R.J.C. WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information WYNNE, C.G., Field Correctors for Astronomical Telescopes WYROWSKI, F., see Bryngdahl, 0. WYROWSKI, F., see Bryngdahl, 0. YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements using Laser Light YAW, K., Design of Zoom Lenses
11, 131 XVII, 239
XM. 139
XV, 245
VI, 259 XXXVII, 57 XIV, 245 XXXIII, 261 XXXIX. 63 XXVIII, XXV, XIV, XXIX, XXXIV,
181 279 89 293 333 IV,241 XIII, 267 XXVII, 161
XXXIX, X, XVII, XXVII, XXXI,
63 89 163 161 263
I, 155 X, 137 XXVIII, 1 XXXIII, 389
XXII, 271 VI, 105
510
CUMULATIVE INDEX - VOLUMES I-XXXIX
YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy YAMAMOTO, Y., S. MACHIDA,S. SAITO,N. IMOTO, T. YANAGAWA, M. KITAGAWA, G. BJORK,Quantum Mechanical Limit in Optical Precision Measurement and Communication YANAGAWA, T., see Yamamoto, Y. YAROSLAVSKY, L.P., The Theory of Optimal Methods for Localization of Objects in Pictures YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques Yu, F.T.S., Principles of Optical Processing with Partially Coherent Light Yu, F.T.S., Optical Neural Networks: Architecture, Design and Models ZALEVSKY, Z., see
Lohmann, A.W. ZAVOROTNY, VU., see Charnotslui, M.I. VU., see Tatarskii, VI. ZAVOROTNYI, ZUIDEMA,P., see Bouman, M.A.
W I , 295
XXVILI, 87 XXVIII. 87
XXXII, XI, XXIII, XXXII,
145 77 221 61
XXXVIII, 263 XXXII, 203 X W I , 204 XXII, 77
